Influence of Geometric Equations in Mixed Problem of Porous Micromorphic Bodies with Microtemperature
Abstract
:1. Introduction
2. Notations and Basic Equations
- -
- the geometric equations, representing the strain tensors respectively the vector in compliance with [14],
- -
- the equations of motion
- -
- the balance of equilibrated forces
- -
- the energy equation, in the form of the following relation
- -
- the additional energy equations, as a result of the existence of the microtemperatures, introduced by the next relations, according to [40]
- -
- are the stress tensor components, the microstress tensor components and the stress moment tensor components,
- is the reference mass density,
- are the components of the body force vector,
- are the body moment tensor components,
- are the components of the microinertia tensor, endowed with the property of symmetry
- are the equilibrated stress vector components related to the voids,
- represents the intrinsic equilibrated body force,
- ℓ represents the extrinsic equilibrated body force corresponding to the voids,
- k is the equilibrated inertia coefficient,
- represents the internal rate of entropy production per mass unit,
- are the entropy flux vector components,
- are the mean entropy flux vector components,
- represents the specific entropy per mass unit,
- s represents the external rate of entropy supply,
- are the first entropy moment vector components,
- are the first entropy flux moment tensor components,
- are first moment of the external rate of entropy supply.
3. Results
3.1. Auxiliary Results
- (i)
- (ii)
- the quadratic form expressed through the relation (13), is positively semi-defined;
- (iii)
- the conductivity coefficients and the constitutive ones are the components of positively defined tensors, i.e., and such that
3.2. Main Results
3.2.1. Uniqueness
3.2.2. Existence
3.2.3. Continuous Dependence of the Solution
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Codarcea-Munteanu, L.; Marin, M. Influence of Geometric Equations in Mixed Problem of Porous Micromorphic Bodies with Microtemperature. Mathematics 2020, 8, 1386. https://doi.org/10.3390/math8081386
Codarcea-Munteanu L, Marin M. Influence of Geometric Equations in Mixed Problem of Porous Micromorphic Bodies with Microtemperature. Mathematics. 2020; 8(8):1386. https://doi.org/10.3390/math8081386
Chicago/Turabian StyleCodarcea-Munteanu, Lavinia, and Marin Marin. 2020. "Influence of Geometric Equations in Mixed Problem of Porous Micromorphic Bodies with Microtemperature" Mathematics 8, no. 8: 1386. https://doi.org/10.3390/math8081386