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ELASTICITYAIRY’S STRESS FUNCTION

BYJONNALAGADDA SRI HARSHA

VISWESWARA R MUDIAM

UNDER GUIDANCE OFDr. LARRY D. PEEL

(ASSOCIATE PROFESSOR)

TEXAS A&M UNIVERSITY, KINGSVILLE

MEEN 5330CONTINUUM MECHANICS

NOV, 14 2005

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INTRODUCTION

One of the main difficulties in solving elasticity problems is that we are required to calculate a vector field or if we are solving for stress we need to calculate a tensor field. This requires to solve at least three, possibly six partial differential equations.

A promising approach is to find a way to reduce the coupled partial differential equations to a single partial differential equations for scalar valued function, which is then used later to deduce the stresses and strains. This can’t be done by 3D, but the Airy stress function is one way to do this for a plane stress or strain problem[1].

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Elasticity Definition

“An elastic body is defined as one which regains its original dimensions after the forces acting on it are removed”.

Elasticity of a substance depends on the material possessing linear stress and strain relations. The range of stress and strain for which the behavior is linearly elastic will be known as elastic range.

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Stress in 1D, 2D & 3D

Stress is the internal distribution of forces within a body that balances and react to the loads applied to it. It is a tensor quantity.

Stress in One Dimension

The definition of Normal stress, , is sometimes called engineering stress and is used for rating the strength of material loaded in one dimensional.

Stress is a simplified definition of stress that includes the change in cross

sectional area.Stress in Two Dimensions

Two dimensional state of stress is also know as Plane stress or plane strain. This two dimensional state models with the state of stresses in a flat thin plate loaded in the plane of the plate. shows the stresses on the x-and y-faces of a Differential element.

A

F

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Stress in a plane

Stresses normal and tangential to faces

Since moment equilibrium of the differential element show that the shear stresses on the perpendicular faces are equal, the 2D of stresses is characterized by three in depended stress components( ). xyyx ,,

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Stress in Three Dimension As the behavior of the body does not depend on the coordinates used to measure it, stress can be described by a tensor. The stress tensor is symmetric and can always be resolved into the sum of two symmetric tensors.Generalized notation:

In the generalized stress tensor notation, the tensor components are written as where i and j are in

{1, 2, and 3}.The first step is to number the sides of Cube. When the lines are parallel to a vector base ( ), then: 1). The sides perpendicular to are called ”j” and ”-

j”; 2). Point from the center of cube points towards the j

side, the –j is at the opposite.

Stress components in 3d

Components of stress tensor

333231

232221

131211

ij

321 ,, eee

je

je

ij

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Strain

Stain is the deformation of a body when force or load is applied on it. It can be measured by calculating the change in length of a line. The change in length of a line is termed the stretch and may be given by

where l is the change in length. (eq-1) l0 is the original undeformed length. is elongation.strain tensor

The strain tensor [ ] is a symmetric used to quantify the strain of an object undergoing a 3-dimensional deformation.

The diagonal coefficients are the relative change in length in the direction of the ii direction (along the -axis). The other terms ( ) are the variation of the right angle (assuming a cube before deformation).

0l

l

ji ij

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Hooke’s Law

It states that if a force (F) is applied to an elastic substance; its extension is linearly proportional to its tensile stress and modulus of elasticity (E).

(eq-2) where L is Length A is Area

The law holds up to a limit, called elastic limit or limit of elasticity, after which the metal will enter a condition of a yield and the substance will suffer plastic deformation up to the plastic limit or limit of plasticity, after which it will eventually break if the force is further increased .

AE

FLL

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Stress Strain graph

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Hooke’s law Cont’d…

In the three dimensional state, a order tensor ( ) containing 81 elastic coefficients must be defined to link the stress tensor ( ) and the strain tensor

( ).

= (eq-3)

where Cijkl are constants

But due to symmetry of the stress and strain tensor, only 36 elastic co-

efficient are independent.

th4 ijklC

ij

klijklC

kl

ij

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Derivation of the Airy Function

To solve a linear elasticity problem, we need to satisfy the following equations:

Strain-Displacement relation (eq-4)

Stress-strain relation (eq-5)

Equilibrium Equation (eq-6)

E- Young’s modulus, - Poison’s ratio & F- Body force

where we have neglected thermal expansion, for simplicity.

i

j

j

iij x

u

x

ul

2

1

ijkkijij E

v

E

vl

1

0

ij

ij Fx

v

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Derivation of the Airy Function cont’d..

The Airy’s function is chosen so as to satisfy the equilibrium equations automatically. For plane stress or plane strain conditions, the equilibrium equations reduce to

011

12

1

11

Fxx

02

2

22

1

12

Fxx

Substitute for the stresses in terms of - body force potential function

;0121

2

222

2

1

Fxxxxx

0221

2

22

1

2

1

Fxxxxx

where ),( 21 xx -is ascalar function of the position

eq-7 & 8

eq-9 & 10

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Derivation of the Airy Function cont’d..

The strain—displacement relation is satisfied provided that the strains obey the compatibility conditions

(eq-11)

The last two of these equations are satisfied automatically by any plane strain or plane stress field. We substitute into the first equation in terms of

stress to see that

where is a constant, if = 0 for plane stress (eq-12)

= 1 for plane strain

022

2

2

2

2

ji

ij

i

jj

j

ii

xxxx

01

2)()1(1

21

122

221122

2

21

2

21

222

22

112

xxE

v

xxv

E

v

xxE

v

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Derivation of the Airy Function cont’d..

Finally, substitute into the equation for stress in terms of and rearrange

to get

(eq-13)

A few more algebra reduces this to which is the result we were looking for

(eq-14)

022)1(1 2

21

2

4

22

2

1

2

22

2

21

2

1

2

41

4

22

2

42

4

xxxxxxv

v

v

xxxx

021

12

22

2

21

2

2

2

41

4

22

21

4

41

4

xxvv

v

xxxx

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Airy stress functions in Rectangular Coordinates

Consider a 2D region (plane strain or plane stress) subjected to a prescribed distribution of traction t on its surface.

To compute the stress fields in the solid, begin by finding a scalar function

known as the Airy potential which satisfies:

(eq-15)

Choose so that it also satisfies the following traction boundary conditions on the surface of the solid

(eq-16 & 17)

Where are the components of unit normal to the boundary.

),( 21 xx

0241

4

22

21

4

41

44

xxxx

1221

2

122

2

tnxx

nx

2121

2

221

2

tnxx

nx

),( 21 nn

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Airy stress functions in Rectangular Coordinates contd.

The stress field within the region of interest is then given by

22

2

11 x

21

2

22 x

21

2

2112 xx

If the strains are needed, they may be computed using the elastic stress—strain relations. If the displacement field is needed, it may be computed by integrating

the strains.

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Limitations of Airy’s Stress functions

The Airy’s Stress function is applicable only to plane strain or plane stress problem [3].

The Airy’s Stress function can only be used if the body force has a special form [3].

Specifically, the requirement is

where is a scalar function of position, F1 & F2 are body forces.

The Airy’s Stress function approach works best for problems where a solid is subjected to prescribed tractions on its boundary, rather than prescribed displacements [3].

1

1 xF

2

2 xF

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Home Work Problem Determine the necessary relationship between the constants A and B if

is to serve as an Airy’s stress function52

32

21 BxxAx

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References

[1] Pie Chi Chou and Nicholas J.Pagano, “Elasticity Tensor, dyadic, and engineering Approaches,” New York, Dover publications, Inc., 1992.

[2] Timoshenko and Goodier, “Theory of Elasticity,” New York, McGraw-Hill, 1970

[3] Adel S. Saada, “Elasticity Theory and applications,” Florida, Krieger Publication Company, 1993.

[4] George E. Mase, “Theory and problems of Continuum Mechanics,” New york,Schaum’s outline series of McGraw- Hill, 1970.

[5] Daniel Frederick and Tien Sun Chang, “Continuum Mechanics,” Bostan, Allyn And Beacon, Inc. 1965.

[6] www.engin.brown.edu/courses/en175/notes/airy/airy.htm