One-Dimensional Solutions of Elastodynamics
In this chapter a number of typical one-dimensional initial-boundary value problems of homogeneous isotropic isothermal and nonisothermal elastodynamics are solved in a closed-form using the Laplace transform technique. The isothermal solutions include: (
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One-Dimensional Solutions of Elastodynamics
In this chapter a number of typical one-dimensional initial-boundary value problems of homogeneous isotropic isothermal and nonisothermal elastodynamics are solved in a closed-form using the Laplace transform technique. The isothermal solutions include: (a) one-dimensional displacement waves in a semispace subject to a uniform dynamic boundary pressure, (b) one-dimensional displacement waves in a semispace subject to the initial disturbances, and (c) one-dimensional stress waves in an infinite space composed of two homogeneous isotropic elastic semispaces of different material properties. The nonisothermal solutions include: (i) one-dimensional dynamic thermal stresses produced by a plane source of heat that varies harmonically with time in an infinite elastic solid, (ii) one-dimensional dynamic thermal stresses produced by a plane nucleus of thermoelastic strain in an infinite elastic solid, and (iii) one-dimensional dynamic thermal stresses in a semispace due to the action of a plane internal nucleus of thermoelastic strain.
12.1 One-Dimensional Field Equations of Isothermal Elastodynamics The one-dimensional field equations of isothermal elastodynamics describe an elastic process p = [u, E, S] that depends on a single space variable x = x1 and on time t. Such a process corresponds to the data that depend on x and t only. In the following we let u(x, t) = [u(x, t), 0, 0] (12.1) and b(x, t) = [b(x, t), 0, 0]
(12.2)
where b = b(x, t) is the body force vector field.
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_12, © Springer Science+Business Media Dordrecht 2013
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12 One-Dimensional Solutions of Elastodynamics
The strain tensor E has only one component: E 11 = E 11 (x, t), and the straindisplacement relation reads ∂u (12.3) E 11 = ∂x The equation of motion takes the form ∂ S11 ∂ 2u +b =ρ 2 ∂x ∂t
(12.4)
∂u ∂x
(12.5)
The constitutive relation reads S11 = (λ + 2μ)
provided the body is homogeneous and isotropic. By letting S11 = S(x, t) and eliminating E 11 = E 11 (x, t) and S = S(x, t) from Eqs. (12.3)–(12.5), the one-dimensional displacement equation of motion is obtained
∂2 1 ∂2 − ∂x2 c2 ∂t 2
where c=
u=−
b λ + 2μ
λ + 2μ ρ
(12.6)
(12.7)
Also, by eliminating u = u(x, t) and E 11 = E 11 (x, t) from Eqs. (12.3)–(12.5), the one-dimensional stress equation of motion is obtained
∂2 1 ∂2 − 2 2 2 ∂x c ∂t
S=−
∂b ∂x
(12.8)
12.2 One-Dimensional Field Equations of Nonisothermal Elastodynamics A one-dimensional thermoelastic process p = [u, E, S] can be associated with a pair (, T ) in which = (x, t) represents a thermoelastic displacement potential and T = T (x, t) is a temperature field. In a dimensionless setting, the displacement u = u(x, t) and the stress S = S(x, t), respectively, are computed from the formulas u(x, t) =
∂Φ ∂x
(12.9)
12.2 One-Dimensional Field Equations of Nonisothermal Elastodynamics
and S(x, t) = where
and
∂2
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