Formulation of Problems of Elasticity
In this chapter both the basic boundary value problems of elastostatics and initial-boundary value problems of elastodynamics are recalled; in particular, the mixed boundary value problems of isothermal and nonisothermal elastostatics, as well as the pure
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Formulation of Problems of Elasticity
In this chapter both the basic boundary value problems of elastostatics and initial-boundary value problems of elastodynamics are recalled; in particular, the mixed boundary value problems of isothermal and nonisothermal elastostatics, as well as the pure displacement and the pure stress problems of classical elastodynamics are discussed. The Betti reciprocal theorem of elastostatics and Graffi’s reciprocal theorem of elastodynamics together with the uniqueness theorems are also presented. An emphasis is made on a pure stress initial-boundary value problem of incompatible elastodynamics in which a body possesses initially distributed defects. [See also Chap. 16.]
3.1 Boundary Value Problems of Elastostatics Field Equations of Isothermal Elastostatics The strain-displacement relation = 1 (∇u + ∇uT ) E = ∇u 2
(3.1)
The equations of equilibrium div S + b = 0, S = ST
(3.2)
S = C [E]
(3.3)
The stress-strain relation
By eliminating E and S from Eqs. (3.1)–(3.3) we obtain the displacement equation of equilibrium div C [∇u] + b = 0 (3.4)
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_3, © Springer Science+Business Media Dordrecht 2013
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3 Formulation of Problems of Elasticity
For a homogeneous isotropic body the displacement equation of equilibrium (3.4) reduces to (3.5) μ ∇ 2 u + (λ + μ) ∇(div u) + b = 0 or ∇2u +
1 b ∇(div u) + = 0 1 − 2ν μ
(3.6)
or (λ + 2μ)∇(div u) − μ curl curl u + b = 0
(3.7)
An equivalent form of the stress-strain relation (3.3) reads E = K [S]
(3.8)
Therefore, by eliminating u and E from Eqs. (3.1), (3.2), and (3.8) the stress equations of equilibrium are obtained div S + b = 0, S = ST
(3.9)
curl curl K[S] = 0
(3.10)
For a homogeneous isotropic body, the stress equations of equilibrium (3.9)–(3.10) reduce to (3.11) div S + b = 0, S = ST ∇2S +
ν 1 =0 ∇∇(tr S) + (div b)1 + 2∇b 1+ν 1−ν
(3.12)
Field Equations of nonisothermal Elastostatics The strain-displacement relation = E = ∇u
1 (∇u + ∇uT ) 2
(3.13)
The equations of equilibrium div S + b = 0, S = ST
(3.14)
The stress-strain-temperature relation S = C [E] + T M
(3.15)
3.1 Boundary Value Problems of Elastostatics
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or, the strain-stress-temperature relation E = K [S] + T A
(3.16)
In Eqs. (3.15) T stands for a temperature change; while M = MT and A = AT are the stress-temperature and thermal expansion tensors, respectively. For an isotropic body Eqs. (3.15) and (3.16), respectively, take the form
and
S = 2μ E + λ (tr E) 1 − (3λ + 2μ) α T 1
(3.17)
λ 1 S− (tr S) 1 + α T 1 E= 2μ 3λ + 2μ
(3.18)
By eliminating E and S from Eqs. (3.13)–(3.15) the displacement-temperature equation of nonisothermal elastostatics is obtained div{C [∇u] + T M} + b = 0
(3.19)
Also, by eliminating u and E from Eqs. (3.13), (3.14), and (3.16), the stresstemperature equations of nonisothermal elastostatics are obtained div S + b = 0, S = ST
(3.20)
curl curl {K[S] + T A} = 0
(3.21)
For an i
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