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TENSION.PDE

This example shows the deformation of a tension bar with a hole. The equations of Stress/Strain arise from the balance of forces in a material medium, expressed as

dx(Sx) + dy(Txy) + Fx = 0
dx(Txy) + dy(Sy) + Fy = 0,

where Sx and Sy are the stresses in the x- and y- directions, Txy is the shear stress, and Fx and Fy are the body forces in the x- and y- directions.

The deformation of the material is described by the displacements, U and V, from which the strains are defined as

ex = dx(U)
ey = dy(V)
gxy = dy(U) + dx(V).

The eight quantities U,V,ex,ey,gxy,Sx,Sy and Txy are related through the constitutive relations of the material. In general,

Sx = C11*ex + C12*ey + C13*gxy - b*Temp
Sy = C12*ex + C22*ey + C23*gxy - b*Temp
Txy = C13*ex + C23*ey + C33*gxy

In orthotropic solids, we may take C13 = C23 = 0.

Combining all these relations, we get the displacement equations:

dx(C11*dx(U)+C12*dy(V)) + dy(C33*(dy(U)+dx(V))) + Fx = dx(b*Temp)
dy(C12*dx(U)+C22*dy(V)) + dx(C33*(dy(U)+dx(V))) + Fy = dy(b*Temp)

In the "Plane-Stress" approximation, appropriate for a flat, thin plate loaded by surface tractions and body forces IN THE PLANE of the plate, we may write

G = E/(1-nu^2),
C11 = G
C12 = G*nu
C22 = G
C33 = G*(1-nu)/2,

where E is Young's Modulus and nu is Poisson's Ratio.

The displacement form of the stress equations (for uniform temperature and no body forces) is then (dividing out G):

dx(dx(U)+nu*dy(V)) + 0.5*(1-nu)*dy(dy(U)+dx(V)) = 0
dy(nu*dx(U)+dy(V)) + 0.5*(1-nu)*dx(dy(U)+dx(V)) = 0

In order to quantify the load boundary condition mechanism, consider the stress equations in their original form:

dx(Sx) + dy(Txy) = 0
dx(Txy) + dy(Sy) = 0

These can be written as

div(P) = 0
div(Q) = 0,

where P = [Sx,Txy] and Q = [Txy,Sy].

The "load" (or "natural") boundary condition for the U-equation defines the outward surface-normal component of P, while the load boundary condition for the V-equation defines the surface-normal component of Q. Thus, the load boundary conditions for the U- and V- equations together define the surface load vector.

On a free boundary, both of these vectors are zero, so a free boundary is simply specified by

load(U) = 0
load(V) = 0.

Here we consider a tension strip with a hole, subject to an X-load.

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