I am interested on simulating the converse Piezoelectric effect of materials and I have given the mechanical (Young's modulus and Poisson's ratio) and electric material properties (Electric Permittivity, Piezoelectric coupling matrix etc). However from the literature I got double index notation, I am struggling to convert double index notation to triple index notation.

The double index notation from literature is as follows:

Double index notation

$$\begin{bmatrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\ d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 0 & 21.3 & 0 \\ 0 & 0 & 0 & 21.3 & 0 & 0 \\ -2.6 & -2.69 & 3.65 & 0 & 0 & 0 \end{bmatrix}$$

I would like to convert them to Triple index notation used by some commercial software providers like Abaqus and MSC Marc etc and is written as follows:

$$\begin{bmatrix} d_{11-1} & d_{11-2} & d_{11-3} \\ d_{22-1} & d_{22-2} & d_{22-3} \\ d_{33-2} & d_{33-2} & d_{33-3} \\ d_{12-2} & d_{12-2} & d_{12-3} \\ d_{23-2} & d_{23-2} & d_{23-3} \\ d_{31-2} & d_{31-2} & d_{31-3} \end{bmatrix} = \begin{bmatrix} 0 & 0 & -2.6 \\ 0 & 0 & -2.6 \\ 0 & 0 & 3.65 \\ 0 & 21.3 & 0 \\ 21.3 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

Am I doing the conversion correctly? Are the values placed in correct positions? Double Index = Triple Index = Coupling Matrix Value

$$\begin{alignat}{2} d_{11} &= d_{11-1} &&= 0 \\ d_{21} &= d_{11-2} &&= 0 \\ d_{31} &= d_{11-3} &&= -2.6 \\ d_{12} &= d_{12-1} &&= 0 \\ d_{22} &= d_{22-2} &&= 0 \\ d_{32} &= d_{22-3} &&= -2.6 \\ \vdots \\ d_{16} &= d_{31-1} &&= 0 \\ d_{26} &= d_{31-2} &&= 0 \\ d_{36} &= d_{31-3} &&= 0 \\ \end{alignat}$$

  • You could try to use latex, enter $d_{16}$ and you get $d_{16}$. – peterh May 15 at 9:27
  • 1
    Thank you very much for your input, I have edited them. Thanks – John May 15 at 10:05
up vote 1 down vote accepted

Caveat: I don't know the convention used in Marc. The following is based on what I can recall about Abaqus and Ansys.

Constitutive relations

The constitutive relations for linear piezoelectric materials are typically expressed in two forms:

  1. the strain and electric field are the independent variables (called the stress-charge or e-form), and
  2. the stress and electric field are the independent variables (called the strain-charge or d-form).

Case 1 : stress-charge or e-form

The coupled equations in this case are $$ \begin{align} \sigma_{ij} &= C_{ijkl} \varepsilon_{kl} - e_{mij} E_m \\ q_i &= e_{ijk} \varepsilon_{jk} + D_{ij} E_j \end{align} $$

Abaqus uses this form because solid mechanics finite element models are more easily solved in strain-driven form.

Case 2 : strain-charge or d-form

The above equations can also be written in the form $$ \begin{align} \varepsilon_{ij} &= S_{ijkl} \sigma_{kl} + d_{mij} E_m \\ q_i &= d_{ijk} \sigma_{jk} + D_{ij} E_j \end{align} $$ Abaqus allows input of the coupling tensor $d$ because these numbers are easily converted into the $e$ tensor using the relation $$ e_{mij} = C_{ijkl} d_{mkl} $$

Converse piezoelectric effect

The converse piezoelectric effect is modeled by the first of the d-form equations while the direct effect is modeled by the second. In particular, in the absence of stresses and external forces, the converse effect is written as $$ \varepsilon_{ij} = d_{mij} E_m $$

Conventions

The symmetry of the strain tensor leads to various ways of representing the components in matrix form. For example, the standard solid mechanics convention is $$ [\boldsymbol{\varepsilon}] = \begin{bmatrix} \varepsilon_{11} & \varepsilon_{22} & \varepsilon_{33} & \varepsilon_{23} & \varepsilon_{31} & \varepsilon_{12} \end{bmatrix}^T \rightarrow \begin{bmatrix} \varepsilon_{1} & \varepsilon_{2} & \varepsilon_{3} & \varepsilon_{4} & \varepsilon_{5} & \varepsilon_{6} \end{bmatrix}^T $$ The Abaqus convention is $$ [\boldsymbol{\varepsilon}] = \begin{bmatrix} \varepsilon_{11} & \varepsilon_{22} & \varepsilon_{33} & \varepsilon_{12} & \varepsilon_{31} & \varepsilon_{23} \end{bmatrix}^T \rightarrow \begin{bmatrix} \varepsilon_{1} & \varepsilon_{2} & \varepsilon_{3} & \varepsilon_{4} & \varepsilon_{5} & \varepsilon_{6} \end{bmatrix}^T $$

Tensor d in matrix form

To express the coupling tensor in matrix form, let us expand out each term: $$ \begin{align} \varepsilon_{11} & = E_1 d_{111} + E_2 d_{211} + E_3 d_{311} \\ \varepsilon_{22} & = E_1 d_{122} + E_2 d_{222} + E_3 d_{322} \\ \varepsilon_{33} & = E_1 d_{133} + E_2 d_{233} + E_3 d_{333} \\ \varepsilon_{12} & = E_1 d_{112} + E_2 d_{212} + E_3 d_{312} \\ \varepsilon_{13} & = E_1 d_{113} + E_2 d_{213} + E_3 d_{313} \\ \varepsilon_{23} & = E_1 d_{123} + E_2 d_{223} + E_3 d_{323} \end{align} $$ Using the Abaqus convention, we can write the above as $$ \begin{align} \varepsilon_{1} & = E_1 d_{11} + E_2 d_{21} + E_3 d_{31} \\ \varepsilon_{2} & = E_1 d_{12} + E_2 d_{22} + E_3 d_{32} \\ \varepsilon_{3} & = E_1 d_{13} + E_2 d_{23} + E_3 d_{33} \\ \varepsilon_{4} & = E_1 d_{14} + E_2 d_{24} + E_3 d_{34} \\ \varepsilon_{5} & = E_1 d_{15} + E_2 d_{25} + E_3 d_{35} \\ \varepsilon_{6} & = E_1 d_{16} + E_2 d_{26} + E_3 d_{36} \end{align} $$ In matrix form, $$ \begin{bmatrix} \varepsilon_{1} \\ \varepsilon_{2} \\ \varepsilon_{3} \\ \varepsilon_{4} \\ \varepsilon_{5} \\ \varepsilon_{6} \end{bmatrix} = \begin{bmatrix} d_{11} & d_{21} & d_{31} \\ d_{12} & d_{22} & d_{32} \\ d_{13} & d_{23} & d_{33} \\ d_{14} & d_{24} & d_{34} \\ d_{15} & d_{25} & d_{35} \\ d_{16} & d_{26} & d_{36} \end{bmatrix} \begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix} $$ or $$ [\boldsymbol{\varepsilon}] = [\mathbf{d}]^T [\mathbf{E}] $$ where $$ [\mathbf{d}] = \begin{bmatrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{bmatrix} $$

Double to triple index map

From the above we see that, in Abaqus, the matrix $[\mathbf{d}]$ is $$ \begin{align} [\mathbf{d}] &= \begin{bmatrix} d_{111} & d_{122} & d_{133} & d_{112} & d_{113} & d_{123} \\ d_{211} & d_{222} & d_{233} & d_{212} & d_{213} & d_{223} \\d_{311} & d_{322} & d_{333} & d_{312} & d_{313} & d_{323} \end{bmatrix} \\ & = \begin{bmatrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{bmatrix} \end{align} $$ Ansys (and possibly Marc) uses a different convention and the numbers will have to be changed accordingly.

If the constants in your initial matrix are arranged according to the standard mechanics convention, the above map changes to $$ \begin{align} [\mathbf{d}] &= \begin{bmatrix} d_{111} & d_{122} & d_{133} & d_{123} & d_{113} & d_{112} \\ d_{211} & d_{222} & d_{233} & d_{223} & d_{213} & d_{212} \\d_{311} & d_{322} & d_{333} & d_{323} & d_{313} & d_{312} \end{bmatrix} \\ & = \begin{bmatrix} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26} \\d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{bmatrix} \end{align} $$ You have to make sure that the convention used in the input 2-index matrix is known before you can write down the correct mapping.

  • Thank you very much for the prompt response and detailed explanation. Its more clear for me now. Thanks again! – John May 16 at 4:46

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