How do you know if Airy function is stressed?

How do you know if Airy function is stressed?

Consider the Airy stress function φ = α1×12 + α2x1x2 + α3×22. (a) Verify that it satisfies the biharmonic equation. (b) Determine the in-plane stresses T11, T12 and T22. (c) Determine and sketch the tractions on the four rectangular boundaries x1 = 0, x1 = b, x2 = 0, x2 = c.

Why is the airy stress function used in elasticity?

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 traction boundary condition?

Traction free boundary condition means that the the surface is free from external stress. We can mathematically express this as > t = σ.n = 0. where, t is the surface traction in the current configuration; σ = Cauchy stress tensor; n = vector normal to the deformed surface.

Which value of the stress function indicate a good fit?

Stress values equal to or below 0.1 are considered fair, while values equal to or below 0.05 indicate good fit.

What is stress free boundary condition?

The stress-free condition means that the tangent component of the stress at the boundary is zero. We shall see that this boundary condition is admissible in general for non-axisymmetric domains, but it yields pathological stability behaviors if the fluid domain is a solid of revolution.

When the fluctuation of stress has zero min stress it is called as?

(c) Reversed stresses (Ref Fig c): The reversed stress varies in a sinusoidal manner with respect to time, but it has zero mean stress.

When mean stress is zero the stress cycle is called?

Explanation: Half cycle is in tensile stress and other half in compressive stress, hence it belongs to Reversed Stresses Category. Explanation: If mean is to be zero, then there must be compressive as well as tensile stresses and hence belongs to reversed stresses category.

What is the condition of plane stress?

Plane stress is defined to be a state of stress in which the normal stress, 0,, and the shear stresses, Orz and Oy z, directed perpendicular to the x-y plane are assumed to be zero. The geometry of the body is essentially that of a plate with one dimension much smaller than the others.

What are plane strain conditions?

Plane strain A stress condition in linear elastic fracture mechanics in which there is zero strain in the direction normal to the axis of applied tensile stress and direction of crack growth. It is achieved in thick plate, along a direction parallel to the plate.

What are the equivalent boundary conditions for Airy stress function?

In order to use the method of Airy stress function, the equivalent boundary conditions should be obtained to solve the compatibility equation by using the DQM. For illustrations, the equivalent boundary conditions for other uniaxial compressive loadings ( Y = 0) are given below.

How to use the Airy stress function in differential quadrature?

Xinwei Wang, in Differential Quadrature and Differential Quadrature Based Element Methods, 2015 In order to use the method of Airy stress function, the equivalent boundary conditions should be obtained to solve the compatibility equation by using the DQM.

What is Pagano’s Airy stress function?

In an early paper [ 2], Pagano used the Airy stress function method to obtain an exact solution in the framework of the linear theory of elasticity for simply supported cross-ply laminates composed of perfectly bonded orthotropic/isotropic layers.

Is the Airy stress function biharmonic or analytic?

The Airy stress function ϕ is biharmonic according to Eq. (3.10). Introduce a function P by This simply says that P is a harmonic function. Hence, where f ( z) is an analytic function and can be expressed as then ψ is also analytic and its derivative is given by