STABILIZED METHODS IN SOLID MECHANICS

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1 FINITE ELEMENT METHODS: 1970 s AND BEYOND L.P. Franca (Eds.) ccimne, Barcelona, Spain 2003 STABILIZED METHODS IN SOLID MECHANICS Arif Masud Department of Civil & Materials Engineering The University of Illinois at Chicago 842 West Taylor Street, (M/C 246) Chicago, Illinois , USA AMasud@uic.edu Abstract. This paper presents a variational multiscale method for developing stabilized finite element formulations for small strain inelasticity. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unresolved) scales. The resulting finite element formulation allows arbitrary combinations of interpolation functions for the displacement and pressure fields, and thus yields a family of stable and convergent elements. Specifically, equal order interpolations that are easy to implement but violate the celebrated B-B condition, become stable and convergent. A nonlinear constitutive model for thesuperelasticbehaviorofshapememoryalloys is integrated in the multiscale formulation. Numerical tests of the performance of the elements are presented and representative simulations of the superelastic behavior of shape memory alloys are shown. Key words: Variational multiscale method, Stabilized formulations, Nonlinear constitutive models This paper is dedicated to Professor Thomas J.R. Hughes on the occasion of his 60 th birthday

2 1. INTRODUCTION Finite Element Methods: 1970 s and beyond The application of finite element methods to the mixed variational formulations of elasticity has been an area of active research for over two decades. Attention has particularly been focused at incompressible elasticity 1 because of its fundamental place in solid mechanics in general and its ability to model a wide class of materials in particular. Volume preserving or isochoric mode of deformation is an important kinematic constraint on the response of several materials 38. Especially, in finite deformation elasto-plasticity, the plastic or the inelastic response of metals and polymers is assumed volume preserving. Standard displacement based techniques for incompressible elasticity show an overly stiff response commonly termed as locking. In order to develop convergent elements within the standard Galerkin finite element framework, special techniques are employed to address the issues related to volumetric constraint. These techniques include the use of mixed-methods 1,23,28, reduced and selective integration techniques 3,16,23, stress projection techniques 35,37,38, and B-bar methods 16. Published literature on the treatment of locking phenomena is exhaustive 1,3,4,16,23,28,34,35,37,38,andthe interested reader is referred to the standard texts by Brezzi et al. 6 and Hughes 14 for an overview of the various techniques. Amongst the prominent contributions of Tom Hughes in the arena of solid mechanics are the various methods that address the incompressibility issue and yield stable and convergent elements. In this paper we show an application of yet another contribution from Tom Hughes that leads to stabilized mixed displacementpressure formulations for solid mechanics. We first provide a brief account of the various stabilized methods, which in their progression, have lead to the developments that we are presenting in this paper. A literature review reveals that the applications of stabilized methods to mixed-field problems can be traced back to the early 80 s when Hughes and colleagues 7,19 22,24 proposed a generalization of the SUPG method for the Stokes flow problem. The SUPG method turned out to be the fore runner of a new class of stabilization schemes, namely the Galerkin/Least- Squares (GLS) stabilization method 22. Both methods yield formulations that circumvent the Babuska-Brezzi (BB) inf-sup conditions that restrict the use of many convenient interpolations. GLS stabilization was followed by the Unusual Stabilized Finite Element Method proposed by Franca and co-workers, and the Residual-Free Bubble Method introduced by Brezzi and coworkers 2,5 wherein the idea of augmenting the Galerkin method with virtual bubble functions was employed. Even though the similarity between the mixed u p form of incompressible elasticity and the Stokes flow equations was pointed out in Hughes et al. 20,the application of stabilized methods to incompressible elasticity lagged behind its application to incompressible fluid dynamics. In the mid 90 s Hughes revisited the origins of the stabilization schemes from a variational multiscale view point and presented the variational multiscale method. Since solid and structural mechanics literature is heavily dominated by variational methods, the variational multiscale method seems to possess a natural affinity for application in solid mechanics. In this work we employ Hughes variational multiscale method 15 andpresentanewmixedstabilized formulation for computational inelasticity. The new mixed stabilized formulation allows arbitrary combinations of interpolation functions for the displacement and the pressure fields, thus circumventing the Babuska-Brezzi (BB) inf-sup conditions that restrict the use of

3 A. Masud / Stabilized Methods in Solid Mechanics many convenient interpolations. A novelty of the present method is that, unlike the SUPG or the GLS methods, the present method is free of user defined or user designed stability parameters. Rather a canonical expression for the socalled stability parameter appears naturally in the developments presented herein. Secondly, the proposed formulation yields a stable response even in the incompressible limit, which is an important kinematic constraint when modeling the response of several materials 38. Specifically, in metals, in polymers and in SMAs, inelastic or the plastic flow is considered isochoric. Relatively recently, attempts have been made to employ Galerkin/least-squares method for elastoplasticity 10, and multiscale method for modeling weak discontinuities in solids 12. An outline of the paper is as follows. In Section 2, we present a mixed displacement-pressure form, and in Section 3 we develop a stabilized formulation for inelasticity. We then integrate in the stabilized form a kinetic cosine model which is a generalization of the linear kinetic model for SMA macrobehavior, presented in Masud et al. 29. In Section 5, we present a set of numerical examples to verify the theoretical developments and show the range of applicability of the proposed formulation. We present concluding remarks in Section A MIXED DISPLACEMENT-PRESSURE FORM FOR INELAS- TICITY Let n sd be an open bounded region with piecewise smooth boundary Γ. The number of space dimensions, n sd, is equal to 2 or 3. The unit outward normal vector to Γ is denoted by n =(n 1,...n sd ). The mixed displacement-pressure form of the governing equations is as follows: σ + b = 0 in (1) (ε ε I ) δ p K =0 in (2) u = g on Γ g (3) σ n = t on Γ h (4) where u(x) isthedisplacementfield and p(x) is the scalar pressure field representing the volumetric stress or the hydrostatic pressure in the incompressible limit. σ is the Cauchy stress tensor, ε I is the inelastic or the plastic strain tensor, and K is the bulk modulus defined as K = E 3(1 2ν). Equation (1) represents equilibrium in body, equation (2) is the volumetric constitutive equation in, equation (3) is the Dirichlet boundary condition on Γ g, and equation (4) is the Neumann boundary condition on Γ h. 2.1 The standard variational form The standard weighted residual form for the equilibrium and the volumetric constitutive equations is as follows: s w : σ d = w b d + w t d Γ (5) Γ q(ε ε I ) δ d q p d =0 (6) K

4 Finite Element Methods: 1970 s and beyond where s ( ) represents the symmetric gradient operator, and w and q are the weighting functions corresponding to displacement u and pressure p, respectively. This is a system of nonlinear equations because σ is a nonlinear function of strain, while equation (6) explicitly contains the inelastic strain ε I. 3. THE VARIATIONAL MULTISCALE METHOD 3.1 Multiscale decomposition This section presents an application of Hughes variational multiscale method 15, hereon termed as the HVM method, for the development of a stabilized/multiscale finite element formulation for small strain computational inelasticity. We consider the bounded domain discretized into non-overlapping regions e (element domains) with boundaries Γ e,e=1, 2,...n umel,suchthat = n umel e=1 e (7) We denote the union of element interiors and element boundaries by and Γ, respectively. n umel n umel = (int) e (elem. interiors) Γ = Γ e (elem. boundaries) (8) e=1 We assume an overlapping sum decomposition of the displacement field and its weighting function into coarse scales or resolvable scales and fine scales or the subgrid scales. u(x) = ū(x) coarse scale + u (x) fine scale (9) w(x) = e=1 w(x) + w (x) coarse scale fine scale We further make an assumption that the subgrid scales although non-zero within the elements, vanish identically over the element boundaries. (10) u = w = 0 on Γ (11) Substituting (10) in (5) and exploiting the linearity of the weighting function slot we can split equation (5) in a coarse scale and a fine scale problem. Since we have assumed fine scale pressure p = 0, therefore q = 0. Accordingly, equation (6) only contributes to the coarse scale problem. Since the trial solutions and weighting functions are implied to be functions of h, therefore to keep the notation simple, explicit dependence on h is suppressed. Coarse scale sub-problem: s w : σ d = w b d + w t d Γ (12) Γ q(ε ε I ) δ d q p K d = 0 (13)

5 A. Masud / Stabilized Methods in Solid Mechanics Fine scale sub-problem: s w : σ d = w b d + Γ w t d Γ (14) 3.2 Solution of the fine scale problem Let us first consider the fine scale sub-problem. Because of the assumption on the fine scale w as given in (11), the second term on the right hand side of (14) drops out. It is important to note that because of the assumption in (11), (14) is defined over the sum of element interiors. The general idea at this point is to solve (14) and extract an expression for the fine scale fields that can then be substituted in the coarse scale problem defined in (12)-(13), thereby eliminating the explicit presence of the fine scales, yet retaining their effects. Since we are dealing with nonlinear sub-problems, in order to solve the fine scale problem we need to linearize it 33,36. One way of linearization is to substitute the incremental value of σ in (14). Substituting the current state of stress σ i+1 n+1 = σ i n+1 + σ we get s w : σ d = w b d s w : σn+1 i d (15) In general 31, the linearized incremental stress σ is related to the incremental strain and incremental pressure as: σ = D uu ε + D up p. Substituting σ in (15) and employing the definition of the incremental strain, that, because of (9) can be split into coarse and fine scale strain fields ε = ε+ ε, and rearranging the terms we obtain s w : D uu ε d = w b d s w :[σn+1 i + Duu ε + D up p] d In order to solve the linearized fine scale sub-problem, without loss of generality, we assume that the incremental fine scale displacement field u is represented via bubble functions N e. Accordingly, the gradient of the fine scale displacement field is represented in terms of the gradient of the bubble functions as follows: (16) u = N e u e s u = s N e u e = ε (17) w = N e we s w = s N e we (18) where ε is the incremental fine-scale strain field. Substituting (17) and (18) in (16) we can extract the coefficients of the fine scale displacement field as where u e = K 1 1 R i n+1 (19) K 1 = ( s N e ) T D uu s N e d (20) R i n+1 = w b d s w :[σn+1 i + D uu ε + D up p] d (21)

6 Finite Element Methods: 1970 s and beyond The fine scale incremental displacement field u can now be constructed via recourse to (17). u = N e K1 1 R i n+1 (22) 3.3 The modified coarse-scale problem σ i+1 n+1 Let us now consider equation (12). Substituting the current state of stress and rearranging the resulting equationinaresidualform,weget s w : σ d = w b d + w t dγ s w : σn+1 i d (23) Substituting σ = D uu ε + D up p in (23), substituting the incremental strain ε = ε + ε, and rearranging terms, we get s w : D uu ε d + s w : D up pd + s w : D uu ε d = w b d + w t dγ s w : σn+1 i d (24) ε is the increment in the gradient of the elastic fine scale displacement field that can be obtained from (22). Substituting ε = s u in (24), then substituting K1 1 from (20) and R i n+1 from (21), and rearranging terms we get s w : D uu : ε d + s w : D up pd + s w : D uu : τ 1 [(D uu ε + D up p)] d = w b d + w t dγ s w :(1 + D uu τ 1 )σn+1 i d Γ + s w : D uu τ 2 b d (25) where τ 1 =( s N e τ 2 =( s N e Γ Γ s N e d)[ ( s N e ) T D uu s N e d] 1 (26) N e d)[ ( s N e ) T D uu s N e d] 1 (27) 3.4 The constitutive equation for the pressure field The weak form of the constraint equation for the pressure field is given by equation (13). We employ an iterative solution scheme to linearize it. q1 T (ε i+1 n+1 εi,i+1 n+1 )d q pi+1 n+1 d = 0 (28) K

7 A. Masud / Stabilized Methods in Solid Mechanics We now define the various incremental quantities in (28) as follows: e i+1 n+1 = ei n+1 + e (29) e I,i+1 n+1 = ei,i n+1 + ei (30) p i+1 n+1 = pi n+1 + p (31) where ( ) i n+1 represents the value at the previous iteration in the current time step, while ( ) represents the incremental value between two successive iterates. Substituting (29)-(31) in (28) leads to the following form q1 T ( ε ε I )d q p K d = q 1 T (ε i n+1 ε I,i n+1 ) pi n+1 d (32) K In J 2 theory for inelasticity, the volumetric part of the inelastic strain is zero, consequently the inelastic strain ε I is in fact the deviatoric inelastic strain e I, which is obtained via satisfaction of the Kuhn-Tucker constraint conditions of the corresponding phase transformation model. It results in a relation for ε I of the following general form. ε I = D uu ε + D up p (33) Substituting (33) into (32), and rearranging terms we get a general functional form q1 T D pu ε d qd pp pd = q 1 T (ε i n+1 ε I,i n+1 ) pi n+1 d K (34) where D pu = 1 T 1 T Duu and D pp = 1 K + 1T Dup. We now substitute the definition of the incremental strain ε = ε + ε in (34), and rearrange the terms. q1 T D pu ε d + q1 T D pu ε d qd pp pd = q 1 T (ε i n+1 ε I,i n+1 ) pi n+1 d (35) K Since ε is the increment in the gradient of the elastic fine scale displacement field, ε = s u. Substituting in (34) we get q1 T D pu ε d = qd pp pd + q 1 T (ε i n+1 εi,i n+1 ) pi n+1 K d q1 T D pu : τ 1 [(D uu ε + D up p)] d q1 T D pu :[τ 1 σ i n+1 + τ 2b] d (36) Remark 2: It is important to note that (36) is represented completely in terms of the coarse scale fields.

8 Finite Element Methods: 1970 s and beyond 3.5 The HVM form We can write the resulting HVM form, in a residual form, from (25) and (36). s w : D uu : ε d + s w : D up pd q1 T D pu ε d qd pp pd + ( s w : D uu + q1 T D pu ):τ 1 [(D uu ε + D up p)] d = w b d + w t dγ s w :(1 + D uu τ 1 )σn+1 i d Γ + s w : D uu τ 2 b d q 1 T (ε i n+1 ε I,i n+1 ) pi n+1 d q1 T D pu :[τ 1 σn+1 i + τ 2 b] d (37) K 4. A NONLINEAR CONSTITUTIVE MODEL FOR SMAs In this section we outline a 3D nonlinear kinetic cosine model for superelasticity in SMAs under mechanical loading/unloading conditions at constant high temperature. This model is an extension of a linear kinetic model presented in Masud et al. 29. We assume that the transformations are a function of the deviatoric part of the stress and affect only the deviatoric part of the strain. Figure 1 describes the kinetics of phase transformations for the complete phase change from Martensite to Austenite (M A), from Austenite to Martensite (A M), and alsotheincompletetransformations between these two phases. Fig. 1. Schematic diagram of the constitutive model. For complete phase transformation between austensite and martensite, we first define the stress range for the activation of transformation. S MS S S MF, Ṡ>0, (A M) (38) S AF S S AS, Ṡ<0, (M A) (39)

9 A. Masud / Stabilized Methods in Solid Mechanics where S MS and S MF represent the norms of the martensite-start and martensitefinish deviatoric stresses, respectively. Likewise, S AS and S AF represent the norms of the austenite-start and austenite-finish deviatoric stresses, respectively. The three dimensional S bounds, i.e., S MS,S MF,S AS and S AF are related to the experimentally obtained uniaxial σ bounds through the J 2 type norm. The phase transformation functions or the yield functions for complete and incomplete A M and M A transformations are given by F AM (ξ,s)and F MA (ξ,s), respectively. F AM (ξ,s)=ξ 1 2 cos[a M(S V MF )] 1 2 F MA (ξ,s)=ξ cos[a A(S V AF )] 1 2 (V MS S V MF ) (40) (V AF S V AS ) (41) where ξ is a scalar internal variable representing the martensite fraction. The evolution of the martensite fraction ξ for complete phase transformation is given by 1 2 {cos[a M(S S MF )] + 1} (A M) ξ = (42) 1 2 { cos[a A(S S AF )] + 1} (M A) π where a M = (S MS S MF ) and a π A = (S AS S AF ). For the partial phase transformation the activation criteria is given in Masud et al. 29. We append the Kuhn-Tucker type two-way phase transformation conditions to the model. ξ 0, F AM (S, ξ) 0, ξfam (S, ξ) =0 (A M) (43) ξ 0, F MA (S, ξ) 0, ξfma (S, ξ) =0 (M A) (44) Equation (43) represents the Kuhn-Tucker constraint on the admissible state of stress for austenite to martensite conversion that takes place during loading, i.e., Ṡ>0. Likewise, equation (44) represents the Kuhn-Tucker constraint for martensite to austenite conversion that takes place during unloading, i.e., Ṡ<0. The consistency conditions for loading (A M) and unloading (M A) are given as follows. ξ F AM (S, ξ) =0 (A M) (45) ξ F MA (S, ξ) =0 (M A) (46) Remark 3: We have employed ideas from generalized elastoplasticity for the numerical integration of the constitutive model 36.

10 Finite Element Methods: 1970 s and beyond 5. NUMERICAL RESULTS This section presents various numerical tests to verify the stability and accuracy properties of the proposed formulation. Figure 2a shows 2D elements and Figure 2b shows 3D elements that have been developed based on the formulation presented in Section3.InFigure2(a,b)dotscorrespond to the displacement nodes and circles correspond to the pressure nodes. We have employed the simplest polynomial bubbles in all the element types. Fig. 2-a. 2-D elements. Fig. 2-b. 3-D elements. 5.1 Rate of convergence study It is important to note that if we employ a linear material model in the proposed framework, we recover the stabilized form for nearly-incompressible elasticity presented in Masud and Xia 30. In this section we use the numerical rate of convergence study to establish that the proposed formulation gives rise to convergent elements. A cantilever beam of length-to-depth ratio equal to five is subjected to a parabolically varying end load. Boundary conditions are set in accordance with an exact elasticity solution. The exact solution depends on Poisson s ratio; the value is employed in the calculations to model the nearly incompressbile behavior of the material. In the present study, plane strain conditions are assumed in force. The exact solution to an applied shear force is a third-order polynomial. For linear elements, the theoretical rate of convergence for the displacement field in the energy norm is 1, while the optimal rate for the pressure field in L 2 () norm is 2. Figure 3 presents convergence in the energy norm of the displacement field and optimal rates are attained. The convergence in the L 2 () norm of the pressure field is presented in Figure 4 which is again nearly optimal. 5.2 Test of the incompressibility constraint With the help of this numerical example we show that the proposed stabilized formulation does not suffer from the locking phenomena, an issue that arrises when standard Galerkin methods are used for modeling incompressible materials. In this test case, plane strain conditions are enforced. Figure 5 presents the vertical displacement at the tip of the cantilever beam as a function of the Poisson s ratio. The elements based on the proposed formulation work all the way to the

11 A. Masud / Stabilized Methods in Solid Mechanics Fig. 3. Energy norm for u. Fig. 4. L 2 norm for p. incompressible limit. A comparison is made with the elements that are based on the Galerkin/Least-squares stabilization proposed in Hughes et al. 20. It can be seen that the pure displacement-based standard elements show severe locking in the incompressible limit. Fig. 5. Tip deflection convergence as a function of Poisson s ratio. 5.3 Comparison of the SMA model with the published results In this section we present the response from the nonlinear constitutive model for SMAs and compare it with the published results 8. Figure 6 exhibits a complete hysteresis loop; the material is austenite prior to loading, transforms to martensite during loading and later completes the reverse transformation to austenite upon unloading. A good correlation with the kinetic cosine model of Brinson et. al. 8 and the kinetic linear model of Masud et. al. 29 is obtained. Next simulation is another complete superelastic loading-unloading of the model at 60 0 C and a comparison is made with the cosine model by Brinson et. al. 9 and the linear model in Masud et. al. 29. The material properties used here have been taken from reference 9.

12 Finite Element Methods: 1970 s and beyond 5.4 Study of 8-node brick element To further test (i) the stabilized formulation, (ii) the constitutive model, and (iii) the integration algorithm, a series of uniaxial simulations were performed under displacement control. Table 1 presents the material properties employed in the simulations. In this table, the constants a A and a M are material properties that represent the slopes of the stress verses temperature curves for austenite and martensite, respectively. Table 1: Material properties used in the simulation Youngs Modulus E 7500 MPa Poisson Ratio ν Austenite start stress σ As 70 MPa Austenite finish stress σ Af 55 MPa Martensite start stress σ Ms 75 MPa Martensite finish stress σ Mf 90 MPa Austenite slope a A 1MPa 1 Martensite slope a M 1MPa 1 Width of the hysteresis loop e T L 0.06 Figure 7-a presents the stress-strain plot of the internal loops within the hysteresis loop during complete A M but incomplete M A transformation. The strain is increased so that the material now lies in the martensite regime, followed by partial unloading of the specimen in each cycle. The specimen is unloaded to zero stress state only at the end of the final loading cycle. Figure 7-b presents the stress-strain plot of incomplete A M and incomplete M A transformation. A random strain loading is applied as a function of the pseudo-time parameter and the specimen shows strong path dependence as shown in the figure. 5.5 Study of 4-node tetrahedra element Figures 8(a,b) present the partial phase transformations for the 4-node tetrahedra elements. Once again we see a stable constitutive response for the three different test cases in their entire range of deformation. 6. CONCLUSIONS In this paper we have presented an application of the Hughes Variational Multiscale (HVM) method 15 for developing stabilized finite element formulations for small strain inelasticity. Specifically, a stabilized formulation that accommodates nonlinear material behavior and the incompressibility constraint has been developed. This formulation allows arbitrary combinations of interpolation functions for the displacement and pressure fields, and thus yields a family of stable and convergent elements. The novelty of the present method is that the definition of the stability parameter appears naturally in the derivations. Another important feature is that the fine computational scales are consistently represented in terms of the computable scales via the stabilization terms in the modified variational equation. Consequently, the proposed method naturally possesses better accuracy

13 A. Masud / Stabilized Methods in Solid Mechanics Figs. 6a,b. (a) Complete transformations. (b) Partial A M transformations. Figs. 7a,b. 8-node bricks: (a) Partial M A, and (b) Random transformations. properties. Based on the formulation, a family of 2-D and 3-D elements is developed that shows a stable and convergent response while capturing the highly nonlinear constitutive behavior of SMAs. Acknowledgments The author wishes to acknowledge Kaiming Xia for the numerical simulations presented here. This work was supported by NSF grant NSF-CMS and ONR grant N This support is gratefully acknowledged. REFERENCES [1] D.N. Arnold, F. Brezzi and J. Douglas, Jr., PEERS: A new mixed finite element for plane elasticity, Japan Journal of Applied Mathematics 1984; 1: [2] C. Baiocchi, F. Brezzi and L. Franca, Virtual bubbles and galerkin-leastsquares type methods (Ga.L.S.), Comput. Methods App. Mech. Engrg. 1993; 105:

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