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NONLOCAL PLASTICITY FORMULATION INCORPORATING GRADIENT OF

VIEWS: 6 PAGES: 8

									                                                                                                             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.




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