Work Done by Surface and Body Forces on
Elastic Solids Is Stored Inside the Body in
the Form of Strain Energy
Tn
U
F
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Uniaxial Extensional Deformation
u
u
dx
x

y

dy
dz
dx
u
x
x
z
dU  
x
0
d (u 
u

 ex  x
x
E

x
x
u
u
dx )dydz   dudydz   d ( )dxdydz
0
0
x
x
dU  
x
0
d
2x
 dxdydz
dxdydz
E
2E
 2x Eex2 1
dU
U


  x ex . . . Strain Energy Density
dxdydz 2 E
2
2
Elasticity
U = Area
Under Curve
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
ex
e
Shear Deformation
y
u
dy
y
yx
xy
dy
dx
x
v
dx
x

xy
2xy  2xy
1
U   xy  xy 

2
2
2
U = Area
Under Curve
xy
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island

General Deformation Case
U
1
1
(  x e x   y e y   z ez   xy  xy   yz  yz   zx  zx )   ij eij
2
2
T otalStrain Energy UT   Udxdydz
V
In Terms of Strain
1
U (e )  e jj ekk  eij eij
2
1
1
1
1
 (ex  e y  ez ) 2  (ex2  e 2y  ez2   2xy   2yz   2zx )
2
2
2
2
In Terms of Stress
1 

ij ij 
 jj  kk
2E
2E
1  2


( x   2y   2z  2 2xy  2 2yz  2 2zx ) 
( x   y   z ) 2
2E
2E
U (σ ) 
Note Strain Energy Is Positive Definite Quadratic Form
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
U 0
Example Problem
y
P
2c
x
L
Stress Field from Exercise 8 - 2 :
3P
3P  y 2 
1   ,  y   z   yz   zx  0
 x   3 xy ,  xy  
2c
4c  c 2 
1  2
 2
1 2 1  2
 x  2 2xy 
x 
x 
 xy
2E
2E
2E
E
1 c
L 1
1  2 
U T  UdV     
 2x 
 xy dxdydz
0 c 0
E
 2E


U

c

L
c 0

 1 2 1  2 
x 
 xy dxdy

2
E
E


2
1 c L 9P 2 2 2
1   c L 9P 2  y 2 
1   dxdy

x y dxdy 
2 E c 0 4c 6
E c 0 16c 2  c 2 
P 2 L2 9 P 2 L(1  )


4 Ec 3
Ec
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Derivative Operations on Strain Energy
For the Uniaxial Deformation Case:
U (e )


ex
ex
 Eex2 

  Eex   x
2


U (σ )


 x
 x
  2x   x

 
 ex
2
E
E


For the General Deformation Case:
ij 
U (e )
U (σ )
, eij 
eij
ij
ij
ekl
eij
 kl


 kl
eij
ekl
ij
Cijkl  Cklij
Therefore Cij = Cji, and thus there are only 21 independent
elastic constants for general anisotropic elastic materials
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Decomposition of Strain Energy
Strain Energy May Be Decomposed into Two Parts Associated With
Volumetric Deformation Uv , and Distortional Deformation, Ud
U  Uv  Ud
Uv 
Ud 
1~ ~
1
1  2
1  2
 ij eij   jj ekk 
 jj  kk 
( x   y   z ) 2
2
6
6E
6E
1
[( x   y ) 2  ( y   z ) 2  ( z   x ) 2  6(  2xy   2yz   2zx )]
12
Failure Theories of Solids Incorporate Strain Energy of
Distortion by Proposing That Material Failure or Yielding Will
Initiate When Ud Reaches a Critical Value
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Bounds on Elastic Constants
Simple Tension
Pure Shear
Hydrostatic
Compression
0 
 p 0
 ij   0  p 0    p ij


0  p 
 0
  0 0
 ij   0 0 0


 0 0 0
0  0
 ij    0 0


0 0 0
1  2
 2 2
U
 
 
2E
2E
2E
1 
2
2
U
(2 )  (1  )
2E
E
E0
Elasticity
1    0    1
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
U
1  2

3p2
3p 
( 3 p ) 2 
(1  2)
2E
2E
2E
1  2  0   
1   
1
2
1
2
k 0 ,0
Related Integral Theorems
Clapeyron’s Theorem The strain energy of an elastic solid in static equilibrium is equal to
n
one-half the work done by the external body forces Fi and surface tractions Ti
2 UdV   Ti n ui dS   Fi ui dV
V
S
V
Betti/Rayleigh Reciprocal Theorem If an elastic body is subject to two body and surface
force systems, then the work done by the first system of forces {T(1), F(1) } acting through the
displacements u(2) of the second system is equal to the work done by the second system of
forces {T(2), F(2) } acting through the displacements u(1) of the first system
T
S
i
(1)
ui( 2) dS   Fi (1) ui( 2) dV  Ti ( 2) ui(1) dS   Fi ( 2) ui(1) dV
V
S
V
Integral Formulation of Elasticity - Somigliana’s Identity Represents an integral statement
of the elasticity problem. Result is used in development of boundary integral equation methods
(BIE) in elasticity, and leads to computational technique of boundary element methods (BEM)
cu j ()   [Ti ( x)Gij ( x, )  ui Tikj ( x, )nk ]dS   Fi Gij ( x, )dV
S
V
Gij is the displacement Green’s function to the elasticity equations and
Tijk (x, )  Glk,l ij  (Gik , j  G jk,i )
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
 1,  in V
1
c   ,  on S
2
0 ,  outsideV