Numerical Prediction of Macroscopic Material Failure

Lidija Stankovic∗ and Jorn Mosler

Institute of Mechanics, Ruhr University Bochum, Universitatsstr. 150, 44780 Bochum, Germany

The aim of this contribution is the numerical determination of macroscopic material properties based on constitutive rela-tionships characterising the microscale. A macroscopic failure criterion is computed using a three dimensional finite elementformulation. The proposed finite element model implements the Strong Discontinuity Approach (SDA) in order to include thelocalised, fully nonlinear kinematics associated with the failure on the microscale. This numerical application exploits furtherthe Enhanced–Assumed–Strain (EAS) concept to decompose additively the deformation gradient into a conforming part cor-responding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale.This finite element formulation is then used for the modelling of the microscale and for the discretisation of a representativevolume element (RVE). The macroscopic material behaviour results from numerical computations of the RVE.

1 Prediction of Macroscopic Material Failure

Based on the finite element formulation proposed in [1], microscopic material properties (traction–separation laws) are usedfor the computation of the macroscopic material failure. The main idea is to define a Representative Volume Element (RVE)with pre-existing discontinuities which are the result of some previous evolution process. For the sake of simplicity, eachelement is assigned a stochastically distributed surface ∂sΩ. More specifically, the pre-existing internal surface is definedelement-wise by a stochastically distributed normal vector N as well as an initial material strength. If the RVE is subjected toloading, the resulting ultimate load can be interpreted as a macroscopic failure stress. The transformation of the mechanicalbehavior on the micro-scale to the macroscale can be achieved by applying standard homogenisation techniques. It is, however,not part of this contribution.

2 Multiple Localisation Surface Approach

The location of the stochastically generated internal surfaces ∂sΩ is assumed to be constant and therefore, they do not spana continuous global surface. This can lead to locking. A way to enrich the space of admissible internal surfaces is to usemore than one discontinuity per element. This leads to so-called Multiple Localisation Surface Approaches (MLSA). In thefollowing subsections, the fundamentals of such an approach are explained. In the case of a geometrically linearised theory,further details may be found in [2].

2.1 Kinematics

Similarly to the single fixed crack approach, a displacement field of the following type is adopted

u = u +n∑

i=1

(H(i)s − ϕ(i)) [[u]](i) . (1)

As a consequence, the deformation gradient as defined in [1] changes to

F = 1 + GRADu −n∑

i=1

([[u]](i) ⊗ GRADϕ(i)

)+

n∑i=1

([[u]](i) ⊗ N (i)

)δ(i)s . (2)

Here, N (i) represents the normal of the crack surface i, [[u]](i) the corresponding displacement discontinuity and δ(i)s the

DIRAC delta distribution associated with localisation surface i, respectively.

2.2 Constitutive Equations and the Numerical Implementation

The material response is governed by means of traction-separation laws. Following identical lines as proposed in [1], theseinterface laws capturing the softening response associated with the localisation surface i are based on admissible stress statesdefined as

E(i)

T:= (T (i)

, q(i)) ∈ R3+n|φ(i)(T (i)

, q(i)) ≤ 0, T(i) =

(F

T · P · N (i))+

(3)

∗ Corresponding author: e-mail: lidija@tm.bi.ruhr-uni-bochum.de, Phone: + 49 234 3222036, Fax: + 49 234 3214229

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

PAMM · Proc. Appl. Math. Mech. 6, 197–198 (2006) / DOI 10.1002/pamm.200610079

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

and on the evolution equations for the material displacement jump J and the internal displacement-like variable α

J(i)

= λ(i) ∂T (i)g(i), α(i) = λ(i) ∂q(i)h(i) (4)

respectively. Since the restriction (T (i), q(i)) ∈ E

(i)

Thas to be fulfilled for all i ∈ 1, . . . , n, the admissible stress space ET

is defined as the intersection of all subspaces E(i)

T.

The numerical implementation of the MLSA follows the algorithm discussed in [1]. The main property of the numericalmodel presented here is that the number of possible localisation surfaces per element is set to four. Each of the four orientationsof ∂sΩ is chosen such that it is parallel to one of the sides of the tetrahedron. As a consequence N = GRADϕ/||GRADϕ||and hence, N · GRADϕ = ||GRADϕ||. Thus, the resulting formulation is symmetric, which improves the numerical imple-mentation of the proposed model significantly.

3 Numerical Example

mesh I mesh II

mesh II

mesh I

Displacement u [cm]

Rea

ctio

nfo

rce

F[k

N]

0.250.20.150.10.050

250

200

150

100

50

0

Fig. 1 Numerical model of a triaxial compression test: Resulting distribution of the internal variable αmax and load–displacement diagramsfor both meshes

The numerical procedure is now used to simulate the behaviour of a soft rock in the triaxial compression test. For the sakeof simplicity, a hyperelastic neo-HOOKE-type model as given in [3] is adopted for the material in Ω±. The inelastic materialbehaviour at each of the localisation surfaces ∂sΩ(i) is modeled using a VON MISES type yield function

φ(T , q) = ||(T )m||2 − q(α), with (T )m := T − (T · N) N . (5)

The softening response characterised by the internal variable q has an exponential evolution.Two different unstructured discretisations, one with 3588 and the other with 5884 constant–strain tetrahedral elements

are considered for the determination of the maximal load. Both models were displaced downward at the top boundary, atprescribed increments and up to the ultimate load. The deformed meshes and the distribution of the internal variable αmax,which represents the maximal value of the relative displacements in each of the four possible localisation directions, are givenin Figure 1. The resulting load–displacement diagrams, showing the reaction force at the top F with respect to the verticaldisplacement at the top u, are also displayed in Figure 1.

According to the diagrams, the estimated ultimate load is practically the same for both meshes. Thus, the overall mechanicalresponse is almost independent of the considered discretisation of the micro-scale.

Acknowledgements This work was completed under the financial support of the Deutsche Forschungsgemeinschaft (DFG) throughproject BR 580/30-1. The authors wish to express their sincere gratitude for this support.

References

[1] J. Mosler. Modeling strong discontinuities at finite strains - a novel numerical implementation. Computer Methods in Applied Mechanicsand Engineering, 2005. in press.

[2] J. Mosler. On advanced solution strategies to overcome locking effects in strong discontinuity approaches. International Journal forNumerical Methods in Engineering, 63:1373–1401, 2005.

[3] P. Ciarlet. Mathematical elasticity. Volume I: Three-dimensional elasticity. North-Holland Publishing Company, Amsterdam, 1988.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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