A monolithic finite element approach using high-order schemes in time and space applied to finite strain thermo-viscoelasticity

Netz, Torben; Hartmann, Stefan

doi:10.1016/j.camwa.2015.03.030

Cited by 16 publications

(7 citation statements)

References 51 publications

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“…In addition to that, temperature formula has been used along with constitutive Equation for the material heat flux vector. In essence, weak form of the temperature evolution equation lies at the heart of numerous works on finite‐strain thermomechanics such as that of Miehe, Balzani et al, and Netz and Hartmann . Alternatively, differentiating with respect to time, one arrives at the relation

\partial_{θ} \overset{true‾}{u} \dot{θ} = θ \dot{\overline{η}} - θ \partial_{C} \overline{η} : \dot{C} .

Inserting the last equation into , we obtain the entropy form of given by

\int_{scriptB} ({}, v_{θ} θ \overset{\overset{η}{true}}{true} + \nabla v_{θ} \cdot boldK \nabla θ) normald V + \int_{\partial_{q} scriptB} v_{θ} \overset{q}{true} normald A = 0 .

In essence, variational formulation lies at the origin of alternative temperature‐based numerical methods such as in the work Holzapfel and Simo and Hesch and Betsch .…”

Section: Large‐strain Thermoelastictymentioning

confidence: 99%

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Energy‐momentum‐entropy consistent numerical methods for large‐strain thermoelasticity relying on the GENERIC formalism

Betsch

Schiebl

2019

Numerical Meth Engineering

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Summary In the present paper, structure‐preserving numerical methods for finite strain thermoelastodynamics are proposed. The underlying variational formulation is based on the general equation for nonequilibrium reversible‐irreversible coupling (GENERIC) formalism and makes possible the free choice of the thermodynamic state variable. The notion “GENERIC consistent space discretization” is introduced, which facilitates the design of Energy‐Momentum‐Entropy (EME) consistent schemes. In particular, three alternative EME schemes result from the present approach. These schemes are directly linked to the respective choice of the thermodynamic variable. Numerical examples confirm the structure‐preserving properties of the newly developed EME schemes, which exhibit superior numerical stability.

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\partial_{θ} \overset{true‾}{u} \dot{θ} = θ \dot{\overline{η}} - θ \partial_{C} \overline{η} : \dot{C} .

Inserting the last equation into , we obtain the entropy form of given by

\int_{scriptB} ({}, v_{θ} θ \overset{\overset{η}{true}}{true} + \nabla v_{θ} \cdot boldK \nabla θ) normald V + \int_{\partial_{q} scriptB} v_{θ} \overset{q}{true} normald A = 0 .

In essence, variational formulation lies at the origin of alternative temperature‐based numerical methods such as in the work Holzapfel and Simo and Hesch and Betsch .…”

Section: Large‐strain Thermoelastictymentioning

confidence: 99%

“…In essence, weak form (41) of the temperature evolution equation lies at the heart of numerous works on finite-strain thermomechanics such as that of Miehe, 24 Balzani et al, 25 and Netz and Hartmann. 26 Alternatively, differentiating (12) with respect to time, one arrives at the relation…”

Section: Generic-based Weak Form and More Common Descriptionsmentioning

confidence: 99%

Energy‐momentum‐entropy consistent numerical methods for large‐strain thermoelasticity relying on the GENERIC formalism

Betsch

Schiebl

2019

Numerical Meth Engineering

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“…Johlitz [27] further simplifies this model by adding the thermal and mechanical-volumetric part of the free energy, reformulates it and thus omits the thermal intermediate configuration for the representation of thermal strains. Netz and Hartmann [44] show in their model, among other things, the approximation of a highly nonlinear heat capacity and therefore the reduction of the thermoelastic coupling term. These models were further developed by Guo et al [18], among others, in order to integrate further effects, such as the reorganisation of the network.…”

Section: Introductionmentioning

confidence: 99%

Numerical studies on the self-heating phenomenon of elastomers based on finite thermoviscoelasticity

2021

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Due to their typical material characteristics, elastomer components are used in almost all areas of engineering. In many cases, these components are subject to large cyclic deformations which result in hysteresis and dissipation-induced self-heating. Further they are exposed to varying ambient temperatures. Increased component temperatures can lead to the loss of a function or to total failure. Therefore, it is important to understand the causes and influences of critical temperatures and to identify them early in the development process under the condition of efficient applicability. In addition to the calculation time and accuracy, this also includes the experimental effort required to identify the material parameters and perform validation measurements. Within this work, the phenomenon of dissipative heating in elastomers is investigated in a numerical study using a modified model of the finite thermoviscoelasticity. For this purpose, a sufficiently simple material model was formulated and implemented under the assumption of the quasi-incompressible material behaviour. Based on this, the type and the characteristic features of the self-heating effect are specifically considered, and its dependence on thermal and mechanical initial and boundary conditions is studied. Thus, a new suitable parameter is introduced, which is particularly useful to identify critical loads. Analogously, the identification of dissipation-sensitive temperature ranges is presented. The utility of the general steadystate equilibrium condition as initial condition is also shown. Furthermore, the influence of the material properties on the steadystate equilibrium is demonstrated for the first time through parameter studies. Based on these findings, recommendations for modelling, calculation and experimental parameterisation are proposed.

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“…In these works, EM consistent integrators have been successfully implemented. The big drawback of the entropy form is that absolute temperature is preferred as a thermodynamical state variable in terms of material modeling because it can be measured directly. A Legendre transformation would be necessary to gain entropy‐based thermodynamical potentials, which may involve a Newton procedure when the expression for temperature cannot be inverted analytically, which then needs to be performed at each Gauss point.…”

Section: Introductionmentioning

confidence: 99%

An energy momentum consistent integration scheme using a polyconvexity‐based framework for nonlinear thermo‐elastodynamics

Franke

Janz

Schiebl

et al. 2018

Numerical Meth Engineering

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The present contribution provides a new approach to the design of energy momentum consistent integration schemes in the field of nonlinear thermo-elastodynamics. The method is inspired by the structure of polyconvex energy density functions and benefits from a tensor cross product that greatly simplifies the algebra. Furthermore, a temperature-based weak form is used, which facilitates the design of a structure-preserving time-stepping scheme for coupled thermoelastic problems. This approach is motivated by the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework for open systems. In contrast to complex projection-based discrete derivatives, a new form of an algorithmic stress formula is proposed. The spatial discretization relies on finite element interpolations for the displacements and the temperature. The superior performance of the proposed formulation is shown within representative quasi-static and fully transient numerical examples. KEYWORDSfinite element method, nonlinear thermo-elastodynamics, polyconvexity-based framework, structure-preserving discretization, tensor cross product INTRODUCTIONThe present contribution aims for the consistent discretization of nonlinear thermo-elastodynamics. The emphasis of this paper is on both theoretical and numerical aspects. In recent decades, thermomechanical constitutive models have been addressed in numerous works (see, eg, the works of Miehe, 1 Holzapfel and Simo, 2 and Reese and Govindjee 3 as well as textbooks by Holzapfel 4 and Gonzalez and Stuart 5 ). Classically, the hyperelastic Helmholtz free-energy density function depends only on the deformation gradient and the temperature. Furthermore, the weak form is deduced from its strong form. Dependent on the chosen material model, eg, for a Mooney-Rivlin model, the consistent linearization may lead to cumbersome expressions. In contrast to the classic approach, the present work is inspired by the concept of polyconvexity (see, eg, the works of Ball 6 and Ciarlet 7 ), where the Helmholtz free energy is a convex function of the deformation gradient, its cofactor, and its determinant and is concave with respect to absolute temperature. In addition to the polyconvexity-based framework, the present work relies on the cross product between second-order tensors, as introduced in the work of de Boer 8 (see also Appendix B 4.9.3 in another work of de Boer 9 ). This tensor cross product has been used in the context of large-strain Int J Numer Methods Eng. 2018;115:549-577. wileyonlinelibrary.com/journal/nme 549 550 FRANKE ET AL.hyperelasticity in the works of Bonet et al 10,11 and remarkably simplifies the algebraic manipulation of the large-strain continuum formulation and is extended herein to the thermomechanical formulation. Furthermore, the polyconvexity-based framework makes possible a wide range of advanced finite element technologies in the context of continuum mechanics. For example, the introduction of mixed finite elements, 10,12 phase-field fracture models, 13 anisotropic...

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A monolithic finite element approach using high-order schemes in time and space applied to finite strain thermo-viscoelasticity

Cited by 16 publications

References 51 publications

Energy‐momentum‐entropy consistent numerical methods for large‐strain thermoelasticity relying on the GENERIC formalism

Energy‐momentum‐entropy consistent numerical methods for large‐strain thermoelasticity relying on the GENERIC formalism

Numerical studies on the self-heating phenomenon of elastomers based on finite thermoviscoelasticity

An energy momentum consistent integration scheme using a polyconvexity‐based framework for nonlinear thermo‐elastodynamics

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