Multi-scale biothermal and biomechanical behaviours of biological materials

Xu, Feng; Lu, Tianjian

doi:10.1098/rsta.2009.0249

Cited by 3 publications

(3 citation statements)

References 8 publications

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“…if and only if δ y Ω{y} exists and is equal to zero for all y consistent with (38). Now, we will establish a variational principle on the linear theory of thermoelasticity for an anisotropic and inhomogeneous medium under the present heat conduction model by given by Leseduarte and Quintanilla [40] on the basis of the alternative formulation and the theorem established in the previous section.…”

Section: Variational Theoremmentioning

confidence: 91%

“…We consider a functional Ω defined on Y . Let for y, y ∈ Y , y + λ y ∈ Y , for all real λ (38) and we define the variation of the functional Ω{y} as…”

Section: Variational Theoremmentioning

confidence: 99%

“…The mathematical consequences of such theories do not agree with what one would expect a priori [26]. Subsequently, a great interest has been developed to study the different Taylor approximations to these heat conduction equations where continuous dependence and also stability can be obtained [27][28][29][30][31][32][33][34][35][36][37][38]. Recently, Quintanilla [39] has reformulated the three-phase-lag model in an alternative way and investigated the stability and spatial behavior of the new proposed model.…”

Section: Introductionmentioning

confidence: 99%

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Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay

Kumari

Mukhopadhyay

2016

Mathematics and Mechanics of Solids

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The present work is concerned with a very recently proposed heat conduction model—an exact heat conduction model with a delay term for an anisotropic and inhomogeneous material—and some important theorems within this theory. A generalized thermoelasticity theory was proposed based on the heat conduction law with three phase-lag effects for the purpose of considering the delayed responses in time due to the micro-structural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in the Hadamard sense. Subsequently, a proposal was made to reformulate this constitutive equation of heat conduction theory with a single delay term and the spatial behavior of the solutions for this theory have been investigated. A Phragmen–Lindelof type alternative was obtained and it has been shown that the solutions either decay in an exponential way or blow-up at infinity in an exponential way. The obtained results are extended to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction to the delay term and a Phragmen–Lindelof type alternative was obtained for the forward and backward in time equations. In the present work, we consider the basic equations concerning this new theory of thermoelasticity for an anisotropic and inhomogeneous material and make an attempt to establish some important theorems in this context. A uniqueness theorem has been established for an anisotropic body. An alternative characterization of the mixed initial-boundary value problem is formulated and a variational principle as well as a reciprocity principle is established.

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Section: Variational Theoremmentioning

confidence: 91%

“…We consider a functional Ω defined on Y . Let for y, y ∈ Y , y + λ y ∈ Y , for all real λ (38) and we define the variation of the functional Ω{y} as…”

Section: Variational Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay

Kumari

Mukhopadhyay

2016

Mathematics and Mechanics of Solids

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Some solutions for a family of exact phase-lag heat conduction problems

Quintanilla

2011

Mechanics Research Communications

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Phragmén-Lindelöf alternative for an exact heat conduction equation with delay

Leseduarte¹,

Quintanilla²

2013

Communications on Pure &Amp; Applied Analysis

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In this paper we investigate the spatial behavior of the solutions for\ud a theory for the heat conduction with one delay term. We obtain a Phragm én-\ud Lindelöf type alternative. That is, the solutions either decay in an exponential\ud way or blow-up at in nity in an exponential way. We also show how to obtain\ud an upper bound for the amplitude term. Later we point out how to extend\ud the results to a thermoelastic problem. We nish the paper by considering\ud the equation obtained by the Taylor approximation to the delay term. A\ud Phragm én-Lindelöf type alternative is obtained for the forward and backward\ud in time equations.Peer ReviewedPostprint (published version

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Multi-scale biothermal and biomechanical behaviours of biological materials

Cited by 3 publications

References 8 publications

Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay

Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay

Some solutions for a family of exact phase-lag heat conduction problems

Phragmén-Lindelöf alternative for an exact heat conduction equation with delay

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