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Martin Kruzik

Publications and source records attributed to Martin Kruzik.

5 recordsLinked to original sources
arXiv2018

On the passage from nonlinear to linearized viscoelasticity

We formulate a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. We identify weak solutions in the nonlinear framework as limits of time-incremental problems for vanishing time increment. Moreover, we show that linearization around the identity leads to the standard system for linearized viscoelasticity and that solutions of the nonlinear system converge in a suitable sense to solutions of the linear one. The same property holds for time-discrete approximations and we provide a corresponding commutativity result. Our main tools are the theory of gradient flows in metric spaces and Gamma-convergence.

math.AP↗
arXiv2018

A phase-field approach to Eulerian interfacial energies

We analyze a phase-field approximation of a sharp-interface model for two- phase materials proposed by M. Silhavy [32, 33]. The distinguishing trait of the model resides in the fact that the interfacial term is Eulerian in nature, for it is defined on the deformed configuration. We discuss a functional frame allowing for existence of phase- field minimizers and Γ-convergence to the sharp-interface limit. As a by-product, we provide additional detail on the admissible sharp-interface configurations with respect to the analysis in [32, 33].

math-ph↗
arXiv2013

Quasistatic adhesive contact delaminating in mixed mode and its numerical treatment

An adhesive unilateral contact between visco-elastic bodies at small strains and in a Kelvin-Voigt rheology is scrutinized, neglecting inertia. The flow-rule for debonding the adhesive is considered rate independent, unidirectional, and non-associative due to dependence on the mixity of modes of delamination, namely Mode I (opening) needs (=dissipates) less energy than Mode II (shearing). Such mode-mixity dependence of delamination is a very pronounced (and experimentally confirmed) phenomenon typically considered in engineering models. An efficient semi-implicit-in-time FEM discretization leading to recursive quadratic mathematical programs is devised. Its convergence and thus the existence of weak solutions is proved. Computational experiments implemented by BEM illustrate the modeling aspects and the numerical efficiency of the discretization.

math.AP↗
arXiv2012

Sequential weak continuity of null Lagrangians at the boundary

We show weak* in measures on $\barØ$/ weak-$L^1$ sequential continuity of $u\mapsto f(x,\nabla u):W^{1,p}(Ø;\R^m)\to L^1(Ø)$, where $f(x,\cdot)$ is a null Lagrangian for $x\inØ$, it is a null Lagrangian at the boundary for $x\in\partialØ$ and $|f(x,A)|\le C(1+|A|^p)$. We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why $u\mapsto \det\nabla u:W^{1,n}(Ø;\R^n)\to L^1(Ø)$ fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians at the boundary. The paper closes with an example indicating that a well-known result on higher integrability of determinant \cite{Mue89a} need not necessarily extend to our setting. The notion of quasiconvexity at the boundary due to J.M. Ball and J. Marsden is central to our analysis.

math.AP↗