4.8 Existence and Uniqueness of Solution (Uniqueness Theorem)
117
∂σ
x
∂ x
+
∂τ
xy
∂ y
+
∂τ
xz
∂z
+ x = 0; (x, y, z)
and
σ
x l + τ
xy m + τ
xz n = F x ; (x, y, z)
where (x, y, z) means that there are two more equations obtained by changing the
suffixes y for x and z for y, in a cyclic order.
Similarly, if there is another set of stresses σ
x , σ
y , . . . .τ
zx which also satisfies the
boundary conditions and governing equations we have,
∂σ
x
∂ x
+
∂τ
xy
∂ y
+
∂τ
xz
∂z
+ x = 0; (x, y, z)
and
σ
x l + τ
xy m + τ
xz n = F x ; (x, y, z)
By subtracting the equations of the above set from the corresponding equations
of the previous set, we get the following set,
∂
∂ x
σ
x − σ
x
+
∂
∂ y
τ
xy − τ
xy
+
∂
∂z
τ
xz − τ
xz
= 0; (x, y, z)
and
σ
x − σ
x
l +
τ
xy − τ
xy
m +
τ
xz − τ
xz
n = 0; (x, y, z)
In the same way, it is shown that the new strain components (ε
x −ε
x ), (ε
y −ε
y ) . . . .,
etc. also satisfy the equations of compatibility. A new solution (σ
x − σ
x ), (σ
x −
σ
y ), . . . ..(τ
xz − τ
xz ) represents a situation where body forces and surface forces
both are zero. The work done by these forces during loading is zero, and it follows
that the total strain energy vanishes, i.e.
˚
V o dxdydz = 0
where
V o =
σ x ε x + σ y ε y + σ z ε z + τ xy γ xy + τ yz γ yz + τ zx γ zx
The strain energy per unit volume V o is always positive for any combination of
strains and stresses. Hence for the integral to be zero, V o must vanish at all the points,