MECH 303 Chapter 2

2.6 Strain at a point

The strain components x, y ,xy at a point completely determine the strain state ( deformation state ) of the material at this point.


The normal strain N of any line element PN

The change of angle between PN and PN

principal strain, principal plane.

2.7Boundary conditions

(1)8 basic equations for plane problems:

equilibrium

geometrical

physical , (plane stress)

Solve the 8 unknown functions under proper boundary conditions.

(2)3 kinds of boundary conditions :


displacement boundary problem (displacement boundary condition)

s= ,s=, and are the prescribed functions of coordinates on the surface.

Stress boundary problem ( stress boundary condition)

Surface forces acting on the boundary of a body are prescribed, which can be expressed as a condition about the stress components at the boundary:

(relations between boundary stress components and the surface force) when the boundary is normal to a coordinate axis, the above boundary conditions are simplified

Mixed boundary conditions

2.8Saint-Venant’s principal

If a system of forces acting on a small portion of the surface of an elastic body is replaced by another statically equivalent system of forces acting on the same portion of the surface, the redistribution of loading produces substantial changes in the stresses only in the immediate neighborhood of the loading, and the stresses are essentially the same in the parts of the body which are at large distances in comparison with the linear dimension of the surface on which the forces are changed. By “ statically equivalent systems” we mean that the two systems have the same resultant force and the same resultant moment.


2.9 Solution of plane problem in terms of displacements

Equilibrium equation in terms of stresses:

express stress by strain via  (physical equation)


Equilibrium equation in terms of strain :

express strain by displacement
components (geometrical equations) :

Equilibrium equation in terms of displacements: (plane stress)

Solve the above equations of u(x,y) and v(x,y) with the following boundary conditions

(1) displacement boundary condition : ,

(2) stress boundary condition ( in term of displacements) :

Thus, to solve a plane stress problem in terms of displacements, we have to solve two differential equations simultaneously and the obtained u (x, y) and v ( x, y) must satisfy the displacement boundary condition or stress boundary condition or mixed boundary condition at the boundary.

Once we have u (x, y) and v (x, y), then

x (x, y) ,y (x, y),xy (x, y)

x (x, y),y (x, y),xy (x, y) are completely uniquely determined.

For plane strain problem, the formulation is the same as before, only replace E by and  by .

The above method is suitable to solve any plane problems with any boundary conditions.

2.10 Solution of plane problem in terms of stress

Take the 3 stress components x , y , xy (xy ) as the basic unknown functions.

(1)2 equilibrium equations contain 3 functions

(2)The 3rd equation is the compatibility equation in terms of strain (or stress):

The necessary and sufficient condition for the existence of single-valued continuous functions u ( x, y) and v (x, y) is simple

,,

(by using physical equations xx ,……...)

(plane stress)

(plane strain)

(3)The obtained stress solution should satisfy stress boundary condition (B.C.)

2.11 Case of constant body forces

In the case of constant body forces: , , then the 3 stress components x , y , xy are determined by equations

and B. C. on S

The general solution of consists of:

(1)The general solution of

(homogeneous system)

(2)A particular solution of

(non-homogeneous system)

When body forces are constant , the particular solution may be taken as

, ,

or: , y=0 ,


The general solution of is

always satisfy the homogeneous equations

 Airy’s stress function for plane problems.

The complete solution of the equilibrium equations are

depends on 

To determine  , substituting x, y, xy, into compatibility condition, we have

Thus, in the solution of plane problems in terms of stress when body forces are constant, it is necessary only to solve for stress function  from the single differential equation =0, and x, y, xy satisfy stress boundary conditions.

2.12Airy’s stress function. Inverse method and semi-inverse
method

Direct method  usually impossible

Inverse method  take some function  satisfying the compatibility equation, then obtain stress components and find the surface force components, thus we know what problem they can solve.

Semi-inverse method  we assume the solution for stress or displacements in a given problem, then proceed to show that all the differential equations and boundary conditions are satisfied. If some of the boundary conditions are not satisfied, then we have to modify the assumptions made.

(Numerical method ---- Finite element method, directly solve the equilibrium equation in terms of displacements.)

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