A two-dimensional finite element method for simulating the constitutive response and microstructure of polycrystals during high temperature plastic deformation

We describe a finite element method designed to model the mechanisms that cause superplastic deformation. Our computations account for grain boundary sliding, grain boundary diffusion, grain boundary migration, and surface diffusion, as well as thermally activated dislocation creep within the grains themselves. Front tracking and adaptive mesh generation are used to follow changes in the grain structure. The method is used to solve representative boundary value problems to illustrate its capabilities.     

A fundamental model of cyclic instabilities in thermal barrier systems

Cyclic morphological instabilities in the thermally grown oxide (TGO) represent a source of failure in some thermal barrier systems. Observations and simulations have indicated that several factors interact to cause these instabilities to propagate: (i) thermal cycling; (ii) thermal expansion misfit; (iii) oxidation strain; (iv) yielding in the TGO and the bond coat; and (v) initial geometric imperfections. This study explores a fundamental understanding of the propagation phenomenon by devising a spherically symmetric model that can be solved analytically. The applicability of this model is addressed through comparison with simulations conducted for representative geometric imperfections and by analogy with the elastic/plastic indentation of a half space. Finite element analysis is used to confirm and extend the model. The analysis identifies the dependencies of the instability on the thermo-mechanical properties of the system. The crucial role of the in-plane growth strain is substantiated, as well as the requirement for bond coat yielding. It is demonstrated that yielding of the TGO is essential and is, in fact, the phenomenon that differentiates between cyclic and isothermal responses.     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.     

A three-dimensional crystal plasticity model for duplex Ti–6Al–4V

A rate dependent crystal plasticity model for the α/β Ti–Al alloy Ti–6Al–4V with duplex microstructure is developed and presented herein. Duplex Ti–6Al–4V is a dual-phase alloy consisting of an hcp structured matrix primary α-phase and secondary lamellar α + β domains that are composed of alternating layers of secondary α laths and bcc structured residual β laths. The model accounts for distinct three-dimensional slip geometry for each phase, anisotropic and length scale dependent slip system strengths, the non-planar dislocation core structure of prismatic screw dislocations in the primary α-phase, and crystallographic texture. The model is implemented in the general purpose finite element code (ABAQUS, 2005. Ver 6.5, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI) via a UMAT subroutine.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

A finite-deformation-based phenomenological theory for shape-memory alloys

In this work we develop a finite-deformation-based, thermo-mechanically-coupled and non-local phenomenological theory for polycrystalline shape-memory alloys (SMAs) capable of undergoing austenite ↔$\leftrightarrow$ martensite phase transformations. The constitutive model is developed in the isotropic plasticity setting using standard balance laws, thermodynamic laws and the theory of micro-force balance (Fried and Gurtin, 1994). The constitutive model is then implemented in the ABAQUS/Explicit (2009) finite-element program by writing a user-material subroutine. Material parameters in the constitutive model were fitted to a set of superelastic experiments conducted by Thamburaja and Anand (2001) on a polycrystalline rod Ti–Ni. With the material parameters calibrated, we show that the experimental stress-biased strain–temperature-cycling and shape-memory effect responses are qualitatively well-reproduced by the constitutive model and the numerical simulations. We also show the capability of our constitutive mode in studying the response of SMAs undergoing coupled thermo-mechanical loading and also multi-axial loading conditions by studying the deformation behavior of a stent unit cell.     

Debonding failure and size effects in micro-reinforced composites

Failure in micro-reinforced composites is investigated numerically using the strain-gradient plasticity theory of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52 (6) 1379–1406] in a plane strain visco-plastic formulation. Bi-axially loaded unit cells are used and failure is modeled using a cohesive zone at the reinforcement interface. During debonding a sudden stress drop in the overall average stress–strain response is observed. Adaptive higher-order boundary conditions are imposed at the reinforcement interface for realistically modeling the restrictions on moving dislocations as debonding occurs. It is found that the influence of the imposed higher-order boundary conditions at the interface is minor. If strain-gradient effects are accounted for a void with a smooth shape develops at the reinforcement interface while a smaller void having a sharp tip nucleates if strain-gradient effects are excluded. Using orthogonalization of the plastic strain gradient with three corresponding material length scales it is found that, the first length scale dominates the evaluated overall average stress–strain response, the second one only has a small effect and the third one has an intermediate effect. Finally, studies of reinforcement having elliptical cross-sections show rather significant gradients of stress which is not seen for the corresponding circular cross-sections. Also, an increased drop in the overall load carrying capacity is observed for cross-sections elongated perpendicular to the principal tensile direction compared to the corresponding circular cross-sections.     

Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals

In order to model the effects of grain boundaries in polycrystalline materials we have coupled a crystal-plasticity model for the grain interiors with a new elastic-plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to study the deformation and fracture response of nanocrystalline nickel. The results from the simulations reflect the macroscopic experimentally observed tensile stress-strain curves, and the dominant microstructural fracture mechanisms in this material. The macroscopically observed nonlinearity in the stress-strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress-strain response, and the overall ductility. In nanocrystalline nickel, the high-yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension.     

Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation

The effects of applied voltage on the electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators were examined in a combined experimental and numerical investigation. Experiments were performed to measure the strain near internal and surface electrodes at various electrical loading conditions. The finite element method was also used to solve the coupled electro-elastic boundary value problem. The strain, stress and electric displacement concentrations were calculated and a non-linear behavior induced by localized polarization switching was discussed. A comparison of strain concentration was made between experiment and simulation.     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback

The constitutive model for the unusual asymmetric hardening behavior of magnesium alloy sheet presented in a companion paper (Lee, M.G., Wagoner, R.H., Lee, J.K., Chung, K., Kim, H.Y., 2008. Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheet, Int. J. Plasticity 24(4), 545–582) was applied to the springback prediction in sheet metal forming. The implicit finite element program ABAQUS was utilized to implement the developed constitutive equations via user material subroutine. For the verification purpose, the springback of AZ31B magnesium alloy sheet was measured using the unconstrained cylindrical bending test of Numisheet (Numisheet ’2002 Benchmark Problem, 2002. In: Yang, D.Y., Oh, S.I., Huh, H., Kim, Y.H. (Eds.), Proceedings of 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Jeju, Korea) and 2D draw bend test. With the specially designed draw bend test the direct restraining force and long drawn distance were attainable, thus the measurement of the springback could be made with improved accuracy comparable with conventional U channel draw bend test. Besides the developed constitutive models, other models based on isotropic constitutive equations and the Chaboche type kinematic hardening model were also considered. Comparisons were made between simulated results by the finite element analysis and corresponding experiments and the newly proposed model showed enhanced prediction capability, which was also supported by the simple bending analysis adopting asymmetric stress–strain response.     

Elastic properties of model random three-dimensional open-cell solids

Most cellular solids are random materials, while practically all theoretical structure-property relations are for periodic models. To generate theoretical results for random models the finite element method (FEM) was used to study the elastic properties of open-cell solids. We have computed the density (ρ) and microstructure dependence of the Young's modulus (E) and Poisson's ratio (ν) for four different isotropic random models. The models were based on Voronoi tessellations, level-cut Gaussian random fields, and nearest neighbour node-bond rules. These models were chosen to broadly represent the structure of foamed solids and other (non-foamed) cellular materials. At low densities, the Young's modulus can be described by the relation E∝ρⁿ. The exponent n and constant of proportionality depend on microstructure. We find 1.3<n<3, indicating a more complex dependence than indicated by periodic cell theories, which predict n=1 or 2. The observed variance in the exponent was found to be consistent with experimental data. At low densities we found that ν≈0.25 for three of the four models studied. In contrast, the Voronoi tessellation, which is a common model of foams, became approximately incompressible (ν≈0.5). This behaviour is not commonly observed experimentally. Our studies showed the result was robust to polydispersity and that a relatively large number (15%) of the bonds must be broken to significantly reduce the low-density Poission's ratio to ν≈0.33.     

Geometrically necessary dislocation structure organization in FCC bicrystal subjected to cyclic plasticity

Crystal plasticity finite element analysis of cyclic deformation of compatible type FCC bicrystals are performed. The model specimen used in the analysis is a virtual FCC bicrystal with an isotropic elastic property; therefore, the effect of constraint due to elastic incompatibility does not appear. The results of the analysis show the strain-amplitude-dependence of both the organization of the GND structure and the stress–strain behavior. The calculated stress–strain curve with the largest strain amplitude shows additional cyclic hardening. The microscopic mechanisms of the strain-amplitude-dependent organization of the GND structure and additional cyclic hardening behavior are discussed in terms of the activation of secondary slip system(s). Finally, the effects of the elastic anisotropy, the lattice friction stress and the interaction between dislocations are also argued.     

An analytical model to predict diffusion induced intermetallic compounds growth in Cu-Sn-Cu sandwich structures

A mass diffusion model is developed to describe the growth kinetics of Cu_6Sn_5 intermetallic compounds(IMC) in the Cu-Sn-Cu sandwich structure. The proposed model is based on the local interfacial mass conversation law where interfacial Cu/Sn reactions and atomic diffusion are considered. Theoretical analysis shows that the IMC thickness growth is proportional to the square root of the product of the diffusion coefficient and time. The proposed model can explain the polarity effect of electromigration on kinetics of IMC growth where all the parameters have clear physical meaning. The theoretical predictions are compared with experimental results and show reasonable accuracy.     

Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: II—applications

In Part I of this work, an improved “second-order” homogenization theory was developed. This new theory makes use of generalized secant moduli that are intermediate between the standard secant and tangent moduli of the nonlinear phases, and that depend not only on the averages, or first-moments of the fields in the phases, but also on the second-moments of the field fluctuations, or phase covariance tensors. In this article, the theory, which is known to be exact to second-order in the heterogeneity contrast, is applied to the special cases of rigidly reinforced and porous materials. These are cases corresponding to infinite contrast where fairly explicit analytical expressions of the Hashin-Shtrikman and self-consistent-type may be generated for nonlinear composites. The results show that the new theory improves on the earlier theory (Ponte Castañeda, J. Mech. Phys. Solids 44 (1996) 827) in at least two ways. First, the new theory satisfies rigorous bounds, even near the percolation limit, where field fluctuations become important, and the earlier second-order theory had been found to fail. Second, the new theory predicts fully compressible behavior for porous materials with an incompressible matrix phase, where the earlier theory had also been found to fail. In addition, the new estimates are found to be in better agreement with numerical simulations available from the literature.     

Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—theory

This paper is concerned with the development of an improved second-order homogenization method incorporating field fluctuations for nonlinear composite materials. The idea is to combine the desirable features of two different, earlier methods making use of “linear comparison composites”, the properties of which are chosen optimally from suitably designed variational principles. The first method (Ponte Castañeda, J. Mech. Phys. Solids 39 (1991) 45) makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain field over the phases, and delivers bounds, but these bounds are only exact to first-order in the heterogeneity contrast. The second method (Ponte Castañeda, J. Mech. Phys. Solids 44 (1996) 827) makes use of the “tangent” moduli, evaluated at the phase averages (or first moments) of the strain field, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in some special cases. These special cases turn out to correspond to situations, such as percolation phenomena, where field fluctuations, which are captured less accurately by the second-order method than by the bounds, become important. The new method delivers estimates that are exact to second-order in the contrast, making use of generalized secant moduli incorporating both first- and second-moment information, in such a way that the bounds are never violated. Some simple applications of the new theory are given in Part II of this work.     

A Finite Element approach with patch projection for strain gradient plasticity formulations

Several strain gradient plasticity formulations have been suggested in the literature to account for inherent size effects on length scales of microns and submicrons. The necessity of strain gradient related terms render the simulation with strain gradient plasticity formulation computationally very expensive because quadratic shape functions or mixed approaches in displacements and strains are usually applied. Approaches using linear shape functions have also been suggested which are, however, limited to regular meshes with equidistanced Finite Element nodes. As a result the majority of the simulations in the literature deal with plane problems at small strains. For the solution of general three dimensional problems at large strains an approach has to be found which has to be computationally affordable and robust.