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EM 15th ASCE Engineering Mechanics Conference June 2-5, 2002, Columbia University, New York, NY 2002 NONLOCAL PLASTICITY FORMULATION INCORPORATING GRADIENT OF KINEMATIC HARDENING George Z. Voyiadjis1, Fellow ASCE and Robert J. Dorgan1, Member ASCE ABSTRACT In this proposed work, nonlocal behavior is introduced through the second gradient of kinematic hardening models in order to introduce a microstructural characteristic length into the model and in order to introduce long-range microstructural interactions that allow the response of a material point to depend on the state of its neighborhood in addition to the state of the point itself. It is intended to develop a consistent and systematic framework for the gradient approach that will enable one to better understand the nonlocal effects of material inhomogeneity on the macroscopic behavior and the material instabilities. The internal state variables and the corresponding gradient terms are assumed to be independent internal state variables with respect to each other with different physical interpretations and initial conditions which allows these two different physical phenomena to be identified separately. The second order gradient of the kinematic hardening is introduced through the Helmholtz free energy and through the plastic potential function. Computational issues of the gradient approach are introduced in a form that can be applied using the finite element approach. Keywords: nonlocal, gradient, plasticity INTRODUCTION Materials with microstructure are nonlocal in behavior in the interplay of characteristic lengths including sizes or spacing of micro defects, grain sizes, fiber spacing, etc. The microstructural characteristic lengths are significant in the analysis of the material at a scale where the microstructure characteristic length is greater than the required resolution length, or where the size of the representative volume element is significant compared to the specimen size. For example, defects in engineering materials lead in some cases to specific pattern formation due to a coupling of inelastic mechanisms of microcrack and microvoid growth with plastic flow. Initial loading of heterogeneous materials causes non-interacting microcracks and microvoids; however, experimental observations indicate that further loading will cause failure mechanisms to occur at localized zones of plasticity where increased interaction and coalescence of microcracks and microvoids take place, leading to a degradation of the global stiffness and to a subsequent decrease of the load carrying capacity of the material. The localization over a narrow region of 1 Dept. of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70820. E-mail: Voyiadjis@eng.lsu.edu, RDorgan@lsu.edu. the continuum causes the characteristic length scale to fall far below the scale of the state variables of strain used to describe the response of the continuum. Classical plasticity does not adequately capture a decreased length scale, and it is therefore necessary to look to alternative strategies. Aifantis (1982) suggested a gradient approach to deformation to describe plastic instabilities including dislocation patterning and spatial characteristics of shear bands. In this work, nonlocal behavior was introduced into the plasticity model through gradient terms in the yield function. Voyiadjis, Deliktas, & Aifantis (2001), Voyiadjis, Dorgan, & Dorroh (2001), and Voyiadjis & Dorgan (2001) proposed a gradient dependent theory of plasticity and damage over multiple scales that incorporated internal variables and the corresponding gradients at both the macro and mesoscales in the plasticity and damage potential functions as well as the yield and damage criteria. In this work, a formulation is given in which a second order gradient of the plasticity kinematic hardening at the macroscale is introduced through the Helmholtz free energy and through the plasticity potential function. The gradient approach used will enable one to better understand the nonlocal effects of material inhomogeneity on the macroscopic behavior and the material instabilities. The kinematic hardening and the corresponding gradient are assumed to be independent internal state variables with respect to each other with different physical interpretations and initial conditions which allows these two different physical phenomena to be identified separately. Computational issues of the gradient approach are introduced in a form that can be applied using the finite element approach. NONLOCAL MEASURE OF KINEMATIC HARDENING Aifantis (1982) introduced first introduced the used of a gradient continuum enhancement as a special case of the general concept of nonlocal continua. In order to introduce long-range microstructural interaction, the stress response at a material point was assumed to depend on the state of its neighborhood in addition to the state of the point itself. Voyiadjis, Deliktas, & Aifantis (2001), Voyiadjis, Dorgan, & Dorroh (2001), and Voyiadjis & Dorgan (2001) investigated the use of nonlocal measures to describe nonlocal behavior of damage and plastic hardening through the damage and plasticity potential functions. In this work, kinematic hardening flux variable will be considered and is denoted by αij, where the subscripted letters indicate the tensorial nature. Kinematic hardening deals with anisotropic hardening and is modeled by the movement and distortion of the yield surface. Based on simplifications of the formulations given in Voyiadjis & Dorgan (2001) in which only kinematic hardening with gradients taken at the macroscale is considered, the nonlocal measure of the kinematic hardening flux (which is not necessarily an internal state variable) is denoted with an over bar as α ij and can be expressed as: α ij = α ij + A ∇ 2α ij (1) In this equation, A is a coefficient proportional to a length squared and weights each component of the gradient term identically. It is through this coefficient that an internal length scale is introduced into the plasticity model. Since stresses and strains are macro-variables that are computed using the internal state variables of the material, gradient effects are not introduced directly through the strains and stresses by introducing terms such as εij and σ ij . The second order gradient of the hardening measures will be introduced through the Helmholtz free energy and through the plastic potential function, and thus nonlocal effects for the stresses and strains will indirectly be introduced. 2 THERMODYNAMICALLY CONSISTENT GRADIENT THEORY The kinematic hardening of plasticity, α ij , is defined as the cumulative effect from the flux related backstresses Yet, the gradient term referring to the backstress may have a different physical interpretation characterizing the internal embedded stress variations introduced by dislocation pile-ups, etc. In this proposed work, the kinematic hardening and the corresponding gradient term are assumed to be independent internal state variables with respect to each other with different physical interpretations and initial conditions which allows these two different physical phenomena to be identified separately. Thus, the thermoelastic Helmoltz free energy may now be expressed in terms of these independent internal state variables as: Ψ = Ψ ( εij ,T,α ij , ∇ 2α ij ) ′ (2) ′ ′′ Additive decomposition of the strain is assumed with ε ij being the elastic component and ε ij being the corresponding plastic component such that (Nemat-Naser, 1983): ′ ′′ ε ij = ε ij + ε ij (3) Since the internal state variables are selected independently of one another, one can express the analytical form of the Helmholtz free energy given by Eq. (2) as the quadratic form in terms of its internal state variables as: ρΨ = 1 2 (ε ij − ε ij′′ ) Eijkl (ε kl − ε kl ) + 1 a1 αij αij + 1 a2 ∇ 2 αij ∇ 2 αij ′′ 2 2 (4) where the matrix Eijkl is the fourth-order elastic stiffness tensor. In Eq. (4), the coefficients a1 and a2 are dependent on material and geometrical properties. In the case of composites the geometrical properties may include size, shape, and spacing of the fibers. In the case of the gradient theory these coefficients become also dependent on the gradient of the fiber size and fiber spacing variation. The functional dependency of these coefficients can be obtained by studying the interaction problem of an inclusion embedded in an infinite homogeneous matrix subjected to a macroscopic stress rate and corresponding strain rate at infinity (Kroner, 1967) One can express the time derivative of Eq. (2) as follows: ∂Ψ ∂Ψ ∂Ψ ∂Ψ Ψ= ′ ε ij + T+ α ij + ∇ 2 αij (5) ′ ∂ε ij ∂T ∂α ij ∂∇ 2 α ij Using this equation in the Clausus-Duhem inequality gives the following thermodynamic state laws: σ ij = ρ ( ∂Ψ ∂ε ij ) ′ (6) s = − ∂Ψ ∂T (7) X ij = ρ ( ∂Ψ ∂α ij ) ∇ 2 X ij = ρ ( ∂Ψ ∂∇ 2 α ij ) (8) 3 where X ij and ∇ 2 X ij are defined as the thermodynamic conjugate forces corresponding to the nonlocal internal state flux variables, α ij and ∇ 2 α ij , respectively. Substitution of Eq. (4) into Eqs. (8) gives the definitions for the local thermodynamic conjugate forces describing the kinematic hardening and gradient of kinematic hardening as: X ij = a 1 α ij ∇ 2 X ij = a2 ∇ 2 α ij (9) Another method of deriving the gradient of the thermodynamic conjugate force for kinematic hardening is to directly take the gradient of Eq. (9)-1. Assuming that ∇ 2 a1 = 0 , this procedure and Eq. (9)-2 give the same result only if a1 = a2 = a so that: X ij = a α ij ∇ 2 X ij = a ∇ 2 αij (10) The nonlocal measure of the thermodynamic conjugate force is assumed to be of the same form as the conjugate forces given in Eq. (9)-1: X ij = a α ij = X ij + A ∇ 2 X ij (11) The value of the thermodynamic conjugate force can be obtained through the evolution relations of the internal state variables, which are obtained by assuming the physical existence of the dissipation potential at the macroscale. The dissipation processes are given as the product of the thermodynamic conjugate force with the respective flux variable as follows: ′′ Π = σij ε ij − X ij α ij − ∇ 2 X ij ∇ 2 α ij (12) YIELD CRITERIA AND EVOLUTION EQUATIONS The theory of functions of several variables is used with the Lagrange multiplier λ to construct the objective function Ω in the following form: Ω=Π−λ F (13) where, in order to be consistent and satisfy the generalized normality rule of thermodynamics, the plastic potential function, F, is defined in the following form: X ij X ij = f + ( X ij X ij + 2 A X ij ∇ 2 X ij + A2 ∇ 2 X ij ∇ 2 X ij ) k k F= f + (14) 2 2 In this equation, k is a constant used to adjust the units of the equation, and f is a yield function of Von Mises type and is defined such that: 1 3 f = ( sij − X ij )( sij − X ij ) 2 − σ yp ≡ 0 (15) 2 4 where sij is the deviatoric component of the stress tensor, σ ij . It is noted that yield function is in terms of the local conjugate force X ij . This form of the yield function is introduced due to the fact that the authors do not introduce a “nonlocal” stress such as σij . Thus, it would be meaningless to subtract a nonlocal term from the local stress quantity. The evolution equations for the plastic strain and the internal state variables are obtained using the generalized normality rule of thermodynamics such that: ∂F ′′ ε ij = λ (16) ∂σij ∂F ∂f α ij = −λ = −λ + k ( X ij + A ∇ 2 X ij ) (17) ∂X ij ∂X ij ∂F ∇ 2 αij = −λ = −λ kAX ij (18) ∂∇ 2 X ij where, from Eqs. (14) and (15), it is seen that the following relations can be obtained: −∂f ∂f ∂f ∂F = = = (19) ∂X ij ∂sij ∂σ ij ∂σ ij Using Eq. (16), a modified form of the Frederick-Armstrong equation can be obtained as: ′′ α ij = ε ij − λ kX ij − λ kA ∇ 2 X ij (20) In order to obtain the plastic multiplier, λ , the consistency condition for plasticity ( f = 0) is used: ∂f ∂f f ≡ : σ ij + : X ij = 0 (21) ∂σ ij ∂X ij COMPUTATIONAL ISSUES OF THE GRADIENT APPROACH In this work, the set of differential equations involve macroscale second order gradients of the internal state variable for plasticity. In order to solve this higher order problem, the finite element approach is used here in a similar fashion as the models proposed by de Borst et al. (1995), where the yield condition is satisfied only in the weak form. The discretization procedure for the displacement field u and the plasticity multiplier, λ , will require Co continuous interpolation functions assembled in the shape function N, such that u = Na , λ = NΛ (22) where a is the nodal displacement vector and Λ denotes a vector of the nodal degrees of freedom for the plastic multiplier field. 5 Discretization of the Displacement Field and the Yield Condition To begin with, we will assume that the loads are applied slow enough to be considered static, and that the inertial forces are small enough to be neglected. From these assumptions, the equilibrium equation, strains, and incremental stresses can be given as follows: ∇ ⋅σ + ρb = 0 (23) ε = LT u = LT Na = Ba (24) dσ = C e dε ′ (25) where b is the vector of body-forces, ρ is the density, Ce is the elastic stiffness tensor, and L is a tensor of differential operators, the superscript T is the transpose symbol, and where B is a matrix that relates the strain and the displacement.. The divergence term is denoted by ∇, from which the term ∇ ⋅σ can be written in indicial notation as ∂σ ji ∂x j . From the assumption of additive decomposition of strains, and also assuming associative plasticity, Eq. (25) can be written as: ∂f dσ = C e dε − dλ (26) ∂σ If we ignore body forces, we can take the weak form of Eq. (22)-1 to be: ∫δ u ( ⋅ σ j +1 ) dV = 0 T ∇ (27) where δ denotes the variation of a term. The standard boundary conditions are defined in the following equations: ∑ νs = t u = us (28) where ∑ is the stress tensor in matrix form, ν s is the outward normal to a surface S , and t is the boundary traction vector. Integrating by parts, and using these standard boundary conditions Eq. (27) can be rewritten as follows: e ∂f ∫δ ε C dε − dλ dV = ∫ δ u t j +1 dS − ∫ δ ε σ j dV T T T (29) V ∂σ S V Using Eqs. (22) and (24) in this equation, the discretized equilibrium equation is written as follows: δ aT ∫ {BT Ce B da − BT Ce mNdΛ} dV = δ aT ∫ NT t j +1 dS − ∫ BT σ j dV (30) V S V where: 6 ∂F ∂f mj = = (31) ∂σ j ∂σ j A second set of linear system of equations may be obtained by using the yield condition that is satisfied in a distributed sense such that ∫ δλ F ( σ V j +1 , X j +1 , ∇ 2 X j +1 ) dV = 0 (32) After expanding the yield potential, F j +1 , around [ σ j , X j , ∇ 2 X j ] by using a Taylor series expansion, and using the evolution equations for X and ∇ 2 X are found from Eqs. (10) with Eqs. (17) and (18), the yield potential is written in terms of the plastic multiplier, d λ , as: T T T ∂F ∂F ∂f ∂F F j +1 = F j + dσ j + a d λ − kX − d λ akA 2 X j (33) ∂σ j ∂X j ∂σ j ∂∇ X j such that, after simplifications of this yield potential, the discretized yield condition can be written in the following form: V { −δ ΛT ∫ NT mTj Ce Bda + a − ( mTj − kXTj ) ( m j − kX j ) + k 2 A2 XTj X j NdΛ dV } (34) = δ ΛT ∫ NT Fj dV which is valid provided that the non standard boundary conditions for plasticity given in the following expressions are valid on the elastic-plastic boundary S p : δλ = 0, or ( ∇d λ )ν p = 0 (35) A more detailed explanation for the non standard boundary conditions of plasticity can be found in the work of de Borst et al. (1995). Combined Discretization Equations Combining Eqs. (30) and (34), one can obtain a set of algebraic equations in terms of the variations da and dΛ : K aa K a λ da f e + f a K = K λλ dΛ fλ (36) λa where the diagonal and off diagonal matrices are defined as follows K aa = ∫ BT Ce BdV (37) V 7 { } K λλ = −a ∫ −NT ( mTj − kXTj ) ( m j − kX j ) N + k 2 A2 NT XTj X j N dV (38) K aλ = − ∫ BT Cem j NdV , K λ a = − ∫ NT mTj Ce BdV (39) V V The corresponding external force vector and the nodal force vector equivalent to internal stresses is given by fe = ∫ NT t j +1dS , f a = − ∫ BT σ j dV , fλ = ∫ NT Fj dV (40) S V S CONCLUSIONS Thermodynamically consistent constitutive equations have been derived which may be used to investigate issues such as size effect on the strength of the composite or strain localization effects on the macroscopic response of composites. The proposed capability of the model is to simulate properly size dependent behavior of the materials together with localization problems through the incorporation of an internal material length scale. Calibration for the different material parameters in the proposed approach may be difficult, or impossible for certain cases. Ultimately, a hybrid computational simulation framework in finite elements that will cover multiple length scales will need to be used in order to properly associate the physics of the material behavior with the gradient theory and its corresponding parameters. REFERENCES Aifantis, E.C. (1982), “Some thoughts on degrading materials,” NSF Workshop on Mechanics and of Damage and Fracture (Edited by S. N. Atluri and J. E. Fitzerald), Georgia Tech., Atlanta, pages 1-12. de Borst, R., Pamin, J., and Sluys, L.J. (1995), “Computational issues in gradient plasticity,” In Continuum Models for Materials with Microstructure (Edited by H.-B. Mühlhaus), John Wiley & Sons Ltd., pages 159-200. Kroner, E. (1967), “Elasticity theory of materials with long range cohesive forces,” International Journal of Solid and Structures, 3: 731-742. Nemat-Naser, S. (1983), “On finite plastic flow of crystalline solids and geomaterials,” Journal of Applied Mechanics, 50: 1114-1126. Voyiadjis, G.Z., Deliktas, B., Aifantis, E.C. (2001), “Multiscale analysis of multiple damage mechanisms coupled with inelastic behavior of composite materials,” Journal of Engineering Mecanics, 127(7): 636. Voyiadjis, G.Z., Dorgan, R.J. and Dorroh, J.R. (2001), “Bridging length scales between macroscopic response and microstructure through gradient anisotropic damage for MMCs,” Proceedings of the 2001 Energy Sources Technology Conference & Exhibition, February 2001, Houston, Texas, ASME Publishing Company. Voyiadjis, G.Z. and Dorgan, R.J. (2001), “Gradient formulation in coupled damage-plasticity,” Archives of Mechanics, 53(4-5): 565-597. 8