Large strain FE-analyses of localized failure in snow C.H. Liu, G. Meschke, H.A. Mang Institute for Strength of Materials, Technical University of Vienna, A-1040 Karlsplatz 13/202, Vienna, Austria ABSTRACT An algorithm for finite deformation plasticity theory, recently proposed by Simo [5] and Simo and Meschke [6], is applied to numerical simulations of localized failure in snow. Within the algorithmic framework forfinitestrain plasticity the classical Drucker-Prager model and the modified Cam-Clay model are employed for the representation of snow. The respective material parameters are calibrated according to results from hydrostatic and shear-box tests of snow specimen. The effectiveness of the FE-model is demonstrated by means of re-analyses of the shear-box test, characterized by the formation of a sharp, localized failure zone. The traction mechanism of a single automobile tread moving over a snow-covered road surface is numerically simulated. The predicted failure mechanism of snow in consequence of the movement of the tread agrees relatively well with respective experimental observations. INTRODUCTORY REMARKS This paper is concerned with the numerical investigation of the traction behavior of automobile tires on snow-covered road surfaces by means of the finite element method (FEM). This type of analysis involves realistic modelling of rubber, snow and the contact at the rubber-snow interface. The traction behavior of tires for winter conditions is characterized by a complex mechanical interaction between the treads of the tires and the snow. The snow in the vicinity of the treads is subjected to large plastic deformations associated with the appearance of localized failure zones. The occurance of large non-recoverable deformations suggests using the of classical plasticity theory, embedded in the geometrically nonlinear theory offinitestrain
462 Localized Damage plasticity, for the modelling of snow. Relatively little research work concerning 3D constitutive models for snow is found in the open literature. In a recent work by Mohamed, Yong and Murcia [1] it is assumed that snow follows a,/2-type of plasticity law. However, this simplified model does not account for the considerable plastification of snow when subjected to hydrostsatic pressure. Salm [4] proposed a viscoelastic constitutive law restricted to uniaxial stress states. The present numerical investigation has been accompanied by an experimental investigation of snow specimens involving hydrostatic and shear-box tests. The results of this series of tests suggest the 'critical state concept', widely used in geomechanics, to be an adequate representation of the constitutive behavior of snow. In the present analyses, the (modified) Cam-Clay model, recast within a recently proposed algorithmic framework for finite strain plasticity [5], [6] is employed. This algorithm is formulated on the basis of hyperelasticity in the current configuration, with the elastic region being defined in terms of Kirchhoff stresses. The key advantage of this method is that the structure of the return mapping algorithm of the infinitesimal theory can be taken over to the nonlinear theory without modification. A fully implicit algorithmic treatment of the nonlinear elastic constitutive law along with an implicit integration of the flow rule on the basis of the return mapping algorithm is used. The Cam-Clay model is calibrated by means of the hydrostatic and the shear-box tests, respectively. For comparison, an elastic, ideally-plastic Drucker-Prager model is adopted. The important issue of viscosity of snow is not considered in this paper. It will be addressed in a follow-up publication. Both models for snow are applied in the re-analyses of shear-box tests. As far as the Drucker-Prager model is concerned, a comparison between the new plasticity algorithm with the standard incremental hypoelastic formulation, implemented in the multipurpose program MARC is presented. The obtained results are compared with experimental data. Numerical simulations of the traction mechanism of a single automobile tread moving on a snow-covered surface, based on the Cam-Clay model for snow, are performed. REMARKS ON FINITE STRAIN PLASTICITY The extension of the classical (modified) Cam-Clay plasticity model to the finite strain regime is based on an algorithm for finite strain plasticity recently proposed by Simo [5] and Simo and Meschke [6]. The key ingredients of this algorithm are briefly outlined below: The stress response is governed by a hyperelastic law, characterized by a function of stored energy, W(F^) involving the elastic part of the deformation gradient, F*. The restriction to isotropy implies that W(F*) = M^(b^), where b* is the elastic part of the left Cauchy-Green tensor. The total de-
Localized Damage 463 formation gradient F is decomposed into a plastic part F? and an elastic part F^ by means of the standard multiplicative decomposition F = FT?, where (F*)~* is associated with the stress-free, unloaded configuration. The evolution equation for the plastic flow is obtained from consideration of the priniciple of maximum plastic dissipation as -iw-^b-, «^,,1, [5], where 7 is the plastic consistency parameter, a is the plastic internal variable and denotes the Lie-derivative. f(t,q) is the yield function depending on KirchhofF stresses r acting on the current configuration and on the plastic hardening parameter q. Employing an exponential approximation to Equation (1), formulating the algorithmic flow rule in principal axes and making use of logarithmic strain measures (see [5], [6] for a detailed description), the algorithmic flow rule for the geometrically nonlinear case has a form completely identical to the infinitesimal theory: ^ = er-7^, A =1,2, 3, a = a,+7^, (2) where c^ (CA*) are the elastic principal (trial) strains, defined as the logarithmic (Hencky-) strains. Hence, the structure of the return map algorithm of the infinitesimal theory, now formulated in the principal stress space, takes over to the nonlinear theory without any modification [5]. The elastic principal trial strains t * are defined as t * InX*'** where the principal elastic trial stretches A*'**" are obtained from the spectral decomposition of the elastic trial left Cauchy- Green tensor A-\ fta,tr (^4 = 1,2,3) denote the principal directions of b^**" which, for i sot ropy coincide with the principal directions of the KirchhofF stress tensor T. The consistent linearization of the algorithm involves the standard elasto-plastic moduli formulated in principal axes and geometric moduli following from linearization of the term n^'^ x n^'**" in Equation 3 [5]. THE CAM-CLAY MODEL The functional form of the ellipsoidal yield surface of the modified Cam- Clay model proposed by Roscoe and Burland [3] is recast in the following modified, convex format [2] (3), q) = J2 + /, + <7 - q < 0, (4)
464 Localized Damage where /i = /i 3. t denotes the hydrostatic strength of the soil, /i and J^ are invariants of the KirchhofF stress tensor and M is the slope of the critical state line. A specific feature of the Cam- Clay model is the nonlinear elastic law, represented by a function of stored energy of the form = U(detF) + G ln((detv)-*'* \^ (5) A=l [6], where G denotes the constant shear modulus. The evolution of the plastic strains and of the plastic internal variable a is governed by an associative flow rule along with the hardening rule A =1,2, 3. (6) q=-h(p,q)a, H = ^.q, a = 7^, p = p-t, p = A/3, (7) where v(p) = N Xln(-p) denotes the specific volume with N as the specific volume at unit pressure and A (AC) is the consolidation (recompression) index. The hardening law (7)s is non associative, and consequently, the resulting algorithmic tangent matrix is unsymmetric [2]. Since q changes sign with df I dp, stress states on the subcritical (supercritical) side of the critical state line are associated with hardening (softening). In the numerical analyses contained in this paper, a fully implicit algorithm for Cam- Clay as proposed in [2] is employed. THE DRUCKER PRAGER MODEL The yield condition for the ideally-plastic Drucker-Prager is given as /(A, ^2) = \/J2 + adp/i - ^ < 0, (8) where a DP and &y are material parameters. RE-ANALYSES OF HYDROSTATIC AND SHEARBOX TESTS The material parameters for the Cam-Clay model and the Drucker-Prager model have been calibrated from results of hydrostatic and shear-box tests of snow specimens. In order to assess the suitability of the different plasticity models for the representation of snow and to verify the calibrated material parameters, the hydrostatic and the shear box tests are re-analyzed in this section. A nonsymmetric solver is used because of the unsymmetry of the elasto- plastic tangent matrix. Re- Analysis of Hydrostatic Tests A snow-cube (1= 7.71 cm), represented by one singlefiniteelement, is subjected to purely hydrostatic compression. The initial density of the sped-
Volume [cm"*] 450.00 \ Loading Localized Damage 465 350.00 Un- and Reloading P \N/cm*\ 10.000 20.000 30.000 40.000 50.000 60.000 Figure 1: Volume-pressure relationship for hydrostatic test: Comparison between Cam-Clay model prediction with experimental result men is p ~ QAg/cm^. The Cam-Clay material parameters are chosen as G = 700 TV/cm*, A = 0.39, K = 0.02, TV = 3.262, M = 2.88, 90 = 7.0 TV/cm*, qres = 2.5 TV/cm* and t = 1.0 TV/cm*, qo is the preconsolidation pressure, qres denotes a lower limit for q in case of pronounced softening. Figure 1 contains a comparison of the numerical and the experimental results, showing a close agreement of both results. Numerical Simulations of a Shear-Box Test Fig. 2 contains the geometry, the loading configuration and FE-discretization of a snow specimen considered in the numerical simulation of a shear box test. The FE-mesh includes 150 bilinear plane strain elements. The initial density of the specimen is p = 0.52#/cm^. This requires a modification of the parameters related to the prconsolidation. They are chosen as go = 15.07V/cm*, $. = 9.07V/cm* and t = 3.57V/cm*. The numerical simulation starts with a uniform normal pressure p = 2.5 TV/cm* applied on the top of the specimen, followed by a horizontal motion of the upper part of the snow specimen. A plot of the deformed configuration along with the distribution of equivalent deviatoric plastic strains, corresponding to a horizontal displacement u = 2.5 mm, is contained in Figure 3. A sharp shear zone, indicating the localized failure of snow, is observed. Figure 4(a) shows a comparison of the experimentally and numerically obtained relation between the average shear stress, defined as r^ TI Area, where T is the shear force and Ares is the area of the residual failure plane of the snow specimen, and the displacement u. Good agreement between the numerical prediction and the experimental result, including the ultimate shear stress, the postpeak branch and the residual shear stress, is obtained. For comparison, the
466 Localized Damage Steel p = 25 kpa i n HIM mn mm mm 14 mm Snow** 18 mm 14 mm 18 mm 60 mm Figure 2: FE-model for the numerical simulation of a shear-box test of a snow specimen Figure 3: Deformed configuration of the snow specimen and distribution of equivalent deviatoric plastic strains at u= 2.5 mm ideally-plastic Drucker-Prager plasticity model is used for the representation of snow in the simulation of the shear-box test. In addition to the new finite strain algorithm, characterized by hyperelasticity and a Kirchhoff-stress based yield condition, the standard updated Lagrange formulation based on hypoelasticity in conjunction with a Cauchy-stress-based yield criterion, as implemented in the multi-purpose FE-package MARC, is employed. The material parameters, are chosen as a^p = 0.277 and ay = 3.8N/cm*. In the numerical simulations, the uniform pressure on the top of the specimen is applied in 10 steps; subsequently a total horizontal displacement of 4 mm is applied in 400 steps. The numerical results obtained from both algorithmic formulations are shown in Figure 4(b). Note that the algorithm described in this paper results in a post-peak branch of the load-displacement curve. This phe-
140 140 Localized Damage 467 35 standard algorithm (MARC) multiplicative plasticity algorithm u [mm] 1.2 2.4 3.6 Figure 4: Simulation of shear-box test: Average shear-stress r^ vs. horizontal displacement u. (a): Comparison between Cam-Clay model prediction and experimental result, (b): Comparison between different algorithms on the basis of the Drucker-Prager model nomenon is induced by the use of the Kirchhoff-stress-based yield condition. Because of considerable plastic dilatation induced by the associative Drucker-Prager plasticity model, a Kirchhoff-stress-based ideal plasticity yield condition implies a shrinkage of the corresponding Cauchy-stress based yield surface. Increment 100 MARC Algorithm.100E+01.112E+00.657E-01.115E-01.275E-02.126E-02.758E-03.360E-03.191E-03.940E-04 New Algorithm.100E+01.998E-02.222E-04.637E-09.192E-12 Increment 400 MARC Algorithm.100E+01.822E-01.474E-01.275E-01.181E-01.926E-02.540E-02.370E-02.252E-02.171E-02 New Algorithm.100E-h01.372E-02.426E-05.383E-10 Table 1: Comparison of convergence ratio obtained from different algorithms The numerical efficiency of the new algorithm for multiplicative plasticity is demonstrated in Table 1 by means of a comparison with the standard
468 Localized Damage hypoelasticfinitestrain plasticity algorithm implemented in MARC. Table 1 contains the convergence ratio, defined as the ratio of the maximum displacement change in the current iteration and the maximum displacment change in the current increment, obtained for both algorithms. Table 1 clearly shows the computational superiority of the new algorithm, resulting from consistent linearization yielding a quadratic rate of convergence. SIMULATION OF AUTOMOBILE TREADS RUNNING ON SNOW The traction mechanism of a single automobile tread block moving over a snow-covered road surface is numerically simulated. The Cam-Clay plasticity model is employed for the representation of snow. The constitutive behavior of the rubber tread is represented by a lienar elastic law, characterized by the material parameters K 750.0 TV/cra^, G = 5007V/cra^. A 1 cm x 1 cm rubber block is discretized by 36 bilinear plane strain elements. The snow-covered road surface is represented by a block of 20 cm length and a depth of 8 cm. It is discretized by 1105 bilinear plane strain elements. Horizontal (vertical) boundary conditions are applied at both sides (the bottom) of the block. At first, vertical displacements are applied at the top nodes of the rubber block. The corresponding maximal pressure is ~ 497V/cra^. Subsequently, the tread block moves horizontally. Figure 5 contains the deformed configurations of the snow specimen and the distribution of the plastic volumetric strain at three different stages of the movement. According to the numerical predictions, failure of the snow is induced by the formation of vertical cracks under the front tip of the tread (Figure5(c)). This failure mechanism was also observed in experimental investigations accompanying the numerical simulations. CONCLUSIONS An efficient algorithm for multiplicative plasticity was applied to numerical simulations of localized failure of snow, including the mechanical interaction between automobile treads and snow-covered road surfaces. The modified Cam-Clay Model and the classical Drucker-Prager plasticity model are employed for the representation of snow. Re-analyses of a shear-box test demonstrate the numerical effectiveness of the new algorithm. A pronounced softening behavior is obtained as a consequence of the definition of the yield criterion in terms of Kirchhoff stresses. A simulation of the movement of a single rubber tread over a block of snow was perfomed. The failure mechanism of the snow, characterized by the opening of a pronounced vertical crack under the front tip of the tread, and the deformation of the rubber tread agree well with observations from experiments.
Localized Damage 469 Figure 5: Deformed configurations of the snow specimen in the vicinity of the tread block and distributions of volumetric plastic strains at different stages of the movement REFERENCES 1. Mohamed, A.M.O, Yong, R..N. and Murcia. A.J. 'Evaluation of the Performance of Deep Snowpack under Compression Loading using Finite Element Analysis', Journal of Terramechanics, Vol. 30, pp.219-257, 1993. 2. Meschke, G. and Liu, C.H. 'The Cam-Clay Model at Finite Strains: Algorithmic Aspects and Finite Element Analysis of Snow', Proceedings of IACMAG 94, West Virginia University, May 22-28, 1994, to appear. 3. Roscoe, K.H. and Burland, J.B. 'On the Generalized Stress-Strain Behavior of 'Wet' Clay' in: Hey man, J.; Leckie, F.A., eds., Engineering Plasticity, Cambridge University Press, 535-609, 1968. 4. Salm, B. 'On the Rheological Behavior of Snow', Contributions from the Inst. of Low Temperature Science, Hokkaido University, Sapporo, Japan, 1971. 5. J.C. Simo: "Algorithms for Static and Dynamic Multiplicative Plasticity that Preserve the Classical Return Mapping Schemes of the Infinitesimal Theory", Comp. Meth. Appl. Mech. Eng., Vol. 99, pp. 61-112, 1992. 6. J.C. Simo and G. Meschke: "A New Class of Algorithms for Classical Plasticity Extended to Finite Strains. Application to Geomaterials", Computational Mechanics, Vol. 11, pp. 253-278, 1993.