Instytut Podstawowych Problemów Techniki

Streszczenie:

A one-dimensional nonlinear homogeneous isotropic thermoelastic model with an elastic heat flow at low temperatures and small strains is analyzed using the method of weakly nonlinear asymptotics. For such a model, both the free energy and the heat flux vector depend not only on the absolute temperature and strain tensor but also on an elastic heat flow that satisfies an evolution equation. The governing equations are reduced to a matrix partial differential equations, and the associated Cauchy problem with a weakly perturbed initial condition is solved. The solution is given in the form of a power series with respect to a small parameter, the coefficients of which are functions of a slow variable that satisfy a system of nonlinear second-order ordinary differential transport equations. A family of closed-form solutions to the transport equations is obtained. For a particular Cauchy problem in which the initial data are generated by a closed-form solution to the transport equations, the asymptotic solution in the form of a sum of four traveling thermoelastic waves admitting blow-up amplitudes is presented.

Słowa kluczowe:

low temperatures, nonlinear thermoelasticity, small strains, weakly nonlinear asymptotics

Afiliacje autorów:

Streszczenie:

A one-dimensional nonlinear hyperbolic homogeneous isotropic rigid heat conductor proposed by Coleman is analyzed using the method of weakly nonlinear geometric optics. For such a model the law of conservation of energy, the dissipation inequality, the Cattaneo's equation, and a generalized energy-entropy relation with a parabolic variation of the energy and entropy along the heat-flux axis, are postulated. First, it is shown that the model can be described by a non-homogeneous quasi-linear hyperbolic matrix partial differential equation of the first order for an unknown vector u = (θ, Q) T, where θ and Q are the dimensionless absolute temperature and heat-flux fields, respectively. Next, the Cauchy problem for the matrix equation with a weakly perturbed initial condition is formulated, and an asymptotic solution to the problem in terms of the amplitudes σα (α = 1, 2) that satisfy a pair of nonlinear first order partial differential equations, is obtained. The Cauchy problem is then solved in a closed form when the initial data are suitably restricted. Numerical examples are included.

Słowa kluczowe:

Asymptotic methods, Blow-up heat waves, Coleman heat conductor, Hyperbolic, Nonlinear geometrical optics, Rigid heat conductor

Afiliacje autorów:

Streszczenie:

A one -dimensional nonlinear homogeneous isotropic thermo-elastic model with an elastic heat flow at low temperatures and small strains is analyzed using the method of weakly nonlinear asymptotics. For such a model both the free energy and the heat flux vector depend not only on the absolute temperature and strain tensor but also on an elastic heat flow that satisfies an evolution equation. The governing equations are reduce d to a matrix PDE, and the associated Cauchy problem with a weakly perturbed initial condition is solved. The solution is given in the form of a power series with respect to a small parameter the coefficients of which are functions of a slow variable that satisfy a system of nonlinear second-order ordinary differential transport equations. For a particular Cauchy problem in which the initial data are generated by a closed-form solution to the transport equations, the principal part of the asymptotic solution is a sum of four travelling thermo-elastic waves admitting blow-up amplitudes.

Słowa kluczowe:

nonlinear thermo-elasticity, low temperatures, small strains, weakly nonlinear asymptotics

Afiliacje autorów: