arXiv subjects

Lennart Machill

Publications and source records attributed to Lennart Machill.

7 recordsLinked to original sources

arXiv2026

Frame-indifferent discretization in nonlinear thermoviscoelasticity: Analysis and numerical simulations

We consider a quasi-static nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin-Voigt rheology where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. We refine the discretization schemes in [Badal-Friedrich-Kru\v{z}\'ik '23, Mielke-Roub\'{\i}\v{c}ek '20] by imposing frame indifference already at a time-discrete level. This is justified both analytically and numerically.

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arXiv2025

The energy scaling behaviour of singular perturbation models of staircase type in linearized elasticity for higher order laminates

We investigate the scaling behaviour of a singular perturbation model within the geometrically linearized theory of elasticity involving data of higher lamination order. We study boundary data which are of staircase type and show rather general lower scaling bounds, both in the setting of prescribed Dirichlet data and for periodic configurations with a mean value constraint. In contrast to the setting without gauge invariances, these lower scaling bounds depend on \emph{two} parameters -- the order of lamination of the boundary data as well as the number of involved (non-)degenerate symmetrized rank-one directions. By discussing upper bounds in specific geometries and for a specific constellation of wells, we give evidence of the sharpness of these lower bound estimates. Hence, it is necessary to keep track of the outlined \emph{two} parameters in deducing scaling laws within the geometrically linearized theory of elasticity.

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arXiv2024

Nonlinear relations of viscous stress and strain rate in nonlinear Viscoelasticity

We consider a Kelvin-Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame indifference. Using a rigidity estimate by [Ciarlet-Mardare '15], existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows.

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arXiv2024

Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models

According to the Nernst theorem or, equivalently, the third law of thermodynamics, the absolute zero temperature is not attainable. Starting with an initial positive temperature, we show that there exist solutions to a Kelvin-Voigt model for quasi-static nonlinear thermoviscoelasticity at a finite-strain setting [Mielke-Roub\'i\v{c}ek '20], obeying an exponential-in-time lower bound on the temperature. Afterwards, we focus on the case of deformations near the identity and temperatures near a critical positive temperature, and we show that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. Our result extends the recent linearization result in [Badal-Friedrich-Kru\v{z}\'ik '23], as it allows the critical temperature to be positive.

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arXiv2023

Derivation of a von K\'arm\'an plate theory for thermoviscoelastic solids

We derive a von K\'arm\'an plate theory from a three-dimensional quasistatic nonlinear model for nonsimple thermoviscoelastic materials in the Kelvin-Voigt rheology, in which the elastic and the viscous stress tensor comply with a frame indifference principle [Mielke-Roub\'{i}\v{c}ek '20]. In a dimension-reduction limit, we show that weak solutions to the nonlinear system of equations converge to weak solutions of an effective two-dimensional system featuring mechanical equations for viscoelastic von K\'arm\'an plates, previously derived in [Friedrich-Kru\v{z}\'ik '20], coupled with a linear heat-transfer equation. The main challenge lies in deriving a priori estimates for rescaled displacement fields and temperatures, which requires the adaptation of generalized Korn's inequalities and bounds for heat equations with $L^1$-data to thin domains.

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arXiv2022

One-dimensional viscoelastic von K\'{a}rm\'{a}n theories derived from nonlinear thin-walled beams

We derive an effective one-dimensional limit from a three-dimensional Kelvin-Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via [Friedrich-Kru\v{z}\'ik '20] and [Friedrich-Machill '22] by a successive dimension reduction, first from 3D to a 2D theory for von K\'{a}rm\'{a}n plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static $\Gamma$-convergence in [Freddi-Mora-Paroni '13], on the abstract theory of metric gradient flows, and on evolutionary $\Gamma$-convergence by Sandier and Serfaty.

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arXiv2021

Derivation of a one-dimensional von K\'{a}rm\'{a}n theory for viscoelastic ribbons

We consider a two-dimensional model of viscoelastic von K\'arm\'an plates in the Kelvin's-Voigt's rheology derived from a three-dimensional model at a finite-strain setting. As the width of the plate goes to zero, we perform a dimension-reduction from 2D to 1D and identify an effective one-dimensional model for a viscoelastic ribbon comprising stretching, bending, and twisting both in the elastic and the viscous stress. Our arguments rely on the abstract theory of gradient flows in metric spaces by Sandier and Serfaty and complement the $\Gamma$-convergence analysis of elastic von K\'{a}rm\'{a}n ribbons in [Freddi et al., 2018]. Besides convergence of the gradient flows, we also show convergence of associated time-discrete approximations, and we provide a corresponding commutativity result.

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