Formulation of Two-Dimensional Problems
In this chapter a class of problems is discussed in which an elastic state depends on two space variables only, or an elastic process depends on two space variables and time only. In particular, problems related to a plane strain state and a generalized p
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Formulation of Two-Dimensional Problems
In this chapter a class of problems is discussed in which an elastic state depends on two space variables only, or an elastic process depends on two space variables and time only. In particular, problems related to a plane strain state and a generalized plane stress state of homogeneous isotropic elastostatics, a plane strain process, and a generalized plane stress process of homogeneous isotropic elastodynamics are discussed. The problems related to a two-dimensional homogeneous isotropic elastodynamics described in terms of stresses only are also considered. [See also Chaps. 16 and 17].
7.1 Two-Dimensional Problems of Isothermal Elastostatics A state of plane strain. An elastic body is said to be in a state of plane strain corresponding to a body force b = (b1 , b2 , 0) if the elastic state s = [u, E, S] complies with the two-dimensional field equations u α = u α (x1 , x2 ) for (x1 , x2 ) ∈ C0
(7.1)
1 (u α,β + u β,α ) on C0 2
(7.2)
Sαβ,β + bα = 0 on C0
(7.3)
Sαβ = 2μ E αβ + λ E γ γ δαβ on C0
(7.4)
E αβ =
In Eqs. (7.1)–(7.4), and in all plane problems, the Greek subscripts α, β, and γ take values 1 and 2; and C0 is a domain in the x1 , x2 plane. In Eq. (7.3) bα = bα (x1 , x2 ), and the remaining components of s = [u, E, S] are given by
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_7, © Springer Science+Business Media Dordrecht 2013
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7 Formulation of Two-Dimensional Problems
u 3 = 0,
E 13 = E 23 = E 33 = 0 on C0
(7.5)
S13 = S23 = 0, S33 = ν Sαα on C0
(7.6)
An alternative form of the constitutive relation (7.4) is E αβ =
1 (Sαβ − ν Sγ γ δαβ ) on C0 2μ
(7.7)
By eliminating the fields E αβ and Sαβ from Eqs. (7.2)–(7.4) we obtain the displacement field equations for a plane strain problem μ u α,γ γ + (λ + μ)u γ ,γ α + bα = 0 on C0
(7.8)
A generalized plane stress state. A generalized plane stress state s = [u, E, S] corresponding to a body force b = (b1 , b2 , 0) is defined as an elastic state which complies with the two-dimensional field equations u α = u α (x1 , x2 ) for (x1 , x2 ) ∈ C0 1 E αβ = (u α,β + u β,α ) on C0 2 S αβ,β + bα = 0 on C0
(7.10)
S αβ = 2μ E αβ + λ E γ γ δαβ on C0
(7.12)
(7.9)
(7.11)
The remaining components of s = [u, E, S] are given by u 3 = 0,
E 13 = E 23 = 0,
S 13 = S 23 = S 33 In Eq. (7.12) λ=
λ E γ γ on C0 λ + 2μ = 0 on C0
E 33 = −
2μ λ λ + 2μ
(7.13) (7.14)
(7.15)
and an alternative form of (7.12) reads E αβ =
1 (S αβ − ν S γ γ δαβ ) on C0 2μ
(7.16)
By eliminating the fields E αβ and S αβ from Eqs. (7.10)–(7.12) we obtain the displacement equations for a body subject to generalized plane stress conditions μ u α,γ γ + (λ + μ) u γ ,γ α + bα = 0 on C0
(7.17)
7.2 Two-Dimensional Problems of Nonisothermal Elastostatics
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7.2 Two-Dimensional Problems of Nonisothermal Elastostatics A nonisothermal plane strain state in the x1 , x2 plane corresponding to zero body forces and a temperature change T = T (x1 , x2 )
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