28 research outputs found
On Computational Modelling of Strain-Hardening Material Dynamics
In this paper we show that entropy can be used within a functional for the stress relaxation time of solid materials to parametrise finite viscoplastic strain-hardening deformations. Through doing so the classical empirical recovery of a suitable irreversible scalar measure of work-hardening from the three-dimensional state parameters is avoided. The success of the proposed approach centres on determination of a rate-independent relation between plastic strain and entropy, which is found to be suitably simplistic such to not add any significant complexity to the final model. The result is sufficiently general to be used in combination with existing constitutive models for inelastic deformations parametrised by one-dimensional plastic strain provided the constitutive models are thermodynamically consistent. Here a model for the tangential stress relaxation time based upon established dislocation mechanics theory is calibrated for OFHC copper and subsequently integrated within a two-dimensional moving-mesh scheme. We address some of the numerical challenges that are faced in order to ensure successful implementation of the proposed model within a hydrocode. The approach is demonstrated through simulations of flyer-plate and cylinder impacts
Solid-fluid dynamics of yield-stress fluids
On the example of two-phase continua experiencing stress induced solid-fluid
phase transitions we explore the use of the Euler structure in the formulation
of the governing equations. The Euler structure guarantees that solutions of
the time evolution equations possessing it are compatible with mechanics and
with thermodynamics. The former compatibility means that the equations are
local conservation laws of the Godunov type and the latter compatibility means
that the entropy does not decrease during the time evolution. In numerical
illustrations, in which the one-dimensional Riemann problem is explored, we
require that the Euler structure is also preserved in the discretization.Comment: 51 pages, 7 figure
Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations
Continuum mechanics with dislocations, with the Cattaneo type heat
conduction, with mass transfer, and with electromagnetic fields is put into the
Hamiltonian form and into the form of the Godunov type system of the first
order, symmetric hyperbolic partial differential equations (SHTC equations).
The compatibility with thermodynamics of the time reversible part of the
governing equations is mathematically expressed in the former formulation as
degeneracy of the Hamiltonian structure and in the latter formulation as the
existence of a companion conservation law. In both formulations the time
irreversible part represents gradient dynamics. The Godunov type formulation
brings the mathematical rigor (the well-posedness of the Cauchy initial value
problem) and the possibility to discretize while keeping the physical content
of the governing equations (the Godunov finite volume discretization)
Two-phase hyperbolic model for porous media saturated with a viscous fluid and its application to wavefields simulation
We derive and study a new hyperbolic two-phase model of a porous deformable
medium saturated by a viscous fluid. The governing equations of the model are
derived in the framework of Symmetric Hyperbolic Thermodynamically Compatible
(SHTC) systems and by generalizing the unified hyperbolic model of continuum
fluid and solid mechanics. Similarly to the unified model, the presented model
takes into account the viscosity of the saturating fluid through a hyperbolic
reformulation. The model accounts for such dissipative mechanisms as
interfacial friction and viscous dissipation of the saturated fluid. Using the
presented nonlinear finite-strain SHTC model, the governing equations for the
propagation of small-amplitude waves in a porous medium saturated with a
viscous fluid are derived. As in the conventional Biot theory of porous media,
three types of waves can be found: fast and slow compression waves and shear
waves. It turns out that the shear wave attenuates rapidly due to the viscosity
of the saturating fluid, and this wave is difficult to see in typical test
cases. However, some test cases are presented in which shear waves can be
observed in the vicinity of interfaces between regions with different porosity
Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity
The aim of this paper is to compare a hyperelastic with a hypoelastic model
describing the Eulerian dynamics of solids in the context of non-linear
elastoplastic deformations. Specifically, we consider the well-known
hypoelastic Wilkins model, which is compared against a hyperelastic model based
on the work of Godunov and Romenski. First, we discuss some general conceptual
differences between the two approaches. Second, a detailed study of both models
is proposed, where differences are made evident at the aid of deriving a
hypoelastic-type model corresponding to the hyperelastic model and a particular
equation of state used in this paper. Third, using the same high order ADER
Finite Volume and Discontinuous Galerkin methods on fixed and moving
unstructured meshes for both models, a wide range of numerical benchmark test
problems has been solved. The numerical solutions obtained for the two
different models are directly compared with each other. For small elastic
deformations, the two models produce very similar solutions that are close to
each other. However, if large elastic or elastoplastic deformations occur, the
solutions present larger differences.Comment: 14 figure
First-order hyperbolic formulation of the pure tetrad teleparallel gravity theory
Motivated by numerically solving the Einstein field equations, we derive a
first-order reduction of the second-order -teleparallel gravity field
equations in the pure-tetrad formulation (no spin connection). We then restrict
our attention to the teleparallel equivalent of general relativity (TEGR) and
propose a 3+1 decomposition of the governing equations that can be used in a
computational code. We demonstrate that for the matter-free space-time the
obtained system of first-order equations is equivalent to the tetrad
reformulation of general relativity by Estabrook, Robinson, Wahlquist, and
Buchman and Bardeen and therefore also admits a symmetric hyperbolic
formulation. The structure of the 3+1 equations resembles a lot of similarities
with the equations of relativistic electrodynamics and the recently proposed
dGREM tetrad-reformulation of general relativity
On thermodynamically compatible finite volume schemes for continuum mechanics
In this paper we present a new family of semi-discrete and fully-discrete
finite volume schemes for overdetermined, hyperbolic and thermodynamically
compatible PDE systems. In the following we will denote these methods as HTC
schemes. In particular, we consider the Euler equations of compressible
gasdynamics, as well as the more complex Godunov-Peshkov-Romenski (GPR) model
of continuum mechanics, which, at the aid of suitable relaxation source terms,
is able to describe nonlinear elasto-plastic solids at large deformations as
well as viscous fluids as two special cases of a more general first order
hyperbolic model of continuum mechanics. The main novelty of the schemes
presented in this paper lies in the fact that we solve the \textit{entropy
inequality} as a primary evolution equation rather than the usual total energy
conservation law. Instead, total energy conservation is achieved as a mere
consequence of a thermodynamically compatible discretization of all the other
equations. For this, we first construct a discrete framework for the
compressible Euler equations that mimics the continuous framework of Godunov's
seminal paper \textit{An interesting class of quasilinear systems} of 1961
\textit{exactly} at the discrete level. All other terms in the governing
equations of the more general GPR model, including non-conservative products,
are judiciously discretized in order to achieve discrete thermodynamic
compatibility, with the exact conservation of total energy density as a direct
consequence of all the other equations. As a result, the HTC schemes proposed
in this paper are provably marginally stable in the energy norm and satisfy a
discrete entropy inequality by construction. We show some computational results
obtained with HTC schemes in one and two space dimensions, considering both the
fluid limit as well as the solid limit of the governing partial differential
equations