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This is the
reciprocal theorem of Betti
2
.
It can be stated as the following:
If a linearly elastic body is subjected to two force systems, the work done by the first
system of forces in moving through the displacements produced by the second system of
forces is equal to the work performed by the second system of forces in moving through
the displacements due to the first system of forces.
A theory of elasticity form of Betti's theorem is also derived in a straightforward manner.
Because of the already complex index notation of the theory of elasticity, we choose to
abandon the 1 and 2 subscripts used above and to replace them with no index for the first
system and with a superscript
∗
for the second system. Begin with the equilibrium condition
σ
+
p
Vi
=
0in
V
(3.26)
ij, j
Follow the reasoning of Chapter 2, Section 2.2.1, and form the global condition
u
i
V
(σ
ij, j
+
p
Vi
)
dV
=
0
(3.27)
This process can be considered as having weighted the equilibrium equations for one set of
forces with displacements
u
i
due to a second set of forces and having orthogonalized the
product. From Gauss' integral theorem (Appendix II),
V
(σ
p
j
u
j
dS
ij
a
i
u
j
dS
ij
u
j
)
ij
u
j
)
=
S
σ
=
S
(σ
a
i
dS
=
,i
dV
(3.28)
S
Since (Chapter 2, Problem 2.13)
σ
ij
u
i, j
=
σ
ij
ij
(3.29)
ij
dV
ij
u
i, j
dV
ij
u
i
)
ij, j
u
i
dV
V
σ
=
V
σ
=
V
(σ
,j
dV
−
V
σ
(3.30)
ij
From Eqs. (3.28) and (3.30) ,
ij
dV
ij, j
u
i
p
j
u
j
dS
V
σ
+
V
σ
dV
=
(3.31)
ij
S
Since, from Eq. (3.26),
σ
ij, j
=−
p
Vi
Eq. (3.31) becomes
V
σ
ij
ij
dV
u
i
p
j
u
j
dS
−
+
p
Vi
dV
=−
(3.32)
V
S
Use
§
σ
ij
ij
=
σ
ij
ij
(3.33)
2
Enrico Betti (1823-1892) was an Italian mathematician who for many years was a professor of mathematical
physics at the University of Pisa. He made major contributions to algebra, topology, and elasticity. Betti's reciprocal
theorem appeared in 1872.
§
To show that
σ
ij
ij
=
σ
ij
ij
,
note that
σ
T
∗
is a scalar quantity, so that
σ
T
∗
=
(
σ
T
∗
)
T
=
∗
T
σ
=
∗
T
E
For a symmetric
E
,
∗
T
E
=
∗
T
E
T
=
(
E
∗
)
T
=
σ
∗
T
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