Variational Formulation of Elastostatics
In this chapter the variational characterizations of a solution to a boundary value problem of elastostatics are recalled. They include the principle of minimum potential energy, the principle of minimum complementary energy, the Hu-Washizu principle, and
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Variational Formulation of Elastostatics
In this chapter the variational characterizations of a solution to a boundary value problem of elastostatics are recalled. They include the principle of minimum potential energy, the principle of minimum complementary energy, the Hu-Washizu principle, and the compatibility related principle for a traction problem. The variational principles are then used to solve typical problems of elastostatics.
4.1 Minimum Principles To formulate the Principle of Minimum Potential Energy we recall the concept of the strain energy, of the stress energy, and of a kinematically admissible state. By the strain energy of a body B we mean the integral U C E
1 2
E CE dv
(4.1)
B
and by the stress energy of a body B we mean UK S
1 2
S KS dv
(4.2)
B
Since S CE, therefore, UK S UC E By a kinematically admissible state we mean a state s
(4.3) u, E, S
that satisfies
(1) the strain-displacement relation
u E
1 ( u uT ) on B 2
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_4, © Springer Science+Business Media Dordrecht 2013
(4.4) 103
104
4 Variational Formulation of Elastostatics
(2) the stress-strain relation S C E on B
(4.5)
(3) the displacement boundary condition u on ∂B1 u
(4.6)
u is prescribed on ∂B1 . where The Principle of Minimum Potential Energy is related to a mixed boundary value problem of elastostatics [see Chap. 3 on Formulation of Problems of Elasticity]. The Principle of Minimum Potential Energy Let R be the set of all kinematically admissible states. Define a functional F on R by F s UC E
u,
F .
(4.7)
∂B2
B
for every s Then
s u da
b u dv
E, S R. Let s be a solution to the mixed problem of elastostatics. F s F s for every s R
(4.8)
and the equality holds true if s and s differ by a rigid displacement. u in ( 4.7) an alternative form of the Principle of Minimum By letting E Potential Energy is obtained. Let R1 denote a set of displacement fields that satisfy the boundary conditions ( 4.6), and define a functional F1 . on R1 by F1 u
1 2
( u) C u dv B
b u dv
s u da
u
R1
(4.9)
∂B2
B
If u corresponds to a solution to the mixed problem, then F1 u F1 u
u
R1
(4.10)
To formulate the Principle of Minimum Complementary Energy, we introduce a concept of a statically admissible stress field. By such a field we mean a symmetric second-order tensor field S that satisfies (1) the equation of equilibrium div S b 0 on B
(4.11)
(2) the traction boundary condition s on ∂B2 Sn
(4.12)
4.1 Minimum Principles
105
The Principle of Minimum Complementary Energy Let P denote a set of all statically admissible stress fields, and let G functional on P defined by s u da
G S UK S
S
G . be a
P
(4.13)
∂B1
If S is a stress field corresponding to a solution to the mixed problem, then G S G S
S
P
(4.14)
and the equality holds if S S.
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