Development of X-FEM methodology and study on mixed-mode crack propagation

Transcription

1 Acta Mech. Sin. (2011) 27(3): DOI /s x RESEARCH PAPER Development of X-FEM methodology and study on mixed-mode crack propagation Zhuo Zhuang Bin-Bin Cheng Received: 2 February 2010 / Revised: 30 April 2010 / Accepted: 13 June 2010 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2011 Abstract The extended finite element method (X-FEM) is a novel numerical methodology with a great potential for using in multi-scale computation and multi-phase coupling problems. The algorithm is discussed and a program is developed based on X-FEM for simulating mixed-mode crack propagation. The maximum circumferential stress criterion and interaction integral are deduced. Some numerical results are compared with the experimental data to prove the capability and efficiency of the algorithm and the program. Numerical analyses of sub-interfacial crack growth in bi-materials give a clear description of the effect on fracture made by interface and loading condition. Keywords X-FEM Mixed-mode fracture Bi-material Sub-interfacial crack 1 Introduction Computational mechanics has had a profound impact on science and technology over the past three decades. It has transformed much of classical Newtonian theory into practical tools for prediction and understanding of complex engineering and science system, which have some limitations by using the classical analysis solutions. Finite element method (FEM), which is one of the most powerful computation tools, has been widely used in the simulation-based engineering and science. There are a number of methods using FEM in dealing with the problem of crack growth in materials. These methods include node force release [1], moving finite element [2], etc. However, the first method requires the crack Z. Zhuang ( ) B.-B. Cheng School of Aerospace, Tsinghua University, Beijing, China zhuangz@tsinghua.edu.cn path to be given in advance and the second one incurs high computation cost. For the crack arbitrary growth in the materials, the most suitable solution is the extended finite element method (X-FEM). The X-FEM was first proposed by Belytschko et al. [3, 4] in Its idea was summarized in an article by Moes and Belytschko [5] in After that, X-FEM has been developed into a vigorous numerical method by the efforts of many scientists. Its mathematical property was discussed by Réthoré et al. [6, 7]. The possibility of modeling inclusions and voids with X-FEM was discussed by Sukumar et al. [8] in Belytschko and Chen [9] imposed a novel criterion for fracture which introduced a hyperbolicity indicator to determine the path and velocity of the crack. The idea of enriching shape function was further extended to time axis by Chessa and Belytschko [10, 11], where the method is called space-time extended finite element method (STX- FEM). It seems to be a prospective way in dealing with the time discontinuity. Réthoré and Gravouil [12] adopted a similar formula of STX-FEM and obtained remarkable results. In the work by Belytschko [13] in 2006, X-FEM was connected with the mesh-free method, which requires no additional unknowns. Frictional contact between crack surfaces was considered in the work by Gravouil [14] in In the work of Fang and Jin [15], the X-FEM algorithm was coupled with commercial software ABAQUS and a subtriangle integration algorithm was improved. Xie and Feng [16] also implemented X-FEM in ABAQUS with UEL and mode- I crack and mixed mode crack were numerically studied. The kernel idea of X-FEM is to represent the discontinu ities with enriched shape function and thus it allows the mesh to be independent of the discontinuities. X-FEM can simulate not only cracks but also inclusions and voids. Cracks are those on which displacements are discontinuous while on inclusions they change to be the derivatives of displacements. There are two kinds of discontinuities classified: strong and weak, but in X-FEM the only difference between them is the

2 Development of X-FEM methodology and study on mixed-mode crack propagation 407 form of enriched shape function. Some major advantages of this method are: (1) As discontinuities can cross the element and occur in it, crack with actual shape can be captured precisely on regular mesh; (2) Mesh keeps unchanged during discontinuity evolution, so it avoids re-meshing required and turns out to be an efficient solution to the problem of dynamic crack growth or phase transition; (3) With asymptotic enriched shape function, more accurate results can be expected on coarse meshes near the crack tip. Due to these advantages, X-FEM has been used in various fields, for examples, dynamic crack propagation, shear band evolution and phase transition, etc. Thus, the X-FEM is recognized to be a great potential methodology for using in multi-scale computation and multi-phase coupling problems. In this paper, the formulas and program based on X- FEM algorithm have been developed and applied to several numerical examples. A brief review and governing equations of X-FEM are given in Sects. 2 and 3, respectively. The maximum circumferential stress criterion and interaction integral used for computing the stress intensity factor are illustrated in Sect. 4. Several numerical results compared with the experimental data are presented in Sect. 5. Different fracture modes of sub-interfacial crack propagation in bi-materials are discussed in Sect. 6 2 Governing equation and weak form Consider a domain Ω with a boundary Γ, as shown in Fig. 1. Γ c denotes the strong discontinuity and Γ int denotes the weak discontinuity in the domain. The conservation of momentum in terms of the nominal stress P and the Lagrangian coordinates X gives P ji X j + ρ 0 b i = ρ 0 ü i, in Ω 0, (1) Fig. 1 Initial configuration of domain Ω 0 where the superposed dots denote the material time derivative, b is the body force and ρ 0 is the mass corresponding to the initial configuration Ω 0. The kinematics equation using Green strain is E i j = 1 2 (F ki F k j δ i j ). (2) The constitutive equation gives PK2 stresses resulting form the Green strain S i j = C i jkl E kl. (3) To complete the description of the problem, the boundary conditions and initial conditions are given by u i = ū i, on Γ u 0, (4a) n 0 j P ji = t 0 i, on Γt 0, (4b) and u i (0) = u i0, P i j (0) = P i j0. (5) For the strong discontinuity in the domain, there is the traction free on crack surfaces n 0+ j P + ji = n0 j P ji = 0, on Γc 0. (6) The contact between crack surfaces is not concerned here. Otherwise, Eq. (6) should be taken a more complex form. For the weak discontinuity in the domain, there is the traction consistent on the interface n j P ji = 0, on Γ int 0. (7) For deducing the weak form, the space of trial and test functions are defined as follows { U = u (X, t) u (X, t) C 0, u (X, t) = ū (t), on Γ u 0, U 0 = u discontinuous on Γ c 0, u } discontinuous on Γint X 0, (8) { δu (X ) δu (X ) C 0, δu (X ) = 0, on Γ u 0, δu discontinuous on Γ c 0, (δu ) X discontinuous on Γint 0 }, (9) No other difference can be found between the spaces of standard FEM and the above, but that discontinuity across a crack and derivative discontinuity across an interface are allowed here, which is the kernel idea of X-FEM. Multiply δu on both sides of Eq. (1) and integrate on initial configuration. This gives ( P ) ji δu i + ρ 0 b i ρ 0 ü i dω 0 = 0, (10) Ω 0 X j in which the first term can be deduced by P ji (δu i ) δu i dω 0 = (δu i P ji )dω 0 P ji dω 0 Ω 0 X j Ω 0 X j Ω 0 X j = (δu i P ji )dω 0 δf i j P ji dω 0. (11) X j Ω 0 Ω 0 By using Gauss theorem, the first term on the right-hand side can be converted into a contour integral (δu i P ji )dω 0 = δu i n 0 j X P jidγ 0 j Γ 0 Ω 0

3 408 Z. Zhuang, B.-B. Cheng + + Γ c 0 Γ int 0 (δu + i n0+ j P + ji +δu i n0 j P ji )dγ 0 δu i n 0 j P ji dγ 0. (12) The first term on the right side is reduced to an integral only on Γ t as δu i = 0 on Γ u. The second and the third terms both vanish according to Eqs. (6) and (7), respectively. By combining Eqs. (10) (12), the weak form is obtained to find u U, such that (δf i j P ji δu i ρ 0 b i + δu i ρ 0 ü i )dω 0 δu i t i0 dγ 0 = 0, Ω 0 δu U 0. (13) It looks quite similar with that of standard FEM while the trial and test functions are changed here. The expression is simplified because of the assumption that no interaction between the crack surfaces exists. More complicated terms will be appended to the weak form if the contact and friction are included. 3 Discretization The enriched shape function for discontinuity is adopted to build the suitable trial and test function in Eqs. (8) and (9), respectively. The finite element trial function resulting from the nodal displacements u ii is given by u i = N I (X )u ii + φ J (X )a ij, (14) I J where N I (X ) is the standard FEM shape function, φ J (X ) is the enriched shape function and a ij is the enriched degree of freedom associated with φ J (X ). For element-crossed crack, φ J (X ) takes the following forms φ J (X ) = N J (X )H( f (X )), (15) where H(x) is Heaviside step function 1, x 0, H(x) = 1, x < 0, Γ t 0 (16) and f (X ) is signed distance function defined as below f (X ) = min X X ( sign n + (X X )), (17) X Γ c 0,Γint 0 where n + is the unit normal vector to discontinuity Γ c 0 (Γint 0 ). For any point X outside Γ c 0 (Γint 0 ), f (X ) is the shortest distance from X to Γ c 0 (Γint 0 ). It is positive when X lies on the same side with the direction to which n + points and it is negative when X lies on the opposite side. For embedded crack, φ J (X ) could be a linear combination of the function basis. [ r θ Φ(X ) = sin 2, r sin θ sin θ, 2 θ r cos 2, r cos θ ] 2 sin θ, (18) where r and θ are defined in crack-tip polar coordinates. These functions span the near crack-tip asymptotic solution for linear elastic fracture, so they are chosen not only for building a shape function discontinuous behind the crack tip but for improving accuracy as well. The typical enriched shape functions for element-crossed crack and embedded crack are illustrated in Fig. 2 and Fig. 3, respectively. Fig. 2 Shape function for an element-crossing crack Fig. 3 Shape function for embedded crack It is suggested that we can put whatever we know beforehand about a solution, such as its physical character or proved theoretical results, into FEM in terms of enriched shape functions. This is a brilliant breakthrough of X-FEM which efficiently reduces the computation cost. Accordingly, the shape function of weak discontinuity can be written as φ J (X ) = N J (X ) f (X ), (19) where f (X ) is defined in Eq. (17) and it can be easily proved that φ J (X ) is continuous across Γ int 0 while φ J (X )/ X is not, as shown in Fig. 4.

4 Development of X-FEM methodology and study on mixed-mode crack propagation 409 Fig. 4 Shape function for element-crossed interface To discretize the weak form (13) with Eq. (14), we obtain K uu K T ua = K ua K aa f ext q ext u a + M uu M T ua M ua M aa ü ä. (20) This is the equation which should be solved to get the final results. 4 Maximum circumferential stress criterion and Interaction integral In this section, the maximum circumferential stress criterion and interaction integral used for computing the stress intensity factor are illustrated. The asymptotic near-tip stress field for a planar mixed-mode crack is given by σ θθ σ rθ 3 cos θ = K I 4 2 2πr sin θ 2 + K 3 sin θ II 4 2 2πr + cos 3θ 2 + sin 3θ 2 sin 3θ 2 cos θ. (21) 3θ + 3 cos 2 2 Assuming that σ rθ is equal to zero, we can obtain a propagation angle [ 1 ( KI θ c = 2 arctan sign(k II ) 4 K II 8 + Meanwhile the propagation speed is 0, if K Ieq K Ic, v c = ( c r 1 K ) Ic, otherwise, K Ieq ( KI K II ) 2 )]. (22) (23) where K Ic is the critical mode-i stress intensity factor (SIF), K Ieq is the equivalent mode-i SIF and c r is Rayleigh wave speed. The interaction integral is defined as below [ I = W (1,2) δ 1 j σ (1) (u (2) i ) i j σ (2) (u (1) i )] i j n j dγ, (24) x 1 x 1 Γ where (σ (1) i j, ε(1) (σ (2) i j, ε(2) i j, u(2) i i j, u(1) i ) is the actual deforming field and ) is the auxiliary deforming field. The auxiliary deforming field can be anyone satisfying the kinematics and constitutive equations but here it is elaborately selected, which will be revealed latter. The interaction integral is computed for obtaining the SIF. It has the following relationship between the SIFs of the actual and auxiliary deforming fields when contour Γ approaches to the crack tip. I = 2 E (K(1) I K (2) I + K (1) II K (2) II ). (25) { E/(1 ν) where E = 2, plane strain, E plane stress. Now, it is clear that this kind of the auxiliary deforming field should be selected for computing K (1) I. It is chosen by satisfying K (2) I = 1, K (2) II = 0 and similarly K (2) I = 0, K (2) II = 1 ensures to get K (1) II. The definition (24) is usually converted to a domain integral in the practical use (Fig. 5). Fig. 5 Interaction integral I = A [ W (1,2) δ 1 j σ (1) i j u (2) i σ (2) u (1) ] i i j qm j dγ, (26) x 1 x 1 where q is a smooth weight function taking the value of zero on C 0 and one on Γ. In the following section, several numerical results compared with the experimental data are presented for the static and dynamic cracks. 5 Mixed-mode crack propagation 5.1 Static crack propagation A static loading is applied on the specimen depicted in Fig. 6, which engenders a curved-crack propagation. Geometrical symmetry is used and only the upper half is considered during the computation. This experiment was conducted by Kalthoff in 1988, the only difference between the experiment and our simulation is that Kalthoff applied a dynamic impact

5 410 Z. Zhuang, B.-B. Cheng on the specimen not static loading. In his experiment, the propagating angle of the dynamic crack was around 70. The crack growth process in our simulation is illustrated in Fig. 7. The propagating angle is around 67 to 75. In this example, the configuration of the specimen is illustrated in Fig. 8. An initial crack is located at the distance d = cl/2 from the middle axis. An impact is applied on the middle point on the upper edge and the mixedmode crack can be observed propagating towards the impact point. Some relevant geometric and material parameters are: L = mm, D = 76.2 mm, a = mm, thickness B = 25.4 mm, density ρ = kg/m 3, Young s module E = GPa, Poisson ratio ν = 0.2. John and Shah [17] carried out the same experiment. It is simulated by Belytschko and Tabbara [18] with meshfree method and by Song et al. [19] with X-FEM. John and Shah observed that the angle between the growing path of the crack and the vertical line is 22 when c equals to 0.5. Fig. 6 The curved crack growth model Fig. 8 Dynamic propagation of mixed-mode crack The impact is applied in the form of velocity 0.065t/t 1, t t 1, v 0 = m/s, t > t 1, (27) in which t 1 = 196 µs. Figure 9 illustrates the contour of the Mises stress during the propagation. The crack starts to propagate at the time of 647 µs and the simulation ends at 720 µs, during this period of time, the crack propagates a distance of about 28.5 mm, so the average velocity of the crack is about 390 m/s. Fig. 7 The process of the curved-crack growth 5.2 Dynamic crack propagation Fig. 9 Mises stress during the propagation

6 Development of X-FEM methodology and study on mixed-mode crack propagation The crack propagates towards the impact point from the beginning, which shows the character of mixed-mode crack propagation. Regular mesh is used in computation while the crack growing path is arbitrary and independent of the grids this should be attributed to the merit of X-FEM. The angle between the crack path and the vertical line α is measured as 21 which is close to the experiment (Fig. 10). 411 the interface and the initial crack, the different crack growth paths are observed. For each crack path, the magnitude of the q SIF K = KI2 + KII2 and the phase angle φ = tan 1 (KII /KI ) are recorded and plotted. Some parameters of the experiment are: specimen width w, height h and thickness t; distances between the initial crack and left and right sustaining points L1 and L2 ; distance between the interface and the crack d; distance between the interface and the loading point s and initial crack length a. The experimental configuration is illustrated in Fig. 11. The two materials on either side of the interface are PMMA (E = 3.24 GPa, ν = 0.35) and Al6061 (E = 69 GPa, ν = 0.3). Fig. 10 Propagating angle 6 Sub-interfacial crack growth in bi-materials Fig. 11 Experiment configuration Quasi-static propagation of a sub-interfacial crack parallel to the interface of a bi-material is implemented in the experiment of Lee and Krishnaswamy [20]. The effect on the mixed mode crack propagation which is made both by the interface of bi-materials and loading conditions is studied in detail. In the experiment, as shown in Fig. 11, a specimen with an initial crack is subject to three-point quasi-static increasing loading and the interface is adjacent to the crack. By changing the boundary condition and the distance between Fig. 12 Configuration and results of Case I of the experiment [20] In Case I of the experiment, as shown in Fig. 12 (w = 300 mm, h = 150 mm, t = 9 mm, L1 = L2 = 150 mm, a = 30 mm, d = s = 10 mm), the crack deflects leftward from the interface initially and then it finds a path almost parallel to the interface. K is around MPa m1/2 which is the fracture toughness of homogeneous PMMA. The phase angle φ rises from 15 to zero and maintains at zero during the rest propagation period.

7 412 In Case II of the experiment, as shown in Fig. 13 (w = 300 mm, h = 150 mm, t = 9.5 mm, L1 = 50 mm, L2 = 150 mm, a = 30 mm, d = s = 10 mm), the crack is attracted to the interface initially and then grows along the interface. The K increases MPa m1/2 while the phase angle φ descends from 30 to 10. Case III (Fig. 14) considers an interesting question: as Z. Zhuang, B.-B. Cheng the crack paths of Cases I and II have different curvatures, is there an equilibrium status in which the crack propagates straightly while the effect of the interface and the loading counteracts each other? This is achieved by selecting the proper parameters (w = 300 mm, h = 150 mm, t = 9 mm, L1 = 100 mm, L2 = 150 mm, a = 70 mm, d = 16 mm, s = 10 mm). Fig. 13 Configuration and results of Case II [20] Fig. 14 Configuration and results of Case III [20] This experiment visualizes the mixed mode crack growth in bi-materials under different loading conditions and crack positions. It gives a clear picture that how a crack path can be strongly affected by the geometric condition and material mismatch. Along with the experimental data, Lee and Krishnaswamy [20] also provided numerical results of the crack path simulated by FEM and auto-remeshing. The numerical results accord very well with the experimental ones in Cases I and III. In Case II, the numerical path is also observed to initially turn towards the interface, but the actual crack trajectory itself does not match very well with that ob- served in the experiment [20]. They also pointed out that Further mesh refinement does not improve the simulation. Maybe this is caused by the fact that the standard FEM has some limitations in capturing complicated crack path, so the crack must grow along the element edges. That s why XFEM is chosen to carry out further simulation in this paper, which can capture arbitrary crack path on regular meshes. Instead of a steadily increasing loading, a constant loading value is applied on the specimen for simplification. For the three cases, loading values are 360 kn, 800 kn and 400 kn. The results of SIF K and phase angles φ are also

8 Development of X-FEM methodology and study on mixed-mode crack propagation 413 calculated and plotted compared with the experimental data. In Case I, as shown in Fig. 15, the numerical crack path grows leftward from the interface initially and then stays parallel to the interface, which is in accordance with the experiment (Fig. 12). As crack length increases, K ascends as expectation. Since the loading value is different from the experiment, K is much larger than the experiment data, but the phase angle φ shows the same tendency, which increases from 6 and maintains zero afterwards (the solid line is the least-square fitting of the discrete data). Though the magnitudes of SIF K differ from the experiment data, the ratio between K I and K II is less sensitive to the magnitude of loading, as given in Fig. 16. In Case II, the crack propagates towards the interface initially and bounces from it when it gets very close to the interface, as shown in Fig. 16, then the crack grows along the interface. The whole procession is also the same with the experiment. It is worth noting that the crack path obtained by X-FEM is closer to the experiment result than that by the standard FEM. On coarser-mesh of X-FEM, the crack propagates towards the interface initially but does not bounce afterwards. The better results are obtained when the meshes are refined as shown in Fig. 16. The mesh shown here is This improvement should be attributed to the merit of X-FEM which allows arbitrary crack path not limited by the mesh. The phase angle φ decreases from 36 to 20 when the crack propagates 20 mm (the end of the experiment data) and continues decreasing until it reaches to zero, which implies a mode-i fracture with a straight crack path, as illustrated in Fig. 16. Case III is much simpler than the former two. The crack propagates along a straight line, as shown in Fig. 17. The phase angle φ remains around zero which implies a mode-i fracture. It is interesting that the interface and loading asymmetry counteract each other in this case. Figure 17 shows the results obtained by X-FEM. Both the crack path and the phase angle are close to the experiment data. When the loading values are changed, another set of results are obtained. The loading values are modified by 500 kn, 900 kn and 500 kn at this time for the three cases. The phase angle φ is plotted as shown in Fig. 18. The results are similar to the former cases and the tendency of φ shows a pleasing accordance with the experiment data. Fig. 15 Simulation for Case I of the experiment Fig. 16 Simulation for Case II of the experiment

9 414 Z. Zhuang, B.-B. Cheng Fig. 17 Simulation for Case III of the experiment Fig. 18 Numerical results of different loading 7 Conclusions meant to be appended in the later work. In this paper a program based on X-FEM is developed and applied to the simulation of two dimensional mixed mode crack propagation. Both quasi-static and dynamic examples show pleasing accordance with the experiment observation. Due to the merit of X-FEM, the crack can evolve independently of the mesh and more accurate results can be obtained on coarse mesh compared with the standard FEM. Numerical analyses of the crack growth in bi-materials give a clear description of the effect on fractures made by the interface and loading conditions. The computational results show that the fracture mode and hence the crack growth path can be strongly affected by the interface of bi-materials and the loading asymmetry, which has also been revealed by the experiment. Numerical solutions accord well with the experimental data. Further work needs to be carried out, as the current simplied simulations do not consider contact and friction of crack surfaces nor cohesive zone ahead of crack tip. They are not yet applied to elastic-plastic materials and heterogeneous materials where fracture criterion, rather than maximum circumferential stress criterion, should be used. All these are References 1 Zhuang, Z., Guo, Y.J.: Analysis of dynamic fracture mechanism in gas pipelines. Engng. Fracture Mech. 64, (1999) 2 Nishioka, T.: Computational dynamic fracture mechanics. Int. J. Fract. 86, (1997) 3 Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Engng. 45, (1999) 4 Moes, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Engng. 46, (1999) 5 Belytschko, T., Moes, N.: Arbitrary discontinuities in finite elements. Int. J. Numer. Meth. Engng. 50, (2001) 6 Réthoré, J., Gravouil, A.: An energy-conserving scheme for dynamic crack growth using the extended finite element method. Int. J. Numer. Meth. Engng. 61, (2004) 7 Menouillard, T., Réthoré, J., Combescure, A., et al.: Efficient explicit time stepping for the extended finite element method (X-FEM). Int. J. Numer. Meth. Engng. 68, (2006) 8 Sukumar, N., Chopp, D.L., Moes, N., et al.: Modeling holes

10 Development of X-FEM methodology and study on mixed-mode crack propagation 415 and inclusions by level sets in the extended finite-element method. Comput. Methods Appl. Mech. Engrg. 190, (2001) 9 Belytschko, T., Chen, H.: Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int. J. Numer. Meth. Engng. 58, (2003) 10 Chessa, J., Belytschko, T.: Arbitrary discontinuities in space time finite elements by level sets and X-FEM. Int. J. Numer. Meth. Engng. 61, (2004) 11 Chessa, J., Belytschko, T.: A local space time discontinuous finite element method. Comput. Methods Appl. Mech. Engrg. 195, (2005) 12 Réthoré, J., Gravouil, A.: A combined space time extended finite element method. Int. J. Numer. Meth. Engng. 64, (2005) 13 Belytschko, T., Fries, T.P.: The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int. J. Numer. Meth. Engng. 68, (2006) 14 Gravouil, A., Ribeaucourt, R., Baietto-Dubourg: A new fatigue frictional contact crack propagation model with the coupled X- FEM/LATIN method. Comput. Methods Appl. Mech. Engrg. 196, (2007) 15 Fang, X.J., Jin, F.: Extended finite element method based on ABAQUS. Engineering Mechanics. 24(7), 6 10 (2007) 16 Xie, H., Feng, M.L.: Implementation of extended finite element method with ABAQUS user subroutine. Journal of Shanghai Jiaotong University. 43(10), (2009) (in Chinese) 17 John, R., Shah, S.P.: Mixed mode fracture of concrete subjected to impact loading. Journal of Structural Engineering (ASCE) 116, (1989) 18 Belytschko, T., Tabbara, M.: Dynamic fracture using elementfree Galerkin methods. Int. J. Numer. Meth. Engng. 39, (1996) 19 Song, J.H., Areias, P., Belytschko, T.: A method for dynamic crack and shear band propagation with phantom nodes. Int. J. Numer. Meth. Engng. 67, (2006) 20 Lee, H., Krishnaswamy, S.: Quasi-static propagation of subinterfacial cracks. ASME J. Appl. Mech. 67, (2000)