![2 Computational Methods for Fracture in Porous Media
Figure 1.1 Fracture in the intervertebral disc, a fluid-saturated human tissue [Courtesy:
J.M.R. Huyghe].
Figure 1.2 Simplified diagram of hydraulic fracturing on the horizontal part of a shale
gas well [http://bbc.com/news].
porous materials occur frequently, indicating that there is a large practical
relevance. The existence and propagation of cracks in porous materials can
be undesirable, like those that form in human tissues, Fig. 1.1, or when the
storage of waste or CO2 in rocks or salt domes is concerned. But cracking
can also be a pivotal element in an industrial process, for example hydraulic
fracturing in the oil and gas industry, Fig. 1.2. Another important applica-
tion area is the rupture of geological faults, where the change in geometry
of a fault can drastically affect pore pressures and local fluid flow as the faults
can act as channels in which the fluid can flow freely (Rudnicki and Rice,
2006).
The first approaches to analyze the propagation of fluid-saturated cracks
were of an analytical nature (Perkins and Kern, 1961; Nordgren, 1972;](https://image.slidesharecdn.com/3491835-250622074101-a1b75bf8/85/Computational-Methods-For-Fracture-In-Porous-Media-Isogeometric-And-Extended-Finite-Element-Methods-1st-Edition-Ren-De-Borst-18-320.jpg)
![September the guardianship of the infant school was accepted under
the name of the Iowa Conference Seminary, by the Methodist
Episcopal church.
In this highly democratic manner Cornell College was founded by the
people as an institution of higher learning, which should ever be of
the people and for the people. It was born on the anniversary of the
nation's natal day, and was to remain one of the highest expressions
of patriotism and civic life. Christened by the head of the educational
interests of the young commonwealth, supported by its citizens,
protected by a charter from the state, and exempt as a beneficent
institution of the state from contributing by taxation to the support
of other institutions, the college was thus begun as a state school in
a very real sense.
One can not read the early archives of the college without the
profoundest admiration for the pioneers, its founders. Avid of
education to a degree pathetic, they depended on no beaurocracy of
church or state; they waited for no foreign philanthropy to supply
their educational needs. They laid the foundations of their colleges
with the same free, independent, self-sufficing spirit with which they
laid their hearthstones, and they laid both at the same time.
THE IOWA CONFERENCE SEMINARY
In January, 1853, the first meeting of the board of trustees was
held, and in the fall of the same year the school was opened in the
old Methodist church at Mount Vernon. Before the end of the term a
new edifice on the campus was so far completed that it was
available for school purposes and on the morning of November 14,
1853, the school met for the last time in the old church and after
singing and prayer the students were formed in line and walked in
procession with banners flying, led by the teachers, through the
village, and took formal possession of what was then declared to be
a large and commodious building.[J]](https://image.slidesharecdn.com/3491835-250622074101-a1b75bf8/85/Computational-Methods-For-Fracture-In-Porous-Media-Isogeometric-And-Extended-Finite-Element-Methods-1st-Edition-Ren-De-Borst-67-320.jpg)
![sending no less than twelve, although the entire population of
Dubuque county was then, less than 16,000. Considering the
difficulty of communications, the poverty of the pioneers, the wide
extent of the sphere of influence of the school is remarkable.
Students were drawn this first year from as far to the northeast as
Elkader and Garnavillo. They came from Dyersville and
Independence, from Quasqueton and Vinton, from Marengo,
Columbus City, West Liberty, and Burlington. Muscatine alone sent
seven students. This town was at the time the point of supply for
Mount Vernon, and the materials for the first building of the college
except such as local saw mills and brick kilns could supply were
hauled from that river port.[K] Students came also from Davenport,
Le Claire, Princeton, and Blue Grass in Scott county, from Comanche,
and from the pioneer settlements of La Motte and Canton in Jackson
county. The eight hundred students of Cornell today reach the
school from all parts of the state and the adjacent portions of our
neighboring states by a few hours swift and comfortable ride by rail.
But who shall picture in detail the long and adventurous journeys in
ox cart and pioneer wagon and perchance often on foot of the boys
and girls of 1853—the climbing of steep hills, the fording of rivers,
the miring in abysmal sloughs, the succession of mile after mile of
undulating treeless prairie carpeted with gorgeous flowers stretching
unbroken to the horizon, the camp at night illuminated by distant
prairie fires, until at last a boat shaped hill surmounted by a lonely
red brick building lifts itself above the horizon, and the goal of the
long journey is in view!
No doubt there were other hardships awaiting these students after
their arrival. Rule No. 1 of the new school compelled their rising at
five o'clock in the morning. They were expected to furnish their own
beds, lights, mirrors, etc., when boarding in Seminary Hall. It is
interesting to note that they paid for tuition $4.00 and $5.00 per
quarter, and for board from $1.50 to $1.75 per week. The next year
the steward's petition to the board of trustees that he be allowed to
put three students in each of the little rooms was granted with the
proviso that he furnish suitable bunks for the same. The catalog's](https://image.slidesharecdn.com/3491835-250622074101-a1b75bf8/85/Computational-Methods-For-Fracture-In-Porous-Media-Isogeometric-And-Extended-Finite-Element-Methods-1st-Edition-Ren-De-Borst-69-320.jpg)
![college classes, and at the commencement of this year all upon the
program were women except a delicate youth unfit for war and a
boy of sixteen years. This commencement was unique in the history
of the college. On commencement day the audience of peaceful folk
seated in the grove quietly listening to the student orations was
suddenly transformed to an infuriated mob, when one girl visitor
attempted to snatch from another a copperhead pin she was
wearing. So strong was the excitement, that the college buildings
were guarded by night for some time afterward for fear that they
might be burned in revenge by sympathisers with the south.[L]
Near the close of the war it was seen that many of the soldier
students of the college would be unable to complete their education
because of the sacrifices they had made in the service of their
country. A fund of fourteen thousand dollars was therefore
contributed by patriot friends at home and in part by Iowa regiments
in the field for the education of disabled soldiers and soldiers'
orphans. No gift to the school has ever been more useful than this
foundation, which aided in the support of hundreds of the most
worthy students of the college.
Two of the students of Cornell were enrolled in the armies of the
Confederacy. Of these one became a lieutenant in a Texas regiment.
At one time learning that one of his prisoners was a Cornell boy and
a member of his own literary society, the Texas lieutenant found
Cornell loyalty a stronger motive than official duty. He took his
prisoner several miles from camp, gave him a horse and started him
for the Union lines.
THE SOCIAL ORGANIZATION
From the beginning Cornell college has been coëducational. In the
earliest years of her history some concessions were made in the
courses of study to the supposed weakness of woman's intellect,
and ornamental branches, such as Grecian painting, which seems
to have been a sort of transfer work, ornamental hair work and wax](https://image.slidesharecdn.com/3491835-250622074101-a1b75bf8/85/Computational-Methods-For-Fracture-In-Porous-Media-Isogeometric-And-Extended-Finite-Element-Methods-1st-Edition-Ren-De-Borst-86-320.jpg)
Computational Methods For Fracture In Porous Media Isogeometric And Extended Finite Element Methods 1st Edition Ren De Borst
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- 8. COMPUTATIONAL METHODS FOR FRACTURE IN POROUS MEDIA Isogeometric and Extended Finite Element Methods René de Borst University of Sheffield, Department of Civil and Structural Engineering, Mappin Street, Sir Frederick Mappin Building, Sheffield S1 3JD, UK
- 9. Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2018 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-100917-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Joe Hayton Acquisition Editor: Ken McCombs Editorial Project Manager: Jennifer Pierce Production Project Manager: Sruthi Satheesh Designer: Christian Bilbow Typeset by VTeX
- 11. CONTENTS About theAurhor xi Preface xiii Acknowledgment xv 1. Introduction 1.1. Fracture in Porous Media 1.2. The Representation of Cracks and Fluid Flow in Cracks 3 ,.3. Purpose and Scope 7 References 8 2. Fractured or Fracturing, Fully or Partially Saturated Porous Media 13 2.1. Mass and Momentum Balance in a Porous Medium 13 2.2. A Saturated Porous Medium 15 2.2.1. Balance Equations and Constitutive Equations 15 2.2.2. Weak Forms 20 2.3. Unsaturated Porous Medium 22 2.3.1. Balance Equations and Constitutive Equations 22 2.3.2. Weak Forms 27 2.4. Modeling of Mass Transport Within Cracks 2.4.1. Fully Open Cracks 2.4.2. Partially Open Cracks 2.4.3. Fluid Flow Normal to the Crack References 3. Fracture Mechanics 3.1. Linear Elastic Fracture Mechanics 3.2. Cohesive Zone Models References 4. Interface Elements 4.1. Standard Interface Elements 4.1.1. Interface Kinematics 4.1.2. Constitutive Relation for the Interface 4.1.3. Internal Force Vector and Tangential Stiffness Matrix 4.1.4. Numerical Integration of Interface Elements 4.2. Poromechanical lnterface Elements 4.2.1. Interface Elements With a Continuous Pressure 4.2.2. Interface Elements With a Discontinuous Pressure 4.2.3. An Independent Pressure in the Interface 29 29 31 33 34 3S 3S 41 44 47 47 47 49 49 S2 S3 S4 60 61 vii
- 12. viii Contents 4.3. Remeshing Techniques 65 References 66 5. The Extended Finite Element Method 69 5.1. The Partition-of-Unity Concept 69 5.2. Extension to Fluid-Saturated Porous Media 79 5.2.l. Continuous Pressure Field 80 5.2.2. Discontinuous Pressure Field 84 5.3. Extension to Dynamics 89 5.4. Large Deformations 94 5.4.l. Nonlinear Kinematics 94 5.4.2. Balance Equations 96 5.4.3. Constitutive Equations 98 5.4.4. Weak Forms and Discretization 98 5.4.5. Example Calculations 101 References 106 6. Fracture Modeling Using Isogeometric Analysis 109 6.l. Basis Functions in Isogeometric Analysis 109 6.1.1. Univariate B-Splines 110 6.1.2. Univariate Non-Uniform Rational B-Splines 113 6.1.3. Multivariate 8-Splines and NURBS Patches 114 6.2. Isogeometric Finite Elements 115 6.3. Isogeometric Analysis for Poroelasticity 120 6.3.l. Formulation Using Bezier Extraction 120 6.3.2. Local Mass Conservation and Minimum Time Step 122 6.3.3. Unequal Orders of Interpolation 124 6.4. Discontinuities in B-Splines and NURBS 126 6.5. An Isogeometric Interface Element 132 6.5.l. Bezier Extraction 134 6.5.2. Spatial Integration 138 6.5.3. An Isogeometric Interface Element for Porous Media 141 6.6. Cohesive Crack Propagation 146 References 151 7. Phase-Field Methods for Fracture 155 7.1. The Phase-Field Approach to Brittle Fracture 155 7.1.1. The Phase-Field Approximation 156 7.1.2. Brittle Fracture 158 7.1.3. Discretization and Linearization 160 7.1.4. Internal Length Scale and Degradation Function 161 7.1.5. r-Convergence 164 7.2. A Phase-Field Method for Cohesive Fracture 166
- 13. 7.2.1. Kinematics 7.2.2. Discretization and Linearization 7.2.3. Order of the Interpolations 7.3. Phase�Fjeld Approaches for Fracture in Porous Media References Index Contents ix 167 170 171 175 182 185
- 14. ABOUT THE AUTHOR René de Borst received an MSc. in Civil Engineering and a PhD. in En- gineering Sciences from Delft University of Technology. He has been a Distinguished Professor at the Delft University of Technology and at the Eindhoven University of Technology, as well as the Regius Professor of Civil Engineering and Mechanics at the University of Glasgow. Currently, he is the incumbent of the Centenary Chair of Civil Engineering at the University of Sheffield. He has held visiting professorships in Albuquerque, Tokyo, Barcelona, Milan, Cachan, Metz, Lyon, has been a visiting Di- recteur de Récherche at CNRS in France, a Marie-Curie Distinguished Researcher in Lublin, the John Argyris Visiting Professor in Stuttgart, and MTS Visiting Professor of Geomechanics at the University of Minnesota. René de Borst has authored more than 250 articles and book chapters, edited 13 books, and is Editor-in-Chief of the International Journal for Numerical Methods in Engineering, Editor of the International Journal for Numerical and Analytical Methods in Geomechanics, Editor-in-Chief of the Encyclopedia of Computational Mechanics, and Associate Editor of the Aeronautical Journal. He is the recipient of several honours and awards, including the Com- posite Structures Award, the Max-Planck Research Award, the IACM Computational Mechanics Award, the NWO Spinoza Prize, the Royal So- ciety Wolfson Research Merit Award, and the JSCES Grand Prize. He has been inducted in the Royal Netherlands Academy of Arts and Sciences, the Royal Society of Edinburgh, the European Academy of Sciences and Arts, and the Royal Academy of Engineering in London. He is an Officer in the National Order of Merit in France, and holds an honorary doctorate from the Institut National des Sciences Appliquées de Lyon. xi
- 15. PREFACE Computational approaches for fracture and mass transport in fluid-saturated porous media are currently enjoying much attention. It is a fascinat- ing research topic, where some of the grand challenges of computational mechanics come together: multi-scale phenomena, multi-physics, i.e. the interaction between mechanical phenomena and one or more diffusion problems, and uncertainty and stochasticity. In spite of these challenges, or perhaps because of the inherent difficulty involved in their solution, not so much attention has been given to the subject as one would have ex- pected, especially considering the huge economic and societal relevance of the topic, being prominent in such different areas as petroleum engineering, waste disposal in the underground and clean water supply, and biomedical engineering. However, after a rather long dormant period, research activities have picked up rapidly in the last few years. As a result, this book would probably have looked differently if it had been written five years later, as the field is far from mature, and shows a rapid development. It is inevitable that the book as a whole, through the choice of topics, but also the treatment of the methodologies will suffer from a certain bias. Nevertheless, the author hopes that the book will help the reader to get a proper overview of which techniques are currently available, and where the challenges and obstacles lie. René de Borst Sheffield May 2017 xiii
- 16. ACKNOWLEDGMENT Many people have contributed to generate the knowledge that has enabled me to write this book and I wish to thank them collectively. Two of my former PhD. students I would like to thank in particular: Christian Michler, currently at Shell Global Solutions, Rijswijk, Netherlands, for his meticu- lous reading of the entire manuscript, and Clemens Verhoosel, currently at Eindhoven University of Technology, for the joint work on isogeometric analysis and phase field methods. René de Borst Sheffield May 2017 xv
- 17. CHAPTER 1 Introduction 1.1 FRACTURE IN POROUS MEDIA Fracture lies at the heart of many failures in natural and man-made materi- als. Fracture mechanics, as a scientific discipline in its own right, originated in the early 20th century with the pioneering work of Griffith (1921). Driven by some spectacular disasters in the shipbuilding and aerospace in- dustries, and building on the seminal work of Irwin (1957), linear elastic fracture mechanics (LEFM) has become an important tool in the analysis of structural integrity. Linear elastic fracture mechanics applies when the dissipative processes remain confined to a region in the vicinity of the crack tip that is small compared to the structural dimensions. When this condition is not met, e.g. when considering cracking in more heterogeneous materials like soils, rocks, concrete, ceramics, or many biomaterials, cohesive-zone models are to be preferred (Dugdale, 1960; Barenblatt, 1962). Cohesive-zone models remove the stress singularity that exists in linear elastic fracture mechanics. Fracture is then a natural outcome of the constitutive relations in the bulk and the interface, together with the balances of mass and momentum. Rice and Simons (1976) have provided compelling arguments in favor of the use of cohesive-zone models in fluid-saturated porous media by analyzing shear crack growth. Other arguments based on experimental evidence have been given in Valkó and Economides (1995). The vast majority of the developments in fracture relate to solid ma- terials. Occasionally, porous materials have been considered, but studies of crack initiation and propagation in porous materials, where the pores can be filled with fluids, are rather seldom found, at least until fairly recently. Indeed, the theory of fluid flow in deforming porous media has been prac- tically confined to intact materials (Terzaghi, 1943; Biot, 1965; Coussy, 1995, 2010; Lewis and Schrefler, 1998; de Boer, 2000), and this holds a fortiori for numerical studies on fracture in porous media. At the same time, fracture in heterogeneous, (partially) fluid-saturated porous media is a challenging, multi-scale problem with moving internal boundaries, characterized by a high degree of complexity and uncertainty. Moreover, fracture initiation and propagation in (partially) fluid-saturated Computational Methods for Fracture in Porous Media DOI: http://dx.doi.org/10.1016/B978-0-08-100917-8.00001-0 Copyright © 2018 Elsevier Ltd. All rights reserved. 1
- 18. 2 Computational Methods for Fracture in Porous Media Figure 1.1 Fracture in the intervertebral disc, a fluid-saturated human tissue [Courtesy: J.M.R. Huyghe]. Figure 1.2 Simplified diagram of hydraulic fracturing on the horizontal part of a shale gas well [http://bbc.com/news]. porous materials occur frequently, indicating that there is a large practical relevance. The existence and propagation of cracks in porous materials can be undesirable, like those that form in human tissues, Fig. 1.1, or when the storage of waste or CO2 in rocks or salt domes is concerned. But cracking can also be a pivotal element in an industrial process, for example hydraulic fracturing in the oil and gas industry, Fig. 1.2. Another important applica- tion area is the rupture of geological faults, where the change in geometry of a fault can drastically affect pore pressures and local fluid flow as the faults can act as channels in which the fluid can flow freely (Rudnicki and Rice, 2006). The first approaches to analyze the propagation of fluid-saturated cracks were of an analytical nature (Perkins and Kern, 1961; Nordgren, 1972;
- 19. Introduction 3 Khristianovic and Zheltov, 1955; Geertsma and de Klerk, 1969). Idealized geometries of a single, fluid-filled crack were considered, the surround- ing medium was taken as linear elastic, homogeneous and impervious, and an ad hoc leak-off term was introduced to account for the fluid loss into the surrounding medium (Carter, 1957). Linear elastic fracture mechanics was used to derive a crack propagation criterion. Invoking scaling laws, Detournay (2004) has put these works on a solid basis, and has identified that, depending on, inter alia, the values for the fracture toughness and the fluid viscosity, different propagation regimes can be distinguished. In case of viscosity-dominated propagation the classical square-root singularity at the crack tip no longer holds, and is replaced by a weaker singularity. Differen- tiation is made between four regimes: almost no leak-off vs. much leak-off, and viscosity vs. toughness dominated (Adachi et al., 2007). 1.2 THE REPRESENTATION OF CRACKS AND FLUID FLOW IN CRACKS Ever since the first attempts to simulate fracture using the finite element method, there has been a debate on the most efficient and physically real- istic method to model cracking. Essentially, there are two approaches: one can either represent cracks in a discrete manner, which dates back to Ngo and Scordelis (1967), or use a smeared or continuum approach (Rashid, 1968), see de Borst et al. (2004) for an overview and evolution of both approaches. Because of their relative simplicity and ability to simulate complex crack patterns, at least in principle, smeared models have gained much popularity. However, this comes at a price. First, the introduction of decohesion ren- ders continuum models ill-posed at a generic stage of crack propagation. In addition to this mathematical deficiency there is the physical argument that it is difficult, if possible at all, to translate the strains in the continuum model into discrete quantities like crack opening and crack sliding. In- deed, gradient-damage models (Peerlings et al., 1996; Frémond and Nedjar, 1996) and phase-field models (Francfort and Marigo, 1998) overcome the mathematical deficiency, but do not necessarily resolve the issue of quan- tifying discrete quantities like the crack opening. With the need to use cohesive fracture models which employ the crack opening and sliding as essential components in the constitutive relation in the crack, the difficul- ties to properly represent the crack opening only become a more pressing issue (Verhoosel and de Borst, 2013). The issue is also prominent when
- 20. 4 Computational Methods for Fracture in Porous Media Figure 1.3 Two scales at which fluid can flow in a fractured porous medium: a micro- scopic scale with interstitial fluid between particles, and (nearly) free fluid within the fractures. considering fluid transport in cracked porous media, as the possible dif- ference between the fluid velocities inside and outside the cracks makes it difficult to quantify mass transport. In the spirit of distributing discontinuities over a finite width, a model to capture fluid flow in a porous medium, which is intersected by multiple cracks, was proposed by Barenblatt et al. (1960). Fig. 1.3 shows the two different scales at which flow in fractured porous media is then considered: a microscopic scale at which we have interstitial pore fluid between grains, and a mesoscopic scale where fluid can flow almost freely in the cracks or faults. This idea was generalized to a deformable porous medium, which resulted in the double porosity model (Aifantis, 1980; Wilson and Aifan- tis, 1982; Khaled et al., 1984; Beskos and Aifantis, 1986; Bai et al., 1999), wherein Biot’s theory for deformable porous media (Biot, 1941) was ex- ploited. The double porosity model describes the effects of cracks on fluid flow and vice versa in a homogenized sense, but as in any distributed ap- proach, the local interaction between crack propagation and fluid flow is not captured. Returning to discrete crack models, it is noted that these have first been implemented by a simple nodal release technique (Ngo and Scordelis, 1967), and later, in a more elegant and versatile manner, using interface ele- ments. Remeshing has been introduced to decouple the crack propagation
- 21. Introduction 5 path from the original mesh layout (Ingraffea and Saouma, 1985). Espe- cially in three dimensions this can lead to complications and a considerable amount of remeshing. The extended finite element method (Belytschko and Black, 1999; Moës et al., 1999) has been proposed as an alternative, accommodating linear elastic fracture mechanics as well as cohesive frac- ture (Wells and Sluys, 2001; Moës and Belytschko, 2002; Remmers et al., 2003). It decouples the crack propagation path from the underlying dis- cretization, and has been a main carrier of numerical approaches to fracture for more than a decade. While being elegant in nature, the extended finite element method has a few drawbacks. Also here, crack propagation in three dimensions poses challenges regarding a robust implementation, although the use of level set methods has alleviated this issue somewhat. Complications may also ensue from the numerical quadrature used to evaluate the internal force vector and stiffness matrix for enriched elements. Completely arbitrary cracks that traverse an element can create highly irregular subdomains within an element. Subdividing enriched elements into triangles or tetrahedra, within which higher-order integration schemes are employed, is effective but cumbersome. However, there is a fundamental difficulty that numerical accuracy imposes a lower limit on the size of the subelements: when the crack is such that a very small part of the element is cut off, ill-conditioning of the stiffness matrix results (Remmers et al., 2008). Another drawback is that the location of the additional degrees of freedom is intrinsically tied to the original mesh. Hence, the results depend on the original mesh, and, for linear elastic fracture mechanics calculations, the computed stress intensity factors may be less accurate. Indeed, the stress prediction around the crack tip can be poor when using finite element methods. To a lesser extent this also applies when the crack tip is enriched using tailor-made functions to capture the stress sin- gularities or high stress gradients that typically occur at the tip when using linear elastic fracture mechanics (Fleming et al., 1997), or when smooth- ing techniques are added. For fluid-saturated porous media, standard finite element methods also suffer from the fact that the fluid velocity, which, assuming Darcy’s relation is proportional to the pressure gradient, is discon- tinuous at element boundaries. This can cause the local mass balance not to be satisfied unless special degrees of freedom are introduced (Malakpoor et al., 2007). The underlying cause is the same in both cases. The primary variables (displacement, pressure) are only C0-continuous across element
- 22. 6 Computational Methods for Fracture in Porous Media boundaries, causing jumps in the derived quantities as strains and fluid ve- locities. A promising solution is the use of isogeometric analysis (IGA), origi- nally proposed to obtain a seamless connection between Computer-Aided Design (CAD) tools and analysis tools, with the aim of bypassing the elab- orate and time-consuming meshing phase (Kagan et al., 1998; Kagan and Fischer, 2000; Hughes et al., 2005; Cottrell et al., 2009). The consequence is that Non-Uniform Rational B-Splines (NURBS), which are the pre- dominant functions in CAD-packages, are also used in the analysis phase. Since this spline technology results in C1 and higher-order continuity of the primary variables – depending on the degree of the interpolation – this approximation renders derived quantities like strains or fluid velocities con- tinuous across element boundaries (Irzal et al., 2013b). Local mass balance is automatically satisfied and the stress prediction is vastly improved. It is noted that around (discrete) crack tips, the continuity can be reduced, thus locally sacrificing the higher smoothness of isogeometric analysis, e.g. May et al. (2016). Building on the seminal work of Boone and Ingraffea (1990) on fluid- driven crack propagation – see also Sousa et al. (1993), Carter et al. (2000) – Schrefler et al. (2006), Secchi et al. (2007), Secchi and Schrefler (2012), Simoni and Schrefler (2014) have applied remeshing to model the propagation of cohesive cracks in a fluid-saturated porous medium. Inter- face elements enhanced with pressure degrees of freedom were considered by Segura and Carol (2008a,b); Carrier and Granet (2012) and Jha and Juanes (2014), enabling fluid flow within a crack. A first step towards the application of isogeometric analysis to crack propagation in fluid-saturated porous media was made in Irzal et al. (2014) and Vignollet et al. (2016), using isogeometric interface elements. Exploiting the partition-of-unity property of finite element shape func- tions, de Borst et al. (2006), Réthoré et al. (2007a) and Irzal et al. (2013a) have decoupled the crack propagation path in a (partially) fluid-saturated porous medium from the underlying discretization, see also Mohammad- nejad and Khoei (2013b,a); Khoei (2015) and Faivre et al. (2016). In an alternative approach a meshless method has been adopted (Samimi and Pak, 2016). Fig. 1.4 shows two scales that are now involved: a macroscopic scale at which the discretization is applied and the computation is performed, and the mesoscopic scale at which the mass transport within the crack is considered. From Fig. 1.3 we recall that the third scale is the microscopic level, where we have flow of the interstitial fluid between the grains.
- 23. Introduction 7 Figure 1.4 Fluid-saturated porous medium with discretization and a crack. The zoom shows the mass balance at the mesoscopic level. At the mesoscopic scale a model has been developed for the transport and storage of fluids in pre-existing or propagating cracks (Réthoré et al., 2007b, 2008; Irzal et al., 2013a). Fig. 1.4 shows the basic idea. Starting from the local mass and momentum balances for the fluid in the crack and exploiting the fact that the width of the crack is small compared to the other dimensions of the crack, the mesoscopic scale model can be coupled to the mass and momentum balances at the macroscopic level. This provides the possibility to analyze deformation and fluid flow in large formations that contain multiple cracks. 1.3 PURPOSE AND SCOPE After what can perhaps be called a late start, the development of methods for large-scale simulations of (multiple) fracture(s) in fluid-saturated porous media is now receiving an increasing amount of attention, partly due to its practical relevance in such diverse application areas as biomedical engineer- ing and petroleum engineering, and partly due to the intrinsic scientific challenges that are posed. In this book, we will first give a concise review of a basic and estab- lished theory of (partially) fluid-saturated porous media, and enhance this by including mass transport within cracks, thus arriving at a three-scale approach: the macroscopic level at which the discretization is applied and the computations are performed, the mesoscopic level where mass trans- port within the cracks is considered, and the microscopic level, at which we have fluid flow between the grains, and which is modeled, rather than resolved explicitly. Next, we will briefly recapitulate some basic notions of fracture mechanics, paying attention to linear elastic fracture mechanics, as well as to cohesive fracture.
- 24. 8 Computational Methods for Fracture in Porous Media The major part of this book is devoted to discretization techniques that allow the modeling of fracture in fluid-saturated porous media, in- cluding mass transport within the cracks. Chapter 4 treats standard inter- face elements. First, the purely mechanical case is discussed, followed by poromechanical interface elements of an increasing complexity: nodes at the interface with a single, a double, and a triple pressure degree of freedom. These different discretizations have implications for the physics that can be modeled, and the possibilities and constraints will be explained. A succinct discussion on remeshing techniques completes the chapter. Unless remesh- ing is used, interface elements can only be used to capture discontinuities with a path that is known a priori. To relax this restriction, the extended finite element method, detailed in Chapter 5, provides a versatile tool. It contains interface elements as a special case, but, as said before, it decou- ples the crack path from the underlying discretization. The basic concept is discussed, and extended to poromechanical conditions with a single and a double pressure degree of freedom at the discontinuity. The extension to dynamics and large strains is made as well. As briefly touched upon, the C0 interpolation of the displacements and pressures in standard finite elements can lead to a local loss of mass conser- vation, which is less straightforward to solve. Isogeometric finite elements, which are discussed in Chapter 6, provide an elegant solution. After an in- troduction into the basic concepts, including the important topic of Bézier extraction which enables the use of standard finite element data structures, it is explained how isogeometric analysis can be used in poroelasticity, and how cracks can be modeled. The final chapter is devoted to phase-field methods for fracture. This technique has recently enjoyed much attention, and although many ques- tions remain to be solved, for instance related to the issue of quantifying the crack opening and the mass transport in cracks properly, it has a large potential, especially for modeling cracks in three-dimensional structures. The application to brittle fracture, to cohesive fracture, and the extension to include interstitial fluid flow are discussed. REFERENCES Adachi, J., Seibrits, E., Peirce, A., Desroches, J., 2007. Computer simulation of hydraulic fractures. International Journal of Rock Mechanics and Mining Sciences 44, 739–757. Aifantis, E.C., 1980. On the problem of diffusion in solids. Acta Mechanica 37, 265–296.
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- 27. Introduction 11 Moës, N., Belytschko, T., 2002. Extended finite element method for cohesive crack growth. Engineering Fracture Mechanics 69, 813–833. Moës, N., Dolbow, J., Belytschko, T., 1999. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46, 131–150. Mohammadnejad, T., Khoei, A.R., 2013a. An extended finite element method for fluid flow in partially saturated porous media with weak discontinuities: the convergence analysis of local enrichment strategies. Computational Mechanics 51, 327–345. Mohammadnejad, T., Khoei, A.R., 2013b. Hydro-mechanical modelling of cohesive crack propagation in multiphase porous media using the extended finite element method. International Journal for Numerical and Analytical Methods in Geomechanics 37, 1247–1279. Ngo, D., Scordelis, A.C., 1967. Finite element analysis of reinforced concrete beams. Journal of the American Concrete Institute 64, 152–163. Nordgren, R.P., 1972. Propagation of a vertical hydraulic fracture. SPE Journal 12, 306–314. Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M., de Vree, H.P.J., 1996. Gradient- enhanced damage for quasi-brittle materials. International Journal for Numerical Meth- ods in Engineering 39, 3391–3403. Perkins, T.K., Kern, L.R., 1961. Widths of hydraulic fractures. Journal of the Petroleum Technology 13, 937–949. Rashid, Y.R., 1968. Analysis of reinforced concrete pressure vessels. Nuclear Engineering and Design 7, 334–344. Remmers, J.J.C., de Borst, R., Needleman, A., 2003. A cohesive segments method for the simulation of crack growth. Computational Mechanics 31, 69–77. Remmers, J.J.C., de Borst, R., Needleman, A., 2008. The simulation of dynamic crack propagation using the cohesive segments method. Journal of the Mechanics and Physics of Solids 56, 70–92. Réthoré, J., de Borst, R., Abellan, M.A., 2007a. A discrete model for the dynamic propa- gation of shear bands in a fluid-saturated medium. International Journal for Numerical and Analytical Methods in Geomechanics 31, 347–370. Réthoré, J., de Borst, R., Abellan, M.A., 2007b. A two-scale approach for fluid flow in frac- tured porous media. International Journal for Numerical Methods in Engineering 75, 780–800. Réthoré, J., de Borst, R., Abellan, M.A., 2008. A two-scale model for fluid flow in an un- saturated porous medium with cohesive cracks. Computational Mechanics 42, 227–238. Rice, J.R., Simons, D.A., 1976. The stabilization of spreading shear faults by coupled deformation–diffusion effects in fluid-infiltrated porous materials. Journal of Geophys- ical Research 81, 5322–5334. Rudnicki, J.W., Rice, J.R., 2006. Effective normal stress alteration due to pore pressure changes induced by dynamic slip propagation on a plane between dissimilar materials. Journal of Geophysical Research 111, B10308. Samimi, S., Pak, A., 2016. A fully coupled element-free Galerkin model for hydro- mechanical analysis of advancement of fluid-driven fractures in porous media. Interna- tional Journal for Numerical and Analytical Methods in Geomechanics 40, 2178–2206. Schrefler, B.A., Secchi, S., Simoni, L., 2006. On adaptive refinement techniques in multi- field problems including cohesive fracture. Computer Methods in Applied Mechanics and Engineering 195, 444–461.
- 28. 12 Computational Methods for Fracture in Porous Media Secchi, S., Schrefler, B.A., 2012. A method for 3-D hydraulic fracturing simulation. Inter- national Journal of Fracture 178, 245–258. Secchi, S., Simoni, L., Schrefler, B.A., 2007. Mesh adaptation and transfer schemes for discrete fracture propagation in porous materials. International Journal for Numerical and Analytical Methods in Geomechanics 31, 331–345. Segura, J.M., Carol, I., 2008a. Coupled HM analysis using zero-thickness interface elements with double nodes. Part I: Theoretical model. International Journal for Numerical and Analytical Methods in Geomechanics 32, 2083–2101. Segura, J.M., Carol, I., 2008b. Coupled HM analysis using zero-thickness interface ele- ments with double nodes. Part II: Verification and application. International Journal for Numerical and Analytical Methods in Geomechanics 32, 2103–2123. Simoni, L., Schrefler, B.A., 2014. Multi-field simulation of fracture. Advances in Applied Mechanics 47, 367–519. Sousa, J.L.S., Carter, B.J., Ingraffea, A.R., 1993. Numerical simulation of 3D hydraulic frac- ture using Newtonian and power-law fluids. International Journal of Rock Mechanics and Mining Sciences 30, 1265–1271. Terzaghi, K., 1943. Theoretical Soil Mechanics. Wiley & Sons, New York. Valkó, P., Economides, M.J., 1995. Hydraulic Fracture Mechanics. Wiley & Sons, Chich- ester. Verhoosel, C.V., de Borst, R., 2013. A phase-field model for cohesive fracture. International Journal for Numerical Methods in Engineering 96, 43–62. Vignollet, J., May, S., de Borst, R., 2016. Isogeometric analysis of fluid-saturated porous media including flow in the cracks. International Journal for Numerical Methods in Engineering 108, 990–1006. Wells, G.N., Sluys, L.J., 2001. A new method for modelling cohesive cracks using finite elements. International Journal for Numerical Methods in Engineering 50, 2667–2682. Wilson, R.K., Aifantis, E.C., 1982. On the theory of consolidation with double porosity – III: A finite element formulation. International Journal of Engineering Science 20, 1009–1035.
- 29. CHAPTER 2 Fractured or Fracturing, Fully or Partially Saturated Porous Media 2.1 MASS AND MOMENTUM BALANCE IN A POROUS MEDIUM We consider a multi-phase porous medium subject to the restriction of small variations in the concentrations and small displacement gradients, where it is noted that the latter restriction will be relaxed in Chapter 5, see also Irzal et al. (2013). Further, the assumptions are made that there is no mass transfer or chemical interaction between the constituents and that the processes which we consider occur isothermally. The latter assumption can be dropped without major consequences, e.g. Khoei et al. (2012), in which an extension towards thermo-hydro-mechanical coupling has been made. With the above assumptions, the balances of linear momentum for the individual phases read: ∇ · σπ + p̂π + ρπ g = ∂(ρπ u̇π ) ∂t + ∇ · (ρπ u̇π ⊗ u̇π ) , (2.1) with σπ the partial, or apparent stress tensor of constituent π, i.e. the force Fπ carried by constituent π divided by the total load carrying area A: σπ = Fπ A , (2.2) ρπ its apparent mass density, i.e. the mass mπ of constituent π per unit volume V: ρπ = mπ V , (2.3) and u̇π the absolute velocity of constituent π. The gravity acceleration is denoted by g, and p̂π is the momentum source for constituent π from the other constituents. This source term for instance takes into account the Computational Methods for Fracture in Porous Media DOI: http://dx.doi.org/10.1016/B978-0-08-100917-8.00002-2 Copyright © 2018 Elsevier Ltd. All rights reserved. 13
- 30. 14 Computational Methods for Fracture in Porous Media possible local drag interaction between a solid and a fluid. Evidently, the source terms must satisfy the momentum production constraint: π p̂π = 0 . (2.4) Neglecting convective terms, the momentum balance of constituent π re- duces to: ∇ · σπ + p̂π + ρπ g = ρπ ∂u̇π ∂t . (2.5) Summing the momentum balances of the individual phases, noting that the mass density of the mixture is the sum of the apparent mass densities, ρ = π mπ V = π mπ V = π ρπ , (2.6) and taking into account Eq. (2.4), one obtains the momentum balance for the mixture: ∇ · σ + ρg = π ρπ ∂u̇π ∂t , (2.7) where the total stress in the medium is the sum of the partial stresses σπ : σ = π σπ . (2.8) In a similar fashion, one can write the mass balance for each phase as: ∂ρπ ∂t + ∇ · (ρπ u̇π ) = 0 . (2.9) Consistent with the derivation of the balance of linear momentum, varia- tions in the mass density gradients are neglected, and the equations for the mass balance can be simplified to yield (after dividing by the apparent mass density ρπ ): 1 ρπ ∂ρπ ∂t + ∇ · u̇π = 0 . (2.10) Defining ρ π as the true, or intrinsic mass density of constituent π: ρ π = mπ Vπ , (2.11)
- 31. Fractured or Fracturing, Fully or Partially Saturated Porous Media 15 so that ρπ = nπ ρ π , (2.12) with nπ = Vπ V (2.13) the volumetric ratio of constituent π, and Vπ the volume occupied by constituent π, we can multiply the mass balance of constituent π by its volumetric ratio nπ to yield: 1 ρ π ∂ρπ ∂t + nπ ∇ · u̇π = 0 . (2.14) Summing the mass balances of the individual constituents π and exploiting the constraint π nπ = 1 (2.15) yields the overall mass balance: π 1 ρ π ∂ρπ ∂t + π nπ ∇ · u̇π = 0 . (2.16) 2.2 A SATURATED POROUS MEDIUM In this section we narrow the focus to the case of a solid and a fluid phase only, and we denote these phases by the subscripts π = s and π = f , re- spectively. This case will be used in the majority of the elaborations and examples. 2.2.1 Balance Equations and Constitutive Equations For the case of a fluid-saturated, two-phase medium the momentum bal- ance of the mixture specializes as: ∇ · σ + ρg = ρs ∂u̇s ∂t + ρf ∂u̇f ∂t , (2.17) and the total stress is composed of a solid and a fluid part: σ = σs + σf . (2.18)
- 32. 16 Computational Methods for Fracture in Porous Media With the Biot coefficient α, which takes into account the compressibility of the solid grains (Lewis and Schrefler, 1998), α = 1 − Kt Ks , (2.19) Kt being the overall bulk modulus of the skeleton and Ks that of the solid grains, the total stress can be written as: σ = σs − αpI , (2.20) with p the (apparent) fluid pressure and I the unit tensor. A model with two separate inertia terms requires the independent inter- polation of three fields: the solid velocity u̇s, the fluid velocity u̇f , and the fluid pressure p. To simplify the ensuing numerical model, the assumption is often made that the accelerations of the solid particles and of the fluid are approximately equal: ∂u̇s ∂t ≈ ∂u̇f ∂t . (2.21) Numerical analyses typically make use of this assumption, cf. Lewis and Schrefler (1998). Especially for relatively slow dynamic loadings it seems to be a reasonable approximation, but its accuracy has seldom been quantified. Results shown in Box 2.1 on page 18 suggest that the influence of two separate inertia terms can indeed be limited. Using Eq. (2.21) the balance of momentum of the solid–fluid mixture, Eq. (2.17), becomes: ∇ · σ + ρg = ρ ∂u̇s ∂t . (2.22) Inserting Eq. (2.20) then gives: ∇ · (σs − αpI) + ρg = ρ ∂u̇s ∂t . (2.23) From Eq. (2.16) we obtain the mass balance for the solid–fluid mix- ture: 1 ρ s ∂ρs ∂t + 1 ρ f ∂ρf ∂t + ns∇ · u̇s + nf ∇ · u̇f = 0 , (2.24) or exploiting the constraint condition, Eq. (2.15), ∇ · u̇s + nf ∇ · (u̇f − u̇s) + 1 ρ s ∂ρs ∂t + 1 ρ f ∂ρf ∂t = 0 . (2.25)
- 33. Fractured or Fracturing, Fully or Partially Saturated Porous Media 17 The governing equations, i.e. the balance of momentum of the saturated medium, Eq. (2.22) or Eq. (2.23), and the balance of mass, Eq. (2.25), are complemented by the kinematic relation, which, for the case of small displacement gradients, reads: ˙ s = ∇s u̇s , (2.26) with ˙ s the strain rate field of the solid, the superscript s denoting the symmetric part of the gradient operator. The effective, true, or intrinsic stress rate in the solid skeleton, σ̇ s, is related to the strain rate ˙ s of the solid phase by a tangential stress–strain relationship: σ̇ s = Dtan s : ˙ s , (2.27) with Dtan s the fourth-order tangent stiffness tensor of the solid material. Since the effective stress in the solid skeleton, σ s, is related to the partial stress σs by σ s = σs/ns , (2.28) Eq. (2.27) can be replaced by σ̇s = Dtan : ˙ s , (2.29) where the denotation Dtan = nsDtan s has been used. In the examples, a linear-elastic behavior of the bulk material will be assumed, so that Dtan = De, the fourth-order linear-elastic stiffness tensor. For most applications that relate to the flow of fluids in porous media, Darcy’s relation can be assumed to hold, which, assuming isotropy, takes the form: nf (u̇f − u̇s) = −kf ∇p + ρf ∂u̇f ∂t , (2.30) with kf the permeability coefficient of the porous medium, and nf the vol- umetric ratio of the fluid, which, for the present solid–single-fluid system, equals the porosity n. Again using the assumption that the accelerations of the solid and fluid particles are equal, cf. Eq. (2.21), Eq. (2.30) can be
- 34. 18 Computational Methods for Fracture in Porous Media approximated as: nf (u̇f − u̇s) = −kf ∇p + ρf ∂u̇s ∂t , (2.31) which comes at the expense of losing symmetry of the system, but has again the benefit of not having to separately interpolate the fluid velocity u̇f , thus avoiding a three-field formulation. Indeed, inclusion of the dynamic seepage term −kf ρf ∂u̇f ∂t in Darcy’s relation has a similar effect as having two separate inertia terms (Schrefler and Scotta, 2001), see also Box 2.1. In most practical cases, neither two separate inertia terms, nor the dynamic seepage term is included in the analyses, so that Darcy’s relation reduces to: nf (u̇f − u̇s) = −kf ∇p . (2.32) Inserting this reduced form of Darcy’s relation into Eq. (2.25) gives: ∇ · u̇s − ∇ · kf ∇p + 1 ρ s ∂ρs ∂t + 1 ρ f ∂ρf ∂t = 0 . (2.33) From Eq. (2.14) with π = s and considering that it is reasonable to assume Kt = nsKs for a nearly incompressible fluid, the mass balance for the solid constituent can be transformed into: 1 ρ s ∂ρs ∂t = − Kt Ks ∇ · u̇s . (2.34) Using the Biot coefficient, defined in Eq. (2.19), this equation can be rewritten as: (α − 1)∇ · u̇s = 1 ρ s ∂ρs ∂t . (2.35) BOX 2.1 Influence of separate inertia terms To assess the effect of including the dynamic seepage term in Darcy’s relation, simulations have been carried out in which this effect has been incorporated. Fig. 2.1 presents the beginning of load-displacement curves for the simulation of shear banding in a biaxial test of a fluid-saturated medium with a Tresca initia- tion criterion (Réthoré et al., 2007). The results show that there is little effect of the inclusion of a dynamic seepage term. This also holds for local quantities like the pressures, since also there the differences are negligible (about 10−6). A sim-
- 35. Fractured or Fracturing, Fully or Partially Saturated Porous Media 19 ulation in which both inertia terms were taken into account yielded very similar results. Figure 2.1 Influence of the dynamic seepage term (zoom on the load- displacement curves after initiation), after Réthoré et al. (2007). For the fluid phase, a phenomenological relation is assumed between the rates of the apparent fluid mass density and the fluid pressure p: 1 ρ f ∂ρf ∂t = 1 M ∂p ∂t , (2.36) with M the Biot modulus, which can be related to the bulk modulus of the solid material, Ks, and the bulk modulus of the fluid, Kf , e.g. Lewis and Schrefler (1998): 1 M = α − nf Ks + nf Kf . (2.37) Inserting Eqs. (2.35) and (2.36) into the overall mass balance, Eq. (2.33), then gives: α∇ · u̇s − ∇ · kf ∇p + 1 M ∂p ∂t = 0 . (2.38) The initial value problem is now closed by specifying the appropriate initial and boundary conditions. The following boundary conditions need to be specified for the solid: n · σ = tp , u̇s = u̇p , (2.39)
- 36. 20 Computational Methods for Fracture in Porous Media Figure 2.2 Body with external boundary and internal boundaries + d and − d . which hold on complementary parts of the boundary ∂t and ∂u, with = ∂ = ∂t ∪ ∂u, ∂t ∩ ∂u = ∅. Herein, n is the outwards pointing normal vector at the external boundary (Fig. 2.2), tp is the prescribed external traction and u̇p is the prescribed velocity. Regarding the fluid, the boundary conditions nf (u̇f − u̇s) · n = qp , p = pp (2.40) hold on complementary parts of the boundary ∂q and ∂p, with = ∂ = ∂q ∪ ∂p and ∂q ∩ ∂p = ∅, and qp and pp being the prescribed outflow of pore fluid and the prescribed pressure, respectively. The initial conditions at t = 0 read: uπ (x,0) = u0 π , u̇π (x,0) = u̇0 π , p(x,0) = p0 , π = s,f . (2.41) 2.2.2 Weak Forms To arrive at the weak form of the balance equations, we multiply the mo- mentum balance, Eq. (2.22), and the mass balance (2.38) by kinematically admissible test functions for the displacements of the skeleton, η, and for the pressure, ζ, respectively. Taking into account the internal boundary, see Fig. 2.2, integrating over the domain , and using the divergence theorem and the boundary conditions, Eqs. (2.39)–(2.40), leads to the correspond- ing weak forms: ∇η : σd + η · ρ ∂u̇s ∂t d − + d η+ · (n+ d · σ+ )d − − d η− · (n− d · σ− )d = t η · tpd (2.42)
- 37. Fractured or Fracturing, Fully or Partially Saturated Porous Media 21 and − αζ∇ · u̇s d − kf ∇ζ · ∇p d − ζ 1 M ∂p ∂t d − + d ζ+ n+ d · q+ d d − − d ζ− n− d · q− d d = q ζn · qpd . (2.43) It is noted that the gravity term has been omitted in the balance of momen- tum. Clearly, it is straightforward to include the term, but inclusion would not add any insight, while making the resulting expression more elaborate. It is emphasized that because of the presence of a discontinuity inside the domain , the power of the external tractions on d and the nor- mal fluid flux through the faces of the discontinuity are essential features of the weak formulation. Indeed, these terms enable the momentum and mass couplings between a discontinuity – the mesoscopic scale – and the surrounding porous medium – the macroscopic scale. In view of d = + d = − d , which defines a zero-thickness interface, the integrals in Eq. (2.42) at the discontinuity can be elaborated as follows. We first define nd = n− d = −n+ d , (2.44) see also Fig. 2.2. Next, we assume equilibrium between the cavity and the bulk: σ+ · n+ d = − σ− · n− d = tloc d − pnd , (2.45) with tloc d the cohesive tractions in a local coordinate system, which van- ish in case of a fully open crack. Using Eq. (2.45), the balance of linear momentum, Eq. (2.42), can then be reworked as: ∇η : σd + η · ρ ∂u̇s ∂t d + d JηK · (tloc d − pnd )d = t η · tpd , (2.46) with JηK the jump in the test function η. Use of Eq. (2.20) subsequently gives the more explicit form: ∇η : (σs − αpI)d + η · ρ ∂u̇s ∂t d + d JηK · (tloc d − pnd )d = t η · tpd . (2.47)
- 38. 22 Computational Methods for Fracture in Porous Media Having assumed equilibrium between the cavity and the bulk, Eq. (2.45), and noting that the (cohesive) tractions tloc d have a unique value, the fluid pressure p has the same value at both faces of the cavity: p = p+ = p− (D’Angelo and Scotti, 2012; Formaggia et al., 2014). Using a Bubnov–Galerkin approach, this implies that also the test function ζ attains the same value at both faces: ζ = ζ+ = ζ− . With this corollary, the weak form of the mass balance is modified as: − αζ∇ · u̇s d − kf ∇ζ · ∇p d − ζ 1 M ∂p ∂t d + d ζnd · JqdKd = ζn · qpd . (2.48) A jump in the flux, JqdK = q+ d − q− d , (2.49) has now emerged in the integral for the discontinuity. This term is multi- plied by the normal nd to d, resulting in a jump of the flow normal to the internal discontinuity. Accordingly, the flow can be discontinuous at d and some of the fluid that flows into the crack can be stored or be transported within the crack. The jump in the flux is therefore a measure of the net fluid exchange between a discontinuity (the cavity) and the surrounding bulk material. The assumption of equilibrium at the faces of the cavity can be relaxed. Although it is less easy to imagine this for the cohesive tractions that are transferred across the crack, this is conceivable for the fluid pressure, e.g. when the cavity is not well permeable in the direction normal to the cavity due to the presence of a diaphragm. Such a case will be considered in Subsection 2.4.3. 2.3 UNSATURATED POROUS MEDIUM 2.3.1 Balance Equations and Constitutive Equations Flow of several fluid phases in a porous medium is often encountered, e.g., the flow of two liquids such as oil and water. The governing equations (mo- mentum and mass balances) of such systems are not necessarily complicated when derived in a systematic manner, but quickly become comprehensive and rather unwieldy. For this reason we will restrict the discussion here to unsaturated soils, where we have a liquid (often water) and a gas phase. In
- 39. Fractured or Fracturing, Fully or Partially Saturated Porous Media 23 the remainder of this section we will denote the liquid phase by a subscript w and the gas by a subscript g. BOX 2.2 The degree of saturation and the capillary pressure A typical, and frequently used function that gives the degree of saturation of the water as a function of the capillary pressure has been proposed by van Genuchten (1980), see also, e.g. Meschke and Grasberger (2003), which reads: Sw(pc) = Sirr + (1 − Sirr) 1 + pc pref (1− )−1 . The degree of saturation for the water is not allowed to decrease to the irreducible saturation Sirr, and the reference pressure pref is used as a scaling factor for the capillary pressure pc. is a porosity index which characterizes the micro-structure of the porous skeleton. A dependence of the permeability on the degree of satu- ration can be included as follows: kπ = k μπ krπ (Sπ ) . The relative permeability for the water is then defined as: krw = Se 1 − (1 − S 1 e ) 2 and that for the gas phase as kra = (1 − Se)2 1 − S 2+3 e , where Se = Sw − Sirr 1 − Sirr is a relative saturation. Another expression for the degree of saturation has been suggested by Brooks and Corey (1966). In the light of the foregoing, the bulk is now considered as a three-phase medium subject to the restrictions of small displacement gradients and small variations in the concentrations. The problem is formulated in terms of the velocity of the solid phase, u̇s, and the water and gas pressures, pw and pg, respectively. The voids of the solid skeleton are partly filled with water and
- 40. 24 Computational Methods for Fracture in Porous Media partly with gas. The degrees of saturation for the fluid phases, Sπ = Vπ Vw + Vg , π = w,g , (2.50) form a partition of unity: Sw +Sg = 1. The degree of saturation of the liquid phase is normally described via a function of the capillary pressure, pc = pg − pw , (2.51) such that (see Box 2.2 for an example): Sw = Sw(pc) . (2.52) For a three-phase medium consisting of solid particles, liquid (water), and gas, Eq. (2.8) can be written explicitly as: σ = σs + σw + σg . (2.53) Assuming immiscibility of both fluid phases this identity can be elaborated as follows: σ = σs + nwσ w + ngσ g = σs − Vw Vw + Vg Vw + Vg V p w + Vg Vw + Vg Vw + Vg V p g I = σs − n Swp w + Sgp g I , (2.54) with n = Vw + Vg V = nw + ng (2.55) the porosity of the three-phase medium. We next define the intrinsic fluid pressure p = Swp w + Sgp g (2.56) and use the porosity n to relate it to the average fluid pressure p = np = n(Swp w + Sgp g) = Swpw + Sgpg . (2.57) Taking into account the dependence of Sw on the capillary pressure pc, Eq. (2.52), the time derivative of the average pressure can be written as: ṗ = Sw − pw ∂Sw ∂pc ṗw + Sg − pg ∂Sw ∂pc ṗg. (2.58)
- 41. Fractured or Fracturing, Fully or Partially Saturated Porous Media 25 Eq. (2.57) can now be used to rewrite Eq. (2.54) as: σ = σs − pI , (2.59) or, using the Biot coefficient α, cf. Eq. (2.19): σ = σs − αpI , (2.60) which is identical to Eq. (2.20) except for the definition of the average fluid pressure p. We finally sum the linear balances of momentum of the individ- ual phases to obtain the balance of momentum of the mixture, Eq. (2.7). Assuming that the accelerations of the individual phases are approximately equal, cf. Eq. (2.21), the linear balance of momentum of the mixture again gives Eq. (2.22): ∇ · σ + ρg = ρ ∂u̇s ∂t . Under the same assumptions as for the balance of linear momentum, one can write the balance of mass for each phase, cf. Eq. (2.14). Trans- forming this equation to exploit the apparent density rather than the true density by using Eq. (2.11), we have: 1 ρπ ∂ρπ ∂t + ∇ · u̇π = 0 . (2.61) Summing the mass balances of the solid phase and of a fluid phase π (note that now π = w,g), the following expression is obtained: 1 − n ρs ∂ρs ∂t + (1 − n)∇ · u̇s + n ρπ ∂ρπ ∂t + n∇ · u̇π = 0 , π = w,g . (2.62) For an unsaturated porous medium, assuming isothermal conditions, small gradients, and no mass exchange between the different phases, which is in line with the assumptions made in Section 2.1, the time derivative of the solid phase reads (Lewis and Schrefler, 1998): 1 ρs ∂ρs ∂t = 1 ns α − n Ks ṗ − (1 − α)∇ · u̇s . (2.63) Recalling that the volume fraction of the solid constituent, ns, and the porosity, n, form a partition of unity, i.e. ns = 1 − n, see Eq. (2.55), this
- 42. 26 Computational Methods for Fracture in Porous Media equation can also be written as: 1 ρs ∂ρs ∂t = 1 1 − n α − n Ks ṗ − (1 − α)∇ · u̇s . (2.64) For the fluid phases, one can write: 1 ρπ ∂ρπ ∂t = 1 Kπ ṗπ , π = w,g , (2.65) although for the gas phase a more tailored expression can be written, for instance assuming an ideal gas (Lewis and Schrefler, 1998). Substitution of Eqs. (2.64) and (2.65) into Eq. (2.62) and rearranging gives: α − n Ks ṗ + n Kπ ṗπ + α∇ · u̇s + n∇ · (u̇π − u̇s) = 0 . (2.66) Substitution of the derivative for the pressure, Eq. (2.58), into Eq. (2.66) then gives: ṗw Mww + ṗg Mwg + α∇ · u̇s + n∇ · (u̇w − u̇s) = 0 (2.67) for the liquid phase, and ṗw Mgw + ṗg Mgg + α∇ · u̇s + n∇ · u̇g − u̇s = 0 (2.68) for the gas phase. The coefficients Mww etc. are defined as: 1 Mww = α − n Ks Sw − pw ∂Sw ∂pc + n Kw 1 Mwg = α − n Ks Sg − pg ∂Sw ∂pc 1 Mgw = α − n Ks Sw − pw ∂Sw ∂pc 1 Mgg = α − n Ks Sg − pg ∂Sw ∂pc + n Kg . (2.69) For flow in a porous medium, Darcy’s relation is assumed to hold in each of the fluid phases, n(u̇π − u̇s) = −kπ ∇pπ , (2.70)
- 43. Fractured or Fracturing, Fully or Partially Saturated Porous Media 27 with kπ the permeability coefficient of the porous medium with respect to the fluid phase π: kπ = k μπ , (2.71) μπ being the viscosity of the fluid phase π, and k the intrinsic permeability, which reflects the microstructure of the solid skeleton. Substitution of this identity into Eqs. (2.67)–(2.68) yields: ṗw Mww + ṗg Mwg + α∇ · u̇s − ∇ · kw∇pw = 0 (2.72) for the liquid phase, and ṗw Mgw + ṗg Mgg + α∇ · u̇s − ∇ · kg∇pg = 0 (2.73) for the gas phase. 2.3.2 Weak Forms To arrive at the weak form of the balance equations, we multiply the momentum balance (2.22) and the mass balances (2.72) and (2.73) by ad- missible test functions for the displacements of the skeleton, η, and for the pressures, ζπ . Integrating over the domain and using the divergence the- orem then leads to the corresponding weak forms: ∇η : σd + η · ρ ∂u̇s ∂t d − + d η+ · (n+ d · σ+ )d − − d η− · (n− d · σ− )d = t η · tpd , and − αζw∇ · u̇s d − kw∇ζw · ∇pw d − ζw ṗw Mww d − ζw ṗg Mwg d − + d ζ+ w n+ d · q+ w d − − d ζ− w n− d · q− w d = ζwn · qwp d (2.74)
- 44. 28 Computational Methods for Fracture in Porous Media for the liquid, and − αζg∇ · u̇s d − kg∇ζg · ∇pg d − ζg ṗw Mgw d − ζg ṗg Mgg d − + d ζ+ g n+ d · q+ g d − − d ζ− g n− d · q− g d = ζgn · qgp d (2.75) for the gas phase. Using the same assumptions as for the fully saturated porous medium, the surface integrals along the internal discontinuity can be reworked, again yielding Eq. (2.46) for the weak form of the momentum balance, ∇η : σd + η · ρ ∂u̇s ∂t d + d JηK · (tloc d − pnd )d = t η · tpd , and the set − αζw∇ · u̇s d − kw∇ζw · ∇pw d − ζw ṗw Mww d − ζw ṗg Mwg d + d ζwnd · JqwKd = ζwn · qwpd (2.76) and − αζg∇ · u̇s d − kg∇ζg · ∇pg d − ζg ṗw Mgw d − ζg ṗg Mgg d + d ζgnd · JqgKd = ζgn · qgpd (2.77) for the mass balances, with JqwK = q+ w − q− w , JqgK = q+ g − q− g . (2.78) In a number of applications it is reasonable to assume that pg is at the at- mospheric pressure. In this case of an unsaturated medium with a so-called passive gas phase, pg, cancels as an independent variable, thus reducing the system to a two-field formulation and simplifying the initial value problem.
- 45. Fractured or Fracturing, Fully or Partially Saturated Porous Media 29 Figure 2.3 Geometry and local coordinate system in the cavity. 2.4 MODELING OF MASS TRANSPORT WITHIN CRACKS 2.4.1 Fully Open Cracks We assume an open cavity which is filled with a Newtonian fluid, where the flow is not disturbed. This case can be associated with linear elastic frac- ture mechanics, where no material exists in the cracks which can support possible (cohesive) tractions between both crack faces. The mass balance for the flow within the cavity reads: ρ̇f + ρf ∇ · u̇f = 0 subject to the assumptions of small changes in the concentrations and no convective terms. Since the fluid velocity in the cavity is usually much higher than the velocity of the interstitial fluid in the surrounding bulk material, the first term is often small compared to the second term, and can be neglected for many practical purposes. Focusing the further derivations on a two-dimensional configuration – note that the extension to three dimensions is straightforward but just involves more lengthy expressions – the mass balance simplifies to: ∂v ∂s + ∂w ∂n = 0 , (2.79) where v = u̇f ·td and w = u̇f ·nd are the tangential and normal components of the fluid velocity in the discontinuity, respectively, nd and td being the vectors normal and tangential to the discontinuity d, see Fig. 2.3, which also shows the local s,n-coordinate system. The difference in the fluid velocity components that are normal to both crack faces is now given by: Jwf K = − h/2 n=−h/2 ∂v ∂s dn . (2.80)
- 46. 30 Computational Methods for Fracture in Porous Media The momentum balance for the fluid in the s-direction reads: ∂τ ∂n = ∂p ∂s (2.81) with τ the shear stress. Together with the assumption of a Newtonian fluid, τ = μ ∂v ∂n , (2.82) with μ the viscosity of the fluid, this gives: μ ∂2v ∂n2 = ∂p ∂s . (2.83) After integration from n = −h/2 to n = h/2, a parabolic velocity profile results: v(n) = 1 2μ ∂p ∂s (n2 − (h/2)2 ) + vf , (2.84) where the essential boundary condition v = vf for the tangential velocity of the fluid has been applied at both faces of the cavity. Assuming a no-slip condition at the faces of the cavity, this boundary condition derives from the relative fluid velocity in the porous medium at n = ±h/2: vf = (u̇s − nf −1 kf ∇p) · td . (2.85) Substituting Eq. (2.84) into Eq. (2.80) and again integrating with respect to n then leads to: Jwf K = 1 12μ ∂ ∂s ∂p ∂s h3 − h ∂vf ∂s . (2.86) This equation gives the amount of fluid attracted in the tangential fluid flow, and can also be written as Jwf K = 1 12μ ∂ ∂s ∂p ∂s h3 − ∂(hvf ) ∂s + vf ∂h ∂s , (2.87) which brings out the similarity with the Reynolds lubrication equa- tion (Reynolds, 1886). This is not surprising as the same assumptions underly both equations. It is noted that Eq. (2.87) involves a dependence of the third order on the width of the cavity, h. This is often referred to as the ‘cubic law,’ although it is just a consequence of reducing the flow
- 47. Fractured or Fracturing, Fully or Partially Saturated Porous Media 31 equations for a viscous fluid to a narrow space between two plates. Ex- perimental evidence that corroborates the theoretical derivation has been provided by Witherspoon et al. (1980). The projection of the jump in the fluid flux, JqdK, in the direction normal to the discontinuity can be expressed as: nd · JqdK = nf Jwf − wsK . (2.88) Since ∂h ∂t = JwsK , (2.89) the mass coupling term in Eq. (2.48) becomes: nd · JqdK = nf Jwf − wsK = nf 1 12μ ∂ ∂s ∂p ∂s h3 − h ∂vf ∂s − ∂h ∂t . (2.90) Substitution of Eq. (2.87) and elaboration then yield: nd · JqdK = nf h3 12μ ∂2p ∂s2 + h2 4μ ∂h ∂s ∂p ∂s − h ∂(u̇s)s ∂s − kf nf ∂2p ∂s2 − ∂h ∂t , (2.91) with (u̇s)s the velocity of the solid particles in the local s-direction. 2.4.2 Partially Open Cracks When the cavity is partially filled with solid material, e.g. in the case that a cohesive-zone model applies, the initial value problem can be closed by assuming that the cavity is a porous material itself, of course with a different permeability than that of the surrounding bulk material. The mass balance for the fluid inside the cavity then reads: αd∇ · u̇s + nfd∇ · (u̇f − u̇s) + 1 Md ∂p ∂t = 0 , (2.92) where the subscript d distinguishes quantities in the discontinuity from those in the bulk. Because the width of the cavity h is negligible compared to its length, the mass balance is again enforced in an average sense over the cross section. For the first term, we can elaborate for a two-dimensional
- 48. 32 Computational Methods for Fracture in Porous Media configuration: h/2 n=−h/2 αd∇ · u̇sdn = h/2 n=−h/2 αd ∂vs ∂s + ∂ws ∂n dn (2.93) = h/2 n=−h/2 αd ∂vs ∂s dn − αdJwsK , where vs and ws are the components of the solid velocity tangential and normal to the crack, respectively. Like the other constants (Md and nfd), αd has been assumed to be constant over the cross section. Also assuming that vs varies linearly over the height of the cavity, and defining vs = 1 2 (vs(h/2) + vs(−h/2)), the integral can be solved analytically: h/2 n=−h/2 αd ∂vs ∂s dn = αdh ∂vs ∂s . (2.94) Repeating these operations for the second term of Eq. (2.92), interchang- ing the order of integration and differentiation, and assuming that the boundary terms can be neglected, the following expression is obtained: h/2 n=−h/2 nfd∇ · (u̇f − u̇s)dn = nfdJwf − wsK + ∂ ∂s h/2 −h/2 nfd(vf − vs)dn . (2.95) We now introduce Darcy’s relation in a one-dimensional sense in the di- rection of the crack, nfd(vf − vs) = −ksd ∂p ∂s , (2.96) with ksd the permeability of the damaged, porous material in the s-direction inside the cavity. In line with the preceding assumptions ksd is assumed not to depend on n. However, the decohesion inside the cavity can affect the permeability, and therefore an assumption like ksd = ksd(h) can be reason- able. Substituting Eq. (2.96) into Eq. (2.95), the following relation ensues: h/2 n=−h/2 nfd∇ · (u̇f − u̇s)dn = nfdJwf − wsK − ∂ ∂s h/2 −h/2 ksd ∂p ∂s dn , (2.97) so that: h/2 n=−h/2 nf ∇ · (u̇f − u̇s)dn = nf Jwf − wsK − h ∂ksd(h) ∂s ∂p ∂s − ksdh ∂2p ∂s2 . (2.98)
- 49. Fractured or Fracturing, Fully or Partially Saturated Porous Media 33 It is noted that the symbol n is used to denote the axis normal to the discontinuity, but also denotes the porosity. The meaning, however, should be clear from the context. Neglecting variations of the pressure over the height of the cavity, the third term can be elaborated as: h/2 n=−h/2 1 Md ∂p ∂t dy = h Md ∂p ∂t . (2.99) The mass coupling term then becomes: nd · JqdK = − h Md ∂p ∂t + αd ∂h ∂t − αdh ∂vs ∂s + h ∂ksd(h) ∂s ∂p ∂s + ksdh ∂2p ∂s2 . (2.100) We finally note that the derivations for the fully open crack and for a crack that is partly filled with rubble are strictly valid only for the case that we have a single fluid phase in the cavity. However, the resulting expres- sions also apply to the case of a passive gas phase, where pg = constant, see also Réthoré et al. (2008). The extension to two fluid phases in the cavity, which is compatible with the assumption of an active gas phase in the bulk, has been pursued by Mohammadnejad and Khoei (2013) for a crack that is partly filled with rubble. 2.4.3 Fluid Flow Normal to the Crack Storage and fluid flow in a direction that is tangential to the discontinuity, as described in the preceding two subsections, is possible when the pres- sure gradient orthogonal to the crack is discontinuous. The pressure can then still be assumed to be continuous. However, when this assumption is relaxed, so that the pressure itself can be discontinuous, for instance when having two pressure degrees of freedom at the crack, p− and p+ , there can also be fluid transport across the discontinuity. This type of modeling has been pursued within the context of extended finite element methods by de Borst et al. (2006) and for interface elements by Segura and Carol (2008a,b). Defining the permeability of a diaphragm that is assumed to co- incide with the discontinuity d as knd, a discrete analog of Darcy’s relation can be postulated: nd · JqdK = −knd(p+ − p− ). (2.101) Evidently, knd = 0 corresponds to an impervious boundary. For the limiting case that knd → ∞ the case of a continuous pressure is retrieved (p+ = p− ).
- 50. 34 Computational Methods for Fracture in Porous Media REFERENCES Brooks, R.H., Corey, A.T., 1966. Properties of porous media affecting fluid flow. ASCE Journal of the Irrigation and Drainage Division 92, 61–88. D’Angelo, C., Scotti, A., 2012. A mixed finite element method for Darcy flow in fracture porous media with non-matching grids. ESAIM: Mathematical Modelling and Numer- ical Analysis 46, 465–489. de Borst, R., Réthoré, J., Abellan, M.A., 2006. A numerical approach for arbitrary cracks in a fluid-saturated porous medium. Archive of Applied Mechanics 75, 595–606. Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P., 2014. A reduced model for Darcy’s problem in networks of fractures. ESAIM: Mathematical Modelling and Numerical Analysis 48, 1089–1116. Irzal, F., Remmers, J.J.C., Huyghe, J.M., de Borst, R., 2013. A large deformation formula- tion for fluid flow in a progressively fracturing porous material. Computer Methods in Applied Mechanics and Engineering 256, 29–37. Khoei, A.R., Moallemi, S., Haghighat, E., 2012. Thermo-hydro-mechanical modelling of impermeable discontinuity in saturated porous media with X-FEM technique. Engi- neering Fracture Mechanics 96, 701–723. Lewis, R.W., Schrefler, B.A., 1998. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, second ed. Wiley Sons, Chichester. Meschke, G., Grasberger, S., 2003. Numerical modeling of coupled hygromechanical degradation of cementitious materials. ASCE Journal of Engineering Mechanics 129, 383–392. Mohammadnejad, T., Khoei, A.R., 2013. Hydro-mechanical modelling of cohesive crack propagation in multiphase porous media using the extended finite element method. International Journal for Numerical and Analytical Methods in Geomechanics 37, 1247–1279. Réthoré, J., de Borst, R., Abellan, M.A., 2007. A discrete model for the dynamic propa- gation of shear bands in a fluid-saturated medium. International Journal for Numerical and Analytical Methods in Geomechanics 31, 347–370. Réthoré, J., de Borst, R., Abellan, M.A., 2008. A two-scale model for fluid flow in an un- saturated porous medium with cohesive cracks. Computational Mechanics 42, 227–238. Reynolds, O., 1886. On the theory of lubrication and its application to Mr. Beauchamp tower’s experiments, including an experimental determination of the viscosity of olive oil. Philosophical Transactions of the Royal Society of London 40, 191–203. Schrefler, B.A., Scotta, R., 2001. A fully coupled dynamic model for two-phase fluid flow in deformable porous media. Computer Methods in Applied Mechanics and Engineer- ing 190, 3223–3246. Segura, J.M., Carol, I., 2008a. Coupled HM analysis using zero-thickness interface elements with double nodes. Part I: Theoretical model. International Journal for Numerical and Analytical Methods in Geomechanics 32, 2083–2101. Segura, J.M., Carol, I., 2008b. Coupled HM analysis using zero-thickness interface ele- ments with double nodes. Part II: Verification and application. International Journal for Numerical and Analytical Methods in Geomechanics 32, 2103–2123. van Genuchten, M.T., 1980. A closed-form equation for predicting the hydraulic conduc- tivity of unsaturated soil. Soil Science of America Journal 44, 892–898. Witherspoon, P.A., Wang, J.S.Y., Iwai, K., Gale, J.E., 1980. Validity of cubic law for fluid flow in a deformable rock fracture. Water Resources Research 16, 1016–1024.
- 51. CHAPTER 3 Fracture Mechanics In this chapter we give a concise summary of fracture mechanics, in partic- ular of linear-elastic fracture mechanics and of cohesive-zone models. It is not intended to present a full treatment of fracture mechanics, but it should serve the purpose of outlining important concepts that have implications for, or will be used in, the ensuing numerical formulations. 3.1 LINEAR ELASTIC FRACTURE MECHANICS The seminal papers of Inglis (1913) and Griffith (1921) mark the start of the development of linear elastic fracture mechanics as a branch of engi- neering mechanics which has contributed much to the understanding of the propagation of initial flaws in structures. The solution of displacements and stresses around elliptical holes in elastic bodies by Inglis (1913) was the basis for the observation of Griffith (1921) that, for the limiting case that the elliptical hole reduces to a crack, the stresses at the crack tip be- come singular, which made him consider crack propagation from an energy perspective. Indeed, the stresses at a sharp crack tip are singular, and a tradi- tional strength criterion, where the maximum stress is compared with the tensile strength of the material, say ft, alone, then no longer suffices to assess whether crack propagation will occur or not. Instead, the fracture energy, that is the energy needed to create a unit area of crack, plays a central role. The idea is simple, and departs from considering an existing crack, say of a finite length a, in an infinite plate which is composed of a linear elastic material, Fig. 3.1, and is subject to a far-field all-around stress σ∞. The difference between the stored elastic energy in the plate and the surface energy at the crack surface equals (Broek, 1989; Hertzberg, 1996; Bažant and Planas, 1998): U = − πσ2 ∞a2t E + 4atγs, (3.1) with t the thickness of the plate and γs the specific surface energy; E = E for plane-stress conditions and E = E/(1 − ν2) for plane-strain conditions, E being Young’s modulus and ν being Poisson’s ratio. An increase in crack Computational Methods for Fracture in Porous Media DOI: http://dx.doi.org/10.1016/B978-0-08-100917-8.00003-4 Copyright © 2018 Elsevier Ltd. All rights reserved. 35
- 52. 36 Computational Methods for Fracture in Porous Media Figure 3.1 A crack of length 2a in an infinite medium under an all-round stress σ∞. length of δa leads to a change of the energy term: δU = ∂U ∂a δa = − 2πσ2 ∞at E δa + 4tγsδa. (3.2) For quasi-static crack growth the condition δU = 0 (3.3) must hold, so that, in view of Eq. (3.2), the critical stress becomes: σ∞ = 2Eγs πa . (3.4) Very few materials are perfectly brittle, and energy is also dissipated in a vanishingly small area close to the crack tip, e.g. due to small-scale plas- tic yielding, micro-cracking, or fiber-bridging. For this reason, the second term in Eq. (3.2) is better modified by introducing the (strain) energy re- lease rate G, which incorporates such effects. With G instead of the specific surface energy γs, the previous equation can be replaced by: σ∞ = EG πa . (3.5) Owing to local inelastic effects, G can be history or specimen size depen- dent. In many practical purposes, such dependencies can be omitted, G is assumed constant and can be conceived as a macroscopic material parame- ter. Under these assumptions it is often denoted by Gc, the critical energy
- 53. Fracture Mechanics 37 release rate, or by Gf , the fracture energy, which is the energy (in J/m2) that is required to create a unit area of crack. From Eq. (3.5) it can be deduced that an internal length scale is present in linear elastic fracture mechanics (as well as in the cohesive approach to fracture, to be discussed in the next section). Indeed, rewriting Eq. (3.5) shows that: a = E Gc πσ2 ∞ . (3.6) Based on this observation, the internal length scale is commonly defined as: ch = EGc f 2 t . (3.7) It is noted that the presence of an internal length scale implies that there is a size effect in fracture mechanics, which is absent in a pure strength-based theory. With reference to Fig. 3.1, the normal stress in the y-direction along the line y = 0 is given by, e.g., Bažant and Planas (1998): σyy = σ∞ x √ x2 − a2 . (3.8) Noting that x2 − a2 = (x − a)(x + a), using the coordinate transformation r = x − a we can rewrite this equation as σyy = σ∞ r + a √ 2r √ r/2 + a . (3.9) Developing (r + a)/ √ r/2 + a in a Taylor series, we obtain σyy = σ∞ √ πa √ 2πr 1 + 3r 4a − 5r2 32a2 + ... . (3.10) For near-tip behavior (r a) the term between the parentheses vanishes, and we obtain: σyy = KI √ 2πr , (3.11) where the customary definition for the mode-I stress intensity factor has been adopted: KI = σ∞ √ πa, (3.12)
- 54. 38 Computational Methods for Fracture in Porous Media which depends on the geometry and loading conditions. Eq. (3.11) shows the √ r-singularity which is typical of a sharp crack in a linear-elastic body. Expression (3.11), which employs the stress intensity factor KI (similarly, the mode-II or mode-III stress intensity factors KII and KIII can be used in the corresponding loading configurations), is a local fracture criterion. It can be related to the global energy approach of Griffith (1921) using the following argument. We substitute the expression for the near-tip normal stress σyy into the work that is needed to advance the crack by δa: δU = 1 2 t δa 0 σyywdr, (3.13) with w = 32 π KI E √ δa − r (3.14) the crack opening, cf. Bažant and Planas (1998), and carry out the integra- tion to yield δU = K2 I E tδa. (3.15) Setting this result equal to the global change in energy, δU = Gctδa, then gives the famous Irwin result: Gc = K2 I E or, equivalently KI = GcE, (3.16) which allows to relate Griffith’s global energy approach (Griffith, 1921) and Irwin’s local approach to fracture (Irwin, 1957). Away from the y = 0 line, the near-tip stresses under mode-I loading can be written as σij = KI √ πr SI ij(r,θ), (3.17) with SI ij(r,θ) a dimensionless function that is independent of the structural size and the load magnitude, r being the distance to the crack tip, and θ the angle between the x-axis and the local r-direction, see Fig. 3.1. Under mode-I loading, SI ij(r,θ) attains the following format: SI xx(r,θ) = cos θ 2 1 − sin θ 2 sin 3θ 2
- 55. Exploring the Variety of Random Documents with Different Content
- 56. formation and alteration of school districts, the records of the county go back as far as 1849, in which records Mr. Brown signed as school fund commissioner. However, this may be attributed to the fact that previous to 1852, Mr. Brown was clerk of the county board of commissioners, and the duties of the school fund commissioner devolved upon that office at that time; consequently the presumption is that when he entered upon his duties as school fund commissioner, and began to make up his records, he naturally took from the records of the clerk of the board of county commissioners the things which belonged to the office. Mr. Brown held this office for three full terms, also about six or eight months additional time, although Albert A. Mason was elected and qualified as county superintendent of schools in the election of April, 1858. Mr. Brown served until January, 1859, as school fund commissioner. This came from the fact that the county superintendent was provided for by the Statute of '58, the election taking place on the first Monday in April, but at this time some of the duties devolved upon the county superintendent. By chapter 36 of the Statutes of 1858, section 1, the office of the school fund commissioner was continued until the county treasurer was elected. The presumption is, therefore, that for about six months we had both a school fund commissioner and a county superintendent of schools in this county. It is possible, also, that Mr. Brown served as a sort of triumvirate, as he was school fund commissioner by election, for the simple reason that Mr. Mason may not have qualified until three or four days after the time set; he was also school fund commissioner by the extension Statute, and county superintendent of schools from the fact that his successor had not qualified; in fact in some of the school reports, he signed as both school fund commissioner and county superintendent. However, Mr. Mason entered upon his duties and served as superintendent of schools for one term, when Ira G. Fairbanks (who by the way, still lives in Mount Vernon) was elected as his successor.
- 57. It is a difficult matter to state who was the first school teacher in the county. In 1839 several schools were in operation. In July of that year Elizabeth Bennett taught in Linn Grove, and later that same year Judge Greene taught at Ivanhoe. One of the noted schools of the early day was the one known as the Buckskin School, in Linn Grove, so named because teacher and scholars alike attended clad in buckskin suits. The first school district was formed in 1840 with Marion as its center. After that school houses sprang up in every direction. The buildings were constructed out of logs; the seats were benches hewn from slabs or logs, and so were the desks. Colleges early sprung up in the county. Of the three that flourished here more or less at one time, the history of two—Cornell and Coe— are given at length. These institutions are now in splendid condition. The third institution that in its day was a power for excellence in educational lines was Western, founded in 1856 on the borders of Johnson county at the little town of Western, in College township. Of this institution the late Jesse A. Runkle, some years ago, wrote as follows: In January, 1856, Iowa City became the western terminus of the only railroad in the state, and no other was built within a couple of years. The fine country surrounding Western, would easily lead one to believe that the early plan was feasible, to make the school an industrial one, where deserving young men could make their way through school by devoting some of their time to agricultural work. But Western was unfortunate in two things: First, none of the railroads that were built in Iowa, ever came near the town. It seems as if a Nemesis had brooded over the place, for even the interurban now being built between Cedar Rapids and Iowa City swerves from a direct line, and misses both Western and Shueyville by about a mile. Second, the surrounding country began to be possessed by a population that in the main had little or no sympathy with religious education, and the older generations were alien in thought and
- 58. temper to our American institutions. These things made the task of maintaining the college at that point a most heroic and arduous work. SHOWING THE TWO CESSIONS AS AT PRESENT DIVIDED After some years of struggle, the college was removed to Toledo, where it now wields an influence second to none in the state.
- 59. One of the early educational centers in Linn county was the private school established in 1850 in the Greene Bros. block, which stood on the corner of First street and First avenue, Cedar Rapids, where now stands the building owned by Sunshine Mission. It was founded by Miss Elizabeth Calder, a native of New York, and who in 1855 married R. C. Rock, the first hardware dealer in the city, who came here from Burlington and whose place of business was located on First street a few doors south of the corner of First avenue. This school prospered and was conducted by Miss Calder for four years when it was discontinued. One of the first, if not the very first, teacher in Cedar Rapids was Miss Susan Abbe, daughter of the old pioneer. She taught in this city in 1846, the superintendent being Alexander Ely. Miss Emma J. Fordyce, at present a teacher in the Cedar Rapids high school, contributes to this work the following sketch of early schools in the county, and more particularly in the city of Cedar Rapids: It is not often in this changing country that a person lives a lifetime in one community and sees the schools grow from their beginning. This has happened to me. Of the early country schools but two memories remain: a visit in the summer, and one in the winter. There remains an impression of very homely school houses, equally homely surroundings, and very little comfort without or within. It is a standing wonder that even now an Iowa farmer is much more likely to provide an up-to-date fine building for his cattle than a beautiful, well-ordered school-house for the education of his children. A little has been done, but by far too little. Early Cedar Rapids was a little village surrounded by groves of oaks, crab-apple, plum, and everywhere the climbing wild grape. Between these groves were the sand hills on which grew vast quantities of sand-burs. Where the Methodist church now stands was a hill which sloped toward the railroad. Where the old Presbyterian church was, the children coasted down 'Pepper Grass Hill;' and where Mr. Crozer's florist establishment is, was a deep and wide pond which,
- 60. on occasions of heavy rain, furnished water for rafts made from bits of sidewalk. The earliest school was on the site of the present Granby building, but of that school I have no personal knowledge. The first school building in my memory was the three-story one which was erected in 1856. It had a white cupola, white trimmings to the windows, with a high, solid board fence, painted red, surrounding it. An iron pump at the side furnished refreshment to the spirit and ammunition for the wetting of people. On the lower floor on the side next the railroad, Miss Elizabeth Shearer taught the children. She was a woman of fine family, fine attainments, and of great patience of spirit. Superintendent Ingalls was in charge of the school at that time. C. W. Burton followed him the next year. His school board was A. C. Churchill, president; Benjamin Harrison, treasurer; J. W. Henderson, vice-president; D. A. Bradley, secretary. These were assisted by three directors, J. F. Charles, W. W. Smith, E. E. Leach. Mr. Harrison had a unique way of collecting taxes from the delinquent foreign citizens to whom our system of collecting them was a dark puzzle; when they refused to pay, he notified them that on a certain day if the taxes were not forthcoming, he would sell everything they had and apply the proceeds to tax payment. The auction was often begun, but never finished, as the taxes were always forthcoming. Mrs. E. J. Lund was one of the earliest of Cedar Rapids teachers. For many years her inspiring example and her patient work developed good children out of bad, and she finished her life's work by taking care of all the poor and unfortunate of the county. The Cedar Rapids superintendents were Professor Humphrey, 1861-4, Professor Ingalls, 1864-5, C. W. Burton, 1865-70, J. E. Harlan, now president of Cornell, 1870-5, F. H. Smith, the latter part of 1875, J. W. Akers, 1875-81, W. M. Friesner, 1881-5, L. T. Weld, 1885-6, J. P. Hendricks, 1886-90, J. T. Merrill, 1890-1901, J. J. McConnell, 1901—, twelve men in thirty-four years. The list shows plainly the growing tendency to keep a superintendent for long periods at a time.
- 61. The high school principals show the same tendency; A. Wetherby, from 1870-1, E. C. Ebersole, 1872-73, W. A. Olmsted, 1871-2, Miss Mary A. Robinson, 1873-86, Miss A. S. Abbott, 1886—. The original high school building contained four rooms. In 1876 it had a corps of three teachers: Miss M. A. Robinson, Miss E. J. Meade, Miss Estella Verden, and had an attendance of 106 pupils; it now has twenty teachers with an attendance of 838 pupils. In 1876 there were five buildings in the city; there are now sixteen. Of the teachers thirty-one in number in 1876, there are two left: Miss Emma Forsythe and Miss Emma J. Fordyce. In 1876 the total number of pupils handled by thirty-one teachers was 1,752. In 1911, with 181 teachers, there are 6,122 pupils, not quite six times as many teachers, but showing a smaller average number to each teacher. Evidently the school-houses have always been crowded, since the superintendent's report of 1876 says: 'We have in the school district five school buildings, and these are taxed to their utmost to accommodate the pupils already enrolled.' He also remarks pensively: 'In your wisdom for the coming year, you have reduced the salaries of your teachers, and in some cases the reduction has been such that some of your best teachers have been compelled to seek employment elsewhere.' Since no following superintendent makes the same complaint, it is evident that school boards do improve. As to salaries, the salary of the superintendent in 1883 is given as $1,000; in 1911 as $3,000, which means the magnificent increase of $42 a year; not a great temptation. The salaries of the teachers increase in the same period about $25 a year. Comment is unnecessary. As to the high school, the graduates of 1873 to 1885 were but eleven pupils, with nine times as many in 1908. Amongst the older and pioneer high school teachers were Mr. Wetherbee, Miss Ella Meade, and Miss Ada Sherman, who afterward decided to doctor bodies instead of minds, as it paid much better. Mr. Olmsted, the principal of 1872, who left Cedar Rapids in 1873 to found a business
- 62. in Chicago, died a hero. He lost his life in his burning building trying to save his bookkeeper. The tendencies in school work are shown by the fact that the reports of the early superintendents are largely lists of members of the school board, while the later reports give large tabulations of expense. It is to be regretted that Iowa has not adopted a series of uniform reports, giving items almost impossible to discover as these reports are at present made out. The older schools report seventy- two pupils to a primary teacher. The newer reports are silent on the subject. Since efficiency comes in handling the right number of pupils, it would certainly be wise to keep a careful account of this item. The courses of the schools show the growth in public service. The courses of the high school in 1876 are twenty; those of the high school in 1910, eighty-three. All of the older and more prominent citizens served as school directors at one time or another. In 1858 J. L. Enos was president of the board, Freeman Smith, secretary, W. W. Smith, vice-president, J. T. Walker, treasurer, W. W. Walker, director. In 1859 the names of R. C. Rock, E. H. Stedman, J. P. Coulter, and J. M. Chambers appear. In 1860, S. C. Koontz, Henry Church, William Stewart, J. H. Camburn, and William Richmond served. In 1861, W. W. Smith, George M. Howlett, Henry Church, William H. Merritt, A. C. Churchill, and S. L. Pollock directed affairs. In 1862 E. G. Brown, A. C. Churchill, J. F. Ely, George M. Howlett elected Mr. Humphrey superintendent of schools. His reputation seems to have been that of a man of great strength and the bad big boys stood in awe of him accordingly. C. W. Burton, the superintendent of 1865, was noted for his cleverness in mathematics, and his deep interest in horticulture. All of these early directors, superintendents, and teachers were hard workers and great optimists. History has confirmed that optimism, and from the services of these men developed a race of ambitious, energetic, moral citizens to whom the present Cedar Rapids owes a great debt of gratitude.
- 63. Through the courtesy of County Superintendent Alderman we are enabled to give below some interesting data regarding our schools: In 1873 the number of school corporations in the county was 42, increased to 87 in 1909. The number of ungraded schools in the former year was 178, and 166 in the latter year. The average number of months the schools were in session has increased from 6.6 in 1873 to 8.9 in 1909, and the average compensation from $39.78 to $73.50 for males, and from $26.33 to $50.85 for females. The number of female teachers employed in 1873 was 244, and in 1909, 503. The number of male teachers was 90 and 40 respectively. In the matter of attendance there has been a vast betterment. In 1873 there were 460 boys and 544 girls between the ages of seven and fourteen not in school. In 1909 these numbers were 29 and 17. The value of school property in 1873 was $240,105; in 1909, $814,300. The value of school apparatus was $2,309.50 in 1873, and in 1909, $20,035.25. There were in 1873 in the school libraries 482 volumes, which was increased to 17,079 in 1909. There are now between twenty-five and thirty fine school buildings in the country districts. They are modern in all respects, being supplied with slate blackboards, hardwood floors, ventilators, cloak rooms, bookcases and cupboards. Several have furnaces and cloak rooms in the basements. Some of the buildings are supplied with telephones, making it possible for the county superintendent and patrons to communicate direct with the school. The plans and specifications for these buildings are owned by the county, and are furnished gratis to the school districts wishing to build. All of these school-houses except two or three are not only provided with libraries, cloak rooms, etc., but are also provided with a good organ. This year there is being installed a hot air ventilating system which keeps the warm air pure, the cold air being taken directly from the
- 64. outside and passed through the hot air radiators before being allowed to enter the school room. CORNELL COLLEGE IN 1865
- 65. CHAPTER XXIII Historical Sketch of Cornell College BY WILLIAM HARMON NORTON, ALUMNI PROFESSOR OF GEOLOGY, CORNELL COLLEGE Linn county may well take pride in the history of her oldest school of higher education, founded in 1853, when the county held but 6,000 people. But the beginnings of Cornell College are of more than local interest; they are thoroughly typical of America and of the West. Cornell was founded in much the same way as were hundreds of American colleges along the ever advancing frontier of civilization from Massachusetts to California—a way which the world had never seen before and will never see again. THE FOUNDATION AND THE FOUNDER Cornell owes its inception to a Methodist circuit rider, the Rev. George B. Bowman, a North Carolinian by birth, who came to Iowa from Missouri in 1841, three years after the territorial organization of the commonwealth. This heroic pioneer, resourceful, far seeing, and sanguine of the future, eminent in initiative and in the power of compelling others to his plans, was one of those rare men to whom the task of building states is intrusted. He was not himself a college man, but with him education was a passion. To found institutions of higher education he considered his special mission. Hardly had he been appointed as pastor of the church at Iowa City in 1841 when he undertook the building of a church school, called Iowa City College. In 1845 Rev. James Harlan, a local preacher of Indiana, was chosen president, and with one assistant opened the school in 1846.
- 66. The next year Mr. Harlan was elected state superintendent of public instruction, and the college was closed never to be re-opened. It had at least served to bring to the state one of its most distinguished citizens, afterward to be honored with the United States senatorship and the secretaryship of the interior. Meanwhile Mr. Bowman had been appointed presiding elder of the Dubuque district, which then included much of east-central Iowa. The failure of the premature attempt at Iowa City had not discouraged him; he awaited the favorable opportunity he still looked for—suitable local conditions for a Christian college in the state. It is a long-told legend, even if it be nothing more than legend, that when Elder Bowman came riding on horseback to the Linn Grove circuit, he stopped on the crest of the lonely hill on which Mount Vernon now stands. From its commanding summit vistas of virgin prairie and primeval forest stretched for ten and twenty miles away. Here there fell upon him, the circuit preacher, the trance and vision of the prophet. He saw the far-off future; he heard the tramp of the multitudes to come. Dismounting, he kneeled down in the rank prairie grass and in prayer to Almighty God consecrated this hill for all time to the cause of Christian education. And it is a matter of authentic history that in the spring of 1851 Elder Bowman and Rev. Dr. A. J. Kynett, in the parsonage at Mount Vernon, planned together for the early founding and upbuilding of a Christian college on this site. With the characteristic initiative of the Iowa pioneer, Bowman did not wait for authority to be given him by anybody, for articles of incorporation to be drawn up, or even for a title deed to the land on which the college was to stand. Early in 1852 he laid his plans for the launching of the school. On the Fourth of July of this year an educational celebration was held at Mount Vernon, which drew the farmers for miles about the town, and other friends of the new enterprise from Marion and Cedar Rapids, Anamosa, Dubuque, and Burlington. The oration of the day was delivered by State Superintendent Harlan on the theme of Education, and at its close ground was broken formally for the first building of the college. A month later a deed was obtained for the land and the following
- 67. September the guardianship of the infant school was accepted under the name of the Iowa Conference Seminary, by the Methodist Episcopal church. In this highly democratic manner Cornell College was founded by the people as an institution of higher learning, which should ever be of the people and for the people. It was born on the anniversary of the nation's natal day, and was to remain one of the highest expressions of patriotism and civic life. Christened by the head of the educational interests of the young commonwealth, supported by its citizens, protected by a charter from the state, and exempt as a beneficent institution of the state from contributing by taxation to the support of other institutions, the college was thus begun as a state school in a very real sense. One can not read the early archives of the college without the profoundest admiration for the pioneers, its founders. Avid of education to a degree pathetic, they depended on no beaurocracy of church or state; they waited for no foreign philanthropy to supply their educational needs. They laid the foundations of their colleges with the same free, independent, self-sufficing spirit with which they laid their hearthstones, and they laid both at the same time. THE IOWA CONFERENCE SEMINARY In January, 1853, the first meeting of the board of trustees was held, and in the fall of the same year the school was opened in the old Methodist church at Mount Vernon. Before the end of the term a new edifice on the campus was so far completed that it was available for school purposes and on the morning of November 14, 1853, the school met for the last time in the old church and after singing and prayer the students were formed in line and walked in procession with banners flying, led by the teachers, through the village, and took formal possession of what was then declared to be a large and commodious building.[J]
- 68. The first catalog—a little time-stained pamphlet of fifteen pages— lists the following faculty: Rev. Samuel M. Fellows, A. M., professor of mental and moral science and belle lettres. Rev. David H. Wheeler, professor of languages. Miss Catherine A. Fortner, preceptress. Miss Sarah L. Matson, assistant. Mrs. Olive P. Fellows, teacher of painting and embroidery. Mrs. Sophia E. Wheeler, teacher of instrumental music. The first board of trustees is also noteworthy: Rev. George B. Bowman, president, Mount Vernon; E. D. Waln, Esq., secretary, Mount Vernon; Rev. H. W. Reed, Centerville; Rev. E. W. Twining, Iowa City; Rev. J. B. Taylor, Mount Vernon; Jesse Holman, North Sugar Grove; Henry Kepler, North Sugar Grove; William Hayzlett, Mount Vernon; A. I. Willits, Mount Vernon. The roster of students enrolls 104 gentlemen, and 57 ladies. Among them are familiar and honored names, some of which are to reappear in all later catalogs of the school, either as students of the second and third generation, or as trustees and members of faculty. Four Rigbys, for example, were students in 1853. In 1910 the catalog lists three Rigbys, one a student and two members of the faculty. The first catalog contains the names of no less than nine Keplers as students, six stalwart young men from North Sugar Grove and their three sisters. Four Walns are enrolled from Mount Vernon, two Farleys from Dubuque and two Reeders from Red Oak. In 1853 the population of the entire state was only about 300,000. Not a railway had been projected west of the Mississippi river. And yet the scattered settlements sent across the unbroken prairie and the unbridged rivers no less than 161 students to the young school on this the first year of its existence. The most important route to Mount Vernon was the military road extending from Dubuque to Iowa City. Both towns contributed their quota of students, Dubuque
- 69. sending no less than twelve, although the entire population of Dubuque county was then, less than 16,000. Considering the difficulty of communications, the poverty of the pioneers, the wide extent of the sphere of influence of the school is remarkable. Students were drawn this first year from as far to the northeast as Elkader and Garnavillo. They came from Dyersville and Independence, from Quasqueton and Vinton, from Marengo, Columbus City, West Liberty, and Burlington. Muscatine alone sent seven students. This town was at the time the point of supply for Mount Vernon, and the materials for the first building of the college except such as local saw mills and brick kilns could supply were hauled from that river port.[K] Students came also from Davenport, Le Claire, Princeton, and Blue Grass in Scott county, from Comanche, and from the pioneer settlements of La Motte and Canton in Jackson county. The eight hundred students of Cornell today reach the school from all parts of the state and the adjacent portions of our neighboring states by a few hours swift and comfortable ride by rail. But who shall picture in detail the long and adventurous journeys in ox cart and pioneer wagon and perchance often on foot of the boys and girls of 1853—the climbing of steep hills, the fording of rivers, the miring in abysmal sloughs, the succession of mile after mile of undulating treeless prairie carpeted with gorgeous flowers stretching unbroken to the horizon, the camp at night illuminated by distant prairie fires, until at last a boat shaped hill surmounted by a lonely red brick building lifts itself above the horizon, and the goal of the long journey is in view! No doubt there were other hardships awaiting these students after their arrival. Rule No. 1 of the new school compelled their rising at five o'clock in the morning. They were expected to furnish their own beds, lights, mirrors, etc., when boarding in Seminary Hall. It is interesting to note that they paid for tuition $4.00 and $5.00 per quarter, and for board from $1.50 to $1.75 per week. The next year the steward's petition to the board of trustees that he be allowed to put three students in each of the little rooms was granted with the proviso that he furnish suitable bunks for the same. The catalog's
- 70. statement regarding apparatus is a guarded one: The Institution is furnished with apparatus for illustrating some of the most important principles of Natural Science. As the wants of school demand, additions will be made to this apparatus. And that regarding the library is wholly prophetic: It is intended to procure a good selection of readable and instructive books, by the commencement of the next academic year, to which the students will have access at a trifling expense. With these books as a nucleus, a good library will be accumulated as rapidly as possible. Donations of good books are solicited from friends of the institution. In the next catalog it is stated that a small but good selection of readable and instructive books has been procured, the remainder of the statement being the same as that of the first year. This statement appeared without change in all succeeding catalogs during the remainder of the first decade. THE FIRST DECADE As early as 1855 the articles of incorporation were amended changing the name of the institution to Cornell College, in honor of W. W. Cornell and his brother J. B. Cornell, of New York City, men prominent in business and widely known for their benevolences to various enterprises of the church. It will be noted that Cornell College was thus named several years before the founding by Ezra Cornell, of Cornell University at Ithaca, N. Y. The first year of the school under the new collegiate régime was that of 1857-1858. Rev. R. W. Keeler of the Upper Iowa Conference was made president, Principal Fellows of the Seminary taking the professorship of Latin. Two years later President Keeler reentered the more congenial work of the ministry, and Principal Fellows was elected president of the college, a position which he held most acceptably until his death on the day after commencement June 26, 1863, thus completing a full decade of years of service in the school.
- 71. President Fellows had come to Cornell from the Rock River Seminary at Mount Morris. His character and the quality of his work left lasting impressions on his pupils at both institutions. Thus Hon. Robert R. Hitt, of Illinois, writes of him as follows: He was a diligent, acute, and active student, and his personal character was admirable. It is the fortune of few men to exercise so wide and prominent an influence from a position which, to the ambitious, is not considered eminent. And Senator Shelby M. Cullom has written: I regard Professor Fellows as one of the best men I ever knew. I said it when I was under him at school, and now that I am over seventy years of age, I say it now. He was strong, honest-hearted, full of kindness, and a splendid teacher. His colleague at Cornell, Dr. David H. Wheeler, described him as a man sweet-spirited, pure-minded, of fine executive ability, a rarely qualified teacher, a patient sufferer, a tireless worker, a model friend. A word may be said as to the members of President Fellows's faculty: Miss Catharine A. Fortner, a graduate of Cazenovia Seminary, N. Y., was sent out in 1851 by Governor Slade, of Vermont, as a missionary teacher to Iowa. Her success near Tipton was so marked that she was chosen as the first preceptress of the institution. In 1857 she resigned to marry Rev. Rufus Ricker, of the Upper Iowa Conference. Wm. H. Barnes, professor of languages in 1854-1855, resigned to accept a professorship in Baldwin University, Ohio, and is known as author of several works in history and politics. His successor, Rev. B. W. Smith, after leaving the school in 1857 became pastor of several of the largest churches in northern Indiana, and president of Valparaiso College. Dr. David H. Wheeler, professor of languages in 1853-1854, and professor of Greek from 1857 to 1861, when he was appointed U. S.
- 72. consul to Genoa, was a brilliant and versatile man, author of a number of books, professor for eight years at Northwestern University, editor for eight years of the New York Methodist, and for nine years president of Allegheny College. The brother of President Fellows, Dr. Stephen N. Fellows, has a large place in the educational history of Iowa. He assisted his brother in laying the foundation of Cornell College, being professor of mathematics from 1854 to 1860, and later occupied the chair of mental and moral science and didactics at the State University of Iowa for twenty years. On account of her long connection with the college, from 1857 to 1890, Miss Harriette J. Cooke exerted a more potent influence on the institution than any of her colleagues of the first decade. Miss Cooke came to Cornell from Hopkinton, Massachusetts, and brought the best culture for women which New England then afforded, as well as an exceptionally forceful personality, and rare natural aptitudes for her profession. From 1860 to the time of her resignation she was dean of women, and her influence for good on the thousands of young women under her care is incalculable. After long service as an instructor she was made a full professor in 1871, the first woman in America, it has been said, to be thus honored. Her chair for fifteen years was history and German, and after 1886 history and the science of government. On leaving the college she studied the methods of deaconess work in England, wrote a book upon the subject, and returning to her native land became one of the leaders in this new department of social service. For many years she has been closely connected with the University Settlement of Boston. On the recent celebration of her eightieth birthday she received hundreds of letters of loving congratulation from her former students of Cornell, and each of these letters was answered by her painstakingly and at length.
- 73. A STREET SCENE IN MARION THE DANIELS HOTEL, MARION
- 74. The first ten years of the institution were marked by a singularly rapid growth, considering the fact that they included the darkest days of the Civil war, when nearly every male student was drawn from the college halls to the service of his country. At the end of the decade the faculty numbered eight professors and instructors, and 375 students were enrolled, fifty-one of whom were in college classes, the largest enrollment of collegiate students in the state, unless at the State University. The assets of the institution amounted to $50,000 in notes and pledges, a campus of fifteen acres, and two brick buildings which compared not unfavorably with other college buildings in the west and with the earlier halls of Harvard. In a large measure this exceptional growth was due to Elder Bowman, to his initiative and wide and powerful influence. The chief problem then as now was one of sustenance, and as a college beggar Bowman was incomparable. He travelled over the settled portions of the state, winning men to his cause by a singular personal charm, and enticing even out of poverty money, promissory notes at altitudinous rates of interest, farm produce, live stock and poultry, household furniture and jewelry. His barnyard at Mount Vernon was continually stocked with horses, cattle, and chickens— votive offerings to the cause of higher education. A citizen of the town once told me how under some mesmeric influence he bought at high price from Elder Bowman an old book case and coal scuttle, begged somewhere for the school. This prince of college beggars once returned from Dubuque with a silver watch which he had plundered off the person of an eminent minister of that city. FROM 1863 TO 1910—GROWTH IN RESOURCES Nothing is so tame as the history of a college once the interesting period of its childhood is over, and the history of Cornell is exceptionally uneventful among colleges. No building has been destroyed by fire or tornado. No famous lawsuit against the school
- 75. has been defended by some Webster among the alumni. None of the faculty has won notoriety by sensational speech or erratic morals. The salient feature of the forty-seven years since 1863 is a marvelous growth unparalleled in some respects in the history of education. The campus has been enlarged by addition after addition until now it measures sixty acres, including the larger part of the long hill and wide athletic fields along its northern base. To the two first buildings, still used, one for the chemical, biological and physical laboratories and the other for class rooms and society halls, there have been added South Hall, built in 1873 and now used for the engineering and geological laboratories; the Chapel, completed in 1882, a stately Gothic structure of stone, containing the auditorium, seating about 1,500, a smaller audience room, the museum, and several music rooms; Bowman Hall, built in 1885, as the well appointed home of ninety-two young women; the library dedicated in 1905, the gift of Andrew Carnegie; the alumni gymnasium in Ash Park, built in 1909, a noble structure, one of the largest of the kind in the state, besides several minor buildings used for allied schools and professors's residences. The material equipment has made a phenomenal growth, until several of the scientific laboratories are reckoned among the best in the Central West, and the library, numbering 35,000 volumes, ranks as third in size among the university and college libraries in the state, and second to but one of the city libraries of Iowa. The museum includes several collections which rank among the largest in the west: the Kendig collection of minerals, the Norton collection of fossils, and the Powers collection in American anthropology. GROWTH IN ATTENDANCE From the beginning Cornell has been a relatively large school measured by the number of its students, and its growth the last decades forbids it longer to be called a small college. Indeed, for many years it has maintained its place as the largest denominational
- 76. college, or among the two or three largest, in the United States west of the Great Lakes, reckoned by the number of students of collegiate rank. The attendance has steadily risen until, in 1909-1910, 741 students were enrolled, 450 of them being in the college of liberal arts. The steady growth in numbers of collegiate students evidences the satisfaction which the school has given to its patrons, and an ever widening influence and power. Moreover, it has increased the efficiency of the school by the inspiration of numbers and the intensity of competition in all departments of college life. By bringing together students from all parts of the state and scores from other states, some with the polish of the city and others with the sturdy strength of the country, it has escaped the narrowness of the provincial and has attained something akin to cosmopolitanism. To make Cornell an institution state-wide in its patronage and influence was the evident purpose of its founders. Nothing was further from their minds than a local college for the students of a town or county, or one drawing its patronage from a few contiguous counties. The trustees have been chosen widely over the state and the attendance from all parts of Iowa has been surprisingly large, considering the many excellent colleges the state supports. In an investigation made a few years since of the geographic distribution of the students it was found that 41 per cent of the collegiate students came from beyond the borders of the patronizing conference, and the counties west and south of the Des Moines river furnished 20 per cent of the students in attendance from the state. The college has thus grown to have a state-wide field. THE STRATEGIC POSITION In explaining the growth of Cornell college we must recognize, of course, that it has grown up with the country. We must relate the growth of the school directly to the material prosperity of this land of corn and swine, to the marvelously fertile soil and to the era of expansion in which our history falls. The fact remains, however, that
- 77. the college has obtained somehow a good deal more than its due share in the general advance. While the population of the state increased 330 per cent from 1860 to 1900, the collegiate attendance at Cornell increased 720 per cent. The college has grown more than twice as fast as has the state, and that notwithstanding the numerous good schools which have sprung up to share its patronage. We can not doubt that much of the success of the school has been due to its strategic position. It is located in a suburban town of the chief railway center of eastern Iowa. From Cedar Rapids long iron ways, like the spokes of a wheel, reach in all directions to the limits of the state and beyond, and bring every portion of the commonwealth and the adjacent parts of our neighboring states within a few hours ride of Cornell college. It is located also in east Central Iowa, an area of the state the first to be settled and developed, an area surpassed by none in the fertility of its soils, and the wealth which has been produced from them. To these geographic factors, advantages shared in like degree by none of the early competitors of the school, we may assign a place similar to that given such factors in explaining the growth of New York city and of Pittsburg. While the college had thus had the city's advantages of communication and markets because of its nearness to Cedar Rapids, it has retained all the peculiar advantages which inhere in a location in a village. Like Bowdoin, Dartmouth, and Oberlin, Cornell has found in the small town, rather than in the city, an ideal college environment. It has never permitted the presence of saloon or other haunt of vice. The citizens with whom the students have made their homes have been people of culture drawn to the town by its educational advantages. In all that makes for the intellectual life, in libraries and collections, in lectures and good music, and church privileges, Mount Vernon has had more to offer than perhaps any city of the state; while the temptations and distractions, the round of low amusements offered by the city, have been fortunately absent.
- 78. THE BOARD OF TRUSTEES More than geographic location, it is great men and great plans that make great schools. Let us give much credit therefore to the men who have administered the college as members of its board of trustees. Our debt to them is like that of Michigan University to its board of regents whose wise plans pushed it early to the fore among the state universities of the west and far in advance of the place to which geographic causes alone would have assigned it. Some of these were pioneers of only local fame, such as Elijah D. Waln, Henry D. Albright, William Hayzlett, Jesse Holman, Noah McKean, and Dr. G. L. Carhart, men whose memory will ever be cherished in Mount Vernon. Others were men of note in the early history of the state, such as Hon. Hiram Price, of Davenport, Jesse Farley, of Dubuque, and A. P. Hosford and W. H. Lunt, of Clinton. Especially to be noted is the long service which the trustees have given to the school. Of the members of the executive committee Col. Robert Smyth, sturdy Scotch Presbyterian, was a member for twenty-eight years until his death in 1896. On the same committee Hon. W. F. Johnston, of Toledo, long president of the board, has already served for thirty-three years. Col. H. H. Rood, another of the members of the executive committee, has served continuously as trustee since 1867, and Capt. E. B. Soper, of Emmetsburg, since 1878. Captain Soper has long been one of the most influential members of the governing board, and it is to his initiative and faith that the alumni gymnasium is due. Dr. J. B. Allbrook has served since 1874. H. A. Collin was treasurer of the college from 1860 to his death in 1892. Hon. D. N. Cooley, of Dubuque, served as trustee for twenty-four years, and Hon. W. J. Young, of Clinton, for twenty-six years, their terms of office being terminated only by death. Of the present board of trustees there may be named as among those longest in service, F. H. Armstrong, of Chicago; Hon. W. C. Stuckslager, of Lisbon; E. J. Esgate, of Marion; Maj. E. B. Hayward, of Davenport; Hon. Eugene Secor, of Forest City; Dr. Edward T. Devine, of New York; T. J. B. Robinson, of Hampton; John H. Blair, of Des Moines; Rev. W. W.
- 79. Carlton, of Mason City; Rev. E. J. Lockwood and John H. Taft, of Cedar Rapids; Hon. Leslie M. Shaw, of Philadelphia; R. J. Alexander, of Waukon; E. B. Willix, of Mount Vernon; Senator Edgar T. Brackett, of Saratoga, N. Y.; O. P. Miller, of Rock Rapids; Rev. Homer C. Stuntz, of Madison, N. J. and N. G. Van Sant, of Sterling, Ill. Among the eminent men who have served the college we must give special mention to Rev. Alpha J. Kynett, one of the pioneers of Methodism west of the Mississippi, who served on the board from 1865 to his death in 1899. Dr. Kynett was the founder of the great Church Extension society and for many years was its chief executive. In this capacity he probably built more churches than any man who has ever lived. For a third of a century he was a close friend and adviser of the college, and all his wide experience and his ability as an organizer and financier were always at its service. THE ADMINISTRATION In 1863 occurred the sad death of President Fellows, under whose superintendence the school had been organized. He was succeeded in office by William Fletcher King, a graduate of the Ohio Wesleyan University and a member of its faculty, who thus brought to Cornell an acquaintance with the scope and methods of one of the best colleges of the middle west. At the time of his election to the presidency Dr. King was professor of Latin and Greek at Cornell, and thus for the second time a president was chosen from the ranks of those actively engaged in the work of higher education rather than, as was then almost universally the custom, from those of another profession. In 1908 Dr. King resigned his office after a term of service of forty-five years. For a number of years he had thus been the oldest college president in the United States in the duration of his office. His administration was essentially a business administration, with little talk but much of doing. There was in it nothing spectacular, and no pretense, or sham. No discourteous act ever strained friendly relations with other schools. Dr. King made no
- 80. enemies and no mistakes. He was ever tactful, poised, discreet, far- seeing, winning men to the support of his wise and well-laid plans but never forcing their acceptance. The college itself is a monument to this successful business administration. For Cornell does not owe its success to any munificent gifts. Like John Harvard, W. W. Cornell and his brother left the college which perpetuates their memories little more than a good name and a few good books. No donation of more than $25,000 was received until more than forty years of the history of the college had elapsed. Whatever excellence the college has attained is due to the skill and patience of its builders and not to any unlimited or even large funds at their disposal. On the resignation of Dr. King, the presidency passed to his logical successor, Dr. James Elliott Harlan, who had served as vice president of the college since 1881. He had long had the management and investment of the large funds of the college and the administration of the school in its immediate relations with the students. Just, sympathetic, patient, he had won the esteem of all connected with the college, and to him was largely due the exceptional tranquillity which the college had enjoyed in all its intimate relations. Dr. Harlan was graduated from Cornell College in 1869. For three years he was superintendent of the schools of Cedar Rapids, and for one year he held a similar place at Sterling, Ill. From here he was called to the alumni professorship of mathematics in Cornell College. The larger part of his life has thus been bound up inextricably with the school. He knows and is known and loved by all the alumni and old students. The first year of his administration was signalized by the erection of the new alumni gymnasium, and the second by the conditional gift by the general educational board of $100,000.00 to its endowment funds.
- 81. REV. SAMUEL M. FELLOWS, A. M. First President Cornell College The dean of the college since 1902 has been Professor H. H. Freer, a graduate of the school of the class of 1869, and a member of the faculty since 1870. Dean Freer was one of the first men in Iowa to see the need of schools of education in connection with colleges and universities and was placed at the head of such a school—the normal department of Cornell—early in the '70s. As has recently been said of him by Pres. H. H. Seerley, of Iowa Teachers College, his connection with teacher education is probably unexcelled in
- 82. Iowa educational history and no tribute that can be paid could do justice to his faithful endeavors. Dean Freer has been most intimately connected with the administration for many years. In 1873 he organized the alumni, with the help of Rev. Dr. J. B. Albrook, for the endowment of a professorship. At that time there were but 108 living graduates, forty-seven of whom were women. Of the men, only thirty-eight had been out of college more than three years. Yet this audacious enterprise was carried through to complete success and was followed by the endowment of a second alumni chair. In all of the great financial campaigns Dean Freer has been indispensible, and the moneys he has secured to the college amount to hundreds of thousands of dollars. More than this, by his wide acquaintance throughout the state and by his cordial friendship with all old students, he has been one of the chief representatives of the college around whom its friends have ever rallied. Since 1887 he has been professor of political economy in the college, and now occupies the David Joyce chair of economics and sociology. THE FACULTY Of the nearly 300 teachers who have been enrolled in the faculties of the college there is space for the mention of but few names: Dr. Alonzo Collin, who began by teaching all the sciences and mathematics in the young school in 1860, and resigned in 1906 as professor of physics; Dr. Hugh Boyd, professor of Latin from 1871 to 1906; Prof. S. N. Williams, head of the school of civil engineering since 1873; Prof. George O. Curme, professor of German from 1884 to 1897, now a member of the faculty of Northwestern University; Dr. W. S. Ebersole, professor of Greek since 1892; Dr. James A. James, professor of history from 1893 to 1897, now teaching in Northwestern University; Prof. H. M. Kelley, professor of biology since 1894; Dr. Thomas Nicholson, professor of the English Bible from 1894 to 1904, now general educational secretary of the M. E. church; Dr. F. A. Wood, professor of German from 1897 to 1903, now member of the faculty of University of Chicago; Prof. Mary Burr
- 83. Norton, alumni professor of mathematics, whose connection with the faculty dates from 1877; Dr. H. C. Stanclift, professor of history since 1899; Dr. Nicholas Knight, professor of chemistry since 1899; Dr. George H. Betts, psychology, who entered the faculty in 1902; Prof. C. D. Stevens, English literature, since 1903; Prof. C. R. Keyes, German, since 1903; Miss Mary L. McLeod, dean of women, since 1900; Prof. John E. Stout, education, since 1903. The continuity, the long terms of service of the administrative officers and the professors, can hardly be too strongly emphasized as a potent factor in the growth of the college. If the history of the school had seen a rapid succession of different presidents and frequent changes of faculty, if there had been changes in plans and purposes, factions and struggles, and the loss of friends which such struggles entail, if the power of the machinery had been wasted in internal friction we may be sure that the story of the college would have been far other than it is. THE ALUMNI The graduates of Cornell now number 1,446. This small army of educated men and women have scattered widely over all the states of the union and to many foreign countries. They have entered many vocations. The profession receiving the largest number is teaching. Of the 1,139 graduates including the class of 1905, reported in the catalog of 1908, ninety-seven have been engaged in teaching in colleges and universities, and 165 in secondary and normal schools. One hundred and forty-nine have entered the law, and 139 have entered the ministry. Business and banking were the employments of 113. Medicine has been the choice of forty-nine, and engineering and architecture of fifty-two. The foreign missionary field has claimed thirty-four, and social service in charity organization societies, deaconess work, social settlements, and the Y. M. C. A. and the Y. W. C. A. have engaged twenty-six. Thirty-two have engaged in farming, and twenty-six in newspaper work. The women
- 84. graduates of the school very largely have been induced to enter the profession of matrimony. Up to 1876, for example, ninety per cent of the alumnae had married. Of later years the larger opportunities for professional service, opening for women, and no doubt other general causes, have decreased the percentage, but of all women graduates up to the year 1900, seventy per cent have married. Of these forty- two per cent have married graduates of the college. The common error that college education lessens the opportunities of woman for her natural vocation is disproved, at least so far as Cornell college is concerned. The marriages of the graduates of Cornell have been singularly fortunate. Among the more than 1,400 alumni, there has been so far as known but two divorces. Considering the high percentages of divorce in the states of the Union, rising as high in some states as one divorce to every six marriages, the divorceless history of the Cornell alumni witnesses the sociologic value of the Christian co-educational college. In numbers the graduating classes have steadily increased. The first class, that of 1858, consisted of two members, Mr. and Mrs. Matthew Cavanaugh, of Iowa City. Classes remained small, never exceeding five, until the close of the Civil war when the young men who had entered the service of their country, and who survived the war, returned to school. In 1867 eleven were graduated, and in 1869 the class numbered twenty-two. The last decade the graduating class from the college of liberal arts has averaged sixty. CORNELL AND THE WAR FOR THE UNION President Charles W. Elliot, in one of his educational addresses, after enumerating what the community must do for the college, asks, And what will the college do for the community? It will make rich returns of learning, of poetry, and of piety, and of that fine sense of civic duty without which republics are impossible. That Cornell has made all these returns in ample measure is shown by the roster of the alumni with its many eminent names in the service of state and
- 85. church. More than fifteen thousand young men and women have left the college halls carrying with them for the enrichment of the community stores of learning, poetic ideals of life, and vital piety. The fine sense of civic duty which the college breeds finds special illustration in the crisis of the Civil war, and here we may quote the eloquent words of Colonel Harry H. Rood in an address delivered at the Semi-Centennial of the college in 1904: The first seven and a half years in the history of this college was a period of struggle and embarrassment. The spring of 1861 seemed to be the beginning of brighter days. A railway had brought it in touch with the outer world, and the effects of the great financial panic of 1857 were passing, enabling the sons and daughters of the pioneers to enter its halls to secure the education they so greatly desired. The sky of hope was quickly overcast, and the storm cloud of the Civil war, which had been gathering for half a century, burst over the land. The students of Cornell were not surprised or alarmed. The winter preceding they had organized a mock congress with every state represented, in which all the issues of the coming conflict were fully discussed and understood.... The first regiment the young state sent out to preserve the Union had in its ranks a company from this county—one-third of the names upon its muster rolls were students from this school. The first full company to go from this township into the three years service had one-third of its membership from this college, and the second full company from the township, in 1862, also had an equal number of Cornell's patriotic sons. In the great crisis of 1864, when President Lincoln asked for men to relieve the veteran regiments and permit them to go to the front, almost a full company were college men. In the class of 1861 only two men were graduated and both entered the service.... The record shows that from 1853 to 1871 fifty-four men were graduated from the college, and of these thirty had worn the blue. During the war the college had much the aspect of a female seminary to which a few young boys and cripples had been admitted by courtesy. In 1863 but twelve male students were registered in
- 86. college classes, and at the commencement of this year all upon the program were women except a delicate youth unfit for war and a boy of sixteen years. This commencement was unique in the history of the college. On commencement day the audience of peaceful folk seated in the grove quietly listening to the student orations was suddenly transformed to an infuriated mob, when one girl visitor attempted to snatch from another a copperhead pin she was wearing. So strong was the excitement, that the college buildings were guarded by night for some time afterward for fear that they might be burned in revenge by sympathisers with the south.[L] Near the close of the war it was seen that many of the soldier students of the college would be unable to complete their education because of the sacrifices they had made in the service of their country. A fund of fourteen thousand dollars was therefore contributed by patriot friends at home and in part by Iowa regiments in the field for the education of disabled soldiers and soldiers' orphans. No gift to the school has ever been more useful than this foundation, which aided in the support of hundreds of the most worthy students of the college. Two of the students of Cornell were enrolled in the armies of the Confederacy. Of these one became a lieutenant in a Texas regiment. At one time learning that one of his prisoners was a Cornell boy and a member of his own literary society, the Texas lieutenant found Cornell loyalty a stronger motive than official duty. He took his prisoner several miles from camp, gave him a horse and started him for the Union lines. THE SOCIAL ORGANIZATION From the beginning Cornell college has been coëducational. In the earliest years of her history some concessions were made in the courses of study to the supposed weakness of woman's intellect, and ornamental branches, such as Grecian painting, which seems to have been a sort of transfer work, ornamental hair work and wax
- 87. flowers were grafted on the curriculum for her special benefit— branches which soon were pruned away. Woman's presence seems to have been regarded in these early years as a menace to the social order, safely permitted only under the most rigorous restrictions. So late as 1869 Rule Number Twelve appeared in the catalog—The escorting of young ladies by young gentlemen is not allowed. This was a weak and degenerate offspring of the stern edict of President Keeler's administration: Young ladies and gentlemen will not associate together in walking or riding nor stand conversing together in the halls or public rooms of the buildings, but when necessary they can see the persons they desire by permission. For many years these blue laws have been abrogated, and the only restrictions found needful are those ordinarily imposed by good society. The association and competition of young men and women in all college activities—an association necessarily devoid of all romance and glamour—has been found sane and helpful to both sexes, and no policy of segregation in any form has ever been as much as suggested. The social life of the college has always been under the leadership of the literary societies. They are now eight in number: The Amphictyon, Adelphian, Miltonian and Star for men and the Philomathean, Aesthesian, Alethean and Aonian for women. The students of the Academy also sustain four flourishing societies, the Irving and Gladstone, Clionian and King. These societies meet in large and rather luxuriously furnished halls in which they entertain their friends each week with literary and musical programs, followed by short socials. Business meetings offer thorough drill in parliamentary practice and often give place to impromptu debates which give facility in extemporaneous speaking. The societies also give banquets and less formal receptions from time to time and in general have charge of the social life of the
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