Download - Elasticity AiryStress
1
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
2
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].
3
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.
4
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
5
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 ,,
6
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
7
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
8
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
9
Stress Strain graph
10
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
11
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
12
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
13
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
14
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
15
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
16
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.
17
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
18
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
19
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