Bifurcation Problems in Constrained Nonlinear Thermoelasticity

  • M. C. Calderer
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 3)

Abstract

I study the problem of inflation under constant pressure of nonlinear ther-moelastic and isotropic spherical shells. The boundary of the shell is kept insulated and the material of the body is supposed to satisfy the internal constraint,
$$ \det \,{\text{F(X,t) = f(}}\tau {\text{(X,t))}} $$
(1.1)
where F denotes the gradient of deformation matrix at the particle X at time t > 0 and τ > 0 denotes the absolute temperature. Moreover, f(x) in (1.1) is assumed to be smooth and to satisfy the following conditions:
$$ {\text{f}}({\tau_0}) = 1,\,{\text{f'}}(\tau)>0,\, $$
(1.2a,b)
where τ0 denotes a constant references temperathure. According to (1.2a), the constraint expressed by equation (1.1) reduces to incompressibility when thermal effects are ignored and the temperature is taken to be τ0 When such conditions are met, we will refer to the problem as ‘purely mechanical’. Another special situation arises when τ̂ is set to be equal to a constant τ̂ in the governing equations. The corresponding problem will be called ‘isothermal’. In this case, the body experiences an increase in volume given by equation (1.1).

Keywords

Entropy Crystallization Rubber Congo Incompressibility 

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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • M. C. Calderer
    • 1
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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