Lab 3: Analysing stresses in anisotropic materials

The objective of this lab is to learn how to analyse anisotropic constitutive equations and stresses defined with respect to a material coordinate system.

Revision

Before starting this lab, please be sure to have completed:
  1. Lab 1: Analysing deformation in isotropic materials, and
  2. Lab 2: Stress transformations.

Section 2 of Lab 2 demonstrated how rotating the material-fibre axis with respect to the reference axes influences the components of the stress tensor. For the model in Section 2 of Lab 2, which considers an isotropic cube subject to equi-biaxial deformation, remind yourself:

  • What happened to the components of the stress tensor as the material-fibre axis was rotated? Why?

All of the analyses in the present lab will be based on the equi-biaxial deformation described in Section 2 of Lab 2. The difference here is that we will now consider anisotropic mechanical properties that describe different stress-strain response alonf the different material axes.

Section 1: Deriving components of the stress tensor

  1. Consider the following exponential constitutive relation, which is used to describe the distortional mechanical response of the cube considered in this lab:
\[W_{I} = \frac{c_{1}}{2}(e^{Q}-1)\]

where

\[\begin{split}Q = & c_{ff}E^{2}_{ff} + c_{ss}E^{2}_{ss} + c_{nn}E^{2}_{nn} + \\ & 2c_{fs}(\frac{1}{2}(E_{fs}+E_{sf}))^{2} + 2c_{fn}(\frac{1}{2}(E_{fn}+E_{nf}))^{2} + 2c_{ns}(\frac{1}{2}(E_{ns}+E_{sn}))^{2}\end{split}\]

Differentiate this strain energy density function with respect to each of the nine Green-Lagrange strain components (\(E_{\alpha \beta}\)), where \(\alpha\) and \(\beta\) each represent one of the microstructural material coordinates, (\(f,s,n\)). Thus, derive generalised analytical expressions for the nine distortional components of the second Piola-Kirchhoff stress tensor in terms of the strain components and the material constants: \(c_1, c_{ff}, c_{ss}, c_{nn}, c_{fs}, c_{fn}, c_{ns}\).

Note

  • At this stage, do not substitute any values for the strain components nor constants.
  • Recognising the similarity of terms should simplify this task.
  • Don’t forget the chain rule when differentiating the exponential.

Section 2: Analysing stresses during equi-biaxial deformation

Analysing stresses with respect to the reference coordinates

  1. Using OpenCMISS, load the stress analysis project and run Model 1. (The procedure for running this simulation in OpenCMISS is outlined in steps 1-3 in Section 2 of Lab 2). See this link for an example on how to open the simulation results pane.

  1. The Model 1 simulation uses the above constitutive equation with the following material constants:
\[\begin{split}c_{1} &= 0.0475~kPa \\ c_{ff} = c_{ss} = c_{nn} &= c_{fs} = c_{fn} = c_{ns} = 15.25 \\\end{split}\]

Substitute the Green-Lagrange strain components (\(\boldsymbol{E}_{ref}\)) for this equi-biaxial deformation into your analytical expressions from Step 1 of this lab to determine values for the distortional components of the second Piola-Kirchhoff stress tensor. Verify that these distortional stresses are:

\[T_{ref}^{ff\_dist} = T_{ref}^{ss\_dist} = 8.59~kPa\]

Note

  • Hint: in the steps below, you will use your your analytical equations from step 1 repeatedly to do number of calculations (with different parameters), so you might consider encoding your equations using a high level programming language (python, C, Matlab, etc) or a spreadsheet.
  • Hint: \(Q=3.74\).
  • Your calculations for \(T_{ref}^{ff\_dist}\) and \(T_{ref}^{ss\_dist}\) could be within \({\pm}0.02~kPa\) of the solution stated above due to round off errors.
  • These distortional components of the second Piola-Kirchhoff stress tensor (e.g. \(T_{ref}^{ff\_dist}\)) do not match the stress values shown in the OpenCMISS results panel because the OpenCMISS results show only the total stress components, which will be considered in the following steps.

  1. Now assume that the material is incompressible, and write down analytical expressions for the total stress components with respect to the reference coordinates: \(T_{ref}^{ff}\) and \(T_{ref}^{ss}\) (these are shown as \(T^{ff}\) and \(T^{ss}\) in Eqn 38 of Nash and Hunter (2007), or Eqn 15 of Nash and Hunter (2000)).

  1. Calculate the total stress components with respect to the reference coordinates: \(T_{ref}^{ff}\) and \(T_{ref}^{ss}\) using the expressions you wrote down in Step 4 above. This requires addition of a dilatational component of stress called the hydrostatic pressure, \(p\), which is a scalar variable with a value that is provided in the simulation results. Check your total stress values against those in the simulation results.

Note

  • \(\{C^{MN}\}\) is the inverse of \(\{C_{MN}\}\). (They are different tensors)
  • It is straightforward to invert a diagonal tensor. Check that \(\{C_{MN}\}^{-1}\{C_{MN}\}=\boldsymbol{I}\).

  1. Using this analysis, what can you infer about the material symmetry of Model 1? Explain your observation.

  1. Now run Model 2, which is similar to Model 1 except that the material constants are set to:
\[\begin{split}c_{1} &= 0.0475~kPa\\ c_{ff} &= 15.25 ~~~~ c_{ss} &= 6.8 ~~~~ c_{nn} &= 8.9 \\ c_{fs} &= 6.95 ~~~~ c_{fn} &= 6.05 ~~~~ c_{ns} &= 4.93\end{split}\]

Re-use your analytical expressions from Step 1 above, now with these new material constants, to calculate distortional components of the second Piola-Kirchhoff stress tensor: \(T_{ref}^{ff\_dist}\) and \(T_{ref}^{ss\_dist}\) with respect to the reference axes.

  1. Re-use your analytical expressions from Step 4 above to calculate, for Model 2, the total stress components: \(T_{ref}^{ff}\) and \(T_{ref}^{ss}\) (use the new hydrostatic pressure value, \(p\), from the simulation results). Check your answers against the simulation results.

  1. Explain the similarities and differences in the total second Piola-Kirchhoff stress components from Steps 5 and 8. What can you infer about the mechanical responses (material symmetries) of the two models?

Stresses with respect to rotated material-fibre axes

  1. Now run Model 5, which uses the same (anisotropic) material constants as in Step 7 above. In this simulation, the material-fibre axis is oriented at \(\theta=45\) degrees with respect to the \(X_{1}\)-axis (in the \(X_{1}\)-\(X_{2}\) plane).

  1. Substitute the fibre strain components (\(\boldsymbol{E}_{fib}\)) from the simulation results, and the material constants from Step 7, into your expressions from Section 2 to determine the components of the total second Piola-Kirchhoff stress tensor with respect to the material-fibre coordinates, \(\boldsymbol{T}_{fib}\) (use the hydrostatic pressure, \(p\), from the simulation results).

  1. Determine the second Piola-Kirchhoff stress components with respect to the reference coordinate axes (\(\boldsymbol{T}_{ref}\)) via an appropriate tensor transformation (see Step 3 of Section 2 of Lab 2a). Check your answers against the simulation results.

  1. How do the stress components of \(\boldsymbol{T}_{fib}\) and \(\boldsymbol{T}_{ref}\) for this model compare to the components of \(\boldsymbol{T}_{ref}\) for the previous model in Step 7 above? Explain the similarities and differences.

  1. Now run Model 6, for which the material-fibre axis is oriented at \(\theta=90\) degrees with respect to the \(X_{1}\)-axis (in the \(X_{1}\)-\(X_{2}\) plane). Repeat the analyses in Steps 11-12.

  1. How do the stress components of \(\boldsymbol{T}_{fib}\) and \(\boldsymbol{T}_{ref}\) for this model compare to the components of \(\boldsymbol{T}_{ref}\) for the previous model in Step 5 above? Explain the similarities and differences.

Extra discussion points for experts

If you have completed the exercises above, you may like to consider the following questions:

  1. What do you notice about the stress tensors, \(\boldsymbol{T}_{fib}\) and \(\boldsymbol{T}_{ref}\), from the above analyses for the isotropic (Model 1) and anisotropic (Models 2,5,6) materials subject to equi-biaxial deformations? Explain this observation.

  1. Model 1 considers equi-biaxial deformation, and there were similarities in some of the stress components. If, instead, a uniaxial stretch was applied along the \(X_{1}\) direction, predict what would happed to the components of stress.

  1. What would you expect if you compared the maximum principal stresses for each of the anisotropic cases (Models 2,5,6)? Justify your amswer.

Note

By completing this lab, you should be able to:

  • derive expressions for the components of the second Piola-Kirchhoff stress tensor.
  • evaluate components of the second Piola-Kirchhoff stress tensor with respect to spatial or material-fibre coordinates.
  • infer the material symmetry of a material described by a specific constitutive equation and a particular set of material constants by analysing the stress components.