ELASTIC MODULII AND THEIR RELATIONSHIP BY CONSIDERING ANY ARBITRARY ANGLE
0 likes279 views
This document presents a derivation of the relationship between three elastic moduli: Young's modulus (E), bulk modulus (K), and shear modulus (G). It begins by introducing Hooke's law and defining the three elastic moduli based on the type of stress and strain. It then presents the main result - an equation relating E, K, and G with an arbitrary angle φ. The derivation considers a two-dimensional body under shear stress and calculates the strain produced. It then extends this to a three-dimensional body under multi-axial stresses to relate the elastic moduli and Poisson's ratio. Eliminating Poisson's ratio produces the final relationship between E, K, and G.
![Nazim A. Khan and Kaleem A. Quraishi
http://www.iaeme.com/IJMET/index.asp 38 editor@iaeme.com
REFERENCES
[1] Chakraborti, M.; Strength of Materials, Second Edition, S. K. Kataria and Sons,
New Delhi, 2001.
[2] Ryder, G. H.; Strength of Materials, Third Edition in S. I. Units, Macmillan
Publishers India Limited, New Delhi, 1969; Reprinted in 2011.
[3] Singh, S.; Strength of Materials, Tenth Edition, Khanna Publishers, New Delhi,
2010.
[4] Praveen Ailawalia and Shilpy Budhiraja, Disturbance In Generalized
Thermoelastic Medium with Internal Heat Source Under Hydrostatic Initial
Stress and Rotation, International Journal of Mechanical Engineering and
Technology, 3(3), 2012, pp. 315-330.](https://image.slidesharecdn.com/ijmet0702004-160312073006/85/ELASTIC-MODULII-AND-THEIR-RELATIONSHIP-BY-CONSIDERING-ANY-ARBITRARY-ANGLE-6-320.jpg)
ELASTIC MODULII AND THEIR RELATIONSHIP BY CONSIDERING ANY ARBITRARY ANGLE
- 1. http://www.iaeme.com/IJMET/index.asp 33 editor@iaeme.com International Journal of Mechanical Engineering and Technology (IJMET) Volume 7, Issue 2, March-April 2016, pp. 33-38, Article ID: IJMET_07_02_004 Available online at http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=7&IType=2 Journal Impact Factor (2016): 9.2286 (Calculated by GISI) www.jifactor.com ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication ELASTIC MODULII AND THEIR RELATIONSHIP BY CONSIDERING ANY ARBITRARY ANGLE NAZIM A. KHAN Department of Mechanical Engineering, Mewat Engineering College (Waqf), Palla, Nuh, Mewat-122107, Haryana (India) KALEEM A. QURAISHI Department of Applied Sciences and Humanities, Mewat Engineering College (Waqf), Palla, Nuh, Mewat-122107, Haryana (India) ABSTRACT In this paper, we obtained a relationship among Young Modulus of Elasticity ,E Bulk Modulus of Elasticity K and Rigid Modulus of Elasticity G by considering any arbitrary angle. Keywords and Phrases: Young Modulus of Elasticity; Bulk Modulus of Elasticity; Rigid Modulus of Elasticity; Relation between elastic moduli. Cite this Article: Nazim A. Khan and Kaleem A. Quraishi, Elastic Modul II and Their Relationship by Considering any Arbitrary Angle, International Journal of Mechanical Engineering and Technology, 7(2), 2016, pp. 33-38. http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=7&IType=2 1. INTRODUCTION Hooke's Law Stress produced in a body is directly proportional to strain in the body, within the elastic limit. elasticityofConstant strain stress Now depending upon the type of stress and strain produced in the body, constant of elasticity is of the following three types:
- 2. Nazim A. Khan and Kaleem A. Quraishi http://www.iaeme.com/IJMET/index.asp 34 editor@iaeme.com (i) Modulus of Elasticity )(E e E n strainNormal stressNormal (1.1) (ii) Modulus of Rigidity )(G tanstrainShear stressShear se G (1.2) (iii) Bulk modulus )(K e K n strainVolumetric stresscHydrostati (1.3)
- 3. Elastic Modul II and Their Relationship by Considering any Arbitrary Angle http://www.iaeme.com/IJMET/index.asp 35 editor@iaeme.com 2. MAIN RESULT The relationship among Young Modulus of Elasticity among ,E Bulk Modulus of Elasticity K and Rigid Modulus of Elasticity G is given by 24 cos23 9 KG GK E (2.1) where is the angle through which face AD tilted from its original position. 3. DERIVATION Consider a two dimensional body subjected to shear stress as its lower surface is fixed and force is applied on the upper surface. Due to produced shear stress the length of the diagonal DB will increase and length of diagonal AC will increase. The surface AB will shift rightward to BA as shown in figure.
- 4. Nazim A. Khan and Kaleem A. Quraishi http://www.iaeme.com/IJMET/index.asp 36 editor@iaeme.com From triangle ADA AA AD AA tan BBtanAA (3.1) 2 BB BD BDBD DBdiagonalinproducedStrain 0 (3.2) From triangle BBB 0 BB BB 24 cos 0 24 cosBBBB0 (3.3) From (3.2) and (3.3), we get 2 24 cosBB DBdiagonalinproducedStrain (3.4) From (3.1) and (3.4), we have 2 24 costan DBdiagonalinproducedStrain (3.5) Linear strain produced in diagonal DB )1( EEE (3.6) From (3.5) and (3.6), we get 2 24 costan )1( E 24 costan )1(2 E From (1.2), we have 24 cos )1(2 G E (3.7) Consider a three dimensional body of length , breadth b and thickness .t Suppose the stress induced in the body along length be .x Similarly stress induced in the body along breadth b and thickness t be y and z respectively. As we know that strain induced in the direction of force applied is longitudinal or linear strain and strain induced in the perpendicular direction of force applied is lateral strain.
- 5. Elastic Modul II and Their Relationship by Considering any Arbitrary Angle http://www.iaeme.com/IJMET/index.asp 37 editor@iaeme.com For calculating the linear strain in the particular direction, lateral strain induced in the other direction by the same force is to be subtracted for that particular direction. Strain produced in the length EEE e zyx EEE e zxy b EEE e yxz t Overall volumetric strain produced is EEEEEE eeee zyxzyx tbV 2 )21( 363 EEE e nnn V )21( 3 V n e E )21(3 KE (3.8) Eliminating from (3.7) and (3.8), we get the main result (2.1).
- 6. Nazim A. Khan and Kaleem A. Quraishi http://www.iaeme.com/IJMET/index.asp 38 editor@iaeme.com REFERENCES [1] Chakraborti, M.; Strength of Materials, Second Edition, S. K. Kataria and Sons, New Delhi, 2001. [2] Ryder, G. H.; Strength of Materials, Third Edition in S. I. Units, Macmillan Publishers India Limited, New Delhi, 1969; Reprinted in 2011. [3] Singh, S.; Strength of Materials, Tenth Edition, Khanna Publishers, New Delhi, 2010. [4] Praveen Ailawalia and Shilpy Budhiraja, Disturbance In Generalized Thermoelastic Medium with Internal Heat Source Under Hydrostatic Initial Stress and Rotation, International Journal of Mechanical Engineering and Technology, 3(3), 2012, pp. 315-330.