Stress In 2D Elements

Strain Definitions

The normal definitions of strain used are as follows

An alternative definition which fits more neatly in tensor form is

with the strain tensor defined as

The calculation of principal strains follows from

The maximum shear strain is calculated from the principal strain as

or

In a similar way to the definitions of average and von Mises stress a volumetric and effective strain can be calculated as

Stress Definitions

Stress can be considered as a tensor quantity whose components can be represented in matrix form as

where each term corresponds to a force per unit area. The following notation for the stress components is common

The principal stresses are calculated as the roots of the cubic

where the terms are stress invariants defined as

Alternatively the principal stress equation can be written

The maximum shear stress is calculated from the principal stress as

Two other stress measures that are used are the average or hydrostatic stress and the von Mises stress; these are defined as

Stress in 2D elements

The stress in 2D elements is calculated via the strains. The strains are calculated from the displacements using the strain displacement relationship (see 2D elements). Using the interpolation functions these can be calculated at any point in the element. Once the strains are calculated the stress can be calculated using the material elastic matrix for example for and elastic isotropic material the material matrix is

Thus the strains are

and the stresses are

This can be used to evaluate the stress at any point in the element. However the stress is based on the strain which in turn is based on the displacement gradients in the element. Thus some of the strain terms in an element that has a parabolic displacement field are linear. It has been found that the best stress results are obtained by evaluating the stress at particular points (the points used for the element integration) and extrapolating the results to the nodes.

In order to have good stress results the mesh will have to be finer that the mesh required for the displacement solution and the stress results are likely to be influenced by high displacement gradients in the element.

Direct extrapolation of results

In the case of direct extrapolation a function is chosen to represent the variation of stress over the element based on the number of Gauss points. In practice this is used when there are 1, 3 or 4 Gauss points. The corresponding polynomial functions are

For 1 Gauss point the values are assumed to be constant over the whole element so the Gauss point values are simply copied to the nodes.

For 3 Gauss points the values are the Gauss points are known so the following set of equations can be set up

This can then be used to calculate the coefficients

For 4 Gauss points a similar approach can be used, but in this case the locations of the Gauss points are at

so the equations can be written in the form

This can then be solved for the coefficients

Once these are established the polynomial functions can be used to establish the values at any position on the element.

Least squares extrapolation of results

In this case a function chosen to fit through the points would imply a higher order polynomial than the one used to interpolate the geometry, so a least squares approach is used to find the polynomial to map the stresses from the Gauss points to the nodes. The interpolation functions used for 6 node and 8 node elements are respectively

The square of the error for any point is then

This is summed over all the Gauss points and then the derivatives with respect to the coefficients are set to zero (selecting the coefficients that minimise the error). This leads to the matrix equation for 8 node elements

The 6 node version is the same except that the and terms are ignored. This can then be solved for the coefficients.