D3PLOT 22.1

Manipulations of Strain Tensor Components

Manipulations of strain tensor components

We have adopted the notation for global strains:

[E] = (Symmetric: Upper triangle = lower triangle.)

Where XX , YY , ZZ are the "direct" strains, and XY , YZ , XZ the shear strains.

Important Note: shear strain tensor terms

In many engineering textbooks the strain component is multiplied by a factor of two (for example see "Voigt" notation):

This is because some equations can be simplified, for example where

are often referred to as "Engineering Shear Strain". (This is not to be confused with "Engineering Strain".)

In conclusion:

  • The shear strain terms written by Ansys LS-DYNA in the strain tensor are "true" strains: XY , YZ , XZ
  • The shear strain terms used in the formulae on this page are the same.
  • There is absolutely no factor of 2 involved !

For Nastran OP2 results:

  • Nastran outputs shear strain terms with a factor of 2, however for consistency D3PLOT applies a factor of 0.5 when they are read.
  • If you want to plot the "Engineering Shear Strains" you can use the GXY_GAMMA_XY_STRAIN , GYZ_GAMMA_YZ_STRAIN and GZX_GAMMA_ZX_STRAIN components which are the "true" shear strain components multiplied by 2.

Computing equivalent strain values

These are all computed in exactly the same way as the stress terms above, substituting [E] for [S] , except that the von Mises strain has a factor of 2/3 applied:

This is given by:
The reasons for applying this factor of 2/3, which make the calculation suitable for the plastic regime, are given below under " What is von Mises strain? "

Note, SAV_AVERAGE_STRAIN is the strain equivalent to PRESSURE .

Computing PEMAG_PLAST_STRN_MAG

This is an averaged scalar term reflecting the maximum plastic strain related to the inelastic strain tensor at a given location.

It is given by: where is the plastic strain component.

Or, in full, the elastic strain tensor is derived from the stress tensor:

and .
where is the elastic strain component,
is Poisson's ratio,
E is Young's modulus, and
is the shear modulus.
The plastic strain tensor is given by: .
Plastic strain magnitude is: .
Note, often the formula is written using engineering shear strain terms, which gives a factor of 0.5: .

For thin and thick shell elements the PEMAG data component should be checked very carefully.  Depending on the element formulation and the value of ISTUPD on *CONTROL_SHELL the through thickness (local Z direction) strain may or may not be calculated and written to the strain tensor.   If this strain value is omitted PEMAG will tend to underestimate the full extent of the strain in the element.

If you wish to use this data component for shells, especially thick shells, it is strongly recommended that you set up a single element "tensile test" model which stretches a representative element well into the plastic regime.   Inspect thickness change and strain tensor to ensure that you see thinning of the shell and the corresponding through thickness strain.   Another good test is that for uniaxial loading of this sort the PEMAG value should be exactly the same as the effective plastic strain as reported by Ansys LS-DYNA.

Computing ENGINEERING STRAIN

For shell elements, as well as 2D principal max and min strain (also referred to as true major and minor strain), D3PLOT derives engineering major and minor strain.

This is given by:
where are 2D principal max and min strains (see Computing 2D PRINCIPAL stresses ).

For shell elements, D3Plot also derives engineering through-thickness strain.

This is given by:
where is the through thickness strain, that is, the strain tensor component in the local Z direction. This can be non-zero to preserve volume of the shell.

Computing INTERNAL ENERGY DENSITY for solids

D3PLOT can derive the internal energy density for solids in the elastic regime. This is not computed by default, users wishing to use it should contact Oasys Ltd Support. The equation implicitly assumes elastic strain, hence there is a factor of 0.5 as the area under the elastic stress/strain curve is a triangle. In the plastic regime this factor needs to be closer to one.

This is given by:


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