slides Chapter 5 Formulation and Solution Strategies

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


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


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


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


0)( nyT

x

y

l

0)( nxT

Fixed Condition u = v = 0

Traction Condition

STT ynyxy

nx )()( ,0

Non-Coordinate Surface Boundary


S


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


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


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


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


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


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


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


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.


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)


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slides Chapter 5 Formulation and Solution Strategies