3.7 Boundary Value Problems
147
The boundary conditions consist of specified tractions on S σ and displacements
on S u , with S = S σ + S u being the total surface of the body. For a problem
formulated in terms of Cauchy stress, these conditions can be written as
On S σ :
n · σ = ¯
T
On S u :
u = ¯
u,
(3.250)
where ¯
T and ¯
u are prescribed Cauchy stress and displacement vectors. The stress
boundary condition is based on Eq. (3.112); equivalent conditions can be written
in terms of Piola-Kirchhoff stresses via Eqs. (3.115). In time-dependent problems,
initial conditions also must be added.
The next chapter deals with solving some fundamental problems in nonlinear
elasticity.
Example 3.22 Consider a rectangular block composed of incompressible, isotropic
material with strain-energy density function given by Eq. (3.217). If the block is free
of all loads, determine the Lagrange multiplier p.
Solution
With I 1 defined by (3.73), Eq. (3.217) becomes
W = c 1
λ
2
1 + λ
2
2 + λ
2
3 − 3
relative to coordinate axes X i oriented normal to the faces of the block. Since the
block is unloaded, all stresses are zero and all stretch ratios are unity. Thus, with
J = 1 for an incompressible material, Eq. (3.249) 1 yields
σ i = λ i
∂W
∂λ i
− p = 0,
where i is not summed. Solving for p and setting λ i = 1 give
p = λ i
∂W
∂λ i
= 2c 1 λ
2
i = 2c 1 .
147
The boundary conditions consist of specified tractions on S σ and displacements
on S u , with S = S σ + S u being the total surface of the body. For a problem
formulated in terms of Cauchy stress, these conditions can be written as
On S σ :
n · σ = ¯
T
On S u :
u = ¯
u,
(3.250)
where ¯
T and ¯
u are prescribed Cauchy stress and displacement vectors. The stress
boundary condition is based on Eq. (3.112); equivalent conditions can be written
in terms of Piola-Kirchhoff stresses via Eqs. (3.115). In time-dependent problems,
initial conditions also must be added.
The next chapter deals with solving some fundamental problems in nonlinear
elasticity.
Example 3.22 Consider a rectangular block composed of incompressible, isotropic
material with strain-energy density function given by Eq. (3.217). If the block is free
of all loads, determine the Lagrange multiplier p.
Solution
With I 1 defined by (3.73), Eq. (3.217) becomes
W = c 1
λ
2
1 + λ
2
2 + λ
2
3 − 3
relative to coordinate axes X i oriented normal to the faces of the block. Since the
block is unloaded, all stresses are zero and all stretch ratios are unity. Thus, with
J = 1 for an incompressible material, Eq. (3.249) 1 yields
σ i = λ i
∂W
∂λ i
− p = 0,
where i is not summed. Solving for p and setting λ i = 1 give
p = λ i
∂W
∂λ i
= 2c 1 λ
2
i = 2c 1 .