Three-dimensional elasticity problems are difficult to solve. Thus we first
develop governing equations for two-dimensional problems, and explore
four different theories:
- Plane Strain
- Plane Stress
- Generalized Plane Stress
- Anti-Plane Strain
Since all real elastic structures are three-dimensional, theories set forth
here will be approximate models. The nature and accuracy of the
approximation will depend on problem and loading geometry.
The basic theories of plane strain and plane stress represent the
fundamental plane problem in elasticity. While these two theories apply
to significantly different types of two-dimensional bodies, their formulations
yield very similar field equations.
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Two vs Three Dimensional Problems
Three-Dimensional
Two-Dimensional
x
y
y
z
z
z
Spherical Cavity
y
x
Elasticity
x
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Plane Strain
Consider an infinitely long cylindrical (prismatic) body as shown. If the body
forces and tractions on lateral boundaries are independent of the z-coordinate
and have no z-component, then the deformation field can be taken in the
reduced form
u  u ( x, y ) , v  v ( x, y ) , w  0
y
x
R
z
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Plane Strain Field Equations
Strains ex 
u
v
1  u v 
, ey 
, exy     , ez  exz  e yz  0
x
y
2  y x 
 x  (ex  e y )  2ex ,  y  (ex  e y )  2e y
Stresses
 z   (e x  e y )   ( x   y )
 xy  2exy ,  xz   yz  0
Equilibrium Equations
 x  xy

 Fx  0
x
y
 xy  y

 Fy  0
x
y
Navier Equations
Elasticity
 2 u  (   )
  u v 
    Fx  0
x  x y 
 2 v  (   )
  u v 
    Fy  0
y  x y 
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Strain Compatibility
2
 2 e xy
 2 ex  e y

2
y 2
x 2
xy
Beltrami-Michell Equation
2 ( x   y )  
1  Fx Fy 



1    x
y 
Examples of Plane Strain Problems
y
P
z
x
x
y
z
Long Cylinders
Under Uniform Loading
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Semi-Infinite Regions
Under Uniform Loadings
Plane Stress
Consider the domain bounded two stress free planes z = h, where h is
small in comparison to other dimensions in the problem. Since the region is
thin in the z-direction, there can be little variation in the stress components
 z ,  xz ,  yz through the thickness, and thus they will be approximately zero
throughout the entire domain. Finally since the region is thin in the zdirection it can be argued that the other non-zero stresses will have little
variation with z. Under these assumptions, the stress field can be taken as
y
 x   x ( x, y )
2h
 y   y ( x, y )
 xy   xy ( x, y )
 z   xz   yz  0
R
z
x
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Plane Stress Field Equations
Strains
1
1
( x   y ) , e y  ( y   x )
E
E


e z   ( x   y )  
(e x  e y )
E
1 
1 
exy 
 xy , exz  e yz  0
E
ex 
Equilibrium Equations
 x  xy

 Fx  0
x
y
 xy  y

 Fy  0
x
y
Navier Equations
Elasticity
 2 u 
E
  u v 
    Fx  0
2(1  ) x  x y 
 2 v 
E
  u v 
    Fy  0
2(1  ) y  x y 
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Strain Displacement Relations
ex 
u
v
w
1  u v 
, ey 
, ez 
, exy    
x
y
z
2  y x 
1  v w 
1  u w 
  0 , exz   
e yz   
0
2  z y 
2  z x 
Strain Compatibility
2
 2 e xy
 2 ex  e y

2
y 2
x 2
xy
Beltrami-Michell Equation
 F Fy 

2 ( x   y )  (1  ) x 

x

y


Examples of Plane Stress Problems
Thin Plate With
Central Hole
Circular Plate Under
Edge Loadings
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Plane Elasticity Boundary Value Problem
Displacement Boundary Conditions
u  ub ( x, y ) , v  vb ( x, y ) on S
Si
So
Stress/Traction Boundary Conditions
Txn  Tx( b ) ( x, y )   (xb ) n x   (xyb ) n y
T T
n
y
(b)
y
( x, y )   n x   n y
(b)
xy
(b)
y
on S
y
R
S = S i + So
x
Plane Strain Problem - Determine inplane displacements, strains and stresses
{u, v, ex , ey , exy , x , y , xy} in R. Out-ofplane stress z can be determined from
in-plane stresses via relation (7.1.3)3.
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Plane Stress Problem - Determine inplane displacements, strains and stresses
{u, v, ex , ey , exy , x , y , xy} in R. Out-ofplane strain ez can be determined from
in-plane strains via relation (7.2.2)3.
Correspondence Between Plane Formulations
Plane Strain
Plane Stress
 2 u  (   )
  u v 
    Fx  0
x  x y 
 2 u 
E
  u v 
    Fx  0
2(1  ) x  x y 
 2 v  (   )
  u v 
    Fy  0
y  x y 
 2 v 
E
  u v 
    Fy  0
2(1  ) y  x y 
 x  xy

 Fx  0
x
y
 xy  y

 Fy  0
x
y
2 ( x   y )  
Elasticity
1  Fx Fy 



1    x
y 
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
 x  xy

 Fx  0
x
y
 xy  y

 Fy  0
x
y
 F Fy 

2 ( x   y )  (1  ) x 
y 
 x
Transformation Between
Plane Strain and Plane Stress
Plane strain and plane stress field equations had identical equilibrium equations
and boundary conditions. Navier’s equations and compatibility relations were
similar but not identical with differences occurring only in particular coefficients
involving just elastic constants. So perhaps a simple change in elastic moduli
would bring one set of relations into an exact match with the corresponding
result from the other plane theory. This in fact can be done using results in the
following table.
Plane Stress to Plane Strain
Plane Strain to Plane Stress
E

E
1  2
E (1  2)
(1  ) 2

1 

1 
Therefore the solution to one plane problem also yields the solution to the other
plane problem through this simple transformation scheme.
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Generalized Plane Stress
The plane stress formulation produced some inconsistencies in particular outof-plane behavior and resulted in some three-dimensional effects where inplane displacements were functions of z. We avoided these issues by simply
neglecting some of the troublesome equations thereby producing an
approximate elasticity formulation. In order to avoid this unpleasant situation,
an alternate approach called Generalized Plane Stress can be constructed
based on averaging the field quantities through the thickness of the domain.
Using the averaging operator defined by  ( x, y ) 
1 h
( x, y , z )dz


h
2h
all plane stress equations are satisfied exactly by the averaged stress, strain and
displacements variables; thereby eliminating the inconsistencies found in the
original plane stress formulation. However, this gain in rigor does not generally
contribute much to applications .
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Anti-Plane Strain
An additional plane theory of elasticity called Anti-Plane Strain involves a
formulation based on the existence of only out-of-plane deformation starting
with an assumed displacement field
u  v  0 , w  w( x, y )
Strains
e x  e y  ez  e xy  0
e xz 
Elasticity
1 w
1 w
, e yz 
2 x
2 y
Stresses
 x   y   z   xy  0
 xz  2exz ,  yz  2e yz
Equilibrium Equations
Navier’s Equation
 xz  yz

 Fz  0
x
y
Fx  Fy  0
 2 w  Fz  0
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Airy Stress Function Method
Numerous solutions to plane strain and plane stress problems can be determined
using an Airy Stress Function technique. The method reduces the general
formulation to a single governing equation in terms of a single unknown. The
resulting equation is then solvable by several methods of applied mathematics,
and thus many analytical solutions to problems of interest can be found.
This scheme is based on the general idea of developing a representation for the
stress field that will automatically satisfy equilibrium by using the relations
 2
 2
 2
 x  2 ,  y  2 ,  xy  
y
x
xy
where  = (x,y) is an arbitrary form called Airy’s stress function. It is easily
shown that this form satisfies equilibrium (zero body force case) and
substituting it into the compatibility equations gives
 4
 4
 4
 2 2 2  4  4   0
4
x
x y
y
This relation is called the biharmonic equation and its solutions are known as
biharmonic functions.
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Airy Stress Function Formulation
The plane problem of elasticity can be reduced to a single equation in terms
of the Airy stress function. This function is to be determined in the twodimensional region R bounded by the boundary S as shown. Appropriate
boundary conditions over S are necessary to complete the solution.
Traction boundary conditions would involve the specification of second
derivatives of the stress function; however, this condition can be reduced to
specification of first order derivatives.
 4
 4
 4
 2 2 2  4  4   0
4
x
x y
y
(n)
x
T
 2
 2
  x nx   xy n y  2 nx 
ny
y
xy
T
(n)
y
y
 2
 2
  xy nx   y n y  
nx  2 n y
xy
x
Si
R
S = S i + So
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
y
xy
x
x
Elasticity
So
Polar Coordinate Formulation
Plane Elasticity Problem
Strain-Displacement
u r
r
u 
1
e   u r   
r
 
1  1 u r u u 
er  

 
2  r 
r
r 
er 
Hooke’s Law
Plane Stress
Plane Strain
 r   ( er  e )  2er
    ( er  e )  2e
 z   ( e r  e )  (  r    )
 r  2er ,  z   rz  0
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
1
(  r   )
E
1
e  (    r )
E


ez   ( r    )  
( e r  e )
E
1 
1 
er 
 r , ez  erz  0
E
er 
Polar Coordinate Formulation
Navier’s Equations
Plane Strain
 2 ur  (  )
  u r u r 1 u 
 

  Fr  0
r  r
r r  
 2 u  (  )
1   u r u r 1 u 
 

  F  0
r   r
r r  
Equilibrium Equations
 r 1  r (  r    )


 Fr  0
r r 
r
 r 1   2 r


 F  0
r
r 
r
Compatibility Equations
Plane Stress
E
  u r u r 1 u 
 

  Fr  0
2(1  ) r  r
r r  
E
1   u r u r 1 u 
 2 u 
 

  F  0
2(1  ) r   r
r r  
 2 ur 
2 1 
1 2
  2 

r
r r r 2 2
2
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Plane Strain
1  Fr Fr 1 F 
 


1    r
r r  
Plane Stress
 2 ( r    )  
 F F 1 F 
 2 ( r    )  (1  ) r  r 

r r  
 r
Polar Coordinate Formulation
Airy Stress Function Approach  = (r,θ)
Airy Representation
1  1  2 
r 

r r r 2  2
 2
  2
r
  1  
 r   

r  r  
Biharmonic Governing Equation
 2 1 
1  2   2 1 
1 2 
    2 
 2 2  2 
 2 2   0

r
r

r
r



r
r

r
r  


4
S
r
r
R
y
Traction Boundary Conditions
Tr  f r ( r, ) , T  f  ( r, )


r

x
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island