Gradient-Dependent Viscoplasticity and Viscodamage

INTRODUCTION

The experimental observations indicated that in general the processes of cold-working, forming, machining of mechanical parts, etc. can cause an initial evolution of defects in the virgin material state in the form of localized zones, such as the nucleation of certain amount of cracks, voids, dislocations, and shear bands. Those localized defects of plasticity and damage induced in the material structure along with the subsequent defects that occur during deformation process leads to a heterogeneous (non-uniform) material behavior. Further loading of materials of this type will cause failure mechanisms to occur at localized zones of plasticity and damage. As the plasticity and damage defects localize over narrow regions of the continuum, the characteristic length-scale governing the variations of those defects and their average interactions over multiple length-scales falls far below the scale of the local state variables of classical plasticity and damage theories used to describe the response of the continuum. This leads to the loss of the statistical homogeneity in the representative volume element (RVE) and causes strong scale effects; in such a way that all the macroscopic response functions of interest are sensitive to the distribution, size, and orientation of the micro-, meso-and macro-structural defects within the RVE. This suggests that the macroscopic inelastic deformations and failure are governed by mechanisms at different scale levels (non-locality) which gives rise to the gradient effects. Thus, the gradient effect is important when the characteristic dimension of the plastic and/or damage deformation zone is of the same order as the material intrinsic length-scale, which is in the order of microns for commonly used materials.

The enhanced gradient plasticity theories with length scales formulate a constitutive framework on the continuum level that is used to bridge the gap between different physical scale levels while it is still not possible to perform quantum and atomistic simulations on realistic time and structures (Voyiadjis et al., 2003a). This will reduce the computational cost, the prototyping costs, and time to the market. A variety of different gradient-enhanced theories are formulated in a phenomenological manner to address the aforementioned size effects through incorporation of intrinsic length-scale measures in the constitutive equations, mostly based on continuum mechanics concepts. Moreover, the gradient theories are non-local in the sense that the interstate variable at any location within a solid does not depend only on the state at that location, as in conventional plasticity theory, but also on the state at any other point within the solid. Gradient approaches typically retain terms in the constitutive equations of higher-order gradients with coefficients that represent length-scale measures of the deformation microstructure associated with the non-local continuum. Aifantis (1984) was one of the first to study the gradient regularization in solid mechanics. The gradient methods suggested by Lasry and Belytschko (1988) and Mühlhaus and Aifantis (1991) provide an alternative approach to the non-local integral equations (Kroner, 1967;Eringen and Edelen, 1972;Pijaudier-Cabot and Bazant, 1987). The gradient terms in several plasticity models are introduced through the yield function (e.g. Mühlhaus and Aifantis, 1991;de Borst and Mühlhaus, 1992;Fleck and Hutchinson, 1997;Gao et al., 1999;Voyiadjis et al., 2003b). The gradient concept has been extended to the gradient damage theory that has been developed for isotropic damage (e.g. Peerlings et al., 1996) and for anisotropic damage (e.g. Voyiadjis et al., 2001Voyiadjis et al., , 2003bVoyiadjis et al., , 2003c.

Abu Al-Rub and and Voyiadjis and Abu Al-Rub (2002) proposed by the analysis of indentation experiments that the intrinsic material length parameter of gradient plasticity theory decreases with increasing strain-rates and increases with temperature decrease. Therefore, the consideration of strain-rate effect and temperature variation on gradient plasticity and damage, particularly in dynamic problems, becomes more necessary. Existing theories of gradient plasticity and damage, however, have failed to explain such behavior. Moreover, although it has been shown that the viscoplasticity theory regularizes the solution by introducing implicitly length-scale through the viscous parameter, the numerical results still show a mesh dependency as shown by Wang and Sluys (2000). Very limited work has been carried out to investigate the influence of strain-rate effect and temperature variation on the size effect problems with the aid of gradient-enhanced plasticity and/or damage theories. In fact very few viscoplastic and/or viscodamage gradient-enhanced models have been proposed until now. Fremond and Nedjar (1996) proposed a combined gradient-and rate-dependent damage model for quasi-brittle materials and performed also two-dimensional analysis. Wang et al. (1998) proposed a gradient viscoplasticity model used to analyze stationary and propagative instabilities. Aifantis et al. (1999) and Oka et al. (2000) proposed a gradient-dependent viscoplastic constitutive model for water-saturated clay, where gradients of the volumetric viscoplastic strain were introduced into the constitutive equations. DiPrisco et al. (2002) modified a pre-existing elasto-viscoplastic constitutive model for granular soils according to gradient and non-local approaches. Gurtin (2002) generalized a rate-independent constitutive theory that includes dependences on a tensorial measure of geometrically necessary dislocations to single-crystal viscoplasticity using the gradient theory. Moreover, Gurtin (2003) developed a theoretical concept for small-deformation viscoplasticity that allows for dependences on plastic strain-gradients.

The objective of the present paper is to develop a consistent and systematic gradient-enhanced model in order to study the influence of strain-rate and temperature variation on the material intrinsic length-scales. We introduce a dipolar (i.e. strain-rate-gradient) material model that gives implicit and explicit length-scale measures in the governing equations through the use of coupled viscoinelastic theory and gradient theory, respectively.

INTERNAL STATE VARIABLES

In this work, thermal, elastic, viscoplastic (rate-dependent plasticity), and viscodamage (rate-dependent damage or creep damage) material behavior is considered. This means that the stress path, strain rate, temperature material dependence, and the nonlinear material response are all considered in this work. Thus the dependent constitutive variables are functions of the elastic strain tensor, e  , the absolute temperature, T , the temperature gradient vector, i T  , and int nof phenomenological internal state variables, k  ( int int 1,..., ; 1 k n n   ). Hence, within the thermodynamic framework and considering the assumption of infinitesimal displacements/strain relationships, the Helmholtz free energy density function can be written as:

Since the main objective is to develop the rate type constitutive equations for a thermoviscoplastic and thermoviscodamage material, the effects of strain rate, viscoplastic strain hardening/ softening, viscodamage strain hardening/ softening, micro-damage mechanisms, and thermomechanical coupling have to be considered. In order to describe such mechanisms, a finite set of internal state variables, k  , representing either a scalar or a tensorial variable are assumed, such that:

where n  is a set of viscoplasticity and viscodamage hardening internal state variables, and 2 n   is the corresponding second-order gradient (Laplacian) of n  . The state variables in this gradient-enhanced approach are no longer independent; therefore, special care must be taken to properly account for state variable coupling between n  and 2 n   . Moreover, setting n  and 2 n   as dependent internal state variables allows one to computationally introduce the effects of the material defects in the mesoscale on the macroscale response. Also, introducing those higher-order variables in the Helmholtz free energy allows the two different physical phenomena in the meso-and macroscales to be identified separately with different evolution equations. This approach is considered in this work. We make use here of the postulate of the isotropic influence (de Borst and Mühlhaus, 1992) of the averaging of the evolution equations of the assumed internal state variables, n  , over a representative volume element, which will be discussed thoroughly in the subsequent sections. Thus, the first-order gradients are disregarded and the second-order gradients (Laplacian) are mainly considered in this work. The set of the macro internal state variables, n  , is postulated as follows:

where p denotes the accumulative equivalent viscoplastic strain and  denotes the flux of the residual stress (backstress). p is associated with the isotropic hardening and  with the kinematic hardening in the viscoplastic flow process. Similarly, r denotes the accumulative viscodamage and  denotes the flux of the residual stress (kinematic hardening) in the viscodamage growth process. These viscoplasticity and viscodamage hardening variables are introduced in the Helmholtz free energy density in order to provide sufficient details of the deformation defects (cracks, voids, mobile and immobile dislocation densities) and their interactions, and to properly (i.e. physically) characterize the material microstructural behavior. Those variables will provide an adequate characterization of these defects in terms of size, orientation, distribution, spacing, interaction among defects, and so forth. However, in order to be able to achieve this explicitly, the macroscale discontinuities influence needs to be addressed and implemented properly in the modeling of the material behavior. The gradient theory introduces in the material constitutive equations higher-order deformation gradients with coefficients that represent length-scale measures that characterize microstructural links with the non-local continuum. An attempt is made here to account for the non-uniform macroscale viscoplastic and viscodamage distribution on the overall macroscale response by assuming the thermoelastic Helmholtz free energy density  to depend not only on the macroscopic response associated with the internal variables n  , but also on its macroscopic spatial higher-order gradients 2 n   , such that:

where   2   denotes the second-order gradient or Laplacian operator. The assumed dependence of the Helmholtz free energy on the distinct variables 2 n   is also motivated by the necessity to include length-scale measures into the equations of state that link the mesoscale interactions to the macroscale viscoplasticity and viscodamage, which can not be captured by n  variables alone. The damage variables  and 2   reflect the long-range microstructural deterioration due to nucleation, growth, and coalescence of voids, cavities, and microcracks. It may also account for internal embedded crack-tip stress variations introduced by crack pile-ups and, moreover, for the lack of a proper statistical distribution of microcracks and microvoids due to viscodamage localization. The determination of the evolution of the assumed internal state variables is the main challenge of the constitutive modeling. This can be effectively achieved, so far, through the thermodynamic principles for the development of a continuum thermo-elasto-viscoplastic and thermo-viscodamage based model. That is, use is made of the balancing laws, the conservation of mass, linear and angular momenta, and the first and second laws of thermodynamics (Coleman and Gurtin, 1967;Valanis, 1971). The Clausius-Duhem inequality can be written as follows:

where  ,  , ext r ,  , and q are the Cauchy stress tensor, the mass density, the density of external heat, the specific entropy, and the heat flux vector, respectively. For small strain problems an additive decomposition of the rate of the total strain tensor,

GENERAL THERMODYNAMIC FORMULATION

According to the definition given above for  , the time derivative of Eq.

(1) with respect to its internal state variables is given by:

Substituting the rate of the Helmholtz free energy density, Eq. (6), into the Clausius-Duhem inequality, Eq. (5), one obtains the following thermodynamic constraint:

Assuming that the axiom of entropy production holds, then the above inequality equation results in the following thermodynamic state laws:

The above equations describe the relations between the state variables (observable and internal) and their associated thermodynamic conjugate forces, where k

), respectively. The stress  is a measure of the elastic changes in the internal structure, while Y and g Y are measures of the elastic-damage changes in the internal structure resulting from crack closure and voids contraction during the unloading process. The conjugate forces R , g R , X and g X are measures of viscoplastic changes in the internal structure, while K , g K ,  and g H are measures of the viscodamage changes in the internal structure. The superscript 'g' indicates the thermodynamic conjugate force corresponding to the second-order gradient or Laplacian of the assumed internal state variables.

Substituting Eqs. (8) into relation (7), one reduces the Clausius-Duhem inequality in order to express the fact that the dissipation energy,  , is necessarily positive as follows:

where F and G are the viscoplastic and viscodamage potential functions, respectively. One now makes use of the maximum viscoinelastic dissipation principle, which states that the actual state of the thermodynamic forces is that which maximizes the inelastic dissipation function over all other possible admissible states. Therefore, we maximize the objective function  by using the necessary conditions as follows:

Substitution of Eq. (10) into Eq. (11) yields the thermodynamic state laws, where Eq. (11) 1 gives the viscoinelastic strain rate as follows:

Eq. (11) 2 gives the viscodamage evolution law as follows:

(1)

and Eq. (11) 3 gives the complementary laws for the evolution of the local hardening variables ( n  , 1,...,5 n  ) which are summarized in Table 1. On the other hand, by adopting the assumption of isotropic influence, the complementary laws for the evolution of the second-order gradients of the assumed local hardening variables ( 2 n   , 1,...,5 n  ) can be directly obtained by operating on the local equations of Table 1 with the Laplacian operator. Similarly, 2    can be obtained from Eq. (13). The resulting evolution laws are listed in Table 1. By doing this we enhance the coupling between the evolutions of n  and 2 n   .

SPECIFIC FREE ENERGY FUNCTION

The definition of  constitutes a crucial point of the thermodynamic formulation. It is possible to decouple the Helmholtz free energy into a potential function for each of the internal state variable in such a way that an analytical expression for the thermodynamic potential is given as a quadratic form of its internal state variables (Voyiadjis et al., 2003b). By adapting this assumption, one can define the constitutive equations for the conjugate forces as:

Superimposed hat in Eq. (17) denotes the spatial non-local operator. The expressions for n  and g n  are outlined in Table 2.

)  E   is the fourth-order damage elastic tensor,  is the second-order tensor of the thermo-mechanical coefficients, c is the coefficient of thermal expansion, r  is the reference entropy, r T is the reference temperature, and k is the heat conductivity second-rank tensor. In Table 2

WEAK-NONLOCAL FORMULATION

As we mentioned earlier, the assumed internal state variables in the current work are no longer independent and special care must be taken to properly account for state variable coupling between n  and 2 n   . In order to enhance this coupling, one can start by defining the evolution of the strong nonlocal variable n   at position x as the weighted average of its local counterpart n   over a surrounding volume V at a small distance C l   from the considered point (Kroner, 1967;Pijaudier-Cabot and Bazant, 1987), such that:

where c l is an intrinsic characteristic length and   h  is a weight function that decays smoothly with distance. By expanding the evolution of the local variable n  around 0   using the Taylor series, truncating it after the quadratic term, and substituting the result into Eq. (17) yields, after lengthy formulation but straightforward implementation, the results outlined in Table 3.

Assuming that dislocations are responsible for the evolution of both plasticity and damage (voids and cracks) morphologies, Abu Al-Rub and derived through a set of dislocation-based considerations evolution law for the length scale parameter different than the phenomenological expression outlined in  is the reference thermal stress.