Three-dimensional elasticity problems are difficult to solve. Thus we first develop governing equations for two-dimensional problems, and explore four

Chapter 7 Two-Dimensional Formulation

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

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.

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.

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Two vs Three Dimensional Problems

x

y

z

x

y

z

Three-Dimensional Two-Dimensional

x

y

zSpherical Cavity


Plane StrainConsider 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

x

y

z R

0,),(,),( wyxvvyxuu


Plane Strain Field Equations0,

2

1,,

yzxzzxyyx eeex

v

y

ue

y

ve

x

ue

0,2

)()(

2)(,2)(

yzxzxyxy

yxyxz

yyxyxyxx

e

ee

eeeeee

Strains

Stresses

Equilibrium Equations

0

0

yyxy

xxyx

Fyx

Fyx

Navier Equations

0)(

0)(

2

2

y

x

Fy

v

x

u

yv

Fy

v

x

u

xu

Strain Compatibility

yx

e

x

e

y

e xyyx

2

2

2

2

2

2

Beltrami-Michell Equation

y

F

x

F yxyx 1

1)(2


Examples of Plane Strain Problems

x

y

z

x

y

z

P

Long CylindersUnder Uniform Loading

Semi-Infinite Regions Under Uniform Loadings


Plane StressConsider 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 through the thickness, and thus they will be approximately zero throughout the entire domain. Finally since the region is thin in the z-direction 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

0

),(

),(

),(

yzxzz

xyxy

yy

xx

yx

yx

yx

x

y

z R

2h

yzxzz ,,


Plane Stress Field EquationsStrains Strain Displacement Relations


0

0

yyxy

xxyx

Fyx

Fyx

Navier Equations

Strain Compatibility

yx

e

x

e

y

e xyyx

2

2

2

2

2

2

Beltrami-Michell Equation

0,1

)(1

)(

)(1

,)(1

yzxzxyxy

yxyxz

xyyyxx

eeE

e

eeE

e

Ee

Ee

02

1,0

2

1

2

1,,,

x

w

z

ue

y

w

z

ve

x

v

y

ue

z

we

y

ve

x

ue

xzyz

xyzyx

0)1(2

0)1(2

2

2

y

x

Fy

v

x

u

y

Ev

Fy

v

x

u

x

Eu

y

F

x

F yxyx )1()(2


Examples of Plane Stress Problems

Thin Plate WithCentral Hole

Circular Plate UnderEdge Loadings


Plane Elasticity Boundary Value Problem

R

So Si

S = Si + So

x

y

Displacement Boundary Conditions

Stress/Traction Boundary Conditions

Sonyxvvyxuu bb ),(,),(

SonnnyxTT

nnyxTT

yb

yxb

xyb

yny

yb

xyxb

xb

xn

x

)()()(

)()()(

),(

),(

Plane Strain Problem - Determine in-plane displacements, strains and stresses {u, v, ex , ey , exy , x , y , xy} in R. Out-of-plane stress z can be determined from in-plane stresses via relation (7.1.3)3.

Plane Stress Problem - Determine in-plane displacements, strains and stresses {u, v, ex , ey , exy , x , y , xy} in R. Out-of-plane strain ez can be determined from in-plane strains via relation (7.2.2)3.


Correspondence Between Plane Formulations

Plane Strain Plane Stress

0)(

0)(

2

2

y

x

Fy

v

x

u

yv

Fy

v

x

u

xu

0

0

yyxy

xxyx

Fyx

Fyx

y

F

x

F yxyx 1

1)(2

0)1(2

0)1(2

2

2

y

x

Fy

v

x

u

y

Ev

Fy

v

x

u

x

Eu

0

0

yyxy

xxyx

Fyx

Fyx

y

F

x

F yxyx )1()(2


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.

E

Plane Stress to Plane Strain 21 E

1

Plane Strain to Plane Stress 2)1(

)21(

E

1

Therefore the solution to one plane problem also yields the solution to the other plane problem through this simple transformation scheme.


Generalized Plane StressThe plane stress formulation produced some inconsistencies in particular out-of-plane behavior and resulted in some three-dimensional effects where in-plane 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

h

hdzzyx

hyx ),,(

2

1),(

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 .


Anti-Plane StrainAn 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

),(,0 yxwwvu

y

we

x

we

eeee

yzxz

xyzyx

2

1,

2

1

0

yzyzxzxz

xyzyx

ee

2,2

0

0

0

yx

zyzxz

FF

Fyx

02 zFw

Strains Stresses

Equilibrium Equations Navier’s Equation


Airy Stress Function MethodNumerous 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

yxxy xyyx

2

2

2

2

2

,,

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

02 44

4

22

4

4

4

yyxx

This relation is called the biharmonic equation and its solutions are known as biharmonic functions.


Airy Stress Function FormulationThe 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 two-dimensional 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.

R

So Si

S = Si + So

x

y

02 44

4

22

4

4

4

yyxx

yxyyxxyn

y

yxyxyxxn

x

nx

nyx

nnT

nyx

ny

nnT

2

22)(

2

2

2)(

x

xy

y


Polar Coordinate FormulationPlane Elasticity Problem

r

u

r

uu

re

uu

re

r

ue

rr

r

rr

1

2

1

1

0,2

)()(

2)(

2)(

Strain Plane

rzzrr

rrz

r

rrr

e

ee

eee

eee

0,1

)(1

)(

)(1

)(1

Stress ane Pl

rzzrr

rrz

r

rr

eeE

e

eeE

e

Ee

Ee

Strain-Displacement

Hooke’s Law


Polar Coordinate Formulation

021

0)(1

r

r

Frrr

Frrr

r

rrr

011

)1(2

01

)1(2

Stress Plane

011

)(

01

)(

Strain Plane

2

2

2

2

Fu

rr

u

r

u

r

Eu

Fu

rr

u

r

u

r

Eu

Fu

rr

u

r

u

ru

Fu

rr

u

r

u

ru

rr

rrr

r

rr

rrr

r


Compatibility Equations

Navier’s Equations

2

2

22

22 11

rrrr

F

rr

F

r

F

F

rr

F

r

F

rrr

rrr

1)1()(

Stress Plane

1

1

1)(

StrainPlane

2

2


Polar Coordinate FormulationAiry Stress Function Approach = (r,θ)

rr

r

rrr

r

r

1

11

2

2

2

2

2

01111

2

2

22

2

2

2

22

24

rrrrrrrr

RS

x

y

r

Airy Representation

Biharmonic Governing Equation

),(,),( rfTrfT rr

Traction Boundary Conditions

r

r


Documents

Three-dimensional elasticity problems are difficult to solve. Thus we first develop governing equations for two-dimensional problems, and explore four