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MCE 571 Theory of Elasticity I
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Chapter 5 Formulation and Solution Strategies
Review of Basic Field Equations
)(2
1,, ijjiij uue
0,,,, ikjljlikijklklij eeee
0, ijij F
ijkkijij
ijijkkij
EEe
ee
1
2
Compatibility Relations
Strain-Displacement Relations
Equilibrium Equations
Hooke’s Law
15 Equations for 15 Unknowns ij , eij , ui
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Boundary Conditions
Mixed Conditions
Symmetry LineRigid-Smooth Boundary Condition Allows Specification of Both Traction and Displacement But Only in Orthogonal Directions
R
S
R
Su
St
R
S
u
Traction Conditions Displacement Conditions
T(n)
0
0)(
nyT
u
x
y
S = St + Su
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Boundary Conditions on Coordinate SurfacesOn Coordinate Surfaces the Traction Vector Reduces to Simply
Particular Stress Components
(Cartesian Coordinate Boundaries) (Polar Coordinate Boundaries)
r
r
r
r
rx
xy
y
x
y x
xy
y
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Boundary Conditions on General SurfacesOn General Non-Coordinate Surfaces, Traction Vector Will Not Reduce
to Individual Stress Components and General Traction Vector Form Must Be Used
sin
cos)(
)(
SnnT
SnnT
yyxxyny
yxyxxnx
Two-Dimensional Example
x
S
y
n
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Example Boundary Conditions
Fixed Condition u = v = 0
Traction Free Condition
0,0 )()( ynyxy
nx TT
x
y
a
b S
Traction Condition
0, )()( xynyx
nx TST
Coordinate Surface Boundaries
Traction Free Condition
0)( nyT
x
y
l
0)( nxT
Fixed Condition u = v = 0
Traction Condition
STT ynyxy
nx )()( ,0
Non-Coordinate Surface Boundary
Traction Free Condition
S
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Interface Boundary Conditions
Embedded Fiber or Rod Layered Composite Plate Composite Cylinder or Disk
Material (1): )1()1( , iij u
)2()2( , iij uMaterial (2):
Interface Conditions: Perfectly Bonded, Slip Interface, Etc.
n
s
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Fundamental Problem ClassificationsProblem 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,
)()( )()()( sii
si
ni xfxT
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.
)()( )()( sii
sii xgxu
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).
R
ST(n)
R
S
u
R
Su
St
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Stress FormulationEliminate Displacements and Strains from
Fundamental Field Equation Set(Zero Body Force Case)
0
0
0
zyx
zyx
zyx
zyzxz
zyyxy
zxyxx
0)()1(
0)()1(
0)()1(
0)()1(
0)()1(
0)()1(
22
22
22
2
22
2
22
2
22
zyxzx
zyxyz
zyxxy
zyxz
zyxy
zyxx
xz
zy
yx
z
y
x
Equilibrium EquationsCompatibility in Terms of Stress: Beltrami-Michell Compatibility Equations
6 Equations for 6 Unknown Stresses
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Displacement FormulationEliminate Stresses and Strains from
Fundamental Field Equation Set(Zero Body Force Case)
Equilibrium Equations in Terms of Displacements:Navier’s/Lame’s Equations
0)(
0)(
0)(
2
2
2
z
w
y
v
x
u
zw
z
w
y
v
x
u
yv
z
w
y
v
x
u
xu
3 Equations for 3 Unknown Displacements
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Summary of Reduction of Fundamental Elasticity Field Equation Set
General Field Equation System(15 Equations, 15 Unknowns:)
Stress Formulation(6 Equations, 6 Unknowns:)
Displacement Formulation(3 Equations, 3 Unknowns: ui)
0},,;,,{ iijiji Feu
)(2
1,, ijjiij uue
0,,,, ikjljlikijklklij eeee
0, ijij F
ijijkkij ee 2)(
},,;{)(iij
t F
ijjikkijijkkkkij FFF ,,,,, 11
1
0, ijij F
},,;{)(ii
u Fu
0)( ,, ikikkki Fuu
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Principle of SuperpositionFor a given problem domain, if the state },,{ )1()1()1(
iijij ue is a solution to the fundamental
elasticity equations with prescribed body forces )1(iF and surface tractions )1(
iT , and the
state },,{ )2()2()2(iijij ue is a solution to the fundamental equations with prescribed body
forces )2(iF and surface tractions )2(
iT , then the state },,{ )2()1()2()1()2()1(iiijijijij uuee
will be a solution to the problem with body forces )2()1(ii FF and surface tractions
)2()1(ii TT .
(1)+(2) (1) (2) = +
},,{ )1()1()1(iijij ue
},,{ )2()2()2(iijij ue},,{ )2()1()2()1()2()1(
iiijijijij uuee
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Saint-Venant’s PrincipleThe stress, strain and displacement fields due to two different statically equivalent force distributions on parts of the body far away from the loading points are approximately the same
(1)
2
P
2
P
(2)
3
P
3
P
3
P
(3)
P
Stresses approximately the same
Boundary loading T(n) would produce 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
S*
T(n) FR
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
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 Saint-Venant’s principle, whereby a complicated boundary condition can be replaced by a simpler statically equivalent distribution.
Elasticity Theory, Applications and NumericsM.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 NumericsM.H. Sadd , University of Rhode Island