Nonlinear thermodynamic phase field theory with application to fracture and dynamic inelastic phenomena in ceramic polycrystals

Abstract

A geometrically nonlinear phase field theory accounting for dissipation, rate effects, nonlinear thermoelasticity, fracture, and other structural changes is constructed in the context of continuum thermodynamics. First, a general framework accommodating arbitrary strain energy potentials, inelastic deformation kinematics, and unlimited order parameters is formulated. Next, the framework is specialized to account for deformation physics pertinent to crystalline ceramics and minerals deformed at high rates and high pressures. Notably, a logarithmic elastic strain tensor referred to an intermediate material configuration enters the nonlinear thermoelastic potential. Order parameters represent fractures, solid–solid phase transformations, deformation twinning, or slip of partial dislocations. Thermodynamically consistent kinetics manifest in equations reminiscent of Ginzburg–Landau dynamics, wherein viscosity coefficients are most generally state- and rate-dependent. Pressure-dependent strength commensurate with frictional resistance is enabled in alternative kinetic equations for dynamic fracture with irreversibility constraints. Linearization of the model suitable for moderate volume changes but small deviatoric elastic strain and rotation is undertaken. The theory is applied to study deformation and failure of polycrystalline forms of boron carbide (B4C), titanium diboride (TiB2), and a B4C-TiB2 ceramic composite. Solutions are derived and evaluated numerically for uniaxial stress tension and compression, uniaxial strain compression, and planar shock compression. The latter analysis yields relationships among viscosity coefficients, gradient regularization lengths, and characteristics of steady plastic waveforms. Results give new insight into high-rate deformation mechanisms previously speculated in these materials.