George Z. Voyiadjis

A plasticity-damage theory for large deformation of solids—I. Theoretical formulation

Abstract A coupled theory of continuum damage mechanics and finite strain plasticity (with small elastic strains) is formulated in the Eulerian reference system. The yield function used is of the von Mises type and incorporates both isotropic and kinematic hardening. An explicit matrix representation is derived for the damage effect tensor for a general state of deformation and damage. Although the theory is applicable to anisotropic damage, the matrix representation is restricted to isotropy. A linear transformation is shown to exist between the effective deviatoric Cauchy stress tensor and the total Cauchy stress tensor. It is also shown that a linear transformation between the deviatoric Cauchy stress tensor and its effective counterpart is not possible as this will lead to plastic incompressibility in damaged materials. In addition, an effective elasto-plastic stiffness tensor is derived that includes the effects of damage. The proposed model is applied to void growth through the use of Gursons yield function. It is also shown how a modified Gurson function can be related to the proposed model. Some interesting results are obtained in this case.

On the coupling of anisotropic damage and plasticity models for ductile materials

In this contribution various aspects of an anisotropic damage model coupled to plasticity are considered. The model is formulated within the thermodynamic framework and implements a strong coupling between plasticity and damage. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. The damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The model considers different interaction mechanisms between damage and plasticity defects in such a way that two-isotropic and two-kinematic hardening evolution equations are derived, one of each for the plasticity and the other for the damage. An additive decomposition of the total strain into elastic and inelastic parts is adopted in this work. The elastic part is further decomposed into two portions, one is due to the elastic distortion of the material grains and the other is due to the crack closure and void contraction. The inelastic part is also decomposed into two portions, one is due to nucleation and propagation of dislocations and the other is due to the lack of crack closure and void contraction. Uniaxial tension tests with unloadings have been used to investigate the damage growth in high strength steel. A good agreement between the experimental results and the model is obtained.

A coupled anisotropic damage model for the inelastic response of composite materials

A coupled incremental damage and plasticity theory for rate-independent and rate-dependent composite materials is introduced here. This allows damage to be path-dependent either on the stress history or thermodynamic force conjugate to damage. This is achieved through the use of an incremental damage tensor. Damage and inelastic deformations are incorporated in the proposed model that is used for the analysis of fiber-reinforced metal‐matrix composite materials. The damage is described kinematically in both the elastic and inelastic domains using the fourth-order damage eAect tensor which is a function of the second-order damage tensor. A physical interpretation of the second-order damage tensor is given in this work which relates to the microcrack porosity within the unit cell. The inelastic deformation behavior with damage is viewed here within the framework of thermodynamics with internal state variables. Computational aspects of both the rate-independent and rate-dependent models are discussed in this work. The Newton‐Rapson iterative scheme is used for the overall laminate system. The constitute equations of both the rate-independent and the rate-dependent plasticity coupled with damage models are additively decomposed into the elastic, inelastic and damage deformations by using the three-step split operator algorithm [J.W. Ju, Internat. J. Solids Struc. 25 (1989) 803‐833]. The main framework return maping algorithm [M. Ortiz, C. Simo, Internat. J. Numer. Meth. Eng. 23 (1986) 353‐366] is used for the correction of the elasto-plastic and viscoplastic states. However, for the case of the damage model these algorithms are redefined according to the governed damage constitutive relations. In order to test the validity of the model, a series of laminated systemsO0O8sUU; O90O8sUU; O0=90U O4sU ; Oˇ45=45U O2sU are investigated at both room and elevated temperatures of 538∞C and 649∞C. The results obtained from the special purpose developed computer program, DVP-CALSET (Damage and Viscoplastic Coupled Analysis of Laminate Systems at Elevated Temperatures), are then compared with the available experimental results and other existing theoretical material models obtained from the work of B.S. Majumdar,

A coupled theory of damage mechanics and finite strain elasto-plasticity—II. Damage and finite strain plasticity

Abstract A constitutive model is formulated here for anisotropic continuum damage mechanics using finite strain plasticity. The formulation is given in spatial coordinates (Eulerian reference frame) and incorporates both isotropic and kinematic hardening. The von Mises yield criterion is modified to include the effects of damage through the use of the hypothesis of elastic energy equivalence. A modified elasto-plastic stiffness tensor that includes the effects of damage is derived within the framework of the proposed model. Numerical implementation of the proposed model includes the finite element formulation where an Updated Lagrangian description is used. The basic example of finite simple shear is solved. The problem of crack initiation is also solved for a thin elasto-plastic plate with a center crack that is subjected to inplane tension. This problem is solved in the companion paper using the coupled theory of elasticity with damage.

A Finite Strain Plastic-damage Model for High Velocity Impacts using Combined Viscosity and Gradient Localization Limiters: Part II - Numerical Aspects and Simulations

In this companion article, we present within the finite element context the numerical algorithms for the integration of the thermodynamically consistent formulation of geometrically nonlinear gradient-enhanced viscoinelasticity derived in the first part of the article. The proposed unified integration algorithms are extensions of the classical rate-independent return mapping algorithms to the rate-dependent problems. An operator split structure is used consisting of a trial state followed by the return map by imposing the generalized viscoplastic and visco-damage consistency conditions simultaneously. Furthermore, a trivially incrementally objective integration scheme is established for the rate constitutive relations. The proposed finite deformation scheme is based on hypoelastic stress-strain representations and the proposed elastic predictor and coupled viscoplastic-viscodamage corrector algorithm allows for the total uncoupling of geometrical and material nonlinearities. A simple and direct computational algorithm is also used for calculation of the higher-order gradients. This algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches for integrating gradient-dependent models. The nonlinear algebraic system of equations is solved by consistent linearization and the Newton-Raphson iteration. The proposed model is implemented in the explicit finite element code ABAQUS via the user subroutine VUMAT. Model capabilities are preliminarily illustrated for the dynamic localization of inelastic flow in adiabatic shear bands and the perforation of a 12 mm thick Weldox 460E steel plates by deformable blunt projectiles at various impact speeds. The simulated shear band results well illustrated the potential of the proposed model in dealing with the well-known mesh sensitivity problem. Consequently, the introduced implicit and explicit length-scale measures are able to predict size effects in localization failures. Moreover, good agreement is obtained between the numerical simulations and experimental results of the perforation problem.

A coupled theory of damage mechanics and finite strain elasto-plasticity—I. Damage and elastic deformations

Abstract A coupled theory of elasticity and continuum damage mechanics is formulated here. It is assumed that the material undergoes damage with small elastic strains. The hypothesis of elastic energy equivalence is used in order to produce the proposed coupling. The damage variable used represents average material degradation which reflects the various types of damage at the microscale level like nucleation and growth of voids, cavities, micro-cracks and other microscopic defects. The proposed model is numerically implemented using finite elements with an Updated Lagrangian description. It is also shown how the model can be applied to problems of ductile frActare. The problem of crack initiation in a thin plate with a center crack that is subjected to uniaxial tension is analyzed using the proposed model. The theory is extended to coupling of damage and finite strain plasticity in the companion paper. It is shown there how the model can be used to solve problems in elasto-plastic ductile fracture.

A Comparative Study of Damage Variables in Continuum Damage Mechanics

In this work, various definitions of the damage variables are examined and compared. In particular, special emphasis is given to a new damage variable that is defined in terms of the elastic stiffness of the material. Both the scalar and tensorial cases are investigated. The scalar definition of the new damage variable was used recently by many researchers. However, the generalization to tensors and general states of deformation and damage is new and appears here for the first time. In addition, transformation laws for various elastic constants are derived. Finally, the cases of plane stress, plane strain, and isotropic elasticity are examined in detail. In these cases it is shown that only two independent damage parameters are needed to describe the complete state of damage in the material. In this work, a physical basis is sought for the damage tensor [M] that is used to link the damage state of the material with effective undamaged configuration. The authors and numerous other researchers have used different paths including fabric tensors (Voyiadjis and Kattan, 2006a; Voyiadjis et al., 2007) to connect the two configurations. However, the approach presented here provides for a strong physical basis for this missing link.

Quantification of damage parameters using X-ray tomography images

Damage in materials can be represented in many forms such as specific void or crack surfaces, the spacing between cracks, scalar representation of damage and general tensorial representation of damage. This paper presents methods to quantify the specific damaged surface area, the specific damaged surface area tensor, the damage tensor, the mean solid path among the damaged surfaces and the mean solid path tensor. The methods are general and use the reconstructed three-dimensional structure from tomography images and a virtual sectioning technique to obtain cross-sectional images needed for the quantification. The paper also presents the relation between the quantified damage parameters and their applications in mechanical modeling. This work is applied in particular to quantify the damage parameters of asphalt concrete specimens of the WesTrack project.

Multiscale simulation of nanosystems

The authors describe simulation approaches that seamlessly combine continuum mechanics with atomistic simulations and quantum mechanics. They also discuss computational and visualization issues associated with these simulations on massively parallel computers. Scientists are combining continuum mechanics and atomistic simulations through integrated multidisciplinary efforts so that a single simulation couples diverse length scales. However, the complexity of these hybrid schemes poses an unprecedented challenge, and developments in scalable parallel algorithms as well as interactive and immersive visualization are crucial for their success. This article describes such multiscale simulation approaches and associated computational issues using recent work as an example.

The kinematics of damage for finite-strain elasto-plastic solids

In this paper the kinematics of damage for finite strain, elasto-plastic deformation is introduced using the fourth-order damage effect tensor through the concept of the effective stress within the framework of continuum damage mechanics. In the absence of the kinematic description of damage deformation leads one to adopt one of the following two different hypotheses for the small deformation problems. One uses either the hypothesis of strain equivalence or the hyphothesis of energy equivalence in order to characterize the damage of the material. The proposed approach in this work provides a general description of kinematics of damage applicable to finite strains. This is accomplished by directly considering the kinematics of the deformation field and furthermore it is not confined to small strains as in the case of the strain equivalence or the strain energy equivalence approaches. In this work, the damage is described kinematically in both the elastic domain and plastic domain using the fourth order damage effect tensor which is a function of the second-order damage tensor. The damage effect tensor is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. Two kinds of second-order damage tensor representations are used in this work with respect to two reference configurations. The finite elasto-plastic deformation behavior with damage is also viewed here within the framework of thermodynamics with internal state variables. Using the consistent thermodynamic formulation one introduces seperately the strain due to damage and the associated dissipation energy due to this strain.