How an Expression Is Evaluated
In order to get optimal quality in the result presentation, it is necessary to have some understanding about how expressions are evaluated.
In structural engineering, the maybe most commonly evaluated result quantities are stresses and strains. Stress expressions can, however, be sensitive to specific settings during result presentation, and are for this reason used as an example in the discussion below.
Consider a linear elastic material with thermal expansion and creep. The stress tensor σ is then computed as
where is the 4th order elasticity tensor, “:” stands for the double-dot tensor product (or double contraction). The elastic strain εel is the difference between the total strain ε and all inelastic strains εinel. In this example, the inelastic strain tensor has two contributions, creep strain εcr and thermal strain εth. There may also be an extra stress term σex with contributions from initial stress and external stress.
When a stress tensor component is to be evaluated, the following happens:
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A position x in an element is identified, where the value is to be computed.
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The total strain ε is computed from the derivatives of the shape functions at x. The strain depends on the values of the displacement degrees of freedom at the nodes and on the shape functions in the element.
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The expression giving the thermal strain εth is evaluated at x. This expression will depend on a temperature, which may be prescribed or computed in another physics interface.
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The creep strain εcr is a state variable, which is stored at the Gauss points in the element. The value is picked from the Gauss point closest to x.
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The inelastic strain εinel (in this example, the creep strains is now subtracted from the total strains, to form the elastic strain εel.
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The elastic strain εel is multiplied by the elasticity tensor , evaluated at x, to give the stress σ. The material properties may depend on the location, either explicitly, or for example by a temperature dependency.
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Any extra stress σex, evaluated at x, is added to the stress tensor.
The subtraction between total and inelastic strains is however a sensitive operation. It is not uncommon that these two terms are close to each other in value, so that the resulting elastic strain is a small number obtained by subtraction of two larger numbers. Since the three types of strain in this example have different types of distribution through the element, there is a risk that this difference will exhibit artificial variations inside the element.
As an example, let’s assume that the temperature has a quadratic variation through the element, and that standard quadratic shape functions are used in Solid Mechanics. The total strain is a linear function within each element, since it contains derivatives of the shape functions. The computed elastic strain is the difference between the linear total strain and the quadratic thermal strain. If there also are creep strains present, they will be piecewise constant, since they are taken from the nearest integration point.
It must, moreover, be realized that the total strain is ‘correct’ only in an average sense. It provides a kind of best fit given by the finite element formulation.