JeeYeon N. Plohr

Linearized analysis of Richtmyer-Meshkov flow for elastic materials

We present a study of Richtmyer-Meshkov flow for elastic materials. This flow, in which a material interface is struck by a shock wave, was originally investigated for gases, where growth of perturbations of the interface is observed. Here we consider two elastic materials in frictionless contact. The governing system of equations comprises conservation laws supplemented by constitutive equations. To analyse it, we linearize the equations around a one-dimensional background solution under the assumption that the perturbation is small. The background problem defines a Riemann problem that is solved numerically; its solution contains transmitted and reflected shock waves in the longitudinal modes. The linearized Rankine-Hugoniot condition provides the interface conditions at the longitudinal and shear waves; the frictionless material interface conditions are also linearized. The resulting equations, a linear system of partial differential equations, is solved numerically using a finite-difference method supplemented by front tracking. In verifying the numerical code, we reproduce growth of the interface in the gas case. For the elastic case, in contrast, we find that the material interface remains bounded: the non-zero shear stiffness stabilizes the flow. In particular, the linear theory remains valid at late time. Moreover, we identify the principal mechanism for the stability of Richtmyer-Meshkov flow for elastic materials: the vorticity deposited on the material interface during shock passage is propagated away by the shear waves, whereas for gas dynamics it stays on the interface.

Dynamically driven phase transformations in heterogeneous materials. I. Theory and microstructure considerations

A theoretical model recently developed for heterogeneous materials undergoing dynamically driven thermodynamic phase transitions [F. L. Addessio et al. J. Appl. Phys. 97, 083509 (2005)] has been extended to allow for complex material microstructures. The model is applied to silicon carbide—titanium (SiC–Ti) unidirectional metal matrix composites where the aligned SiC fibers are filler and Ti is the matrix. Ti is known to undergo a low pressure and temperature solid-solid first-order phase transition. The microstructural analysis uses the generalized method of cells, which partitions a representative volume element into subcells containing the SiC fibers and the Ti matrix. The thermomechanical analysis has been reformulated from the previous work. In the reformulation it is found that thermodynamic quantities are naturally expressed as mass fraction averages over the two coexisting phases while the mechanical quantities are expressed naturally as volume averages. Consequently, the thermomechanical reformul...

Numerical simulation of systems of shear bands in ductile metal with inclusions

We develop a method for numerical simulations of high strain-rate loading of mesoscale samples of ductile metal with inclusions. Because of its small-scale inhomogeneity, the composite material is prone to localized shear deformation (adiabatic shear bands). This method employs the Generalized Method of Cells of Paley and Aboudi [Mech. Materials, vol. 14, pp. 127–139, 1992] to ensure that the micro mechanical behavior of the metal and inclusions is reflected properly in the behavior of the composite at the mesoscale. To find the effective plastic strain rate when shear bands are present, we extend and apply the analytic and numerical analysis of shear bands of Glimm, Plohr, and Sharp [Mech. Materials, vol. 24, pp. 31–41, 1996]. Our tests of the method focus on the stress/strain response in uniaxial-strain flow, both compressive and tensile, of depleted uranium metal containing silicon carbide inclusions. We use the Preston-Tonks-Wallace viscoplasticity model [J. Appl. Phys., vol. 93, pp. 211–220, 2003], w...

Dynamic Response of PBX‐9501 through the β‐δ Phase Transition

The Gibbs free energies of the β and δ phases of HMX are constructed from zero pressure heat capacity data, specific volume measurements, numerical simulations, and diamond anvil experiments. The free energies provide input into a dynamic phase transition model developed for heterogeneous materials that undergo dynamically driven phase transitions. This model, which uses the method of cells analysis to treat the HMX‐polymer binder composite, is used to study dynamically loaded PBX‐9501 as it transforms from the beta to the delta phase.

Dynamically driven phase transformations in heterogeneous materials. II. Applications including damage

A model, developed for heterogeneous materials undergoing dynamically driven phase transformations in its constituents, has been extended to include the evolution of damage. Damage is described by two mechanisms: interfacial debonding between the constituents and brittle failure micro-crack growth within the constituents. The analysis is applied to silicon carbide-titanium (SiC-Ti) unidirectional metal matrix composites that undergo the following phenomena: Ti has a yield stress of approximately 0.5 GPa and above a pressure of about 2 GPa undergoes a solid-solid phase transformation. The inelastic work from plastic dissipation contributes to the temperature and pressure rise in the Ti. SiC behaves elastically below a critical stress, above which it is damaged by microcrack growth. Finally, under tensile loading, the interface between Ti and SiC debonds according to an interfacial decohesion law. Each process is first examined independently in order to understand how its characteristic behavior is manifest...

Large Deformation Constitutive Laws for Isotropic Thermoelastic Materials

We examine the approximations made in using Hookes law as a constitutive relation for an isotropic thermoelastic material subjected to large deformation by calculating the stress evolution equation from the free energy. For a general thermoelastic material, we employ the volume-preserving part of the deformation gradient to facilitate volumetric/shear strain decompositions of the free energy, its first derivatives (the Cauchy stress and entropy), and its second derivatives (the specific heat, Grueneisen tensor, and elasticity tensor). Specializing to isotropic materials, we calculate these constitutive quantities more explicitly. For deformations with limited shear strain, but possibly large changes in volume, we show that the differential equations for the stress components involve new terms in addition to the traditional Hookes law terms. These new terms are of the same order in the shear strain as the objective derivative terms needed for frame indifference; unless the latter terms are negligible, the former cannot be neglected. We also demonstrate that accounting for the new terms requires that the deformation gradient be included as a field variable

Equilibrium Conditions at a Solid-Solid Interface

We derive the thermodynamic conditions necessary for two elastoplastic solid phases to coexist in equilibrium. Beyond temperature, velocity, and traction continuity, these conditions require continuity of a generalization of the specific Gibbs free energy. We express this quantity in the Eulerian frame as well as the Lagrangian frame. We also show that two approaches in deriving the equilibrium conditions, one on the continuum level and the other on the atomistic scale, yield the same results. Finally, we discuss two possible interpretations for the Gibbs free energy, which lead to distinct generalizations, except in the case of inviscid fluids, where they coincide.

Dynamically driven phase transformations in damaged composite materials

A model developed for composite materials undergoing dynamicaly driven phase transitions in its constituents has been extended to allow for complex material micro‐structure and evolution of damage. In this work, damage is described by interfacial debonding and micro‐crack growth. We have applied the analysis to silicon carbide‐titanium (SiC‐Ti) unidirectional metal matrix composites. In these composites, Ti can undergo a low pressure and temperature solid‐solid phase transition. With these extensions we have carried out simulations to study the complex interplay between loading rates, micro‐structure, damage, and the thermo‐mechanical response of the system as it undergoes a solid‐solid phase transitions.

Simplified Shock Conditions for Large‐Strain Thermo‐Visco‐Plasticity

We consider shock loading of a thermo‐visco‐plastic material; the strain is not restricted to be small. We show how to reduce the Rankine‐Hugoniot jump conditions to a simplified form analogous to that used in fluid dynamics. Just as for fluids, the shock conditions can be separated into purely thermodynamic conditions (the Rayleigh and Hugoniot equations), a condition determining the velocity, and a condition determining the strain, which can be solved sequentially to determine the shock wave.