Elastic-plastic Constitutive Law Research Articles

Microindentation on Gelatin Films with a Spherical Indenter -- A Viscoelastic Analysis

ABSTRACTA finite element model was employed to analyze the microindentation test with a spherical indenter on gelatin films. The deficiency of using elastic-plastic constitutive law to describe indentation response of gelatin film was recognized and a viscoelastic model was proposed for the first time to analyze indentation experiments on polymeric materials. Based on viscoelastic analysis, it was found that gelatin is nonlinear viscoelastic. In addition, modulus in the thickness direction of a gelatin film was determined to be 2.5–2.9 GPa as compared with its tensile modulus of 4.6 GPa in the transverse direction.

Generalized variable finite element modeling and extremum theorems in stepwise holonomic elastoplasticity with internal variables

Reference is made to a broad class of elastic-plastic constitutive laws with internal variables (occuring in pairs of conjugate kinematic and static quantities). Their Euler backward-difference, ‘stepwise-holonomic’ formulation is derived for the time discretization of initial-boundary value problems. The space discretization is performed by systematically using the notion of ‘generalized variables’ and by generating overall, average constitutive laws for finite elements (instead of enforcing the material model in some points). Some categories of finite element models in generalized variables are comparatively discussed. Two pairs of dual extremum characterizations of the solution to the boundary value problem for a finite load step are established.

An Expression of Elastic-Plastic Constitutive Law Incorporating Vertex Formation and Kinematic Hardening

A phenomenological corner theory was proposed for elastic-plastic materials by the authors in the previous paper (Goya and Ito, 1980). The theory was developed by introducing two transition parameters, μ (α) and β (α), which, respectively, denote the normalized magnitude and direction angle of plastic strain increments, and both monotonously vary with the direction angle of stress increments. The purpose of this report is to incorporate the Bauschinger effect into the above theory. This is achieved by the introduction of Ziegler’s kinematic hardening rule. To demonstrate the validity and applicability of a newly developed theory, we analyze the bilinear strain-path problem using the developed equation, in which, after some linear loading, the path is abruptly changed to various directions. In the calculation, specific functions, such as μ (α) = Cos (.5πα/αmax) and β (α) = (αmax- .5π) α/αmax, are chosen for the transition parameters. As has been demonstrated by numerous experimental research on this problem, the results in this report also show a distinctive decrease of the effective stress just after the change of path direction. Discussions are also made on the uniqueness of the inversion of the constitutive equation, and sufficient conditions for such uniqueness are revealed in terms of μ(α), β(α) and some work-hardening coefficients.

Extremum properties of finite-step solutions in elastoplasticity with nonlinear mixed hardening

The class of elastic-plastic constitutive laws assumed herein can be described as follows. We envisage y ⩾ 1 yield criteria (or “modes”) which define the current yield surface; each yield function φα consists of two addends: an effective stress Φα and a yield limit Yα (α = 1,…, y). The former is an order-one, positively homogeneous function of the difference between the stresses σii and reference stresses αii, which are generally nonlinear functional of the plastic strain history, so that kinematic hardening is accounted for. The yield limit Yα is generally a nonlinear function of y nondecreasing internal variables λβ, so that another hardening mechanism (which may reduce to isotropic hardening) is accommodated in the model. The rates of these variables play the role of plastic multipliers in the flow rule.A finite step in the geometrically linear evolutive analysis of such solids is defined according to the backward-difference (or “stepwise holonomic”) strategy for approximate time integration. For the “finite-step” boundary value problem thus arising, various extremum characterizations of solutions are established and the underlying constitutive restrictions are pointed out and discussed.

An expression of elastic-plastic constitutive law incorporating a stress increment dependence. (4th Report; Determination of stress increment dependency parameters at the proportionally prestressed state).

In the first report, a new expression of a constitutive law for elastic-plastic material was proposed using two transition parameters which describe the stress increment dependency. These parameters denote the magnitude and the direction of plastic strain increments. In this report, these are determinded by the biaxial loading test of aluminum specimens using testing machine constructed by trial and by the numerical simulation based on the KBW model. The results show that the stress increment dependency is the most considerable at the simple tension and decreases when the stres vector is turned toward the direction of the pure shear.

An expression of elastic-plastic constitutive law incorporating a stress increment dependence. (3rd Report: Analysis of localized necking of rigid-plastic plates).

In the first report, a new expression op a constitutive law for elastic-plastic material was proposed using two transition parameters which are essential in describing the property of plastic materials. These parameters denote the magnitude and direction of plastic strain increments. Here, Localized necking of rigid-plastic plates under biaxial stretching is studied according to Storen-Rice theory. A constitutive equation for linear-comparison solids is used for Hencky's deformation rule.

An expression of elastic-plastic constitutive law incorporating a stress increment dependence. 2nd Report, A law of linear-comparison solids and analysis of buckling of elastic-plastic tube.

In the first report, a new expression of a constitutive law for elastic-plastic material was proposed using two transition parameters which are essential in describing the property of plastic materials. There parameters denotes the magnitude and direction of plastic strain increments. In this report, sufficient conditions of linear comparison solids are discussed in terms of those transition parameters. The characteristic of buckling of an elastic-plastic tube simultaneously subjected to axial load and internal pressure is then analyzed using Hill's bifurcation theory.

An expression of elastic-plastic constitutive law incorporating a stress increment dependence. (1st report. Initial isotropic materials with mises type-plastic potential).

A constitutive law for elastic-plastic materials is presented on the assumption that deviatoric stress increments affect plastic strain increments. By the use of weighing parameters, the law is expressed as an averaging rule of the usual plastic potential law (or flow rule) and a rule that permits plastic strain increments to depend only on stress increments. The developed law permits the existence of a vertex on the yield surface. The law, however, can be written formally in the same equation as the usual flow rule : this implies that the developed law is applicable to such a solution scheme as the Finite Element Method.

Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem

It is shown that a boundary-value problem based on a holonomic elastic-plastic constitutive law may be formulated equivalently as a variational inequality of the second kind. A regularised form of the problem is analysed, and finite element approximations are considered. It is shown that solutions based on finite element approximation of the regularised problem converge.

Elastic-plastic boundary element analysis as a linear complementarity problem

Elastic-plastic constitutive laws allowing for nonassociation, corners and hardening (or softening) are formulated both in the usual terms of rates and in terms of piecewise-linear approximation (local or global). The direct boundary element method combined with such descriptions of nonlinear material behaviour, is shown to reduce elasto-plastic structural analysis, both by infinitesimal and finite steps, to a linear complementary problem. On this basis, a general extremum characterization of the solution and conditions for solution uniqueness are pointed out. Various solution procedures are envisaged in the new unitary framework.

Computer simulation of the Charpy V-Notch toughness test

We calculated the impact of a striker with the ASTM Type A Charpy V-Notch specimen. We used an elastic-plastic constitutive law with J 2-flow theory and isotropic hardening to model A-533 Grade B class 1 steel at 100°C. Large strain and rotation are accounted for. We found that the notch stress-strain state was independent of the three-point loading type (round- or flat-tip striker or pressure push) and was most strongly correlated with notch-opening displacement. The dynamic stress-strain state at the time of fracture initiation was obtained by comparing the calculated deformed shape with that obtained in interrupted Charpy-V-notch tests where cracking had started. This time agreed with a computed prediction of fracture initiation obtained using a computer model of material damage. Cracking starts at the notch at about 40% of the maximum load found in an instrumented Charpy test. We have also calculated the distribution and partition of specimen energy. Adiabatic heating and strain rate are discussed.

A Plane-Strain Formulation of Elastic-Plastic Constitutive Laws

ABSTRACT Constitutive laws for elastic-plastic materials are derived by eliminating the transverse stress component on the basis of the plane-strain constraint. This leads to a fictitious hardening and temperature dependence of the loading function. For standard elastic-plastic materials the resulting laws are associated; however, the plastic strain state is represented by equivalent plastic-strain measures, which also account for transverse yielding. The new constitutive laws, together with the standard reduced form of the equilibrium and compatibility equations, permit the formulation of the plane-strain elastic-plastic analysis problem in terms of the in-plane stress components only. In the case of perfectly plastic materials, the subsequent plane-strain yield surfaces are contained within a domain bounded by a limit surface which represents the yield condition normally adopted in plane-strain limit analysis.

Study of the perforation of reinforced concrete slabs by rigid missiles —Experimental study, part III

The work described below is a contribution to the assessment of aircraft hazards for nuclear power plants. A series of tests was performed to outline the behaviour of reinforced concrete slabs impinged by rigid missiles. Prediction of results was tried by means of finite element computation with elastic-plastic constitutive law for the concrete. Although results are encouraging, no general correlation was carried out.

An Elastic-Plastic Constitutive Law for Sea Ice

The AIDJEX sea ice model is extended by a careful consideration of the deformation measured relative to a reference configuration which changes according to the plastic flow. This approach makes the strain more meaningful than it was previously. The kinematic relations properly allow both finite stretch and rotation. The resulting kinematic relations reflect the feeling that sea ice response is independent of any fixed reference configuration. This formulation has been achieved by writing an incremental kinematic relation using the Finger deformation tensor. A special relation is presented for the case when elastic strain is small, as with sea ice. The elastic response and yield criterion as well as the plastic flow rule remain unchanged from those presently in use. The flow rule is derived by maximizing the rate at which plastic work is done. This approach unifies the development and also helps us to study the material instabilities that occur when the material softens. The material model satisfies the principle of material frame indifference. The model has been subjected to known deformation histories and the response is satisfactory. In many cases the stress history is indistinguishable from that of the previously used elastic-or-plastic model.

Incremental plastic analysis in the presence of large displacements and physical instabilizing effects

Elastic-plastic constitutive laws are assumed with allowance for the following: lack of normality, corners with interaction between yielding modes and work-softening. Continua are replaced by finite element models which fulfill compatibility throughout. Equilibrium equations refer to the deformed configuration; strains, however, are presumed small. The structural response to rates of loads and dislocations (e.g. thermal strains) is studied and the following results are obtained; 1.(a) six extremum properties of the solutions, which reduce the incremental problem to a quadratic programming problem; two of these properties are valid without restrictions;2.(b) methods for obtaining lower and upper bounds to the local instantaneous compliance with respect to a single load or dislocation;3.(c) criteria for the uniqueness of solution to the incremental problem and for the overall stability of the system.