Kevin P. Walker

Viscoplasticity with creep and plasticity bounds

Abstract A viscoplastic theory is developed that reduces analytically to creep theory under steady-state conditions and becomes plasticity theory at its rate-independent bound. A viscoplastic model is then constructed by defining material functions that have close ties to the physics of inelasticity. As a consequence, this model is characterized easily — only steady-state creep data, saturated hysteresis loops, and monotonic stress/strain curves are required. The general applicability of the model is demonstrated by the characterization of three f.c.c. metals. A variety of validation experiments is also provided.

The viscoplastic behavior of Hastelloy-X single crystal

Abstract A viscoplastic constitutive model for Hastelloy-X single crystal material is developed based on crystallographic slip theory. The constitutive model was constructed for use in a viscoplastic self-consistent model for isotropic Hastelloy-X polycrystalline material, which has been described in a recent publication. It is found that, by using the slip geometry known from the metallurgical literature, the anisotropic response can be accurately predicted. The model was verified by using tension and torsion data taken at 982°C (1800°F). The constitutive model used on each slip system is a simple unified visoplastic power law model in which weak latent interaction effects are taken into account. The drag stress evolution equations for the octahedral system are written in a hardening/recovery format in which both hardening and recovery depend on separate latent interaction effects between the octahedral crystallographic slip systems. The strain rate behavior of the single crystal material is well correlated by the constitutive model in uniaxial and torsion tests, but it is necessary to include latent information effects between the octahedral slip systems in order to obtain the best possible representation of biaxial cyclic strain rate behavior. Finally, it was observed that the single crystal exhibited dynamic strain aging at 871°C (1600°F). Similar dynamic strain aging occurs at 649°C (1200°F) in the polycrystalline version of the alloy.

Microstress analysis of periodic composites

Local elastic fields in the unit cell of a periodic composite are examined numerically with an integral equation approach. Techniques of Fourier series and Greens functions are used to construct the integral equations. Numerical solutions are obtained using the Fourier series approach with rectangular subvolume elements. Specific results are given for a tungsten/copper metal matrix composite.

Equivalence of Green’s Function and the Fourier Series Representation of Composites with Periodic Microstructure

This work is concerned with modeling the nonlinear mechanical deformation of composites comprised of a periodic microstructure under small displacement conditions at elevated temperatures. The practical motivation for such work stems from the need to design and optimize new multiphase materials and to predict their micromechanical and bulk material behavior under in-service thermomechanical loading conditions.

Thermoviscoplastic analysis of fibrous periodic composites by the use of triangular subvolumes

The non-linear viscoplastic behavior of fibrous periodic composites is analyzed by discretizing the unit cell into triangular subvolumes. A set of these subvolumes can be configured by the analyst to construct a representation for the unit cell of a periodic composite. In each step of the loading history the total strain increment at any point is governed by an integral equation which applies to the entire composite. A Fourier series approximation allows the incremental stresses and strains to be determined within a unit cell of the periodic lattice. The non-linearity arising from the viscoplastic behavior of the constituent materials comprising the composite is treated as a fictitious body force in the governing integral equation. Specific numerical examples showing the stress distributions in the unit cell of a fibrous tungsten/copper metal-matrix composite under viscoplastic loading conditions are given. The stress distribution resulting in the unit cell when the composite material is subjected to an overall transverse stress loading history perpendicular to the fibers is found to be highly heterogeneous, and typical homogenization techniques based on treating the stress and strain distributions within the constituent phases as homogeneous result in large errors under inelastic loading conditions.

Accuracy of the Generalized Self-Consistent Method in Modelling the Elastic Behaviour of Periodic Composites

Local stress and strain fields in the unit cell of an infinite, two-dimensional, periodic fibrous lattice have been determined by an integral equation approach. The effect of the fibres is assimilated to an infinite two-dimensional array of fictitious body forces in the matrix constituent phase of the unit cell. By subtracting a volume averaged strain polarization term from the integral equation we effectively embed a finite number of unit cells in a homogenized medium in which the overall stress and strain correspond to the volume averaged stress and strain of the constrained unit cell. This paper demonstrates that the zeroth term in the governing integral equation expansion, which embeds one unit cell in the homogenized medium, corresponds to the generalized self-consistent approximation. By comparing the zeroth term approximation with higher order approximations to the integral equation summation, both the accuracy of the generalized self-consistent composite model and the rate of convergence of the integral summation can be assessed. Two example composites are studied. For a tungsten/copper elastic fibrous composite the generalized self-consistent model is shown to provide accurate, effective, elastic moduli and local field representations. The local elastic transverse stress field within the representative volume element of the generalized self-consistent method is shown to be in error by much larger amounts for a composite with periodically distributed voids, but homogenization leads to a cancelling of errors, and the effective transverse Young’s modulus of the voided composite is shown to be in error by only 23% at a void volume fraction of 75%.

Implementation of a Viscoplastic Model for a Plasma Sprayed Ceramic Thermal Barrier Coating

This paper describes the implementation and modification of a previously proposed unified viscoplastic constitutive model to simulate the behavior of a Yttria Stabilized Zirconia plasma sprayed thermal barrier coating. The model was recast for use in finite strain situations and modified to have a more physically acceptable non-associated flow rule. Temperature dependent material constants were found for a specific material using a novel approach based on Genetic Algorithms.

Closed form solution for rectangular inclusions with quadratic eigenstrains

The problem of a rectangular inclusion in an infinite isotropic elastic domain is solved in closed form for the case of a quadratic eigenstrain within the inclusion. The eigenstrain represents a stress free strain such as would occur with a quadratic temperature field. The problem is formulated in terms of fundamental point load solutions yielding a Greens function based integral expression. This integral is singular for interior points of the inclusion and non-singular for exterior points. For both cases closed form formulae are obtained. A non-uniformly heated rectangular region and a rectangular region with a non-uniform shear distortion are solved as examples.

Stress Versus Temperature Dependence of Activation Energies for Creep

The activation energy for creep at low stresses and elevated temperatures is associated with lattice diffusion, where the rate controlling mechanism for deformation is dislocation climb. At higher stresses and intermediate temperatures, the rate controlling mechanism changes from dislocation climb to obstacle-controlled dislocation glide. Along with this change in deformation mechanism occurs a change in the activation energy. When the rate controlling mechanism for deformation is obstacle-controlled dislocation glide, it is shown that a temperature-dependent Gibbs free energy does better than a stress-dependent Gibbs free energy in correlating steady-state creep data for both copper and LiF-22mol percent CaF2 hypereutectic salt.

Three-dimensional deformation analysis of two-phase dislocation substructures

Pioneering investigations have demonstrated that the back stress term present in plastic, viscoplastic and dislocation models arises from the existence of inhomogeneous stress fields within a deforming solid. These inhomogeneities arise naturally as a result of dislocation substructures formed during the deformation of hard and soft regions within the material. Earlier investigators developed one-dimensional substructure models using Voigt strain compatibility to evaluate the deformation characteristics of the solid. While conceptually simple in format, these one-dimensional models are limited in scope. In this paper the authors extend these approaches to three-dimensional stress-strain fields by using the Budiansky and Wu self-consistent formalism with an Eshelby criterion for strain compatibility between the hard and soft regions. The result is a rate-dependent viscoplastic theory which we call dislocation substructure viscoplasticity (DSV) that incorporates a self-consistent effect of dislocation substructure on material response. An algorithm is presented for its numerical implementation. DSV is comparable to classical viscoplasticity in its complexity of numeric integration.