A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy

Demoulini, Sophia; Stuart, David; Tzavaras, Athanasios E.

doi:10.1007/s002050100137

Cited by 75 publications

(79 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We note that Dafermos ([4], p. 30-31) and Demoulini, Stuart, and Tzavaras ( [7]) proved that (1.9, 1.10, 1.11) hold in the sense of distributions. However we feel that the use of differential forms provides additional understanding of the matter.…”

Section: Introduction: Kinematicsmentioning

confidence: 88%

“…This is the same as the enlarged system derived in [17], and studied in [7]. In terms of the Lagrangian velocity u, the system is:…”

Section: Elasticity and Hyperelasticitymentioning

confidence: 95%

“…In [17], Qin showed that the 19 evolution equations for the deformation gradient, its cofactor matrix, and its determinant, together with the standard equations for conservation of momentum, form a symmetrichyperbolic system when the material is hyperelastic with a polyconvex stored energy. In [7], Demoulini, Stuart, and Tzavaras used a variational approximation scheme to prove the existence of global measure-valued solutions to the symmetrized system of [17].…”

Section: Introduction: Kinematicsmentioning

confidence: 99%

See 2 more Smart Citations

Symmetric-Hyperbolic Equations of Motion for a Hyperelastic Material

Wagner

2009

J. Hyper. Differential Equations

View full text Add to dashboard Cite

Abstract. We offer an alternate derivation for the symmetric-hyperbolic formulation of the equations of motion for a hyperelastic material with polyconvex stored energy. The derivation makes it clear that the expanded system is equivalent, for weak solutions, to the original system. We consider motions with variable as well as constant temperature. In addition, we present equivalent Eulerian equations of motion, which are also symmetric-hyperbolic.

show abstract

Section: Introduction: Kinematicsmentioning

confidence: 88%

“…This is the same as the enlarged system derived in [17], and studied in [7]. In terms of the Lagrangian velocity u, the system is:…”

Section: Elasticity and Hyperelasticitymentioning

confidence: 95%

Section: Introduction: Kinematicsmentioning

confidence: 99%

See 1 more Smart Citation

Symmetric-Hyperbolic Equations of Motion for a Hyperelastic Material

Wagner

2009

J. Hyper. Differential Equations

View full text Add to dashboard Cite

show abstract

“…It has been used by Friesecke and Dolzmann [13] to prove a global existence result for the initial boundary value problem for (1.1) with b = 0 (in fact in multiple space dimensions). Recently Demoulini, Stuart and Tzavaras applied the method in the first-order case a = b = 0 ( [8], see also [9] for the multidimensional case) to construct entropy solutions. The case with a, b = 0 is particularly interesting because in the limit case of vanishing dissipation non-classical shock waves can appear which represent dynamic phase interfaces ( [5,11,20,21,25] and references in [18]).…”

Section: Introductionmentioning

confidence: 99%

Global Existence for Phase Transition Problems via a Variational Scheme

Rohde

Thanh

2004

J. Hyper. Differential Equations

View full text Add to dashboard Cite

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress-strain function which contains a nonlocal convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

show abstract

“…Problem (1.1)-(1.3) has been studied in several analytical contributions [1,5,6,7,8,9,10,12,13,15,16,17]. The existence of weak solutions for ε > 0 has been verified in [8], where the implicit Euler method is used to semidiscretize the problem in time: Let I k = {t j } J j=1 be an equidistant net of mesh width k = t j − t j−1 to discretize [0, T ], with j ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

Convergence of a Finite Element-Based Space-Time Discretization in Elastodynamics

Prohl¹

2008

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

We study a finite element-based space-time discretization of the elastodynamics equation u ε tt − div σ(∇u ε ) − εΔu ε t = 0 for ε ≥ 0, where σ = Dφ and φ is a nonconvex function. The convergence for regularization parameters ε > 0 of iterates to weak solutions and for the limiting problem ε = 0 of iterates towards generalized solutions is shown in a general setting of data. Computational experiments are included to motivate formation and propagation of two-dimensional microstructures for decreasing values of the regularization parameter.

show abstract

A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy

Cited by 75 publications

References 6 publications

Symmetric-Hyperbolic Equations of Motion for a Hyperelastic Material

Symmetric-Hyperbolic Equations of Motion for a Hyperelastic Material

Global Existence for Phase Transition Problems via a Variational Scheme

Convergence of a Finite Element-Based Space-Time Discretization in Elastodynamics

Contact Info

Product

Resources

About