178
2 Variational Problems with Fixed Boundaries
After management, the above equations can be turned into
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u +
1 + μ
1 − μ
(u xx + v xy + w xz ) = 0
v +
1 + μ
1 − μ
(u xy + v yy + w yz ) = 0
w +
1 + μ
1 − μ
(u xz + v yz + w zz ) = 0
Example 2.9.2 According to the theory of mechanics of elasticity, the strain energy
expressed with displacement components for three-dimensional strain problem of
the linear elastic material is
J =
E
2(1 + μ)
˚
V
μ
1 − 2μ
(u x + v y + w z )
2
+ u
2
x + v
2
y + w
2
z +
1
2
(v x + u y )
2
+
1
2
(w y + v z )
2
+
1
2
(u z + w x )
2
dxdydz
where, E is the modulus of elasticity for the material; μ is Poisson’s ratio, both are
constants. Find the Euler equations of the functional.
Solution This problem is also equivalent to m = l = 3, S 1 = S 2 = S 3 = 1,
n 1 = n 2 = n 3 = 1. The Euler equations of the functional are
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
2μ
1 − 2μ
(u xx + v xy + w xz ) + 2u xx + (v xy + u yy ) + (u zz + w xz ) = 0
2μ
1 − 2μ
(u xy + v yy + w yz ) + 2v yy + (v xx + u xy ) + (w yz + v zz ) = 0
2μ
1 − 2μ
(u xz + v yz + w zz ) + 2w zz + (w yy + v yz ) + (u xz + w xx ) = 0
After management, the above equations can be changed into
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u +
1
1 − 2μ
(u xx + v xy + w xz ) = 0
v +
1
1 − 2μ
(u xy + v yy + w yz ) = 0
w +
1
1 − 2μ
(u xz + v yz + w zz ) = 0
If μ of Example 2.9.1 is changed into
μ
1−μ
, and substituting it into the Euler
equations of Example 2.9.1, it can be changed into the Euler equations of the threedimensional strain problem of Example 2.9.2. Similarly, if μ of Example 2.9.2 is
changed into
μ
1+μ
, and substituting it into the Euler equations of Example 2.9.2, the
it can be changed into the Euler equations of the three-dimensional stress problem of
Example 2.9.1. These two examples demonstrate the validity of the complete Euler
equations.
2 Variational Problems with Fixed Boundaries
After management, the above equations can be turned into
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u +
1 + μ
1 − μ
(u xx + v xy + w xz ) = 0
v +
1 + μ
1 − μ
(u xy + v yy + w yz ) = 0
w +
1 + μ
1 − μ
(u xz + v yz + w zz ) = 0
Example 2.9.2 According to the theory of mechanics of elasticity, the strain energy
expressed with displacement components for three-dimensional strain problem of
the linear elastic material is
J =
E
2(1 + μ)
˚
V
μ
1 − 2μ
(u x + v y + w z )
2
+ u
2
x + v
2
y + w
2
z +
1
2
(v x + u y )
2
+
1
2
(w y + v z )
2
+
1
2
(u z + w x )
2
dxdydz
where, E is the modulus of elasticity for the material; μ is Poisson’s ratio, both are
constants. Find the Euler equations of the functional.
Solution This problem is also equivalent to m = l = 3, S 1 = S 2 = S 3 = 1,
n 1 = n 2 = n 3 = 1. The Euler equations of the functional are
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
2μ
1 − 2μ
(u xx + v xy + w xz ) + 2u xx + (v xy + u yy ) + (u zz + w xz ) = 0
2μ
1 − 2μ
(u xy + v yy + w yz ) + 2v yy + (v xx + u xy ) + (w yz + v zz ) = 0
2μ
1 − 2μ
(u xz + v yz + w zz ) + 2w zz + (w yy + v yz ) + (u xz + w xx ) = 0
After management, the above equations can be changed into
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
u +
1
1 − 2μ
(u xx + v xy + w xz ) = 0
v +
1
1 − 2μ
(u xy + v yy + w yz ) = 0
w +
1
1 − 2μ
(u xz + v yz + w zz ) = 0
If μ of Example 2.9.1 is changed into
μ
1−μ
, and substituting it into the Euler
equations of Example 2.9.1, it can be changed into the Euler equations of the threedimensional strain problem of Example 2.9.2. Similarly, if μ of Example 2.9.2 is
changed into
μ
1+μ
, and substituting it into the Euler equations of Example 2.9.2, the
it can be changed into the Euler equations of the three-dimensional stress problem of
Example 2.9.1. These two examples demonstrate the validity of the complete Euler
equations.