![A two-scale approach for propagating cracks
in a fluid saturated porous material
F. Irzal, J.J.C. Remmers, J.M.R. Huyghe, R. de Borst, K. Ito
Numerical Methods in Engineering
Eindhoven University of Technology, Faculty of Mechanical Engineering
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
e-mail: f.irzal@tue.nl
Introduction Strategy
This research focuses on the fracture that may occur in in- Micro-scale The subsequent flow of fluids through the de-
tervertebral discs. The presence of damage in the disc will formable porous material is governed by a classical two-
change the physical behaviour of the discs and affect their phase theory [5] and averaged over the section of the cav-
capacity of absorbing and transmitting load, which may re- ity. A ”heuristic” approach is used to define the normal
sult in severe lower back pains. Figure 1 shows an inter- vector at the discontinuity in the deformed configuration
verbral disc in a human body and the various stages of a [6], see Figure 2.
damaged herniated disc.
-n+
n0 Ω+
0 n∗
d Γ+
d
Γd,0 n− Γ∗
d
Γ−
Ω−
0
d
Figure 2 : ”Heuristic” approach for normal vector representa-
tion in the deformed configuration.
Nucleation of micro cracks in the process zone are mod-
eled by cohesive zone law where the traction of the micro
structure is gradually decreasing with respect to the open-
ing of the crack, see Figure 3.
Γ
macro crack active cohesive zone
Figure 3 : Cohesive zone approach for modeling fracture.
Figure 1 : Scheme of a herniated intervertebral disc in human Macro-scale Finite element equations are derived under
body. total Lagrangian formulation to couple the local momentum
and mass balance from the micro-scale model.
Recently, a two-scale numerical model has been devel-
oped for the simulation of crack propagation in deforming Numerical challenges
fluid-saturated porous material [1,2] subject to small de-
formation. The saturated porous material is modeled as a The resulting discrete equations are nonlinear due to
two-phase mixture composed by the deforming solid skele- the the cohesive crack model, the geometrically non-
ton and the interstitial fluid. The evolution of crack growth linear effect and nonlinearity of the coupling terms. A
is simulated by employing the partition-of-unity property of Newton-Raphson iterative procedure is used to con-
finite element shape functions, which allows for the simu- sistently linearize the derived system while a Crank-
lation of crack growth independently of the finite element Nicholson scheme is used to discretise the system in the
mesh [3,4]. time domain.
References
Objective 1. Kraaijeveld F, et. al. Engng. Fracture Mech., Oct, 2009.
2. Rethore J, et. al, . Arch Appl Mech 2006;75: 595-606.
To extend the current approach into a finite strain frame- 3. Remmers JJC. . Comp. Mech. 2003; 31(1), 69-77.
work such that large deformation of the rubber-like porous 4. Babuska I, Melenk JM. Int. J. Num.Meth.Engng, 40(4): 1997.
material can be captured under the assumption of small 5. Lewis RW, Schrefler BA, 1998.
6. Wells GN, , Sluys LJ. Int. J. Numer.Meth.Engng. 54: 2002.
opening of the cracks and hyperelastic material response.](https://image.slidesharecdn.com/em3-1283341974261-phpapp01/85/Em3-1-320.jpg)
Em3
- 1. A two-scale approach for propagating cracks in a fluid saturated porous material F. Irzal, J.J.C. Remmers, J.M.R. Huyghe, R. de Borst, K. Ito Numerical Methods in Engineering Eindhoven University of Technology, Faculty of Mechanical Engineering P.O. Box 513, 5600 MB Eindhoven, The Netherlands e-mail: f.irzal@tue.nl Introduction Strategy This research focuses on the fracture that may occur in in- Micro-scale The subsequent flow of fluids through the de- tervertebral discs. The presence of damage in the disc will formable porous material is governed by a classical two- change the physical behaviour of the discs and affect their phase theory [5] and averaged over the section of the cav- capacity of absorbing and transmitting load, which may re- ity. A ”heuristic” approach is used to define the normal sult in severe lower back pains. Figure 1 shows an inter- vector at the discontinuity in the deformed configuration verbral disc in a human body and the various stages of a [6], see Figure 2. damaged herniated disc. -n+ n0 Ω+ 0 n∗ d Γ+ d Γd,0 n− Γ∗ d Γ− Ω− 0 d Figure 2 : ”Heuristic” approach for normal vector representa- tion in the deformed configuration. Nucleation of micro cracks in the process zone are mod- eled by cohesive zone law where the traction of the micro structure is gradually decreasing with respect to the open- ing of the crack, see Figure 3. Γ macro crack active cohesive zone Figure 3 : Cohesive zone approach for modeling fracture. Figure 1 : Scheme of a herniated intervertebral disc in human Macro-scale Finite element equations are derived under body. total Lagrangian formulation to couple the local momentum and mass balance from the micro-scale model. Recently, a two-scale numerical model has been devel- oped for the simulation of crack propagation in deforming Numerical challenges fluid-saturated porous material [1,2] subject to small de- formation. The saturated porous material is modeled as a The resulting discrete equations are nonlinear due to two-phase mixture composed by the deforming solid skele- the the cohesive crack model, the geometrically non- ton and the interstitial fluid. The evolution of crack growth linear effect and nonlinearity of the coupling terms. A is simulated by employing the partition-of-unity property of Newton-Raphson iterative procedure is used to con- finite element shape functions, which allows for the simu- sistently linearize the derived system while a Crank- lation of crack growth independently of the finite element Nicholson scheme is used to discretise the system in the mesh [3,4]. time domain. References Objective 1. Kraaijeveld F, et. al. Engng. Fracture Mech., Oct, 2009. 2. Rethore J, et. al, . Arch Appl Mech 2006;75: 595-606. To extend the current approach into a finite strain frame- 3. Remmers JJC. . Comp. Mech. 2003; 31(1), 69-77. work such that large deformation of the rubber-like porous 4. Babuska I, Melenk JM. Int. J. Num.Meth.Engng, 40(4): 1997. material can be captured under the assumption of small 5. Lewis RW, Schrefler BA, 1998. 6. Wells GN, , Sluys LJ. Int. J. Numer.Meth.Engng. 54: 2002. opening of the cracks and hyperelastic material response.