Elastic-plastic Constitutive Relation Research Articles

Analysis of Deformation of Shape Memory Alloy Composite Based on Shear-Lag Model.

Incremental constitutive relation of shape memory alloy composite, in which shape memory alloy short-fibers are dispersed in a matrix, has been developed based on the shear-lag model. In the model, the shape memory alloy is described by one-dimensional constitutive relation proposed by Brinson and the matrix is described by the conventional elastic plastic constitutive relation. Using the proposed constitutive relation, the deformation of NiTi short-fiber dispersed nylon 66 composite is analyzed under thermo-mechanical loading. The shape memory alloy composite exhibits the unique performance such as pseudoelasticity, shape memory effect, contraction by heating, formation of compressive residual stress in matrix, and so on. The constitutive relation is useful in the investigation to explore the potentiality of the shape memory alloy composites.

Numerical modeling of fold initiation at thrust ramps

The movement of the hangingwall over the footwall at thrust ramps produces a variety of structures found commonly at different scales in fold-and-thrust belts. The nature of the structure depends on the relative rigidity of the hangingwall and footwall, friction along the fault, fault ‘dip’, fault displacement, and the boundary conditions of the deformation. Structures include fault-bend style folds, fault-propagation style folds and wedge folds. We investigate the initial stages of development of such structures using the finite-difference code FLAC. The rock layers are represented as continua with elastic-plastic Mohr-Coulomb constitutive relations, and the fault and bounding bedding planes are assigned normal and shear stiffnesses and coefficients of friction. Under conditions with all layers compressed, antisymmetric fault-propagation style folds develop in both the hangingwall and footwall. With rigid footwall and restricted far-field slip in the hangingwall, a single fault-propagation style fold develops. With far-field displacement of the hangingwall allowed, broader antisymmetric wedge folds develop if hangingwall and footwall are deformable, and a single fault-bend style fold develops if the footwall is rigid. All structures become accentuated with increasing slip on the fault. Where both hangingwall and footwall are deformable, the deformation reduces the ramp angle and tends to minimize distortion of the rock adjacent to the fault. Fault-propagation style folds, paired wedge folds and fault-bend style folds are common in nature, and small-scale examples can be found in various stages of development. Continued slip on thrust faults may lead to the mature structures commonly seen in fold-and-thrust belts.

Parametric Description of Elastic-Plastic Deformation Behaviour of Inhomogeneous Material with Inclusions

The macroscopic elastic-plastic constitutive relation of inhomogeneous material is affected not only by the characteristics of respective constituents, but also by the microscopic distribution of stress and strain in heterogeneous regions. In order to describe the deformation behaviour of inhomogeneous material, several parameters are defined. A plane model of inhomogeneous material with inclusions is used for numerical simulation, and the changes in the parameters during elastic-plastic deformation are calculated and compared. Uniaxial loading under the plane-strain condition is assumed. It is concluded that the initial aspect ratio has a marked influence on the deformation behaviour. One of the parameters, called the constraint ratio, remains nearly constant with increase in the applied strain or the degree of inhomogeneity. This characteristic seems to be important for estimating the deformation behaviour of inhomogeneous material. The constitutive relation of inhomogeneous material may also be estimated from the value of the constraint ratio.

An anisotropy theory of viscoplasticity for thermomechanical deformations

An infinitesimal anisotropic theory of viscoplasticity for predicting the behavior of elastic-plastic materials subjected to thermomechanical loading is presented. In this theory, an associated flow rule, in which the plastic potential is assumed to be the anisotropic yield function, is used to define the elastic-plastic incremental constitutive relation. This theory is intended for application to anisotropic materials such as high-temperature composites, directionally solidified alloys, and single crystal superalloys in the aerospace, propulsion and power generation industries.

Elastic-Plastic Deformation of Inhomogeneous Material and Parametric Description of Deformation Behaviour

The macroscopic elastic-plastic constitutive relation of inhomogeneous material is affected by the microscopic distribution of stress and strain in respective heterogeneous regions of the material. Several parameters are reviewed and a new parameter is proposed to describe the deformation behaviour of inhomogeneous material, and their relationship is discussed analytically as well as numerically. A plane model of in. homogeneous material composed of rectangular inhomogeneous regions was used for the numerical simulation, and the changes in the parameters during elastic-plastic deformation are calculated and compared. Uniaxial loading under plane strain condition is assumed. A modified plastic accommodations parameter is proposed, which is shown to be uniquely related to the constraint ratio reported previously. It is shown that the. initial aspect ratio has a marked influence on the deformation behaviour. The process to estimate the constitutive relation of inhomogeneous material is also discussed on the basis of the constraint model.

Adaptive load incrementation algorithm during a process of local unloading in the elastic-plastic finite element analysis.

There are two approaches in the incremental finite element analysis of elastic-plastic problems; one takes an explicit scheme using forward Euler integration formula with adaptive load incrementation, the so-called rmin method, and another implicit scheme with iterative procedures such as the Newton method. This paper discusses an algorithm of determining incremental step size in the former approach. A new additional restriction is proposed for the determination of the loading factor rmin when unloading takes place. A concept of admissible error is introduced where errors in stress increments are defined as the difference between one computed by assuming the elastic-plastic constitutive relation and the other computed by assuming the elastic relation. The error norm should be restricted within specified value. Numerical examples are demonstrated to show that the algorithm is reasonable.

Fracture Energy‐Based Plasticity Formulation of Plain Concrete

A comprehensive constitutive model is developed for the triaxial behavior of plain concrete. The formulation covers the full load-response spectrum in tension as well as in compression. The concrete model is based on the nonassociated flow theory of plasticity with hardening in the prepeak regime and fracture energy-based softening in the post-peak regime. The constitutive development is guided by recent experimental observations on laboratory specimens which are loaded under “mixed” control in tension, and in triaxial compression at different levels of lateral confinements. The constitutive parameters are calibrated from a series of stroke-controlled laboratory experiments which include direct tension and triaxial compression tests at different confinements. The predictive quality of the fracture energy-based plasticity model is ascertained with a number of separate load-history experiments for which triaxial test data are available in the concrete literature. Pertinent remarks on the numerical implementation of the elastic-plastic constitutive relations with comments on relevant issues related to the uniqueness and stability of nonassociated strain-softening computations, are also presented.

Plasticity of metallic polycrystals under complex loading paths

The fundamental relations of self-consistent scheme for elastic-plastic metallic polycrystals are reviewed. A simplified scheme is used to predict the behavior of polycrystal aggregates in uniaxial tension, radial loading in biaxial stretching, and nonproportional complex loading and under cyclic loading. The evolution of subsequent yield surfaces after some radial prestraining, the Bauschinger effect, and the hardening of polycrystals are discussed with respect to the experimental observations. All the fundamental features of real FCC materials under complex loading are well modeled. The results presented herein confirm the ability of the self-consistent scheme for determination of precise elastic-plastic constitutive relations for the cases where the phenomenological approach becomes nonefficient.

On finite element formulations for large elastic-plastic deformations

Some issues arising in the formulation of finite strain plasticity problems are discussed. One issue concerns the formulation of finite deformation elastic-plastic constitutive relations for use in this class of problems. Here, a formulation is outlined with particular attention on the role of elasticity in setting the form of the stress rate-strain rate relation. The dependence of solutions to problems involving large plastic deformations on the plastic flow rule is illustrated by reviewing results on finite pure bending of a plane strain strip. The response of elastic-plastic solids is contrasted with that of a nonlinear elastic solid.

Analysis of the cup-cone fracture in a round tensile bar

Necking and failure in a round tensile test specimen is analysed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of micro-voids. Final material failure by coalescence of voids, at a value of the void volume fraction in accord with experimental and computational results, is incorporated in this constitutive model via the dependence of the yield condition on the void volume fraction. In the analyses the material has no voids initially; but high voidage develops in the centre of the neck where the hydrostatic tension peaks, leading to the formation of a macroscopic crack as the material stress carrying capacity vanishes. The numerically computed crack is approximately plane in the central part of the neck, but closer to the free surface the crack propagates on a zig-zag path, finally forming the cone of the cup-cone fracture. The onset of macroscopic fracture is found to be associated with a sharp “knee” on the load deformation curve, as is also observed experimentally, and at this point the reduction in cross-sectional area stops.

An analysis of ductile rupture in notched bars

Ductile fracture in axisymmetric and plane strain notched tensile specimens is analyzed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of microvoids. Final material failure by void coalescence is incorporated into the constitutive model via the dependence of the yield function on the void volume fraction. In the analyses the material has no voids initially; but as the voids nucleate and grow, the resultant dilatancy and pressure sensitivity of the macroscopic plastic flow influence the solution significantly. Considering both a blunt notch geometry and a sharp notch geometry in the computations permits a study of the relative roles of high strain and high triaxiality on failure. Comparison is made with published experimental results for notched tensile specimens of high-strength steels. All axisymmetric specimens analyzed fail at the center of the notched section, whereas failure initiation at the surface is found in plane strain specimens with sharp notches, in agreement with the experiments. The results for different specimens are used to investigate the circumstances under which fracture initiation can be represented by a single failure locus in a plot of stress triaxiality vs effective plastic strain.

Crack-tip fields for fast fracture of an elastic-plastic material

A n asymptotic analysis of the near-tip fields is given for transient crack propagation in an elastic-plastic material. The material is characterized by J 2 flow theory together with a bilinear effective stress-strain curve. Both plane stress and plane strain conditions have been considered. Explicit results are given for the order of the crack-tip singularity, the angular position at which unloading occurs, and the angular variation of the near-tip stresses, all as functions of the crack-tip speed and the ratio of the slopes of the two portions of the bilinear stress-strain relation. It was found that the results are much more sensitive to the elastic-plastic constitutive relation than to the crack speed. This result is important for numerical analyses of dynamic crack propagation problems.

Some comments on elastic-plastic analysis

Definitions of plastic and elastic deformation are discussed which are directly related to the physical characteristics of the phenomena and which provide essential uncoupling of elastic and plastic variables. Using finite deformation kinematics and appropriate strain-rate variables in order to carry out total incremental theory, analysis is presented concerning the fact that the total strain rate is not equal to the sum of elastic and plastic rates as is universally assumed. The consequences of this circumstance are examined.It is shown that a conflicting approach to the finite-deformation kinematics of elastic-plastic theory by Nemat-Nasser[4], which generates summable elastic and plastic strain rates, exhibits undesirable features and in particular the associated kinetics does not permit completion of the objective of developing an elastic-plastic constitutive relation on this basis.

Quasilinear Integration of Elasto-Plastic Shells

A method of solution for large deflection elasto-plastic shells of revolution is developed using numerical integration of quasilinearized equations. The work presented here uses a suppression scheme, originally developed by Goldberg for linear elastic shells of revolution problem, for the large deflection elastic-plastic analysis of shells of revolution. Sander’s large deflection theory for shells of revolution is adopted with incremental elastic-plastic constitutive relations and the Prandtl-Reuss flow rule associated with isotropic material hardening behavior. The solution method for large-deflection elastic problems is validated by considering a torus under external pressure. To verify the solution for large-deflection elastic-plastic problems, an annular plate subjected to edge loadings is solved. Results obtained compare well with previous theoretical and experimental solutions.

Conditions for shear localization in the ductile fracture of void-containing materials

This paper investigates the possibility that ductile fracture occurs by the McClintock-Berg mechanism of localization of deformation within a narrow shear band, owing to the progressive softening of the material by increasing porosity due to void growth. The ductility predicted for a macroscopically homogeneous sample of a voided material is shown to be unrealistically large and hence an initial inhomogeneity of properties is considered, in the sense of an analysis by Marciniak and Kuczynski in the related problem of local necking in sheet metals. General conditions for a localization bifurcation with an initial inhomogeneity (imperfection), concentrating deformation to allow localization within it, are derived. The initial imperfection is taken in the form of a void-containing, thin slice of a material and is assumed to have a void volume fraction slightly larger than the outside of the imperfection. Elastic-plastic constitutive rate relations for void-containing materials proposed by Gurson are adopted to the conditions for the localization bifurcation. The critical conditions are analyzed numerically to discuss the sensitivity of localization conditions to an initial imperfection, in consideration of the implications for the theory of ductile fracture. The results suggest that the existence of an initial imperfection makes it possible for localization to occur at a reasonable strain, and the predictions from this analysis seem broadly consistent with reported experimental observations.

Analyses of Structural Changes and Elastic-Plastic Stresses during Quenching of Steel

Coupling effects between temperature field and structural changes were considered in the analyses of structural changes and stresses due to quenching of steel from the viewpoint of continuum thermodynamics. A heat conduction equation with latent heat was derived by employing the mass fractions of structural components, and a finite element calculation was carried out to obtain the temperature and structural distributions.An elastic-plastic constitutive relation was presented considering the effect of dilatations due to phase changes as well as temperature variation, and stresses were analysed by the finite element method. The results calculated were compared with the experiments for a cylinder of 0.43%C carbon steel quenched in water.

Re-assessment of the incremental elastic-plastic constitutive relation

Re-assessment of the incremental elastic-plastic constitutive relation