Solutions to Particular Two-Dimensional Initial-Boundary Value Problems of Elastodynamics
The particular solutions discussed in this chapter include: (i) dynamic thermal stresses in an infinite elastic sheet subject to a discontinuous temperature field, and (ii) dynamic thermal stresses produced by a concentrated heat source in an infinite ela
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Solutions to Particular Two-Dimensional Initial-Boundary Value Problems of Elastodynamics
The particular solutions discussed in this chapter include: (i) dynamic thermal stresses in an infinite elastic sheet subject to a discontinuous temperature field, and (ii) dynamic thermal stresses produced by a concentrated heat source in an infinite elastic body subject to plane strain conditions.
11.1 Dynamic Thermal Stresses in an Infinite Elastic Body Under Plane Strain Conditions Subject to a Time-Dependent Temperature Field The dynamic thermal stresses in a homogeneous isotropic infinite elastic body under plane strain conditions and initially at rest, and subject to a time-dependent temperature field T = T (x, t) are computed from the formulas ¨ αβ on E 2 × [0, ∞) Sαβ = 2μ (φ,αβ − φ,γ γ δαβ ) + ρ φδ
(11.1)
where φ = φ(x, t) satisfies the field equation 1 ∂2 2 ∇ − 2 2 φ = m T on E 2 × [0, ∞) c ∂t
(11.2)
subject to the homogeneous initial conditions
where
˙ 0) = 0 for x ∈ E 2 φ(x, 0) = φ(x,
(11.3)
1 1+ν ρ and m = α = c2 λ + 2μ 1−ν
(11.4)
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_11, © Springer Science+Business Media Dordrecht 2013
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11 Solutions to Particular Two-Dimensional Initial-Boundary Value Problems
11.2 Dynamic Thermal Stresses in an Infinite Elastic Body Under Generalized Plane Stress Conditions Subject to a Time-Dependent Temperature Field The dynamic thermal stresses in a homogeneous isotropic infinite elastic body under generalized plane stress conditions are computed from the formulas similar to those of a plane strain state. The stresses S αβ = S αβ (x, t) produced by a temperature T = T (x, t) on E 2 ×[0, ∞) and corresponding to a body initially at rest are given by ¨ 2 S αβ = 2μ (φ ,αβ − φ ,γ γ δαβ ) + ρ φδ αβ on E × [0, ∞)
(11.5)
where φ = φ(x, t) satisfies the field equation 1 ∂2 2 ∇ − 2 2 φ = m T on E 2 × [0, ∞) c ∂t
(11.6)
subject to the homogeneous initial conditions ˙ φ(x, 0) = φ(x, 0) = 0 for x ∈ E 2 where
1
=
c2
ρ λ + 2μ
, λ=
2μλ λ + 2μ
(11.7)
(11.8)
and m = (1 + ν)α
(11.9)
11.3 Problems and Solutions Related to Particular Two-Dimensional Initial-Boundary Value Problems of Elastodynamics Problem 11.1. Find the dynamic thermal stresses in an infinite elastic sheet with a ∗ ∗ quiescent past subject to the temperature T = T (x, t) of the form ∗
T (x, t) = T0 δ(x1 )δ(x2 )δ(t) where T0 is a constant temperature and δ = δ(x) is the delta function. Solution. The dynamic thermal stresses in a homogeneous isotropic infinite sheet initially at rest are computed from the formulas [see Eqs. (11.5)–(11.9) with c ≡ c and m ≡ m = (1 + ν)α]
11.3
Problems and Solutions Related to Particular Two-Dimensional Initial-Boundary
¨ 2 S αβ = 2 μ(φ,αβ − ∇ 2 φδαβ ) + ρ φδ αβ on E × [0, ∞)
261
(11.10)
where φ = φ(x, t) satisfies the field equation 1 ∂2 ∇ 2 − 2 2 φ = mT 0 δ(x1 ) δ(x2 ) δ(t) on E 2 × (0, ∞) c ∂t
(11.11)
subject to the homogeneous initial conditions ˙ φ(x,
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