Stress Update Procedure Research Articles

Strain- and stress-based continuum damage models—II. Computational aspects

In Part I of this work we developed continuum isotropic and anisotropic elastoplastic-damage models, formulated either in strain space on the basis of the effective stress concept, or in stress space and employing the dual notion of effective strain. In Part II we consider in detail the variational formulation and subsequent numerical implementation of these models. The former relies crucially on the notions of maximum plastic and maximum damage dissipation. The latter makes systematic use of the operator splitting methodology to derive unconditionally stable algorithms for the numerical integration of the elastoplastic-damage equations of evolution. Appropriate extensions to treat the proposed rate-dependent (viscous) damage are also presented. For stress-based damage models, our numerical treatment relies on a new three-step operator split. The algorithms developed lead to simple and efficient stress update procedures suitable for large-scale finite element calculations. Application is made to a class of inviscid and rate-dependent cap models with an isotropic strain-based damage mechanism. Remarkably good agreement with existing experimental data that includes complicated stress paths is obtained. Numerical examples are also presented that demonstrate the good performance of the proposed algorithms.

Measurements of the compressibility of solids

In high pressure research, static megabar pressures are typically produced by compression of a thin sample by two diamonds in various types of diamond anvil cells. This process is accompanied by large plastic deformation (sample thickness is reduced by a factor of 30), and finite elastic deformation of a sample and even the diamond. A thermodynamically consistent system of equations for large elastic and plastic deformation of an isotropic material obeying nonlinear elasticity and pressure dependent yield condition is formulated. The Murnaghan elasticity law and pressure-dependent J2 plasticity are utilized. The finite-strain third-order elasticity law for cubic crystals is utilized for diamond. A computational algorithm is presented with emphasis on the stress update procedure and derivation of the consistent tangent moduli. It is implemented as a user material subroutine in the finite element code ABAQUS. Material parameters for a rhenium sample, as an example, and a diamond are calibrated based on the experimental and atomistic simulation results in the literature. The evolution of the stress and strain tensor fields in the sample and diamond is studied up to a pressure of 300 GPa. Good correspondence between numerical and experimental pressure distributions at the diamond-sample contact surface is obtained. Because there is a significant scatter of the magnitude of reported third-order single-crystal elastic moduli for diamond, their effect on strains and stresses is studied in detail. With the smaller third-order elastic moduli, the phenomenon of cupping of the diamond-sample contact surface is reproduced, which plays an important role in increasing maximum pressures for a given anvil geometry. The results provide important insight into the mechanical response in diamond anvil cells, interpretation of materials properties under extreme conditions from heterogeneous fields, and optimum design of cells for reaching the maximum static pressure in a volume sufficient for the desired measurements.