An approach to the adaptive finite element analysis in associated and non-associated plasticity considering localization phenomena

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ELSEVIER Comput. Methods Appl. Mech. Engrg. 176 (1999) 187-202

Computer methods in applied

mechanics and engineering

www.elsevier.com/locate/cma

An approach to the adaptive finite element analysis in associated and non-associated plasticity considering localization phenomena

H. Cramera, R. Findeissb, G. Steinlb, W. Wunderlichb’* “Universitiit Restock, Lehrstuhl fiir BaustatiklBaudynamik, PF 1210, 23952 Wistnm, Germany

bTechnische Universitiit M&hen, Lehrstuhl fiir Statik, Arc&r. 21, 80333 Miinchen, Germany

Received 25 May 1998

Abstract

In an adaptive finite element approach elastoplastic problems with associated and non-associated flow rules are investigated. The first part of the paper deals with the underlying numerical formulation for a classical continuum model and a hierarchical h-adaptive mesh refinement strategy. Essential ingredients of the adaptive process are a suitable error indicator and transfer operations for the mapping of history-dependent state variables between different meshes. These are imbedded in a nonlinear incremental finite element procedure. For non-associated plasticity a standard continuum approach may lead to an ill-posed problem. Therefore, in the second part a generalization in the framework of a Cosserat theory is considered. The underlying equations possess a similar structure, and the adaptive finite element formulation can be extended in a straightforward manner. Numerical examples demonstrate the general applicability of the approach to elastoplastic problems including associated as well as non-associated plasticity. They show the superior behaviour of the Cosserat formulation in the case of localization phenomena also for non-associated plasticity. 0 1999 Elsevier Science S.A. All rights reserved.

Introduction

The quality of finite element solutions of elastoplastic problems is very much influenced on how the structural system is discretized, especially when the load carrying capacity has to be determined. Adaptive mesh refinement strategies provide a good basis to improve the accuracy of the solutions. Essential ingredients of adaptive finite element calculations are the estimation of the discretization error and the design of the refined meshes. In the nonlinear analysis of path-dependent problems the mesh refinement has to be controlled by suitable error measures and integrated into the incremental solution procedure.

In the first part of the paper we present an adaptive finite element formulation for a classical continuum model including associated and non-associated plasticity. A standard displacement formulation with conforming shape functions for the displacements is used as the basis of the underlying finite element approach. The mesh refinement is controlled by an error indicator calculated after each load step in the course of the nonlinear incremental solution. This indicator represents a modification of an error estimator derived in [l] which takes into account the stress residuals within the elements and on their boundaries as well as the incremental strains.

Based on the error distributions successive refined meshes are generated by simply subdividing those elements for which the error exceeds a permissible tolerance. This leads to a hierarchical finite element approach in which the shape functions of the coarser meshes are retained in the basis of the shape functions of the finer meshes. In path-dependent problems this facilitates the mapping of state variables like stresses and internal variables between different meshes.

* Corresponding author.

004%7825/99/$ - see front matter 0 1999 Elsevier Science S.A. All rights reserved. PII: SOO45-7825(98)00336-3

188 H. Cramer et al. I Comput. Methods Appl. Mech. Engrg. 176 (1999) 187-202

The performance of the approach is demonstrated by the results of numerical calculations for different problems. The results illustrate the necessity of adaptive calculations, especially when limit states are investigated. In non-associated plasticity localization phenomena are encountered which may lead to an ill-posed problem when the classical continuum theory is applied. Therefore, in the second part of the paper a generalization of the approach in the framework of a Cosserat continuum model is presented. In this formulation an internal length scale is introduced which leads to strain concentrations in shear bands of finite width. This is illustrated by the results of numerical calculations which also demonstrate the superior behaviour of the extended formulation in contrast to the standard continuum model. The problem common to all formulations including an internal length scale is that they need an extremely fine discretization. Therefore, an adaptive approach is absolutely necessary for the analysis of problems where localization phenomena have to be taken into account.

2. Formulation of the classical elastoplastic finite element model

The governing equations of the standard elastoplastic continuum model are outlined first. The equilibrium is defined by the local equations

q.i.j+fi=O inV, (1)

qinj - t;: = 0 on S, . (2)

A variational or weak form of these equations is given by

- I

Sui(qj +x) dV + I

su&nj - ii) ds = 0 (3) V S,

which is equivalent to the common principle of virtual work:

I “qj qj dV - I

6ui fi dV - V V I

Su; ti dS = 0. St

The constitutive law applied is based on an elastoplastic model with isotropic hardening. The total strain increment is decomposed into an elastic and a plastic part

dEij = d&; + 6 (5)

where dcij, d&E and dgj denote the total, elastic and plastic strain components, respectively. For infinitesimal deformations the kinematic relations are given by

dEij = ; (dui,j + dc+)

The elastic constitutive relation is defined by a hypoelastic stress-strain law

d&; = Cij,[ da;, . (7)

For the description of the plastic response the yield surface is expressed in terms of the stress invariants I,, J, and J3, representing the first invariant of the stress tensor and the second and third invariant of the stress deviator, respectively

flgij, K) =f(ll~ J2, J3’ K) c 0 (8)

where

I, = qjsij ) sij = uij - f I, a,j ,

J, =$ s. ij rj 3 J3 = f sijsjtski .

Isotropic hardening is included by an internal variable K.

H. Cramer et al. I Comput. Methods Appl. Mech. Engrg. 176 (1999) 187-202 195

= J v ((ha - Au~)(AE - AE~) + (Ap - Aph)(Ax - Axh)) dV . (54)

The generalization of the error indicator from the classical continuum, Eq. (24), to the Cosserat model leads to

llell = IlAh, (C,hlkli,_ + c, t I&) + llAxll,,( G~lbdl,z + C, ; lwll,x)

where

ri = gji.j +f, in V, , (56)

Ji = Jj (u,: - u,;)nj on S, , (57)

or J, = (qnjfj) on S, , (58)

and

m, = pji , + eijkUjk in V, , (59)

M,=(&-p,j)nj onS,, (60)

or

M, = yin, on S, . (61)

This error indicator consists of two additive parts in which the first is identical with that for the classical model. In the second part maximum norms of the residual moments of the equilibrium equations within the. elements and of the couple stress jumps on the element boundaries are weighted by the L,-norm of the total micro-curvature increments. Based on this error indicator the mesh refinement including the transfer of the state variables can now be performed in the same manner as described in the previous section.

5. Numerical examples

5.1. Elastoplastic problems based on the classical continuum model

The approach described in this paper was applied to various examples in which different aspects were investigated. As a benchmark problem for several research groups a metal sheet with a hole under uniaxial extension was analyzed (Fig. 4). Several adaptive methods were compared for elastic and elastic-ideally plastic material with a von Mises yield condition and associated flow rule.

Fig. 4. Metal sheet with a hole, system and finite element idealization.


For the results of our research group the reader is referred to the literature [l]. They confirmed that the error indicator as well as the transfer process described are very well suited for the adaptive nonlinear finite element method. In a further example a strip footing on cohesive soil with a small angle of internal friction and an associated flow rule was considered in [1] and [4] (Fig. 5). The yield condition was formulated by a modified Coulomb criterion with a smooth yield surface. In this example results of different discretizations were compared with those obtained by the adaptive process (Fig. 7).

The corresponding load-displacement curves for different meshes are given in Fig. 6. The solution with the finest mesh is almost identical with the adaptive solution. This is not surprising as this discretization represents the final adaptive mesh whereas the solutions with the three other meshes show quite considerable deviations, however. This demonstrates the improvement of finite element computations by adaptive mesh refinements.

As an example for a non-associated flow rule a strip footing near a slope under vertical loading was investigated in [1] and [4]. It was observed that the error indicator is also capable to detect zones with localized shear deformations. A frictional material with nearly no cohesion and a friction angle of 4 = 25” was assumed with a yield condition according to Eq. (9) with a non-associated flow rule (lo), (11).

The load-settlement curve of the footing obtained in the adaptive finite element calculation is given in Fig. 9. The comparison with results of non-adaptive finite element calculations illustrates that a localization will not be noticed at all without adaptive mesh refinements. This may lead to a complete misjudgement of the load-carrying behaviour of the structure. But also numerical difficulties were encountered for further adaptive refinement leading to a non-converged solution. This is due to an unstable material behaviour caused by the non-associated flow rule accompanied by a sharp decrease of the load carrying capacity.

In this case the governing differential equations of the classical continuum loose its ellipticity leading to an ill-posed problem. Therefore, in numerical simulations an extreme mesh sensitivity may be observed which is the crucial point in adaptive finite element calculations.

Thus, the formulation of the localization regime has to be extended to generalized continuum models like the

Fig. 5. Strip footing on cohesive soil, system and initial mesh (a).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 settlement of the fooffng [m]

Fig. 6. Load-displacement curves for different discretizations.


mesh b) mesh c) mesh d) mesh e)

Fig. 7. Sequence of refined meshes for 77 = 0.025.

polar continuum theory with additional rotational degrees of freedom employed in this paper. For illustration of this approach two examples with non-associated flow rule are investigated and described in more detail: the numerical simulation of a plane strain compression test and the strip footing near a slope (Fig. 8) similar to the example mentioned above but differing in the material description.

Fig. 8. Strip footing near a slope under vertical loading.

; ; I 0.4 0.45 0.5 0.55 0.6

settkment of the footing [m]

II meshc) L

mesh d)

Fig. 9. Load-settlement-curves for different discretizations.


5.2. Localization in a plane strain compression test

The different behaviour of the classical finite element model and the Cosserat model as well as the influence of different internal length scales shall be illustrated in this example. A plane strain compression test for a generalized Drucker-Prager material with non-associated flow rule is considered. The geometric and material characteristics are shown in Fig. 10. To evaluate the influence of the internal length scale in the Cosserat model different analyses are performed with the three different parameters 1, = 0.1 mm, 1.0 mm and 10.0 mm. In addition, the system is analysed using the classical continuum model. Different uniform discretizations are considered to illustrate the mesh dependence (see Fig. 11).

The load-displacement curves of the finite element calculations with the classical continuum model are shown in Fig. 12. Localization initiated by a small weak zone with lower strength is achieved for all discretizations. However, the results show the strong mesh dependence leading to different load-displacement characteristics. The width of the observed shearbands depends on the mesh size and no convergence may be achieved with any further mesh refinement.

Results obtained by the Cosserat continuum model for different values of the internal length parameter are presented in Fig. 13. They illustrate the strong influence of the size of the internal length parameter. The load-displacement curves of the different finite element discretizations for the small length parameter I, = 0.1 mm show no convergence of the finite element solution. For the mean value of lc = 1.0 mm however, a converged finite element solution with a continuously decreasing load-displacement curve is achieved when the ratio of the mesh size to the internal length parameter has the order of one in the localization zone. On the other

Generalized Dm&er - Rager criterion

rj = 0.23 (9=30")

k = 6.00 3 (c=5kN/m2)

Cossemt formulation: jz = 0.0 , jJ = 1.0

lC = 0.1, 1.0, 10.0 mm piasticpotentiak

if = 0.16 ($=lo")

Fig. IO. Geometry and material characteristics of the plane strain compression test.

192 elements 768 elemnts 1728 elements 3072 elo?lellts h=lOmm h=Smm h = 3.33 mm h = 2.5 mm

Fig. 11. Finite element discretizations for the plane strain compression test.

7W

6coc classical continuum model I

\ I

H. Cramer et al. / Comput. Methods Appl. Mech. Engrg. 176 (1999) 187-202 199

displacement [mm]

Fig. 12. Load-displacement characteristics and deformations of the classical continuum model for different discretizations.

., .

0 -1 4 .6 4 -10 42 -II 0 -I 4 -6 4 -10 .I? -I,

0 0 4 .6 -8 -10 -12 -II

Fig. 13. Load-displacement characteristics and deformations of the Cosserat model for different discretizations and internal length parameters.

hand, when I, approaches the system size as illustrated by the results for I, = 10.0 mm, a much stiffer behaviour of the system with no decrease in the load-displacement curve is observed. Therefore, it is very essential to specify the internal length parameter in an appropriate order of magnitude. In addition, the finite element size in zones of strain localization should also be of the same order.

5.3. Strip footing near a slope

As a first test of the Cosserat model in a real-life problem an adaptive finite element analysis of a strip footing near a slope under vertical loading is investigated. An elastic ideal plastic behaviour of the soil was assumed with a generalized Drucker-Prager criterion and a non-associated flow rule. The system and material characteristics are shown in Fig. 14. The internal length parameter was specified to 1, = 4.0 cm. The load-displacement curve of the load-controlled calculation is represented in Fig. 15. A convergent adaptive finite element analysis with six hierarchical refinement steps can be performed until a horizontal tangent is reached when the Cosserat model is applied. For comparison the load-displacement curve of an adaptive calculation with the classical continuum model is also represented by the dashed line. In contrast to the Cosserat model it was not possible to obtain a converged solution in this case. The analysis could be performed UP to a


E = 30000 W/m2 , v = 0.20

4 = 25O , 11, = 5O, c = 5.0 k.N/m2

?l = 0.144, fT = 0.11, k = 4.59

x = 1.0

m(e) = 1.0, j2 = 0.0, j, = 1.0

Fig. 14. System and material characteristics of the strip footing near a slope.

:0.55 -0.56 -0.57 -0.58

displacement [m] -0.59 -0.6 -0.61

Fig. 15. Load-displacement curves, Cosserat model and classical model.

distinct load level where an uncontrollable sequence of mesh refinements was initiated. A sequence of the adaptively refined meshes is shown in Fig. 16 which illustrates the evolution of shearbands within the system.

The localized zone may be identified by the effective plastic strain and curvature increments which are shown in Fig. 17. It should be noted that the width of the shearband can be determined by the distance of the two lines of the plastic curvature increments. In our present example this width takes a value of about 50 cm. With respect to the internal length scale chosen to 1, = 4.0 cm, it may be recognized that the width of the shearband is approximately 12 times the Cosserat length parameter. This result is in good accordance with the heuristical experience, giving a ratio of 10 to 20 between the extent of the localized zone and the average grain diameter.

Our next goal is to study the influence of the internal length scale I, which was varied from 2.0 cm up to 8.0 cm. In Fig. 18 it may be observed that the deflection curves corresponding to various length parameters show only small differences until they reach the limit load capacity. This means that different values of 1, do not change the structure’s stiffness behaviour. It may also be recognized that in the prefailure regime the effect on the load-carrying capacity due to a variation of the internal length parameter from I, = 2.0 cm up to 8.0 cm is in


Fig. 16. Sequence of adaptively refined meshes of the Cosserat model.

Fig. 17. Effective plastic strain and effective plastic curvature increments.

the range of about 20%. However, the limit load state is reached earlier for small values of 1, (Fig. 19). Furthermore, the decrease of the load-displacement curve in the post-localization regime is not so distinct for higher values of 1,.

6. Conclusions

In this paper the extension of the classical continuum model to the Cosserat model is presented. By introducing micro-rotations as an additional degree of freedom and an internal length scale working as a regularization parameter, the drawbacks of the classical model can be overcome. It is shown that mesh


260

250

240

s ~ 230

B 220

210

200 -0.59 -0.595 -0.6 -0.605 -0.61 -0.615 -0.62 -0.625

displacement [ml

Fig. 18. Load-displacement-curves for different values of I,.

Fig. 19. Displacement increments after limit load state.

dependence of the results may be avoided and that the load-canying capacity of the system can be determined properly using an adaptive technique. The numerical examples show that the a posteriori error estimator which was developed for this nonlinear analysis is capable to detect and refine zones of localization appropriately. To obtain a reliable result the element size has to be reduced to the magnitude of 1, in the region of strain concentration. This was performed with a hierarchical subdivision strategy. The study of different internal length scales shows that the stiffness behaviour of the structure remains unchanged in the prelocalized state. The failure state is reached earlier for small values of 1, and will tend to the behaviour of the classical model for 1, -+ 0.

References

[I] H. Cramer, M. Rudolph, G. Stein1 and W. Wunderlich, A hierarchical adaptive finite element strategy for elastic-plastic problems, in: B.H.V. Topping, ed., Advances in Finite Element Technology (Edinburgh, 1996) 151-159.

[2] C. Johnson and P. Hansbo, Adaptive finite element methods in computational mechanics, Comput. Methods Appl. Mech. Engrg. 101 (1992) 143-181.

[3] M. Ortiz and J.J. Quigley, Adaptive mesh refinement in strain localization problems, Comput. Methods Appl. Mech. Engrg. 90 (1991) 781-804.

[4] W. Wunderlich, H. Cramer and M. Rudolph, Adaptive finite element approach to nonlinear geomechanical problems, in: 3rd Asian-Pacific Conference on Computational Mechanics, Seoul (1996) 215 l-2156.

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An approach to the adaptive finite element analysis in associated and non-associated plasticity considering localization phenomena