Wave Propagation in Rocks – Investigating the Effect of Rheology

Rocks exhibit beyond-Hookean, delayed and damped elastic, behaviour (creep, relaxation etc.). In many cases, the Poynting–Thomson–Zener (PTZ) rheological model proves to describe these phenomena successfully. A forecast of the PTZ model is that the dynamic elasticity coefficients are larger than the static (slow-limit) counterparts. This prediction has recently been confirmed on a large variety of rock types. Correspondingly, according to the model, the speed of wave propagation depends on frequency, the high-frequency limit being larger than the low-frequency limit. This frequency dependence can have a considerable influence on the evaluation of various wave-based measurement methods of rock mechanics. As experience shows, commercial finite element softwares are not able to properly describe wave propagation, even for the Hooke model and simple specimen geometries, the seminal numerical artefacts being instability, dissipation error and dispersion error, respectively. This has motivated research on developing reliable numerical methods, which amalgamate beneficial properties of symplectic schemes, their thermodynamically consistent generalization (including contact geometry), and spacetime aspects. The present work reports on new results obtained by such a numerical scheme, on wave propagation according to the PTZ model, in one space dimension. The simulation outcomes coincide nicely with the theoretically obtained phase velocity prediction

A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity

In Parts I [1] and II [2] of this series, a novel computational framework was presented for the numerical analysis of large strain fast solid dynamics in compressible and nearly/truly incompressible isothermal hyperelasticity. The methodology exploited the use of a system of first order Total Lagrangian conservation laws formulated in terms of the linear momentum and a triplet of deformation measures comprised of the deformation gradient tensor, its co-factor and its Jacobian. Moreover, the consideration of polyconvex constitutive laws was exploited in order to guarantee the hyperbolicity of the system and show the existence of a convex entropy function (sum of kinetic and strain energy per unit undeformed volume) necessary for symmetrisation. In this new paper, the framework is extended to the more general case of thermo-elasticity by incorporating the first law of thermodynamics as an additional conservation law, written in terms of either the entropy (suitable for smooth solutions) or the total energy density (suitable for discontinuous solutions) of the system. The paper is further enhanced with the following key novelties. First, sufficient conditions are put forward in terms of the internal energy density and the entropy measured at reference temperature in order to ensure ab-initio the polyconvexity of the internal energy density in terms of the extended set comprised of the triplet of deformation measures and the entropy. Second, the study of the eigenvalue structure of the system is performed as proof of hyperbolicity and with the purpose of obtaining correct time step bounds for explicit time integrators. Application to two well-established thermo-elastic models is presented: Mie-Gruneisen and modified entropic elasticity. Third, the use of polyconvex internal energy constitutive laws enables the definition of a generalised convex entropy function, namely the ballistic energy, and associated entropy fluxes, allowing the symmetrisation of the system of conservation laws in terms of entropy-conjugate fields. Fourth, and in line with the previous papers of the series, an explicit stabilised Petrov-Galerkin framework is presented for the numerical solution of the thermo-elastic system of conservation laws when considering the entropy as an unknown of the system. Finally, a series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulation

Thermodynamically consistent space-time discretization of non-isothermal mechanical systems in the framework of GENERIC

The present work addresses the design of structure-preserving numerical methods that emanate from the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) formalism. Novel energy-momentum (EM) consistent time-stepping schemes in the realm of molecular dynamics are proposed. Moreover, the GENERIC-based structure-preserving numerical methods are extended to the context of large-strain thermoelasticity and thermo-viscoelasticity

Thermodynamically consistent space-time discretization of non-isothermal mechanical systems in the framework of GENERIC

The present work addresses the design of structure-preserving numerical methods that emanate from the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) formalism. Novel energy-momentum (EM) consistent time-stepping schemes in the realm of molecular dynamics are proposed. Moreover, the GENERIC-based structure-preserving numerical methods are extended to the context of large-strain thermoelasticity and thermo-viscoelasticity

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized. © 2022, The Author(s)