Complete Solutions of Elasticity
In this chapter general solutions of the homogeneous isotropic elastostatics and elastodynamics are discussed. The general solutions are related to both the displacement and stress governing equations, and emphasis is made on completeness of the solutions
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Complete Solutions of Elasticity
In this chapter general solutions of the homogeneous isotropic elastostatics and elastodynamics are discussed. The general solutions are related to both the displacement and stress governing equations, and emphasis is made on completeness of the solutions [See also Chap. 16].
6.1 Complete Solutions of Elastostatics A vector field u
u(x) on B that satisfies the displacement equation of equilibrium
2 u 1 1 2ν (div u) μb
0
(6.1)
is called an elastic displacement field corresponding to b. Boussinesq-Papkovitch-Neuber (B-P-N) Solution. Let u
ψ
4(1 1 ν) (x ψ ϕ)
(6.2)
where ϕ and ψ are fields on B that satisfy Poisson’s equations
and
2 ψ μ1 b
(6.3)
1 bx μ
(6.4)
2ϕ
Then u is an elastic displacement field corresponding to b.
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_6, © Springer Science+Business Media Dordrecht 2013
151
152
6 Complete Solutions of Elasticity
Boussinesq-Somigliana-Galerkin (B-S-G) Solution. Let u be a vector field given by
2 g 2(1 1 ν) (div g)
u where
2 2 g μ1 b
(6.5)
(6.6)
Then u is an elastic displacement field corresponding to b. We say that a representation for the displacement u expressed in terms of auxiliary functions is complete if these auxiliary functions exist for any u that satisfies the displacement equation of equilibrium (6.1). For B-P-N solution such auxiliary functions are the fields ϕ and ψ; while for B-S-G solution an auxiliary function is the field g. Completeness of B-P-N and B-S-G Solutions. Let u be a solution to the displacement equation of equilibrium with the body force b. Then there exists a field g on B that satisfies Eqs. (6.5)–(6.6). Also, there exist fields ϕ and ψ that satisfy Eqs. (6.2)– (6.4). B-P-N solution for axial symmetry. For an axially symmetric problem with b 0 in which x3 z is the axis symmetry of a body, the displacement vector field u u(r, z) referred to the cylindrical coordinates (r, θ , z) takes the form ψk
u
1
4(1 ν)
where
(zψ ϕ)
xk
z
(6.7)
(6.8)
with k being a unit vector along the x3 axis, and with scalar-valued harmonic functions ϕ ϕ(r, z) and ψ ψ(r, z). In components we obtain u where ur
uz and
with
ψ
ur (r, z), 0, uz (r, z)
(6.9)
4(1 1 ν) ∂r∂ (zψ ϕ)
(6.10)
4(1 1 ν) ∂z∂ (zψ ϕ)
2ϕ
0,
2ψ
0
(6.11)
(6.12)
6.1 Complete Solutions of Elastostatics
153
∂2 ∂r 2
2
1r ∂r∂ ∂z∂ 2 2
(6.13)
0 and g χ k is also called Love’s B-S-G solution for axial symmetry with b solution. 1 (k χ ) (6.14) u ( 2 χ )k 2(1 ν) where
22χ
0
(6.15)
In cylindrical coordinates (r, θ, z) ∂ 2(1 1 ν) ∂r∂z χ
(6.16)
0
(6.17)
2
ur
uθ
2(1 ν) 2
1
2(1 ν)
uz
∂2 χ ∂z2
(6.18)
6.2 Complete Solutions of Elastodynamics The displacement equation of motion for a homogeneous isotropic elastic body takes the form c1 2 b 2 2 u 1 (div u) 0 (6.19) c2 μ where 22
2 c12 ∂t∂ 2 , 2
2
1 c21
ρ
λ 2μ
,
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