### Chapter 5 - University of Rhode Island

```Review of Basic Field Equations
eij 
1
(ui , j  u j ,i )
2
Strain-Displacement Relations
eij ,kl  ekl ,ij  eik , jl  e jl ,ik  0 Compatibility Relations
ij , j  Fi  0
ij  ekk ij  2eij
1 

eij 
ij   kk ij
E
E
Equilibrium Equations
Hooke’s Law
15 Equations for 15 Unknowns ij , eij , ui
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Boundary Conditions
T(n)
S
S
St
Su
R
R
R
S = S t + Su
u
Traction Conditions
Displacement Conditions
Rigid-Smooth Boundary Condition Allows
Specification of Both Traction and Displacement
But Only in Orthogonal Directions
Symmetry Line
y
u0
x
Elasticity
Mixed Conditions
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
T y( n )  0
Boundary Conditions on Coordinate Surfaces
On Coordinate Surfaces the Traction Vector Reduces to Simply
Particular Stress Components
r
y
xy
r
x

r

y
r
xy
x
y
x
(Cartesian Coordinate Boundaries)
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
r

(Polar Coordinate Boundaries)
Boundary Conditions on General Surfaces
On General Non-Coordinate Surfaces, Traction Vector Will Not
Reduce to Individual Stress Components and General
Traction Vector Form Must Be Used
y
n
Tx( n )   x nx   xy n y  S cos 
Ty( n )   xy nx   y n y  S sin 
S

x
Two-Dimensional Example
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Example Boundary Conditions
Traction Free Condition
y
Fixed Condition
u=v=0
Traction Condition
Tx( n )   x  S, Ty( n )   xy  0
Traction Condition
Tx( n)   xy  0, Ty( n)   y  S
S
l
x
b
S
Tx( n )  0
Ty( n )  0
a
y
x
Traction Free Condition
Tx( n)   xy  0, Ty(n)   y  0
Coordinate Surface Boundaries
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Fixed Condition
u=v=0
Traction Free Condition
Non-Coordinate Surface
Boundary
Interface Boundary Conditions
Material (1):
 ij(1) , ui(1)
n
s
Material (2):
Embedded Fiber or Rod
Elasticity
Interface Conditions:
Perfectly Bonded,
Slip Interface, Etc.
 ij( 2) , u i( 2)
Layered Composite Plate
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Composite Cylinder or Disk
Fundamental Problem Classifications
Problem 1 (Traction Problem) Determine the distribution of
displacements, strains and stresses in the interior of an elastic
body in equilibrium when body forces are given and the
distribution of the tractions are prescribed over the surface of the
body,
(n)
(s)
(s)
Ti
T(n)
S
R
( xi )  f i ( xi )
Problem 2 (Displacement Problem) Determine the distribution of
displacements, strains and stresses in the interior of an elastic body
in equilibrium when body forces are given and the distribution of
the displacements are prescribed over the surface of the body.
S
R
ui ( xi( s ) )  g i ( xi( s ) )
Problem 3 (Mixed Problem) Determine the distribution of
displacements, strains and stresses in the interior of an elastic body
in equilibrium when body forces are given and the distribution of
the tractions are prescribed as per (5.2.1) over the surface St and
the distribution of the displacements are prescribed as per (5.2.2)
over the surface Su of the body (see Figure 5.1).
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
u
St
Su
R
Stress Formulation
Eliminate Displacements and Strains from
Fundamental Field Equation Set
(Zero Body Force Case)
Equilibrium Equations
 x  yx  zx


0
x
y
z
 xy  y  zy


0
x
y
z
 xz  yz  z


0
x
y
z
Compatibility in Terms of Stress:
Beltrami-Michell Compatibility Equations
2
(1  )  x  2 (  x   y   z )  0
x
2
2
(1  )  y  2 (  x   y   z )  0
y
2
2
(1  )  z  2 (  x   y   z )  0
z
2
2
(1  )  xy 
( x   y   z )  0
xy
2
2
(1  )  yz 
( x   y   z )  0
yz
2
2
(1  )  zx 
( x   y   z )  0
zx
2
6 Equations for 6 Unknown Stresses
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Displacement Formulation
Eliminate Stresses and Strains from
Fundamental Field Equation Set
(Zero Body Force Case)
Equilibrium Equations in Terms of Displacements:
Navier’s/Lame’s Equations
  u v w 
  
0
x  x y z 
  u v w 
0
 2 v  (  )   
y  x y z 
  u v w 
  0
 2 w  (  )   
z  x y z 
 2 u  (  )
3 Equations for 3 Unknown Displacements
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Summary of Reduction of Fundamental
Elasticity Field Equation Set
General Field Equation System
(15 Equations, 15 Unknowns:)
{ui , eij ,  ij ;  , , Fi }  0
1
(ui , j  u j ,i )
2
 ij, j  Fi  0
 ij  (    ) ekk  ij  2eij
eij,kl  ekl ,ij  eik , jl  e jl ,ik  0
eij 
Stress Formulation
Displacement Formulation
(6 Equations, 6 Unknowns:)
(3 Equations, 3 Unknowns: ui)
 ( t ) { ij ;  , , Fi }
 ij,kk
Elasticity
 ij, j  Fi  0
1


 kk ,ij  
 ij Fk ,k  Fi , j  F j ,i
1 
1 
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
 ( u ) {ui ;  , , Fi }
ui ,kk  (    )u k ,ki  Fi  0
Principle of Superposition
For a given problem domain, if the state {ij(1) , eij(1) , ui(1) } is a solution to the fundamental
elasticity equations with prescribed body forces Fi (1) and surface tractions Ti (1) , and the
state {ij( 2) , eij( 2) , ui( 2) } is a solution to the fundamental equations with prescribed body
forces Fi ( 2) and surface tractions Ti ( 2) , then the state {ij(1)  ij( 2) , eij(1)  eij( 2) , ui(1)  ui( 2) }
will be a solution to the problem with body forces Fi (1)  Fi ( 2) and surface tractions
Ti (1)  Ti ( 2) .
(1)+(2)
=
(1)
+
(2)
{ij(1) , eij(1) , ui(1) }
{ij(1)  ij( 2) , eij(1)  eij( 2) , ui(1)  ui( 2) }
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
{ij( 2) , eij( 2) , ui( 2) }
Saint-Venant’s Principle
The stress, strain and displacement fields due to two different
statically equivalent force distributions on parts of the body far
FR
P
(1)
P P
2 2
P P P
3 3 3
(2)
(3)
Stresses approximately the same
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
T(n)
S*
detailed and characteristic effects only in
vicinity of S*. Away from S* stresses would
generally depend more on resultant FR of
tractions rather than on exact distribution
General Solution Strategies Used to
Solve Elasticity Field Equations
Direct Method - Solution of field equations by direct integration. Boundary conditions
are satisfied exactly. Method normally encounters significant mathematical difficulties
thus limiting its application to problems with simple geometry.
Inverse Method - Displacements or stresses are selected that satisfy field equations. A
search is then conducted to identify a specific problem that would be solved by this
solution field. This amounts to determine appropriate problem geometry, boundary
conditions and body forces that would enable the solution to satisfy all conditions on the
problem. It is sometimes difficult to construct solutions to a specific problem of practical
interest.
Semi-Inverse Method - Part of displacement and/or stress field is specified, while the
other remaining portion is determined by the fundamental field equations (normally
using direct integration) and boundary conditions. Constructing appropriate
displacement and/or stress solution fields can often be guided by approximate strength of
materials theory. Usefulness of this approach is greatly enhanced by employing SaintVenant’s principle, whereby a complicated boundary condition can be replaced by a
simpler statically equivalent distribution.
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Mathematical Techniques Used to
Solve Elasticity Field Equations
Analytical Solution Procedures
- Power Series Method
- Fourier Method
- Integral Transform Method
- Complex Variable Method
Approximate Solution Procedures
- Ritz Method
Numerical Solution Procedures
- Finite Difference Method (FDM)
- Finite Element Method (FEM)
- Boundary Element Method (BEM)
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
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