A NUMERICAL STUDY OF A FULLY CONSERVATIVE METHOD FOR HYPERELASTIC-VISCOPLASTIC MATERIALS XIAO LIN, JAMES GLIMM, JOHN GROVE, HYUN-CHEOL HWANG, DAVID SHARP, AND JOHN WALTER Abstract. We present a numerical algorithm for the simulation of the
impact of hyperelastic-viscoplastic materials in two dimensions. There are several distinctive aspects of our approach. The governing equations are based on a fully conservative Eulerian formulation due to Plohr and Sharp and our modi cation of the Steinberg - Lund rate dependent plasticity model. An approximate 2D Riemann solver is constructed in a directionally unsplit manner to resolve the complex elasto-plastic wave structure. The front tracking method provides sharp resolution of interfaces in multimaterial problems while eliminating spurious numerical di usion and the need for mixed material cell constitutive models. Several example problems are presented as a test of our algorithm.
1. Introduction The numerical modeling of large deformation elasto-plastic materials is challenging because the physics is highly nonlinear, leading to complicated wave patterns and ultimately to material fracture and fragmentation. Impact problems in particular involve material boundaries and shock waves as discontinuous solution features. In more than one-dimension the characteristic structure of the equations and the wave structure of the solutions are quite complex. Traditionally, numerical computations for such problems are based on Lagrangian methods, and are not fully conservative owing to the use of stress variables to represent material states. Mesh distortion limits the Lagrangian formulation to problems with small to moderate deformations. Lagrangian remeshing degrades the accuracy of shock propagation and causes numerical di usion. Standard multi-material Eulerian methods (one or more Lagrangian This work was supported in part by: the U. S. Army Research Oce under Grant DAAL-04-94-9510414; the Department of Energy under Grant DE-FG02-90ER25084; and the National Science Foundation under Grant DMS-9500568. DHS is supported by U.S. Dept. of Energy. 1
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steps followed by an advection step) su er from spurious di usion of material interfaces and the use of cell-level mixture models for material response. The purpose of this paper is to report on the rst two-dimensional studies based on a new approach to the simulation of materials undergoing large deformation. Our numerical method is based on a fully conservative Eulerian formulation of the equations of motion for an elasto-plastic materials proposed by Plohr and Sharp [7]. By casting the equations in this form, we can employ high accuracy, direct Eulerian methods of high-order Godunov or total variation diminishing (TVD) type. Fully conservative Godunov methods for elasto-dynamics have been studied by several authors, [1, 3]. Signi cant improvement in solution quality was attributed to the conservative formulation in [1]. In the one-dimensional case, Trangenstein and Colella [10] developed a Godunov method for elasto-plastic deformation using an alternate formulation based upon the inverse rather than the forward deformation gradient. However, they used a rate-independent plasticity model in a partially conservative method. Wang et al. [11] developed a fully conservative Eulerian Godunov scheme for rate-dependent elasto-plasticity; these authors reported good agreement with one dimensional experiments. Our method has several distinctive features. First, it is a directionally unsplit, TVD algorithm which is second-order accurate in smooth regions and rst order-accurate at large gradients or discontinuities. The two-dimensional wave structure is resolved by use of a two-dimensional, unsplit, approximate Riemann solver. This Riemann solver is based on a bicharacteristic analysis in the Lagrangian frame, which is then mapped into the Eulerian formulation. Second, our method utilizes front tracking for material interfaces and free boundaries. This is an important feature because, unlike shocks, the material interfaces are not self-focusing so that they would be handled poorly by an Eulerian nite di erence scheme alone. Moreover, front tracking eliminates the need for cell-level mixture models. In the next section, we recall [7] the conservative Eulerian form of our governing equations. In the Sec. 3, we describe the algorithm for the twodimensional approximate Riemann solver and the TVD ux construction. An interface Riemann solver for tracking of material interfaces is also discussed. The paper concludes in Sec. 4 with numerical examples. 2. The Conservative Eulerian Formulation We consider nite deformation of an elasto-plastic solid body. The motion of the material is speci ed by the map carrying each material point X in the undeformed Lagrangian con guration to its current Eulerian con guration, x = (X; t): The eld F = r is the deformation gradient and its inverse
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G is the inverse deformation gradient. In our formulation, the instantaneous
material state is described by the vector w = (u; v; E; g11; g12; g21; g22; ep11; ep22; ep12; )T and the resulting system of conservation laws with source terms has the form wt + f (w)x + g(w)y = h(w): (2.1) Here is mass density, (u; v) is velocity, E energy density, gij the inverse deformation gradient, epij the plastic strain tensor, and a material hardening parameter. For details of this formulation, see [7, 11].
3. The Finite Difference Scheme The method we have developed to solve Eq. (2.1) is a second order TVD nite di erence scheme. It is well known that a TVD scheme is only rstorder accurate at extreme points and discontinuities. Therefore, it cannot be globally second-order accurate. Our approach to the construction of a TVD scheme can be regarded as a ux limiter method. We reconstruct the numerical ux through an integral average of Riemann solutions from neighboring meshes at the half time step. Recall that if we take the value of the Riemann solution at the t axis as an average, the procedure reduces to the rst-order Godunov method. We now describe a second order scheme. By employing the average of w at the half +1=2 time step wjn+1 =2 in the one-dimensional case, the scheme h i wjn+1 = wjn ? xt f (wjn+1+1==22) ? f (wjn?+11==22) (3.1) is second-order accurate. If the Riemann solution, with initial data wjn and wjn+1, consists of N shock waves with intermediate states wj(k+1) =2 and associated wave speeds k ; (k = 1; : : : ; N ), then this average can be evaluated as wjn+1+1==22 = 21 Nk=1+1(k ? k?1)wj(k+1) =2: (3.2) Here k is a Courant number de ned as k xt , with 0 = ?1 and N +1 = 1. If w is continuous in this interval, then Z n t wj+1 1 n+1=2 n n (3.3) wj+1=2 = 2 (wj + wj+1) ? 2x n dw wj
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LIN, GLIMM, GROVE, HWANG, SHARP, AND WALTER
after integration by parts. Here = x=t and ?x=2 < x < x=2: Notice that, since df (w)dw = dw when w is continuous, Eq. (3.1) is just the second step of the two-step Lax-Wendro method with the rst step t f (wn ) ? f (wn) : wjn+1+1==22 = 21 (wjn + wjn+1) ? 2 j +1 j x
(3.4)
Therefore, we can see the possibility of constructing a scheme having the desired properties by combining two schemes through the proper averaging of Riemann solutions. This idea was rst introduced by Toro [8] for the onedimensional case, and was successfully applied to several problems [9, 6]. We expand this idea to the two-dimensional case in an unsplit manner. To update local wave interactions, we use an approximate, but fully two-dimensional, Riemann solver.
cell-center states w1n+1 : third step
. . . . cell-corner states ..................................................................................... ... ... ... ... ......... rst and second step .... 2 ...... . .4.................... ... . . . . ...........................2......................4................................. ... ......... 1 ... 3 ... . . . . . . . ..................................................................................................... ....1.... ....3.... ... ... ... ... ... ... ... ... ... . ................................................................................ . . . .
Figure 3.1. A sketch of cells and grids for constructing the
numerical scheme.
Our method, based on these ideas, consists of three main steps. As in Fig. 3.1, we have two di erent sets of state vectors in a computational domain: the cell-center and the cell-corner states. Let us assume that states at the beginning of each time step are given as cell-centered states. The rst step is to nd states at all corners at the half time step tn+1=2 , using the approximate Riemann solver. The second step is the construction of the TVD ux at the edge of each cell at the half time step. In the third step, states at the full time step tn+1 at cell-centers, e.g., w1n+1 in Fig. 3.1, are updated implicitly by
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the following unsplit scheme t hf (wn+1=2 ) + f (wn+1=2 ) ? f (wn+1=2 ) ? f (wn+1=2)i w1n+1 = w1n + 2 4 3 2 1 x h i t n+1=2 n+1=2 n+1=2 n+1=2 + g(w4 ) ? g(w3 ) + g(w2 ) ? g(w1 ) 2y h i + 8t h(w1n+1=2) + h(w2n+1=2 ) + h(w3n+1=2) + h(w4n+1=2 ) + 2t h(w1n+1); (3.5)
where win+1=2 (i = 1; : : : ; 4) are TVD states obtained in the second step. An implicit scheme is used for the stable treatment of the source term, and the numerical stability is controlled by a CFL condition such as cmax t (3.6) CFL = x 1: Here cmax = max jcL u vj is the fastest Eulerian wave speed in the entire computational domain, and cL is a longitudinal wave speed. Front tracking is an adaptive computational method in which a lower dimensional moving grid is tted to and follows the dynamical evolution of distinguished waves in material deformation. The method takes advantage of known Riemann solutions to track these waves, while standard nite di erence schemes are used in the interior away from the tracked waves. Finite di erences are never computed across tracked fronts. This method has been used in various uid ow problems, such as gas-dynamics and oil-reservoir simulation [5, 2, 4]. The Riemann problem for a material interface is a special case of the twodimensional Riemann problem whose approximate solution is used in the interior scheme. However, since the materials on the two sides of the interface are di erent the state vector in general su ers a discontinuity. Therefore, we have employed a suitably modi ed version of the algorithm for the solution of the two-dimensional Riemann problem. Details will be presented in a following paper. 4. Numerical Examples We present preliminary numerical results based upon the above algorithm. The rst example is a two dimensional Riemann problem. Consider an in nite body with no pre-stress or initial deformation. The velocity components are all zero except in the second quadrant x < 0; y > 0 where the x-component is 1 km/sec. We calculate the motion and deformation in this body. The material
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LIN, GLIMM, GROVE, HWANG, SHARP, AND WALTER 40.68 (GPa)
hyperelastic
-σxx
hyperelastic 17.17 (GPa)
σxy
30 20 10
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hyperelastic-viscoplastic
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Figure 4.1. Distributions of stresses p = xx and = xy
for the two-dimensional Riemann problem computed by twodimensional TVD method; elastic above, elasto-plastic below.
is tantalum and material parameters are: 0 = 16:69 g/cm3, p0 = 0, p1 = 51:0716 GPa, = 3:8, T0 = 300 K, 0 = 69 GPa, p = 0:0145 1/GPa, t = ?0:13 10?3 1/K, YA = 0:375 GPa, Ymax = 0:45 GPa, = 0:283, = 22, C1 = 0:71 1=s, C2 = 2:4 GPas, Uk =k = 3597:5 K, YP = 0:82 GPa. See Ref. [11] for material parameters. The upper frames in Fig. 4.1 display a hyperelastic material without viscoplastic e ects. Here two kinds of shock waves appear: a longitutional wave propagating in the x-direction, and a transverse wave propagating in the y-direction. The pressure ?p and shear stress at the time step 40 are illustrated in Fig. 4.1. The two-dimensional TVD method with the approximate Riemann solver gives second-order convergence in the continuous region. The shocks are well resolved over about 3-5 zones. The lower frames in Fig. 4.1 show the computational results for the same problem with an elastic viscoplastic model. Signi cant decay e ects from plasticity, applying both to the shock wave fronts and to the smooth regions appear in the solution. Next we apply our method to a high-velocity impact and penetration problem. We consider impact of a circular projectile on a moderately thick target. The material utilized in this simulations is tantalum, for which material
A NUMERICAL STUDY OF A FULLY CONSERVATIVE METHOD
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target 2
Time = 0.15 µs
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Figure 4.2. The picture shows the penetration of a tan-
talum projectile impacting normally on a tantalum target (hyperelastic-viscoplastic model with small shear EOS). Material interfaces at three time steps are shown, as obtained by front tracking techniques.
parameters are given as above. The impact velocity is 2 km/s. Material interfaces at three di erent times are shown in Fig. 4.2. A stress wave generated by the impact of the projectile propagates through the material and is re ected from the back of the target. The material interface and free boundaries are resolved by the front tracking method.
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Acknowledgments
The authors are grateful to Prof. Bradley Plohr for his valuable discussions. References 1. E. Bonnetier, H. Jourdren, and P. Veysseyre, Un modele hyperelastique-plastique eulerien applicable aux grandes Deformations: Quelques Resultats 1-D, Tech. Report preprint, Centre d'Etudes de Limeil-Valenton, 1991. 2. I-L. Chern, J. Glimm, O. McBryan, B. Plohr, and S. Yaniv, Front tracking for gas dynamics, J. Comput. Phys. 62 (1986), 83{110. 3. P. Le Floch and F. Olsson, A second-order Godunov method for the conservation laws of nonlinear elastodynamics, Impact Comput. Sci. Engrg. 2 (1990), 318{354. 4. J. Glimm, W. B. Lindquist, O. McBryan, and L. Padmanabhan, A front tracking reservoir simulator, ve-spot validation studies and the water coning problem, Frontiers in Applied Mathematics, vol. 1, SIAM, Philadelphia, PA, 1983, pp. 107{136. 5. J. W. Grove, Applications of front tracking to the simulation of shock refractions and unstable mixing, J. Appl. Num. Math. 14 (1994), 213{237. 6. X. Lin, Numerical computation of stress waves in solids, Akademie Verlag, Berlin, 1996. 7. B. Plohr and D. Sharp, A conservative formulation for plasticity, Adv. Appl. Math. 13 (1992), 462{493. 8. E. F. Toro, The weighted average ux method for hyperbolic conservation laws, Phil. Trans. Royal Soc. London A. 423 (1989), 401{418. , The weighted average ux method applied to the Euler equations, Phil. Trans. 9. Royal Soc. London A. 341 (1992), 499{530. 10. J. Trangenstein and P. Colella, A higher-order Godunov method for modeling nite deformation in elastic-plastic solids, Comm. Pure Appl. Math. XLIV (1991), 41{100. 11. F. Wang, J. Glimm, J. Grove, B. Plohr, and D. Sharp, A conservative Eulerian numerical scheme for elasto-plasticity and application to plate impact problems, Impact Comput. Sci. Engrg. 5 (1993), 285{308. E-mail address :
[email protected]
Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794-3600
E-mail address : E-mail address : E-mail address :
[email protected] [email protected] [email protected]
Theoretical Division, Los Alamos National Laboratory,, Los Alamos, NM 87545
E-mail address :
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Army Research Laboratory, Aberdeen Proving Ground,, MD 21005
