Due to layered lamellae structure of ID known from the anatomy, due to the physical properties of the collagen fibres and due to their spiral configuration, the intervertebral disc is modelled as a transversally isotropic composite ambient.

Let us define a local Cartesian coordinate system, an ordered triple (*x*_{1}, *x*_{2}, *x*_{3}) = (1, 2, 3) where the axis *x*_{1} ≡ 1 is parallel to the fibres direction, the axis *x*_{2} ≡ *2* is perpendicular to the fibres direction, lying on the tangent plane to the lamella and the axis *x*_{3} ≡ 3 is perpendicular to the (*x*_{1}, *x*_{2}) plane. Let us suppose the global coordinate system (*r*, *t, z*) = $(\overline{1},\overline{2},\overline{3})$ which is rotated from the local coordinate system by the angle *α*, with *x*_{3} = 3 being the axis of rotation, see Fig. 2. While forming the constitutive, i.e. physical equations for the transversally isotropic material we start from the generalized Hook law for anisotropic body. The Cauchy stress tensor *σ*_{ij} is of the form

Figure 2 Intervertebral disc lamella fragment with the “old” local (*x*1,*x*2,*x*3) = (1, 2, 3) and “new” (*r*, *t, z)* coordinate system with *α* being the slope of the fibre ascending

$${\sigma}_{ij}={c}_{ijkl}{\epsilon}_{kl}$$3.5

with *c*_{ijkl} being the elastic coefficients united in the elastic tensor operator with 81 coefficients and *ɛ*_{kl} the Green deformation tensor components. By using the symmetry of the stress tensor *σ*_{ij} = *σ*_{ji}, *(i, j =* 1, 2, 3) and the symmetry of the strain tensor *ɛ*_{kl} = *ɛ*_{lk}, *(k, l =* 1, 2, 3) the number of coefficients decreases to 36. Moreover, subject to thermodynamic laws, it is possible to prove, the validity of the relation, [5]

$${c}_{ijkl}={c}_{klij}$$3.6

$${s}_{ijkl}={s}_{klij}$$3.7

(3.6) performs the double symmetry of the compliance and (3.7) matrix coefficients. In the most general case the terms (3.6) decrease the number of the coefficients to 21. Due to the 2nd law of thermodynamics it can be shown that the tensor operators *c*_{ijkl} = (*s*_{ijkl})^{-1} is a positive definite operator [11]. Relation (3.5) can be then rewritten in the reverse form

$${\epsilon}_{ij}={s}_{ijkl}{\sigma}_{kl},\phantom{\rule{1em}{0ex}}(i,j=1,2,3)$$3.8

Note 1:

a) If an elastic theory problem is elaborated on the atomic level, by using some other assumptions, the number of the coefficient can be even more decreased. This depends on the crystal character and composition of the material [5].

b) Since we have restricted our investigation to the Cartesian coordinate system, we need not to distinguish between covariant and contra-variant tensors. Moreover, as we would like to make the notation more simple, we establish the restriction of the indexes number of stress *σ*_{ij} and strain *ɛ*_{ij} in the following way:

$$11\to 1,22\to 2,33\to 3,12\to 4,23\to 5,31\to 6$$3.9$${\sigma}_{ij\phantom{\rule{thinmathspace}{0ex}}}=\left\{\begin{array}{l}{\sigma}_{11},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{22},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{33};\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{12},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{23},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{31}\\ {\sigma}_{1},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{2},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{3};\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{4},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{5},\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{6}\end{array}\right.$$3.10$${\epsilon}_{ij\phantom{\rule{thinmathspace}{0ex}}}=\left\{\begin{array}{l}{\epsilon}_{11},\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{22},\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{33};\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2{\epsilon}_{12},\phantom{\rule{thinmathspace}{0ex}}2{\epsilon}_{23},\phantom{\rule{thinmathspace}{0ex}}2{\epsilon}_{31}\\ {\epsilon}_{1},\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{2},\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{3};\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{4},\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{5},\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{6}\end{array}\right.$$3.11This restriction of the indexes decreases the order of the tensor twice. In the such a way e.g. a matrix will be aligned to a vector, a tensor of the 4th order will acquire the form of matrix. Hence, we will reduce the number of indexes twice. In the following text we will use *σ*_{5} instead of *σ*_{23}, *ɛ*_{3} instead of *ɛ*_{33}, *ɛ*_{4} instead of 2*ɛ*_{31}. Moreover using the index restriction (3.9) also for tensors *c*_{ijkl} and *s*_{ijkl}, we will write *c*_{11} instead of *c*_{1111}, *s*_{36} instead of *s*_{3331}, *c*_{15} = *c*_{51} instead of *c*_{1123} = *c*_{2311}, *s*_{26} = *s*_{62} instead of *s*_{2231} = *s*_{3122}, etc. Hereinafter we utilize the differentiability of the elastic deformation energy density function 𝒲. Its differential when using the Einstein summation convention and considering the Note 1 can be expressed in the form

$$\text{d}\mathcal{W}={\sigma}_{i}\text{d}{\epsilon}_{i}={c}_{ij}{\epsilon}_{j}\text{d}{\epsilon}_{i}$$3.12

i.e., for entire body of the volume *V*
$$\mathcal{W}=\underset{V}{\int}\mathrm{d}\mathcal{W}({\epsilon}_{i})=\underset{V}{\int}{c}_{ij}{\epsilon}_{j}\mathrm{d}{\epsilon}_{i}$$3.13

I accordance with Assumptions in Chapter III.A, the domain *Ω*, i.e. intervertebral disc is considered as fully elastic, which ensures the linearity herein and integral in (3.13) does not depend on the integration path. Here, the necessary and sufficient condition is dW being the total differential of function W(*ɛ*_{i}), i.e.
$$\mathrm{d}\mathcal{W}=\frac{\mathrm{\partial}\mathcal{W}}{\mathrm{\partial}{\epsilon}_{i}}\mathrm{d}{\epsilon}_{i}$$3.14

By comparing the equations (3.12) and (3.14) we get
$$\frac{\mathrm{\partial}\mathcal{W}}{\mathrm{\partial}{\epsilon}_{i}}={c}_{ij}{\epsilon}_{j}$$3.15

Exploitating the fact that 𝒲 = 𝒲 (*ɛ*_{i}) is a differentiable function we can write
$$\frac{{\mathrm{\partial}}^{2}\mathcal{W}}{\mathrm{\partial}{\epsilon}_{i}\mathrm{\partial}{\epsilon}_{j}}={c}_{ij}$$3.16

respectively
$$\frac{{\mathrm{\partial}}^{2}\mathcal{W}}{\mathrm{\partial}{\epsilon}_{j}\mathrm{\partial}{\epsilon}_{i}}={c}_{ji}$$3.17

The fact, that the order which the derivatives are performed is not dependent on the structure of the material, the symmetry of the elastic coefficients (mathematically) implies

$${c}_{ij}={c}_{ji}$$3.18

That is why the matrix of elastic constants has 21 elements. Hence, the equation (3.13) in the unit volume can be rewritten as
$$\mathcal{W}=\frac{1}{2}{c}_{ij}{\epsilon}_{i}{\epsilon}_{j},\phantom{\rule{1em}{0ex}}(i,j=1,...,6)$$3.19

Similarly, the validity of the relationship
$$\mathcal{W}=\frac{1}{2}{s}_{ij}{\sigma}_{i}{\sigma}_{j},\phantom{\rule{1em}{0ex}}(i,j=1,...,6)$$3.20

can be proved, e.g. [10], whereas *s*_{ij} are the elastic moduli components that read

$${s}_{ij}=[{c}_{ij}{]}^{-1},\phantom{\rule{1em}{0ex}}(i,j=1,\dots ,6)$$3.21

Let us suppose in each point of *Ω* there exists a plane of symmetry, e.g. (1,2) - tangent plane to the lamella of the intervertebral disc, see Fig. 2). Then

$${c}_{ij}=0,\phantom{\rule{1em}{0ex}}(i=1,2,3\wedge j=5,6)$$3.22

In this way the number of coefficients decreases to 13. In each point of the domain *Ω* there exist three planes of symmetry, planes (1,2), (2,3), (3,1), perpendicular each other. That implies *Ω* is an orthotropic body that in addition reads

$${c}_{ij}=0,\phantom{\rule{1em}{0ex}}(i=1,2,3,5\wedge j=4,6)$$3.23

Due to the elastic symmetry *c*_{ij} = c_{ji} the number of elastic coefficient decreases to 9. Finally, the corresponding compliance matrix, i.e the matrix involving the elastic moduli coefficients, is
$$\mathbf{S}=[{s}_{ij}]=\left[\begin{array}{cccccc}\frac{1}{{E}_{1}}& -\frac{{\nu}_{21}}{{E}_{2}}& -\frac{{\nu}_{31}}{{E}_{3}}& 0& 0& 0\\ -\frac{{\nu}_{12}}{{E}_{1}}& \frac{1}{{E}_{2}}& -\frac{{\nu}_{32}}{{E}_{3}}& 0& 0& 0\\ -\frac{{\nu}_{13}}{{E}_{1}}& -\frac{{\nu}_{23}}{{E}_{2}}& \frac{1}{{E}_{3}}& 0& 0& 0\\ 0& 0& 0& \frac{1}{{G}_{12}}& 0& 0\\ 0& 0& 0& 0& \frac{1}{{G}_{23}}& 0\\ 0& 0& 0& 0& 0& \frac{1}{{G}_{31}}\end{array}\right]$$3.24

with $\frac{{\nu}_{ij}}{{E}_{i}}=\frac{{\nu}_{ji}}{{E}_{j}},(i,j=1,2,3)$. In matrix (3.24) the material constants *E*_{1}, *E*_{2}, *E*_{3} are the directional elastic moduli in the 1, 2 and 3 axes direction, see Fig. 2; *ν*_{ij} are Poisson coefficients of the transversal deformation with *ν*_{ij} = –*ɛ*_{j}/*ɛ*_{i} and *G*_{12}, *G*_{23}, *G*_{31} are the shear moduli corresponding to the planes (1,2), (2,3), (3,1). Thus, the orthotropic body reads

$${\sigma}_{1}={c}_{11}{\epsilon}_{1}+{c}_{12}{\epsilon}_{2}+{c}_{13}{\epsilon}_{3}{\sigma}_{2}={c}_{21}{\epsilon}_{1}+{c}_{22}{\epsilon}_{2}+{c}_{23}{\epsilon}_{3}{\sigma}_{3}={c}_{31}{\epsilon}_{1}+{c}_{32}{\epsilon}_{2}+{c}_{33}{\epsilon}_{3}{\sigma}_{4}={c}_{44}{\epsilon}_{4};{\sigma}_{55}={c}_{55}{\epsilon}_{5};{\sigma}_{6}={c}_{66}{\epsilon}_{6}$$3.25

The deformation energy density (3.19) can be rewritten in details $\mathcal{W}=\frac{1}{2}{c}_{11}{\epsilon}_{1}^{2}+{c}_{12}{\epsilon}_{1}{\epsilon}_{2}+{c}_{13}{\epsilon}_{1}{\epsilon}_{3}+\frac{1}{2}{c}_{22}{\epsilon}_{2}^{2}+{c}_{23}{\epsilon}_{2}{\epsilon}_{3}+\frac{1}{2}{c}_{33}{\epsilon}_{3}^{2}+\frac{1}{2}{c}_{44}{\epsilon}_{4}^{2}+\frac{1}{2}{c}_{55}{\epsilon}_{5}^{2}+\frac{1}{2}{c}_{66}{\epsilon}_{6}^{2}$. In our case, regarding to the structure of the intervertebral disc annulus its layered configuration of lamellae and twisted collagen fibres, we will suppose that the plane (2,3) is the isotropy plane, which means the constants *c*_{ij} will read

$${c}_{12}={c}_{13},{c}_{22}={c}_{33},{c}_{44}={c}_{66}$$3.26

So, it remains 9 - 3 = 6 different material constants *c*11, *c*12, *c*22, *c*23, *c*44, *c*55, whereas by using the small deformation tensor invariant it can be proven, see e.g. [10]
$${c}_{55}=\frac{1}{2}({c}_{22}-{c}_{23})$$3.27

Afterwards, taking
$$\mathit{\sigma}=\left[\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\sigma}_{3}\\ {\sigma}_{4}\\ {\sigma}_{5}\\ {\sigma}_{6}\end{array}\right]\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{1em}{0ex}}\mathit{\epsilon}=\left[\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\epsilon}_{3}\\ {\epsilon}_{4}\\ {\epsilon}_{5}\\ {\epsilon}_{6}\end{array}\right]$$

the relationship “stress - strain” representing the matrix form of constitutive equations for the annulus fibrosis with respect to the local coordinate system (LCS) will be of the form

$$\sigma =C\epsilon $$3.28

where **C**[_{6x6}] is the compliance matrix
$$\mathbf{C}=[{c}_{ij}]=\left[\begin{array}{cccccc}{c}_{11}& {c}_{12}& {c}_{12}& 0& 0& 0\\ {c}_{12}& {c}_{22}& {c}_{23}& 0& 0& 0\\ {c}_{12}& {c}_{23}& {c}_{22}& 0& 0& 0\\ 0& 0& 0& {c}_{44}& 0& 0\\ 0& 0& 0& 0& {c}_{55}& 0\\ 0& 0& 0& 0& 0& {c}_{44}\end{array}\right]$$3.29

with the 6 different coefficients: ${c}_{11}={E}_{1}(1-{\nu}_{23}^{2}),{c}_{12}={E}_{2}({\nu}_{12}+{\nu}_{23}{\nu}_{12}),{c}_{22}={E}_{2}(1-{\nu}_{12}^{2}),{c}_{23}={E}_{2}({\nu}_{23}+{\nu}_{12}^{2}),{c}_{44}={G}_{12}D,{c}_{55}={G}_{23}D,D=1-2{\nu}_{12}^{2}{\nu}_{23}-2{\nu}_{12}^{2}-{\nu}_{23}^{2}$, whereas the inverse “strain - stress” relation will read

$$\epsilon =\text{S}\sigma $$3.30

The equation (3.28) expresses the Cauchy stress tensor components *σ*_{j} dependence on the deformations *ɛ*_{i}, *(i =* 1, 2, ...6). In terms of previous assumptions imposed in Chapter III.A while solving our problem, we start from the small deformations theory that provides, e.g. [17]
$${\epsilon}_{ij}=\frac{1}{2}({u}_{i,j}+{u}_{j,i}),\phantom{\rule{1em}{0ex}}(i,j=1,2,3)$$3.31

Table 1 Angles between the coordinates in the “old” and “new” coordinate system and corresponding directional cosines defined in the lamella domain

Table 2 Directional cosines notation between the “old” and “new” coordinate system defined in the lamella domain

where *u*_{i} are the displacement vector **u***(X)* components, *X* ∈ *Ω* and *u*_{•,i} means the derivation of the item *u*_{•} with respect to *x*_{i}. The resulting matrix of elastic moduli, the compliance matrix **s**_{ij} = [*s*_{ij}] with regarding to the GCS of the annulus fibrosis can be obtained from the relationship
$$\overline{\mathbf{s}}=\frac{1}{2}[({\overline{s}}_{ij}(+\alpha )+{\overline{s}}_{ij}(-\alpha ))],\phantom{\rule{1em}{0ex}}(i,j=1,2,...,6)$$3.32

i.e the averaging of the elastic moduli due to annulus fibrosis being layered and to the orientation within the particular lamellae. By using the same procedure, the relationship for the elastic coefficients can be derived, where
$$\overline{\mathbf{c}}=\frac{1}{2}[({\overline{c}}_{ij}(+\alpha )+{\overline{c}}_{ij}(-\alpha ))],(i,j=1,2,...,6)$$3.33

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