Invariant formulation for dispersed transverse isotropy in aortic heart valves

Abstract

Most soft tissues possess an oriented architecture of collagen fiber bundles, conferring both anisotropy and nonlinearity to their elastic behavior. Transverse isotropy has often been assumed for a subset of these tissues that have a single macroscopically-identifiable preferred fiber direction. Micro-structural studies, however, suggest that, in some tissues, collagen fibers are approximately normally distributed about a mean preferred fiber direction. Structural constitutive equations that account for this dispersion of fibers have been shown to capture the mechanical complexity of these tissues quite well. Such descriptions, however, are computationally cumbersome for two-dimensional (2D) fiber distributions, let alone for fully three-dimensional (3D) fiber populations. In this paper, we develop a new constitutive law for such tissues, based on a novel invariant theory for dispersed transverse isotropy. The invariant theory is derived from a novel closed-form ‘splay invariant’ that can easily handle 3D fiber populations, and that only requires a single parameter in the 2D case. The model fits biaxial data for aortic valve tissue as accurately as the standard structural model. Modification of the fiber stress–strain law requires no reformulation of the constitutive tangent matrix, making the model flexible for different types of soft tissues. Most importantly, the model is computationally expedient in a finite-element analysis, demonstrated by modeling a bioprosthetic heart valve.

Appendices

Appendices

The primary reason for adopting a Gaussian distribution to describe fiber splay is that the corresponding stiffness matrix ${\varvec \kappa}$ can be computed analytically. Alternatively, Hurschler et al (1997) have employed a von Mises distribution for splay in conjunction with a Weibull distribution for crimp that they collectively solve numerically.

1. A two-dimensional splay

We recall that our local coordinate system was chosen so that

$$ \left[\kern-0.15em\left[ {{\mathbf{a}}_0 } \right]\kern-0.15em\right]_i = \left\{ {\begin{array}{*{20}c} 1 \\ 0 \\ 0 \\ \end{array} } \right\},\quad \left[\kern-0.15em\left[ {{\mathbf{e}}_f } \right]\kern-0.15em\right]_i = \left\{ {\begin{array}{*{20}c} {\cos \,\theta } \\ {\sin \,\theta } \\ 0 \\ \end{array} } \right\}, $$

(46)

and as such

$$ \left[\kern-0.15em\left[{\bf e}_f \otimes {\bf e}_f \right]\kern-0.15em\right]_{ij} = \left[\begin{array}{*{20}c} \cos^2 \,\theta & \sin\,\theta \cos\,\theta & 0 \\ \sin\,\theta \cos\,\theta & \sin^2 \,\theta & 0\\ 0& 0 &0 \end{array}\right], $$

(47)

which leads to an expression for Eq. 16 that can be solved analytically; it being,

$$ \left[\kern-0.15em\left[ {\varvec \kappa} (\varsigma) \right]\kern-0.15em\right]_{ij} = \left[\begin{array}{*{20}c} \kappa_{11} & 0 &0 \\ 0 & \kappa_{22} & 0 \\ 0 & 0 & 0 \end{array}\right] ,$$

(48)

where the two non-zero stiffness components have values of

$$\kappa_{11} = \frac{1}{2} + \frac{{\hbox{e}^{-2 \varsigma^2}} \left[ \hbox{erf} \left(\left(\pi + i 4\varsigma^2\right)/{2\!\sqrt{2}\varsigma} \right) + \hbox{erf} \left(\left(\pi - i 4\varsigma^2\right)/ {2\!\sqrt{2}\varsigma}\right) \right]}{4 \hbox{erf} \left({\pi}/ {2\!\sqrt{2} \varsigma} \right)} , $$

(49)

and

$$\kappa_{22} = \frac{1}{2} - \frac{{\hbox{e}^{-2 \varsigma^2}} \left[ \hbox{erf} \left(\left(\pi + i 4\varsigma^2\right)/{2\!\sqrt{2}\varsigma} \right) + \hbox{erf} \left(\left(\pi - i 4\varsigma^2\right)/ {2\!\sqrt{2}\varsigma}\right) \right]}{4 \hbox{erf} \left({\pi}/ {2\!\sqrt{2} \varsigma} \right)} , $$

(50)

wherein $i = \sqrt{-1}$ is the unit imaginary number. The sum of two error functions whose arguments are complex conjugates, i.e., erf(x + i y) + erf(x−i y), produces a real result, and as such, κ₁₁ and κ₂₂ are both real numbers.

2. Three-dimensional splay

Here, the formulation is somewhat different; specifically,

$$ \left[\kern-0.15em\left[ {\bf a}_0 \right]\kern-0.15em\right]_i = \left\{ \begin{array}{*{20}l} 1 \\ 0 \\ 0 \end{array} \right\}, \quad \left[\kern-0.15em\left[ {\bf e}_f \right]\kern-0.15em\right]_i = \left\{ \begin{array}{*{20}c} \cos \,\theta \\ \sin\, \theta \cos\, \phi \\ \sin\, \theta \sin\, \phi \end{array}\right\} , $$

(51)

and as such

$$\left[\kern-0.15em\left[ {\bf e}_f \otimes {\bf e}_f \right]\kern-0.15em\right]_{ij} = \left[\begin{array}{*{20}c} \cos^2 \,\theta & \sin\,\theta \cos\,\theta \cos\,\phi & \sin\,\theta \cos\,\theta \sin\,\phi \\ \sin\,\theta \cos\,\theta \cos\,\phi & \sin^2 \,\theta \cos^2\, \phi & \sin^2 \theta \sin\,\phi \cos\,\phi \\ \sin\,\theta \cos\,\theta \sin\,\phi & \sin^2 \,\theta \sin\,\phi \cos\,\phi & \sin^2 \,\theta \sin^2 \phi \end{array}\right],$$

(52)

which leads to an expression for Eq. 17 that can be solved analytically; it being,

$$ \left[\kern-0.15em\left[ {\varvec \kappa} (\varsigma) \right]\kern-0.15em\right]_{ij} = \left[\begin{array}{*{20}c} \kappa_{11}& 0 & 0\\ 0 & \kappa_{22} & 0\\ 0& 0 & \kappa_{33} \end{array} \right],$$

(53)

where the two non-zero stiffness components have values of

$$\kappa_{11} = \frac{1}{2} + \frac{{\hbox{e}^{-2 \varsigma^2}} \left[ \hbox{erf} \left(\left({\pi + i 4\varsigma^2}\right)/{2\!\sqrt{2}\varsigma} \right) + \hbox{erf} \left(\left({\pi - i 4\varsigma^2}\right)/ {2\!\sqrt{2}\varsigma} \right) \right]}{4 \hbox{erf} \left({\pi}/ {2\!\sqrt{2} \varsigma} \right)} , $$

(54)

which is the same as Eq. 49, and

$$\kappa_{22} =\kappa_{33} = \frac{1}{4} - \frac{{\hbox{e}^{-2 \varsigma^2}} \left[ \hbox{erf} \left(\left({\pi + i 4\varsigma^2}\right)/{2\!\sqrt{2}\varsigma} \right) + \hbox{erf} \left(\left({\pi - i 4\varsigma^2}\right)/ {2\!\sqrt{2}\varsigma} \right) \right]}{8 \hbox{erf} \left({\pi}/ {2\!\sqrt{2} \varsigma} \right)} , $$

(55)

where this value for κ₂₂ is exactly half that of the 2D case (Eq. 50, wherein κ₃₃=0) but here $\kappa_{33} = \kappa_{22}.$

3. Collagen model

A micro-structural model based on the physiology of crimped collagen fibers was recently derived by Freed and Doehring (2005) that we have altered to meet our needs (see Algorithm 1); specifically, given a fiber stretch λ, this model returns the engineering stress σ / λ and tangent modulus dσ / dλ of the fiber. There are four physiological parameters (material constants defined at the top of the algorithm) that the user must supply; three if failure is not to be considered.

A typical pair of response curves are plotted in Fig. 9. The discontinuity observed in the dσ/dλ curve indicates that the model presented in Algorithm 1 predicts a stress σ response that is continuous and differentiable in stretch λ up to fiber failure, which is not depicted in this figure. The values assigned to the model to obtain these curves were $H_0/r_0 = 27.5,$ $R_0/r_0 = 2,$ and E _f =45 MPa, which results in a transitional stretch Λ of 1.1. This is but one example of a fiber stress/stretch model.

Fig. 9

Typical soft-tissue response curves for σ / λ and dσ / dλ

Algorithm 1

Given H₀/r₀, E _f and λ _u , where H₀ is the initial wavelength of crimp, R₀ is the initial amplitude of crimp, r₀ is the initial fibril radius, E _f is the elastic modulus of the fiber in the linear region, and λ _u is its ultimate stretch, then:

Set r₀=1 so that H₀≡ H₀/r₀ & R₀≡ R₀/r₀

Compute the constant parameters: $L_0 =\bigl( ( 2 \pi R_0 )^2 + H_0^2 \bigr)^{1/2}, \qquad \Lambda = L_0 / H_0 < \lambda_u,$ $E_s = E_f H_0 / \{ H_0 + [ 1 + 37/6\pi^2 + 2 ( L_0 / \pi r_0 )^2 ] ( L_0 - H_0 ) \},$

where L₀ is the chord length of helix over one wavelength, while Λ is the stretch and E _s is the secant modulus at the transition between the toe and linear regions.

If λ ≤ Λ Then

$$\begin{aligned} \xi &= 6 ( \pi r_0 )^2 [ \Lambda^2 + ( 4\pi^2 - 1 ) \lambda^2 ] \lambda \\ &/ \bigl\{ \Lambda\{ 3 H_0^2 ( \Lambda^2 - \lambda^2 ) [ 3 \Lambda^2 + ( 8 \pi^2 - 3 )\lambda^2 ] + 8 ( \pi r_0 )^2 [ 10 \Lambda^2 + ( 3 \pi^2 - 10 )\lambda^2 ] \} \bigr\},\\ &\hbox{d}\xi / \hbox{d}\lambda = \{ 18 H_0^2 (\pi r_0 )^2 [ 3 \Lambda^6 + ( 28 \pi^2 - 3 ) \Lambda^4 \lambda^2 + ( 32\pi^4 - 8 \pi^2 - 3 ) \Lambda^2 \lambda^4\\ &\quad + ( 32 \pi^4 - 20 \pi^2 + 3 ) \lambda^6 ] + 48 ( \pi r_0 )^4 [ 10 \Lambda^4\\ &\quad+( 117 \pi^2 - 20 ) \Lambda^2 \lambda^2 + ( 12 \pi^4 - 43 \pi^2 + 10 )\lambda^4 ] \}\\ &/ \bigl\{ \Lambda \{ 3 H_0^2 ( \Lambda^2 - \lambda^2 ) [ 3 \Lambda^2 + ( 8 \pi^2 - 3 ) \lambda^2 ] \\ &\quad+8 ( \pi r_0 )^2 [ 10 \Lambda^2 + ( 3 \pi^2 - 10 ) \lambda^2 ] \}^2 \bigr\},\\ &\sigma / \lambda = \xi E_s ( \lambda - 1 ) / \lambda \quad \& \quad \hbox{d}\sigma / \hbox{d}\lambda = E_s [ \xi + ( \lambda - 1 ) \hbox{d}\xi / \hbox{d}\lambda ] \end{aligned} $$

Else If Λ ≤ λ ≤ λ _u Then

$$\sigma / \lambda = E_s ( \Lambda - 1 ) / \lambda + E_f ( \lambda - \Lambda ) / \lambda \quad \& \quad \hbox{d}\sigma / \hbox{d}\lambda = E_f$$

Else Fibril Failure

$$\sigma / \lambda = 0 \quad \& \quad \hbox{d}\sigma / \hbox{d}\lambda = 0.$$

Return σ / λ and dσ / dλ.

4. Voigt notation for dyadic products of tensors

Because the Lagrangian fields for stress S _ij and strain E _ij , i,j=1,2,3, are symmetric tensors, it is customary to express their components as six-dimensional arrays S_α and E_α, α=1,2,...,6 (Belytschko et al. 2000, pp 615–618). These arrays are not vector fields in the sense that they do not obey the tensor-transformation law; nevertheless, they have proven to be very useful in the employ of finite elements.

A linear approximation for stress increment dS _ij relates to the strain increment $\hbox{d}E_{ij} = \frac{1}{2} \hbox{d}C_{ij}$ in the Lagrangian frame through

$$ \hbox{d}S_{ij} ={\mathbb{C}}_{ijk\ell} \hbox{d} E_{k\ell} \quad \hbox{with} \quad {\mathbb{C}}_{ijk\ell} = \frac{\partial S_{\! ij}}{\partial E_{k\ell}} \quad \therefore \quad \hbox{d}S_{\! \alpha} = {\mathbb{C}}_{\alpha\beta} \hbox{d}E_{\beta} , $$

(56)

where the tangent modulus ${\mathbb{C}}$ has a Voigt representation of

$$ \left[\left[ {\mathbb{C}} \right]\right]_{\alpha\beta} = \left[\begin{array}{*{20}l} {\mathbb{C}}_{11} & {\mathbb{C}}_{12} & {\mathbb{C}}_{13} & {\mathbb{C}}_{14} & {\mathbb{C}}_{15} & {\mathbb{C}}_{16}\\ {\mathbb{C}}_{21}& {\mathbb{C}}_{22} & {\mathbb{C}}_{23} & {\mathbb{C}}_{24} & {\mathbb{C}}_{25} & {\mathbb{C}}_{26}\\ {\mathbb{C}}_{31}& {\mathbb{C}}_{32} & {\mathbb{C}}_{33} & {\mathbb{C}}_{34} & {\mathbb{C}}_{35} & {\mathbb{C}}_{36}\\ {\mathbb{C}}_{41}& {\mathbb{C}}_{42} & {\mathbb{C}}_{43} & {\mathbb{C}}_{44} & {\mathbb{C}}_{45} & {\mathbb{C}}_{46}\\ {\mathbb{C}}_{51}& {\mathbb{C}}_{52} & {\mathbb{C}}_{53} & {\mathbb{C}}_{54} & {\mathbb{C}}_{55} & {\mathbb{C}}_{56}\\ {\mathbb{C}}_{61} &{\mathbb{C}}_{62} & {\mathbb{C}}_{63} & {\mathbb{C}}_{64} & {\mathbb{C}}_{65} & {\mathbb{C}}_{66} \end{array} \right] = \left[\begin{array}{*{20}l} {\mathbb{C}}_{1111} & {\mathbb{C}}_{1122} & {\mathbb{C}}_{1133} & {\mathbb{C}}_{1112} & {\mathbb{C}}_{1113} & {\mathbb{C}}_{1123} \\ {\mathbb{C}}_{2211} & {\mathbb{C}}_{2222} & {\mathbb{C}}_{2233} & {\mathbb{C}}_{2212} & {\mathbb{C}}_{2213} & {\mathbb{C}}_{2223} \\ {\mathbb{C}}_{3311} & {\mathbb{C}}_{3322} & {\mathbb{C}}_{3333} & {\mathbb{C}}_{3312} & {\mathbb{C}}_{3313} & {\mathbb{C}}_{3323} \\ {\mathbb{C}}_{1211} & {\mathbb{C}}_{1222} & {\mathbb{C}}_{1233} & {\mathbb{C}}_{1212} & {\mathbb{C}}_{1213} & {\mathbb{C}}_{1223} \\ {\mathbb{C}}_{1311} & {\mathbb{C}}_{1322} & {\mathbb{C}}_{1333} & {\mathbb{C}}_{1312} & {\mathbb{C}}_{1313} & {\mathbb{C}}_{1323} \\ {\mathbb{C}}_{2311} & {\mathbb{C}}_{2322} & {\mathbb{C}}_{2333} & {\mathbb{C}}_{2312} & {\mathbb{C}}_{2313} & {\mathbb{C}}_{2323} \end{array}\right], $$

(57)

with ${\mathbb{C}}_{\alpha\beta} \neq {\mathbb{C}}_{\beta\alpha}$ unless the ${\mathbb{C}}_{ijk\ell}$ possess major symmetry ${\mathbb{C}}_{ijk\ell} = {\mathbb{C}}_{k\ell ij}.$ The tangent modulus always possesses minor symmetries ${\mathbb{C}}_{ijk\ell} = {\mathbb{C}}_{ij\ell k} = {\mathbb{C}}_{jik\ell} = {\mathbb{C}}_{ji\ell k}$ because of the inherent symmetries in S _ij and C _ij . Major symmetry follows automatically if stress S _ij is given by some potential function F such that $S_{ij} = \partial F / \partial E_{ij},$as in hyperelasticity.

Three dyadic products of tensor fields arise naturally when deriving tangent moduli, in the sense of Eqs. D1 and D2. Of these three, there are two fundamental dyadic products that we denote as $\hbox{A} \otimes \hbox{B}$ and $\hbox{A} \boxtimes \hbox{B};$ we call them the outer- and inner-dyadic products, respectively, or the circle- and box-products for short, wherein A and B are taken to be symmetric fields. The third dyadic product, denoted as $\hbox{A} \odot \hbox{A},$ is a special case of the inner-dyadic product $\hbox{A} \boxtimes \hbox{A}$ wherein both arguments are the same tensor field—in this case, A. We know of no publication where $\hbox{A} \boxtimes \hbox{B}$ has been defined as we use it, which is the main reason for writing this appendix.

The circle product $\hbox{A} \otimes \hbox{B}$ is well known and has components

$$(\hbox{A} \otimes \hbox{B})_{ijk\ell} = \frac{1}{4} \bigl( A_{ij} B_{k\ell} + A_{ij} B_{\ell k} + A_{ji} B_{k\ell} + A_{ji} B_{\ell k} \bigr) \equiv A_{ij} B_{k\ell} , $$

(58)

which reduce down to the simple expression $( \hbox{A} \otimes \hbox{B} )_{ijk\ell} = A_{ij} B_{k\ell}$ as a consequence of A and B being symmetric. This dyadic product has the simple Voigt notation of

$$\left[\left[ \hbox{A} \otimes \hbox{B}\right]\right]_{\alpha\beta} = \left[\begin{array}{*{20}c} A_{11} B_{11} & A_{11} B_{22} & A_{11} B_{33} & A_{11} B_{12} & A_{11} B_{13} & A_{11} B_{23} \\ A_{22} B_{11} & A_{22} B_{22} & A_{22} B_{33} & A_{22} B_{12} & A_{22} B_{13} & A_{22} B_{23} \\ A_{33} B_{11} & A_{33} B_{22} & A_{33} B_{33} & A_{33} B_{12} & A_{33} B_{13} & A_{33} B_{23} \\ A_{12} B_{11}& A_{12} B_{22} & A_{12} B_{33} & A_{12} B_{12} & A_{12} B_{13} & A_{12} B_{23}\\ A_{13} B_{11}& A_{13} B_{22} & A_{13} B_{33} & A_{13} B_{12} & A_{13} B_{13} & A_{13} B_{23}\\ A_{23} B_{11}& A_{23} B_{22} & A_{23} B_{33} & A_{23} B_{12} & A_{23} B_{13} & A_{23} B_{23} \end{array}\right]. $$

(59)

Matrix $( \hbox{A} \otimes \hbox{B} )_{\alpha\beta}$ is not symmetric unless A = B; however, $( \hbox{A} \otimes \hbox{B} ) + ( \hbox{B} \otimes \hbox{A} )$ does yield a symmetric matrix in Voigt notation. This is because $\hbox{A} \otimes \hbox{B}$ possesses minor symmetry, but not major symmetry, therefore $( \hbox{A} \otimes \hbox{B} )_{\alpha\beta} \neq ( \hbox{A} \otimes \hbox{B} )_{\beta\alpha},$ in general. The box product $\hbox{A} \boxtimes \hbox{B}$ is a new dyadic product that we define by the sum

$$(\hbox{A} \boxtimes \hbox{B})_{ijk\ell} = \frac{1}{4} \bigl( A_{ik} B_{j\ell} + A_{i\ell} B_{jk} + A_{jk} B_{i\ell} + A_{j\ell} B_{ik} \bigr) ,$$

(60)

and possesses minor and major symmetries, therefore $( \hbox{A} \boxtimes \hbox{B} )_{\alpha\beta} = ( \hbox{A} \boxtimes \hbox{B} )_{\beta\alpha};$ consequently, the box product commutes, viz., $\hbox{A} \boxtimes \hbox{B} = \hbox{B} \boxtimes \hbox{A}.$ A special case that arises frequently in practice is when both tensor arguments of this product are the same. This dyadic product is well known and has its own notation of $\hbox{A} \odot \hbox{A}$ (cf., Holzapfel (2000, pg. 254)) with components

$$(\hbox{A} \odot \hbox{A})_{ijk\ell} = \frac{1}{2} \bigl( A_{ik} A_{j\ell} + A_{i\ell} A_{jk} \bigr) . $$

(61)

The Voigt representation of $( \hbox{A} \boxtimes \hbox{B} )_{\alpha\beta}$ has the following symmetric matrix components:

$$ \left\{ \begin{array}{*{20}l} (\hbox{A} \boxtimes \hbox{B} )_{11} = A_{11} B_{11} ,\cr ( \hbox{A} \boxtimes \hbox{B} )_{12} = A_{12} B_{12} ,\\ ( \hbox{A}\boxtimes \hbox{B} )_{13} = A_{13} B_{13} ,\\ ( \hbox{A} \boxtimes \hbox{B})_{14} = \frac{1}{2} \bigl( A_{11} B_{12} + A_{12} B_{11} \bigr) ,\\ (\hbox{A} \boxtimes \hbox{B} )_{15} = \frac{1}{2} \bigl( A_{11} B_{13} + A_{13} B_{11} \bigr) ,\\ ( \hbox{A} \boxtimes \hbox{B} )_{16} = \frac{1}{2} \bigl( A_{12} B_{13} + A_{13} B_{12} \bigr) ,\\ ( \hbox{A}\boxtimes \hbox{B} )_{22} = A_{22} B_{22} ,\\ ( \hbox{A} \boxtimes \hbox{B} )_{23} = A_{23} B_{23} , \\ ( \hbox{A} \boxtimes \hbox{B} )_{24} = \frac{1}{2} \bigl( A_{12} B_{22} + A_{22} B_{12} \bigr) , \\ ( \hbox{A}\boxtimes \hbox{B} )_{25} = \frac{1}{2} \bigl( A_{12} B_{23} + A_{23} B_{12} \bigr) , \\ ( \hbox{A} \boxtimes \hbox{B} )_{26} = \frac{1}{2} \bigl( A_{22} B_{23} + A_{23} B_{22} \bigr) ,\\ ( \hbox{A} \boxtimes \hbox{B} )_{33} = A_{33} B_{33} ,\\ ( \hbox{A} \boxtimes \hbox{B} )_{34} =\frac{1}{2} \bigl( A_{13} B_{23} + A_{23} B_{13} \bigr) ,\\ ( \hbox{A}\boxtimes \hbox{B} )_{35} = \frac{1}{2} \bigl( A_{13} B_{33} + A_{33} B_{13} \bigr) , \\ ( \hbox{A} \boxtimes \hbox{B} )_{36} = \frac{1}{2} \bigl( A_{23} B_{33} + A_{33} B_{23} \bigr) , \\ ( \hbox{A} \boxtimes \hbox{B} )_{44} = \frac{1}{4} \bigl( A_{11} B_{22} + 2 A_{12} B_{12} + A_{22} B_{11} \bigr) , \\ ( \hbox{A} \boxtimes \hbox{B} )_{45} = \frac{1}{4} \bigl( A_{11} B_{23} + A_{13} B_{12} + A_{12} B_{13} + A_{23} B_{11} \bigr) ,\\ ( \hbox{A} \boxtimes \hbox{B} )_{46} = \frac{1}{4} \bigl( A_{12} B_{23} + A_{13} B_{22} + A_{22} B_{13} + A_{23} B_{12} \bigr) ,\\ ( \hbox{A} \boxtimes \hbox{B} )_{55} = \frac{1}{4} \bigl( A_{11} B_{33} + 2 A_{13} B_{13} + A_{33} B_{11} \bigr) ,\\ ( \hbox{A} \boxtimes \hbox{B} )_{56} = \frac{1}{4} \bigl( A_{12} B_{33} + A_{13} B_{23} + A_{23} B_{13} + A_{33} B_{12} \bigr) , \\ (\hbox{A} \boxtimes \hbox{B} )_{66} = \frac{1}{4} \bigl( A_{22} B_{33} + 2 A_{23} B_{23} + A_{33} B_{22} \bigr) \end{array}\right. ,$$

(62)

while the Voigt representation of $( \hbox{A} \odot \hbox{A})_{\alpha\beta}$ has symmetric matrix components:

$$ \left\{ \begin{array}{*{20}l} ( \hbox{A} \odot \hbox{A} )_{11} = A^2_{11} ,\\ ( \hbox{A} \odot\hbox{A} )_{12} = A^2_{12} , \\ ( \hbox{A} \odot \hbox{A} )_{13} =A^2_{13} ,\\ ( \hbox{A} \odot \hbox{A} )_{14} = A_{11} A_{12} , \\ ( \hbox{A}\odot \hbox{A} )_{15} = A_{11} A_{13} ,\\ ( \hbox{A} \odot \hbox{A} )_{16} = A_{12} A_{13} ,\\ ( \hbox{A} \odot \hbox{A} )_{22} = A^2_{22} , \\ ( \hbox{A} \odot \hbox{A} )_{23} = A^2_{23} ,\\ ( \hbox{A} \odot \hbox{A} )_{24} = A_{12} A_{22} , \\ ( \hbox{A} \odot \hbox{A} )_{25} = A_{12} A_{23} , \\ (\hbox{A} \odot \hbox{A} )_{26} = A_{22} A_{23} , \\ ( \hbox{A} \odot \hbox{A})_{33} = A^2_{33} , \\ ( \hbox{A} \odot \hbox{A} )_{34} = A_{13} A_{23} ,\\ ( \hbox{A} \odot \hbox{A} )_{35} = A_{13} A_{33} ,\\ ( \hbox{A} \odot \hbox{A} )_{36} = A_{23} A_{33} ,\\ ( \hbox{A} \odot \hbox{A} )_{44} = \frac{1}{2} \bigl( A_{11} A_{22} + A^2_{12} \bigr) ,\\ ( \hbox{A} \odot\hbox{A} )_{45} = \frac{1}{2} \bigl( A_{11} A_{23} + A_{13} A_{12} \bigr) ,\\ ( \hbox{A} \odot \hbox{A} )_{46} = \frac{1}{2} \bigl( A_{12} A_{23} + A_{13} A_{22} \bigr) , \\ ( \hbox{A} \odot \hbox{A} )_{55} = \frac{1}{2} \bigl( A_{11} A_{33} + A^2_{13} \bigr) ,\\ ( \hbox{A} \odot \hbox{A} )_{56} = \frac{1}{2} \bigl( A_{12} A_{33} + A_{13} A_{23} \bigr) , \\ ( \hbox{A} \odot \hbox{A} )_{66} = \frac{1}{2} \bigl( A_{22} A_{33} + A^2_{23} \bigr) , \end{array}\right.$$

(63)

wherein terms on the right-hand side are tensor components while those on the left-hand side are the Voigt components. Obviously, $\odot$ and $\boxtimes$ are different products.

6.1 Example

The theory of linear elasticity has stress components

$$ \left\{\begin{array}{*{20}l} S_{1} = S_{11} = {\lambda ( E_1 + E_2 + E_3 ) + 2 \mu E_1 }, \\ S_{2} = S_{22} = {\lambda ( E_1 + E_2 + E_3 ) + 2 \mu E_2} ,\\ S_{3} = S_{33} = {\lambda ( E_1 + E_2 + E_3 ) + 2 \mu E_3} , \\ S_{4} = S_{12} = {\mu E_4} , \\ S_{5} = S_{13} ={\mu E_5 },\\ S_{6}= S_{23} = {\mu E_6 }, \end{array}\right. \quad \hbox{where} \quad \left\{\begin{array}{*{20}l} E_1 = E_{11} , \\ E_2 = E_{22} , \\ E_3 = E_{33} , \\ E_4 = 2E_{12} , \\ E_5 = 2E_{13} , \\ E_6 = 2E_{23} , \\ \end{array}\right. $$

and as such, its tangent modulus is readily determined to be

$$ (\mathbb{C})_{\alpha\beta} = \lambda (\hbox{I} \otimes \hbox{I})_{\alpha\beta} + 2 \mu ( \hbox{I} \odot \hbox{I} )_{\alpha\beta} ,$$

wherein λ and μ are the elastic moduli, and where

$$ \left[\left[ \hbox{I} \right]\right]_{\alpha} = \left\{ \begin{array}{*{20}l} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right\} , \quad \left[\left[\hbox{I} \otimes \hbox{I} \right]\right]_{\alpha\beta} = \left\{ \begin{array}{*{20}l} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] , \quad \left[\left[ \hbox{I} \odot \hbox{I} \right]\right]_{\alpha\beta} = \left\{ \begin{array}{*{20}l} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 &{1 / 2} & 0 & 0 \\ 0 & 0 & 0 & 0 &{1 / 2} & 0 \\ 0 & 0 & 0 & 0 & 0 &{1 / 2} \end{array}\right] $$

The $\frac{1}{2}$’s in $(\hbox{I} \odot \hbox{I})_{\alpha\beta}$ are offset by the 2’s present in E_β.