Thermo‐visco‐elasticity for Norton‐Hoff‐type models with Cosserat effects

Klawe, Filip Z.; Owczarek, Sebastian

doi:10.1002/mma.3890

Cited by 5 publications

(6 citation statements)

References 27 publications

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“…Norton-Hoff or Norton-Hoff-type models were studied e.g. in [15,16,22,29] The function σ : Ω × R + → S 3 is the Cauchy stress tensor. It may be divided into two parts: mechanical (elastic) and thermal one.…”

Section: Introductionmentioning

confidence: 99%

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Thermo-visco-elasticity for Norton–Hoff-type models with homogeneous thermal expansion

Gwiazda

Klawe

Owczarek

2018

Nonlinear Analysis: Real World Applications

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In this work we study a quasi-static evolution of thermo-visco-elastic model with homogeneous thermal expansion. We assume that material is subject to two kinds of mechanical deformations: elastic and inelastic. Inelastic deformation is related to a hardening rule of Norton-Hoff type. Appearance of inelastic deformation causes transformation of mechanical energy into thermal one, hence we also take into the consideration changes of material's temperature.The novelty of this paper is to take into account the thermal expansion of material. We are proposing linearisation of the model for homogeneous thermal expansion, which preserves symmetry of system and therefore total energy is conserved. Linearisation of material's thermal expansion is performed in definition of Cauchy stress tensor and in heat equation. In previous studies, it was done in different way. Considering of such linearisation leads to system where the coupling between temperature and displacement occurs in two places, i.e. in the constitutive function for the evolution of visco-elastic strain and in the additional term in the heat equation, in comparison to models without thermal expansion. The second coupling was not considered previously. For such system of equations we prove the existence of solutions. Moreover, we obtain existence of displacement's time derivative, which has not been done previously.2000 Mathematics Subject Classification. 74C10, 35Q74, 74F05.

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Section: Introductionmentioning

confidence: 99%

“…This method was previously used for continuum mechanic models, e.g. see [21,22,28,29], or for models describing fluid motion, see [12,13]. The main reason to use two level approximation here is low regularity of right-hand side of heat equation.…”

Section: Introductionmentioning

confidence: 99%

Thermo-visco-elasticity for Norton–Hoff-type models with homogeneous thermal expansion

Gwiazda

Klawe

Owczarek

2018

Nonlinear Analysis: Real World Applications

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“…It is worth emphasizing the works of Gwiazda and colleagues and of Klawe and colleagues [22][23][24][25], in which the authors deal with similar types of thermo-visco-elasticity systems. In the problems considered, thermal expansion does not occur, which means that the Cauchy stress tensor does not depend on the temperature function.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

“…It is worth to emphasize the works [29,30,33] and [34], where the authors deal with similar type of thermo-visco-elasticity systems. In considered problems the thermal expansion does not appear, which means that the Cauchy stress tensor does not depend on temperature function.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

On renormalized solutions for thermomechanical problems in perfect plasticity with damping forces

Bartczak

Owczarek

2018

Mathematics and Mechanics of Solids

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We consider the quasi-static evolution of the thermo-plasticity model in which the evolution equation law for the inelastic strain is given by the Prandtl-Reuss flow rule. The thermal part of the Cauchy stress tensor is not linearised in the neighbourhood of a references temperature. This nonlinear thermal part imposed to add a damping term to the balance of the momentum, which can be interpreted as external forces acting on the material.In general the dissipation term occurring in the heat equation is integrable function only and the standard methods can not be applied. Combining truncation techniques and Boccardo-Gallouët approach with monotone methods we prove an existence of renormalised solutions.

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“…It was recently found in Fischle et al that the model with μ c = 0 is capable of producing nonclassical rotation patterns that are at least similar to observations in nanoindentation experiments. Another reason for interest in the model lies in the fact that it provides a regularisation device, in which case μ c is regarded as a penalty parameter …”

Section: Introductionmentioning

confidence: 99%