Karl S. Pister

Variational and projection methods for the volume constraint in finite deformation elasto-plasticity

This paper focuses on the treatment of volume constraints which in the context of elasto-plasticity typically arise as a result of assuming volume-preserving plastic flow. Projection methods based on the modification of the discrete gradient operator B, often proposed on an ad-hoc basis, are systematically obtained in the variational context furnished by a three-field Hu-Washizu principle. The fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume-preserving and dilatational parts, without relying on rate forms of the weak form of momentum balance. This approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient, and enables one to exactly enforce the condition of volume-preserving plastic flow. Within the framework proposed in this paper, rate forms and incrementally objective algorithms are entirely bypassed.

Remarks on rate constitutive equations for finite deformation problems: computational implications

It is explicitly shown that if the (spatial) elasticity tensor of an elastic material is taken as isotropic for all possible configurations, then its coefficients cannot be constants; they must depend nontrivially on the Jacobian determinant of the deformation gradient. Moreover, the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity. It is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity, namely, relating an objective stress rate to the rate of deformation tensor through a fourth-rank constant isotropic tensor, is also incompatible with elasticity, thus furnishing an example of an hypoelastic material which is not elastic.

Implicit-explicit finite elements in nonlinear transient analysis

Abstract Implicit-explicit finite-element “mesh partitions” are developed for transient problems of nonlinear mechanics. The methods are shown to have improved implementational properties and may be easily coded into many existing implicit computer programs. The stability and accuracy properties of the methods are discussed, and techniques for improving the accuracy in the explicit group, without adverseley affecting stability, are described. Implicit-explicit “operator splitting” methods, which enable the efficient treatment of kinematic constraints (e.g. incompressibility) in transient analysis, are also presented.

Consistent linearization in mechanics of solids and structures

Abstract Linearization plays a key role both in formulation as well as numerical analysis of problems in the mechanics of solids and structures. This paper provides a unifying definition of linearization and illustrates some of the operational consequences. Finite motion of elastic plates is chosen to demonstrate how the linearization process may be utilized in the context of motion of initially-stressed, materially nonlinear elastic plates.

Numerical integration of rate constitutive equations in finite deformation analysis

Abstract In analysis of finite deformation problems the use of constitutive equations in rate form is often required. In a spatial setting, these equations may express a relationship between some objective rate of spatial stress tensor and the rate of deformation. Constitutive equations of this type characterize a variety of material models including hyperelasticity, hypoelasticity and elastoplasticity. Employing geometrical concepts, a family of unconditionally stable and incrementally objective algorithms is proposed for the integration of such rate constitutive equations. These algorithms, which are appropriate for finite deformation analysis, are applicable to any choice of stress rate and, in most cases, employ quantities that arise naturally in the context of finite element analysis. Examples illustrate the objectivity and accuracy of the algorithms.

Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids

Abstract The constitutive equation for stress in a hyperelastic body undergoing nonisothermal deformation is derivable from a free energy function. This paper presents a postulate for decomposition of thermomechanical deformation, the properties of which are analogous to those associated with the Duhamel-Neumann postulate for linear thermoelasticity. That is, the deformation gradient is expressed as the product of a free thermal expansion gradient and an “effective” mechanical deformation gradient. This results in the decomposition of the free energy function into an isothermal, “effective” strain energy function and a function depending only on temperature. Some examples of biaxial extension and temperature change of an incompressible thin sheet are included.

On a variational theorem for incompressible and nearly-incompressible orthotropic elasticity

Abstract A mixed variational theorem for linear orthotropic thermoelastic solids is presented. The mechanical state variables are taken to be the displacement vector and a scalar stress variable. The Euler equations of the variational principle are the displacement equations of equilibrium and a condition relating the stress variable to strain and temperature change. An important feature of the principle is that the field equations for both compressible and incompressible solids may be generated. In connection with applications to the development of finite element computer algorithms for the solution of boundary value problems a well-conditioned system of equations is obtained for nearly-incompressible solids.

Variational principles for boundary value and initial-boundary value problems in continuum mechanics

Abstract A procedure for setting up variational principles for a class of linear coupled field problems in continuum mechanics is presented. Some generalizations of the principle and their relationship with existing variational theorems are examined. Alternative schemes useful for direct methods of solution are discussed. Examples of typical application are included.

Identification of nonlinear elastic solids by a finite element method

Abstract The problem of utilizing experimental data to characterize the stress constitutive function for a nonlinear elastic solid is formulated as an inverse boundary value problem. The use of finite element discretization is extended by introducing a technique of material parameterization that utilizes finite elements defined over the domain of the stress constitutive function. The discretized identification problem is then reduced to a system of nonlinear algebraic equations that couples the data set and the discretized boundary value problem. The effect of errors in measured data is minimized by employing a weighted least squares error norm to generate the equations from which the unknown material parameters are obtained. An illustrative numerical experimental is included.

A variational principle for linear, coupled field problems in continuum mechanics

Abstract A variational principle applicable to linear, coupled field problems in continuum mechanics is presented. An important feature for direct methods of calculation, particularly the finite element method, is the inclusion of initial conditions on the field variables as part of the variational principle. Applications to problems in several areas of continuum mechanics are included.