What is airy stress function?

Airy stress function The Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only. This stress function can therefore be used only for two-dimensional problems.

Why is the airy stress function used in elasticity?

By using the Airy stress function representation, the problem of determining the stresses in an elastic body is reduced to that of finding a solution to the biharmonic partial differential equation 3.2. Further, the plane stress and plane strain stress fields are identical.

How do you derive Airy stress function?

Airy stress functions are used to solve 2-D equilibrium problems. The approach will be presented here for the special case of no body forces. Next, propose that a scalar function, ϕ , exists (this is the Airy stress function) and is related to the 2-D stress components by the following cleverly chosen relationships.

What is meant by axisymmetric field problem?

4.3.1 Plane Axisymmetric Problems Some three dimensional (not necessarily plane) examples of axisymmetric problems would be the thick-walled (hollow) cylinder under internal pressure, a disk rotating about its axis1, and the two examples shown in Fig. 4.3.

What is plane stress problem?

A plane stress problem is a specialization of the three-dimensional elasticity problem that contains no through thickness stresses. Specifically, this means that the following conditions hold: Definition of Plane Stress. (3.1) This reduces the general three-dimensional strain energy expression of Lesson 1 to.

Why do we do elasticity theory?

The theory of elasticity treats the relationship between forces applied to an object and the resulting deformations. In practice, the analysis of the elastic behaviour of a material is reduced to the study of simple deformations and the determination of the corresponding elastic constants.

What is the correct condition for the solid axisymmetric problems?

Condition for Axisymmetric Problems. 1. The problem domain must have an axis of symmetry. It is customary to align this symmetry axis with the z-axis of the customary to align this symmetry axis with the z axis of the cylindrical (r,θ,z) coordinate system.

Which are axisymmetric objects?

Axisymmetric Objects in Simple Rotation. As soon as rotation is introduced into the problem, two general complications arise: An extra term due to rotation is introduced into the equation governing hydrostatic equilibrium (in the form of the centrifugal force), breaking the spherical symmetry of the problem.

Why stress is called tensor?

Stress has both magnitude and direction but it does not follow the vector law of addition thus, it is not a vector quantity. Instead, stress follows the coordinate transformation law of addition, and hence, stress is considered as a tensor quantity.

How are Airy stress functions used in 2 d equilibrium?

The use of Airy Stress Functions is a powerful technique for solving 2-D equilibrium problems. They are covered here because the approach was used by several researchers in the mid 1900’s to develop analytical solutions to linear elastic problems involving cracks. The curious fact about Airy stress functions is…

When was the use of Airy stress functions used?

Introduction The use of Airy Stress Functionsis a powerful technique for solving 2-D equilibrium problems. They are covered here because the approach was used by several researchers in the mid 1900’s to develop analytical solutions to linear elastic problems involving cracks.

Are there any problems that are not axisymmetric?

It should be noted that many problems involve axisymmetric geometries but non- axisymmetric loadings, and vice versa. These problems are notaxisymmetric. An example is shown in Fig. 4.3.3 (the problem involves a plane axisymmetric geometry). 1the rotation induces a stress in the disk 2the rest of the cylinder is coming out of, and into, the page

How to find the Airy stress function in Mathematica?

The Airy stress function for specific two-dimensional plane conditions is computed and the stresses and displacements at a given point can be found using Mathematica .