Instytut Podstawowych Problemów Techniki

Streszczenie:

In the present paper, the stress characterization of elastodynamics for a rotating inhomogeneous transversely isotropic cylinder under plane strain conditions is proposed. It is assumed that the geometrical axis of the cylinder coincides with the axis of rotational symmetry of the cylinder. The cylinder rotates together with the Cartesian coordinate system xi (i=1,2,3), in which the geometrical axis of the cylinder coincides with the x3-axis, with a uniform angular velocity Ω in such a way that the acceleration of the cylinder is a sum of three components: (i) classical acceleration, (ii) centripetal acceleration, and (iii) Coriolis acceleration. It is shown that the propagation of an elastic wave in the 3D rotating cylinder can be described by a solution to the associated 2D pure stress initial-boundary value problem. Such a reduction of the 3D problem to the 2D one is based on the theorem on an alternative representation of the displacement vector field u in terms of the stress field S. An example of a complete pure stress formulation of the traction initial-boundary value problem is presented.

Słowa kluczowe:

stress language of elastodynamics, rotating cylinder, classical acceleration, centripetal acceleration, Coriolis acceleration, transversely isotropic inhomogeneous elastic materials, completeness of stress formulation, natural stress traction initial-boundary value problem

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Streszczenie:

Stress-heat flux characterization of linear dynamic coupled thermoelasticityfor an inhomogeneous isotropic infinite cylinder under plane strain conditions and zero heat-flux normal to the plane is presented. It is shown that for a bounded cross-section of the cylinder, a 3D stress-heat flux process is generated by a 2D one, and a uniqueness theorem for the associated 2D initial-boundary value problem is established. In addition, an asymptotic approach to the 2D stress-heat flux initial-boundary value problem, in the form of a power series with respect to a small thermoelastic coupling field, is proposed. Also, Green' s formulas for time-periodic complex-valued solutions to 2D stress-heat flux field equations are obtained; and the existence of a globally constrained real-valued periodic and attenuated on the timeaxis stress-heat flux mode satisfying homogeneous natural boundary conditions is proved. The stress-heat flux characterization covers a large class of FGM's with physical properties smoothly distributed over the cross-section of the infinite cylinder. The results obtained are complementary to those of linear dynamic coupled thermoelasticity published up to date and should prove useful for a number of researchers in the field.

Słowa kluczowe:

existence of a stress-heat flux mode, Green's formulas, isotropic inhomogeneous media, linear dynamic coupled thermoelasticity, stressheat flux characterization, stress-heat flux field equations of linear thermo-elastodynamics, uniqueness theorem

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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

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Streszczenie:

Stress characterization of isothermal elastodynamics for an inhomogeneous transversely isotropic infinite cylinder under plane strain conditions is presented. The cylinder, referred to the Cartesian coordinates xi (i = 1,2,3) and denoted here by View the MathML sourceBi(3), is identified with an infinite solid cylinder of which the rotational symmetry axis coincides with the x3-axis, and the geometrical axis coincides with the xi-axis (i = 1,2,3); and the plane strain conditions exist in the plane xi = 0, (i = 1,2,3). A cross-section of the cylinder View the MathML sourceBi(3) with the plane xi = 0 is denoted by View the MathML sourceCi(3). It is shown that a 3D stress wave Sij (i,j = 1,2,3) propagating in the cylinder View the MathML sourceBi(3) is generated by a solution to a 2D pure stress initial-boundary value problem for View the MathML sourceCi(3), and a uniqueness theorem for the 2D problem is established. In particular, a pure stress initial boundary value problem for View the MathML sourceC3(3) involving only two (out of five) elastic moduli: c11 and c12 is formulated, and it is shown that the problem accommodates two types of the surface stress wave problems for a transversely isotropic semi-space with a traction free boundary and with an inhomogeneity depending on its depth. The first type is obtained when View the MathML sourceC3(3) is the semi-space: |x1|<∞, 0
The results obtained should prove useful for both the analytical and numerical studies of the surface stress waves in an inhomogeneous transversely isotropic elastic semi-space under plane strain conditions.

Słowa kluczowe:

Stress characterization, Isothermal elastodynamics, Transversely isotropic inhomogeneous media, Functionally graded materials (FGM), Plane strain stress equations of motion, Stress-rate energy of linear elastodynamics

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Streszczenie:

Stress characterization of non-isothermal elastodynamics for an anisotropic nonhomogeneous infinite cylinder under plane strain conditions is presented. The cylinder is referred to the Cartesian coordinates x i (i = 1, 2, 3) in which the axis of the cylinder is parallel to the x 3-axis and a cross-section of the cylinder at x 3 = 0, denoted by C, is a domain of the time-dependent stresses S ij = S ij (x α, t), [i, j = 1, 2, 3; α = 1, 2; x α ∈ C; t ≥ 0]. The density of the cylinder ρ, the compliance tensor K ijkl [i, j, k, l = 1, 2, 3], and the stress-temperature tensor M ij depend on x 2 only, while a thermomechanical load that complies with the plane strain conditions, depends on (x 1, x 2) ∈ C and time t ≥ 0 only. It is shown that S ij = S ij (x α, t) is generated by a unique solution S αβ = S αβ(x γ, t), [α, β, γ = 1, 2; t ≥ 0] to a pure stress initial-boundary value problem of nonisothermal elastodynamics on C × [0, ∞), and the in-plane stress components generate the out-of plane stress components provided the inner product of a compliance dependent tensor field and the tensor does not vanish. Also, a body-force analogy for S αβ = S αβ(x γ, t) is formulated from which it follows that S αβ = S αβ(x γ, t) can be identified with a solution to a pure stress initial-boundary value problem of isothermal elastodynamics. The stress characterization presented here should prove useful in a study of stress waves in an infinite cylinder made of an anisotropic functionally graded material within both the isothermal and non-isothermal elastodynamics.

Słowa kluczowe:

Anisotropic and nonhomogeneous bodies, Body force analogy for transient thermal stresses, Functionally graded materials (FGM), Non-isothermal elastodynamics, Plane strain stress equations of motion, Stress characterization, Time-dependent actuation tensor field

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Słowa kluczowe:

Heat transfer, Metal films, Plane progressive heat waves, Third-order derivative-in-time heat conduction equation

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Słowa kluczowe:

Dispersion, Dissipation, Heat transfer, Metal films, Third-order derivative-in-time heat conduction equation, Wave equation

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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

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Streszczenie:

A nonlinear rigid heat conductor obeying the first and second laws of thermodynamics, Cattaneo's law, and a generalized energy-entropy relation in which both the energy and entropy are parabolic functions of the heat flux, is revisited. For a one-dimensional Cauchy problem in which both the temperature and heat flux are time-dependent only, a solution in terms of elementary functions is obtained. Also, for a one-dimensional traveling wave problem, a solution in terms of elementary functions is presented. Graphs illustrating the solutions are included.

Słowa kluczowe:

Closed-form solutions, Heat conductor, Hyperbolic, Nonlinear, Rigid

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Streszczenie:

The second law of thermodynamics asserts that heat will always flow “downhill”, i.e., from an object having a higher temperature to one having a lower temperature. For a parabolic rigid heat conductor with a single temperature T and a single heat-flux q this amounts to the statement that the inner product of q and ∇T must be non-positive for every point x of the conductor and for every non-negative time t. For a homogeneous and isotropic body in which classical Fourier law with a heat conductivity coefficient k is postulated, the second law is satisfied if k is a positive parameter. For ultra-fast pulse-laser heating on metal films, a parabolic two-temperature model coupling an electron temperature Te with a metal lattice temperature Tl has been proposed by several authors. For such a model, at a given point of space x and a given time t there are two different temperatures Te and Tl as well as two different heat-fluxes q e and q l related to the gradients of Te and Tl, respectively, through classical Fourier law. As a result, for a homogeneous and isotropic model the positive definiteness of the heat conductivity coefficients ke and kl corresponding to Te and Tl, respectively, implies that the second law of thermodynamics is satisfied for each of the pairs (Te, q e) and (Tl, q l), separately. Also, the positive definiteness of ke and kl, and of the corresponding heat capacities ce and cl as well as of a coupling factor G imply that a temperature initial-boundary value problem for the two-temperature model has unique solution. In the present paper, an alternative form of the second law of thermodynamics for the two-temperature model with kl = 0 and q l = 0 is obtained from which it follows that in a one-dimensional case the electron heat-flux qe(x, t) has direction that is opposite not only to that of ∂Te(x, t)/∂x but also to that of ∂Tl(x, t + τT)/∂x, where τT is an intrinsic small time of the model. Also, for a general two-temperature rigid heat conductor in which ke, kl, ce, cl, and G are positive, an inequality of the second law of thermodynamics type involving a pair (Te − Tl, q e − q l) is postulated to prove that a two-heat-flux initial-boundary value problem of the two-temperature model has a unique solution. For a one-dimensional case, the semi-infinite sectors of the plane ( q l, q e) over which uniqueness does not hold true are also revealed.

Słowa kluczowe:

Heat conduction, Ultra-fast heating, Metal films, Parabolic two-temperature model, Second law of thermodynamics

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Słowa kluczowe:

harmonic thermoelastic waves, microperiodic composites

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Streszczenie:

A Saint-Venant's principle associated with a one-dimensional dynamic coupled ther moelastic effective modulus theory for a microperiodic layered semispace is presented. In such a theory, the displacement u=u(x,t) and the temperature theta=theta(x,t) (x>=0,t>=0) are approximated by u(x,t)=U(x,t)+h(x)V(x,t) and theta(x,t)=THETA(x,t)+ h(x)PHI(x,t), where U(x,t) and PHI(x,t) represent a macrodisplacement and a macrotemperature, respectively; V(x,t) and PHI(x,t)denote a displacement corrector and a temperature corrector, respectively; h=h(x) is a prescribed periodic microshape function; and the pairs (U,THETA) and (V,PHI) are found by solving an initial boundary value problem described by a system of linear partial differential equations with effective thermoelastic moduli subject to suitable initial and boundary conditions. It is shown that the thermoelastic energy associated with a solution to the problem and stored in the semi-space lying beyond a distance x from the loaded boundary x=0 over the time interval [0,t] decays exponentially as x to infinity and its decay length L depends on the time t, an effective velocity of thermoelastic wave (c*l), an effective time (T*), and an effective thermoelastic coupling parameter (epsilon*). In particular, it is shown that for small (large) times the function L reveals behavior of the decay length for a pure thermal (elastic) energy of a semispace.

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Streszczenie:

A refined averaged theory of a rigid heat conductor with a microperiodic structure is used to solve a one-dimensional initial boundary value problem ofheat conduction in a periodically layeredplate with a largenumber of homogeneous isotropic layers. A uniqueness theorem for the averaged problem is proved, and two closed-form solutions for a periodically layered semispace are obtained. One of the two solutions represents the temperature field in the layered semispace due to a sudden heating of the boundary plane, while the other stands for the temperaturefield in the layeredsemispace produced by laser surface heating. Numerical examples are included.

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: