Kazuyuki Shizawa

A Thermodynamical Theory of Gradient Elastoplasticity with Dislocation Density Tensor. I : Fundamentals

Abstract A thermodynamical theory of gradient elastoplasticity, including kinematic hardening, is developed by introducing the concept of dislocation density tensor. The theory is self-consistent and is based on two fundamental principles, the principle of increase of entropy and the maximal entropy production rate. Thermodynamically consistent constitutive equations for plastic stretching, plastic spin and back stress are rigorously derived. Also, an expression for the plastic spin is obtained from the constitutive equation of dislocation drift rate and an expression for the back stress is given as a balance equation expressing equilibrium between internal stress and microstress conjugate to the dislocation density tensor. Moreover, it is shown that the present gradient theory yields a symmetric stress tensor. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given.

A strain-gradient thermodynamic theory of plasticity based on dislocation density and incompatibility tensors

In this work, we discuss a thermodynamic theory of plasticity for self-organization of collective dislocations in FCC metals. The theory is described by geometrical tensor quantities of crystal defect fields such as dislocation density tensor, representing net mobile dislocation density and geometrically necessary boundaries, and the incompatibility tensor representing immobile dislocation density. Conservation laws for the two kinds of dislocation density are formulated with dislocation products and interactions terms. Based on the second law of thermodynamics, we drive basic constitutive equations for the dislocation flux, production and interaction terms of dislocations. We also derive a set of reaction-diffusion equations for the dislocation density tensor and incompatibility tensor which describes the vein and persistent slip band (PSB) ladder structures. These equations are analyzed using linear stability and bifurcation analysis. An intrinsic mesoscopic length scale is determined which provides an estimate for the wavelength of the PSBs. The basic aspects of the model are motivated and substantiated by analyzing the stress fields of various possible dislocation configurations using discrete dislocation dynamics.

A Thermodynamical Theory of Plastic Spin and Internal Stress With Dislocation Density Tensor

A thermodynamical theory of elastoplasticity including kinematic hardening and dislocation density tensor is developed. The theory is self-consistent and is based on two fundamental principles of thermodynamics, i.e., the principle of increase of entropy and maximal entropy production rate. The thermodynamically consistent governing equations of plastic spin and back stress are rigorously derived. An expression for the plastic spin tensor is obtained from the constitutive equation of dislocation drift rate tensor and an expression for the back stress tensor is given as a balance equation expressing an equilibrium between internal stress and microstress conjugate to the dislocation density tensor. Moreover, it is shown that, in order to obtain a thermodynamically consistent theory for kinematic hardening, the free energy density should have the dislocation density tensor as one of its arguments.

A Polycrystalline Analysis of Hexagonal Metal Based on the Homogenized Method

In this study, a three-dimensional finite element formulation for polycrystalline plasticity model based on the homogenization method has been presented. The homogenization method is one of the useful procedures, which can evaluate the homogenized macroscopic material properties with a periodical microstructure, so-called a unit cell. The present study focuses on hexagonal metals such as titanium or magnesium. An assessment of flow stress by the presented method is conducted and it is clarified how the method can reproduce the behavior of hexagonal metal.

A Phase-Field Simulation of Nucleation from Subgrain and Grain Growth in Static Recrystallization

A new recrystallization phase-field method is proposed, in which the three stages of recrystallization phenomena, i.e., recovery, nucleation and nucleus growth are sequentially taken into account in a computation. From the information of subgrain patterns and crystal orientations in a polycrystal that are obtained by a dislocation-crystal plasticity FE analysis based on a reaction-diffusion model, subgrain groups surrounded by high angle boundary are found out. Next, subgrains in the group are coalesced into a nucleus by rotation of crystal orientation and migration of subgrain boundaries through a phase-field simulation. Then a computation of nucleus growth is performed also using the phase-field method on account of an autonomic incubation period of nucleation, in which stored dislocation energy assumes a role of driving force. It is shown that the present method can numerically reproduce the three stages of recrystallization as a sequence of computational procedure.

Equivalence between higher-order stress power and heat flux in energy equation based on lattice dynamics

The essence of macroscopic quantities in solid mechanics can be grasped by expressin these quantities in terms of kinematic and mechanical quantities of atoms. In this paper, a method is proposed for obtaining the microscopic definitions of internal forces of continua such as stress, higher-order stresses and heat flux. Moreover, the relation between higher-order stress power and heat flux is discussed expressing the first law of thermodynamics with microscopic quantities in the mesodomain. Comparing heat flux with higher-order stress power, it is clarified that the divergence of heat flux is equivalent to the total of each order power due to higher-order stresses.

Investigation on Intragranular Stress of Mg Including Several Twin-Bands Using Dislocation-Based Crystal Plasticity and Phase-Field Models

A coupled model based on crystal plasticity and phase field theories that express both plastic anisotropy of HCP metals and expansion/shrinkage of twin-bands is proposed in the present study. In this model, the difference of the hardening rate in each slip system is expressed by changing their dislocation mobility as a numerical parameter defined in the crystal plasticity framework. The stress calculated via crystal plasticity analysis becomes to the driving force of multi-phase filed equations that express the evolution of twin bands of several variants, which include both the growth and shrinkage. Solving this equation set, the rate of twinning/detwinning and the mirror-transformed crystal basis in the twinned/detwinned phase are obtained and then, crystal plasticity analysis is carried out again. Using the present model, a uniaxial cyclic loading simulation along [0001] direction on the specimen including two variants of twin-bands is carried out by means of finite element method (FEM). The results show that the detwinning stress decreases with increase of the pre-tensioned strain. This is caused by a residual compression stress resulting from the twin shearing that occurs in the vicinity of two twin boundaries approaching each other.

Dislocation-Crystal Plasticity Simulation Based on Self-Organization for Repartition of Dislocation Cell Structures and Subgrain

A self-organization model for repartition of dislocation cell structures and transition of subgrains on a three-stage hardening of single crystal are developed. Stress-effect coefficients models are proposed in order to introduce stress information into the reaction-diffusion equations. A FD simulation for dislocation patterning and a FE one for crystal deformation are simultaneously carried out for an FCC single crystal. It is numerically predicted that a cell structures are repartitioned and the generated dislocation pattern in stage III can be regarded as a subgrain.

A Dislocation-Based Crystal Plasticity Simulation on Kink Band Formation and Evolution in Polycrystalline Mg Alloy with LPSO Phase

A three-dimensional compression analysis is performed by finite element method using a dislocation-based crystal plasticity model to clarify the formation mechanism of kink band in a polycrystalline Mg alloy with a long-period stacking ordered structure (LPSO) phase. The crystalline structure of LPSO phase is regarded as a HCP for simplicity, however, any deformation twinning is not taken into account. In addition, the activities of non-basal systems are considerably limited in the LPSO phase setting the values of their critical resolved shear stresses to large ones. We analyze a simple polycrystalline specimen composed of two α-Mg matrix phases and a LPSO phase both having a rectangular shape and twist grain boundaries are introduced into the interface. The obtained result shows that the kink band formation in the alloy is accomplished by the basal slips with different variants and the non-basal slips are activated on the grain boundary to maintain the continuity of deformation.

Simulation on Nanostructured Metals Based on Multiscale Crystal Plasticity Considering Effect of Grain Boundary

Ultrafine-grained metals whose grain size is less than one micron have attracted interest as high strength materials. Whereas nanostructured metals produced by severe plastic deformation express remarkably peculiar behavior in both material and mechanical aspects, its mechanism has been clarified by neither experimental nor computational approaches. In this study, we develop a multiscale crystal plasticity model considering an effect of grain boundary. In order to express release of dislocation from grain boundaries, information of misorientation is introduced into a hardening law of crystal plasticity. In addition, carrying out FE simulation for FCC polycrystal, the stress-strain responses such as increase of yield stress due to existence of grain boundary are discussed. We investigate comprehensively the effect of dislocation behavior on the material property of nanostructured metal.