On the Problematic Case of Product Approximation in Backus Average

Abstract

Elastic anisotropy might be a combined effect of the intrinsic anisotropy and the anisotropy induced by thin-layering. The Backus average, a useful mathematical tool, allows us to describe such an effect quantitatively. The results are meaningful only if the underlying physical assumptions are obeyed, such as static equilibrium of the material. We focus on the only mathematical assumption of the Backus average, namely, product approximation. It states that the average of the product of a varying function with a nearly constant function is approximately equal to the product of the averages of those functions. We analyse particular problematic case for which the aforementioned assumption is inaccurate. Furthermore, we focus on the seismological context. We examine the inaccuracy’s effect on the wave propagation in a homogenous medium—obtained using the Backus average—equivalent to thin layers. Numerical simulations indicate clearly that the product approximation inaccuracy has negligible effect on wave propagation; irrespective of layers’ symmetries. To give the results a practical focus, we show that the problematic case of product approximation is strictly related to the negative Poisson’s ratio of constituents layers. We discuss the laboratory and well-log cases in which such a ratio has been noticed. Upon thorough literature review, it occurs that examples of so-called auxetic materials (media that have negative Poisson’s ratio) are not extremely rare exceptions as thought previously. The investigation and derivation of Poisson’s ratio for materials exhibiting symmetry classes up to monoclinic become a significant part of this paper. In addition to the main objectives, we also show that the averaging of cubic layers results in an equivalent medium with tetragonal (not cubic) symmetry. We present concise formulations of stability conditions for low symmetry classes, such as trigonal, orthotropic, and monoclinic.

Appendices

Appendix A: Backus Average for Anisotropic Layers

Let us write the strain-stress relations in two dimensions ($x_{3}x_{1}$–plane), namely,

$$ \sigma _{11}=C_{11}\varepsilon _{11}+C_{13}\varepsilon _{33}\,, $$

(61)

$$ \sigma _{33}=C_{13}\varepsilon _{11}+C_{33}\varepsilon _{33}\,, $$

(62)

$$ \sigma _{13}=2C_{55}\varepsilon _{13}\,, $$

(63)

which are the relations valid for the monoclinic, orthotropic, tetragonal, and TI symmetry class. Upon a rearrangement, we get

$$ \sigma _{11}=\left (C_{11}-\frac{C_{13}^{2}}{C_{33}}\right ) \varepsilon _{11}+\left (\frac{C_{13}}{C_{33}}\right )\sigma _{33}\,, $$

(64)

$$ \varepsilon _{33}=-\left (\frac{C_{13}}{C_{33}}\right )\varepsilon _{11} +\left (\frac{1}{C_{33}}\right )\sigma _{33}\,, $$

(65)

$$ \frac{\partial {u_{1}}}{\partial {x_{3}}}=\left (\frac{1}{C_{55}} \right )\sigma _{13}-\frac{\partial {u_{3}}}{\partial {x_{1}}}\,. $$

(66)

Let us treat the above equations as the stress-strain relations that correspond to many individual constituents that we want to average. To perform the averaging process, we use the three following properties: the average of the sum is a sum of the average, the average of the derivative is a derivative of the average, and, finally, the product approximation. We obtain

$$ \overline{\sigma _{11}}=\left [ \overline{\left (C_{11}-\frac{C_{13}^{2}}{C_{33}}\right )}+ \overline{\left (\frac{C_{13}}{C_{33}}\right )}^{2} \overline{\left (\frac{1}{C_{33}}\right )}^{-1}\right ] \overline{\varepsilon _{11}}+ \overline{\left (\frac{C_{13}}{C_{33}}\right )}\, \overline{\left (\frac{1}{C_{33}}\right )}^{-1} \overline{\varepsilon _{33}}\,, $$

(67)

$$ \overline{\sigma _{33}}= \overline{\left (\frac{C_{13}}{C_{33}}\right )}\, \overline{\left (\frac{1}{C_{33}}\right )}^{-1} \overline{\varepsilon _{11}}+ \overline{\left (\frac{1}{C_{33}}\right )}^{-1} \overline{\varepsilon _{33}}\,, $$

(68)

$$ \overline{\sigma _{13}}=\overline{\left (\frac{1}{C_{55}}\right )}^{-1}2 \,\overline{\varepsilon _{13}}\,. $$

(69)

Comparing equations (67)–(69) with equations (61)–(63), we see that the equivalent elasticity parameters are equal to

$$ C_{11}^{\mathrm{{\overline{\mathit{eq}}}}}= \overline{\left (C_{11}-\frac{C_{13}^{2}}{C_{33}}\right )}+ \overline{\left (\frac{C_{13}}{C_{33}}\right )}^{2} \overline{\left (\frac{1}{C_{33}}\right )}^{-1}\,, $$

(70)

$$ C_{13}^{\mathrm{{\overline{eq}}}}= \overline{\left (\frac{C_{13}}{C_{33}}\right )}\, \overline{\left (\frac{1}{C_{33}}\right )}^{-1} \,, $$

(71)

$$ C_{33}^{\mathrm{{\overline{\mathit{eq}}}}}= \overline{\left (\frac{1}{C_{33}}\right )}^{-1}\,, $$

(72)

$$ C_{55}^{\mathrm{{\overline{\mathit{eq}}}}}= \overline{\left (\frac{1}{C_{55}}\right )}^{-1}\,, $$

(73)

and the resulting medium is either monoclinic, orthotropic, tetragonal, or TI. If layers have cubic symmetry, then $C_{33}=C_{11}$. In such a case, $C_{33}^{\mathrm{{\overline{\mathit{eq}}}}}\neq C_{11}^{\mathrm{{\overline{\mathit{eq}}}}}$, which means that the equivalent medium is not cubic. To understand what is the symmetry class of the medium equivalent to cubic layers, we need to derive the analogous equivalent parameters, but for a 3D case. Upon an analogous procedure, shown above, we get

$$ C_{11}^{\mathrm{{\overline{\mathit{eq}}}}}= \overline{\left (C_{11}-\frac{C_{13}^{2}}{C_{11}}\right )}+ \overline{\left (\frac{C_{13}}{C_{11}}\right )}^{2} \overline{\left (\frac{1}{C_{11}}\right )}^{-1}\,, $$

(74)

$$ C_{12}^{\mathrm{{\overline{\mathit{eq}}}}}= \overline{\left (C_{13}-\frac{C_{13}^{2}}{C_{11}}\right )}+ \overline{\left (\frac{C_{13}}{C_{11}}\right )}^{2} \overline{\left (\frac{1}{C_{11}}\right )}^{-1}\,, $$

(75)

$$ C_{13}^{\mathrm{{\overline{\mathit{eq}}}}}= \overline{\left (\frac{C_{13}}{C_{11}}\right )}\, \overline{\left (\frac{1}{C_{11}}\right )}^{-1} \,, $$

(76)

$$ C_{33}^{\mathrm{{\overline{eq}}}}= \overline{\left (\frac{1}{C_{11}}\right )}^{-1}\,, $$

(77)

$$ C_{55}^{\mathrm{{\overline{\mathit{eq}}}}}= \overline{\left (\frac{1}{C_{55}}\right )}^{-1}\,, $$

(78)

$$ C_{66}^{\mathrm{{\overline{\mathit{eq}}}}}=\overline{C_{55}}\,, $$

(79)

where $C_{11}^{\mathrm{{\overline{\mathit{eq}}}}}=C_{22}^{\mathrm{{\overline{\mathit{eq}}}}}$, $C_{13}^{\mathrm{{\overline{\mathit{eq}}}}}=C_{23}^{\mathrm{{\overline{\mathit{eq}}}}}$, and $C_{55}^{\mathrm{{\overline{\mathit{eq}}}}}=C_{44}^{\mathrm{{\overline{\mathit{eq}}}}}$. The equivalent medium has six independent elasticity parameters and exhibits the tetragonal symmetry class.

Appendix B: Backus Procedure for a Trigonal Tensor

First, we write the stress-strain relations in a trigonal medium (expressed in a natural coordinate system) as

$$ \sigma _{11}=C_{11}\varepsilon _{11}+C_{12}\varepsilon _{22}+C_{13} \varepsilon _{33}+C_{15}\frac{\partial u_{1}}{\partial x_{3}}+C_{15} \frac{\partial u_{3}}{\partial x_{1}}\,, $$

(80)

$$ \sigma _{22}=C_{12}\varepsilon _{11}+C_{11}\varepsilon _{22}+C_{13} \varepsilon _{33}-C_{15}\frac{\partial u_{1}}{\partial x_{3}}-C_{15} \frac{\partial u_{3}}{\partial x_{1}}\,, $$

(81)

$$ \sigma _{33}=C_{13}\varepsilon _{11}+C_{13}\varepsilon _{22}+C_{33} \varepsilon _{33}\,, $$

(82)

$$ \sigma _{23}=C_{44}\frac{\partial {u_{2}}}{\partial {x_{3}}}+C_{44} \frac{\partial {u_{3}}}{\partial {x_{2}}}-2C_{15}\varepsilon _{12}\,, $$

(83)

$$ \sigma _{13}=C_{44}\frac{\partial {u_{1}}}{\partial {x_{3}}}+C_{44} \frac{\partial {u_{3}}}{\partial {x_{1}}}+C_{15}\varepsilon _{11}-C_{15} \varepsilon _{22}\,, $$

(84)

$$ \sigma _{12}=(C_{11}-C_{12})\varepsilon _{12}-C_{15} \frac{\partial {u_{2}}}{\partial {x_{3}}}-C_{15} \frac{\partial {u_{3}}}{\partial {x_{2}}}\,. $$

(85)

We can directly rewrite equations (82)–(84) in a manner that the nearly-constant stresses and strains are on the right-hand side, whereas the sole varying function of displacements is on the left-hand side. We get,

$$ \varepsilon _{33}=\sigma _{33} \underbrace{ \left (\frac{1}{C_{33}}\right ) }_{\text{$g_{1}$}}- \underbrace{\left (\frac{C_{13}}{C_{33}}\right )}_{\text{$g_{2}$}} \varepsilon _{11}-\underbrace{\left (\frac{C_{13}}{C_{33}}\right )}_{\text{$g_{3}$}}\varepsilon _{22}\,, $$

(86)

$$ \frac{\partial {u_{2}}}{\partial {x_{3}}}=\sigma _{23} \underbrace{\left (\frac{1}{C_{44}}\right )}_{\text{$g_{4}$}}- \frac{\partial {u_{3}}}{\partial {x_{2}}}- \underbrace{\left (\frac{C_{15}}{C_{44}}\right )}_{\text{$g_{t}$}}2 \varepsilon _{12}\,, $$

(87)

$$ \frac{\partial {u_{1}}}{\partial {x_{3}}}=\sigma _{13} \underbrace{\left (\frac{1}{C_{44}}\right )}_{\text{$g_{5}$}}- \frac{\partial {u_{3}}}{\partial {x_{1}}}-\left (\frac{C_{15}}{C_{44}} \right )\varepsilon _{11}+\left (\frac{C_{15}}{C_{44}}\right ) \varepsilon _{22}\,. $$

(88)

Now, we insert the right-hand side of equation (86) and (88) into equations (80) and (81). Also, we insert the right-hand side of (87) into (85). Upon simple calculations, we obtain

$$ \sigma _{11}= \sigma _{33}\left (\frac{C_{13}}{C_{33}}\right ) + \sigma _{13}\left (\frac{C_{15}}{C_{44}}\right ) + \underbrace{\left (C_{11}-\frac{C_{13}^{2}}{C_{33}}-\frac{C_{15}^{2}}{C_{44}}\right )}_{\text{$g_{6}$}}\varepsilon _{11} + \underbrace{\left (C_{12}-\frac{C_{13}^{2}}{C_{33}}+\frac{C_{15}^{2}}{C_{44}}\right )}_{\text{$g_{7}$}}\varepsilon _{22} \,, $$

(89)

$$ \sigma _{22}= \sigma _{33}\left (\frac{C_{13}}{C_{33}}\right ) - \sigma _{13}\left (\frac{C_{15}}{C_{44}}\right ) + \left (C_{12}- \frac{C_{13}^{2}}{C_{33}}+\frac{C_{15}^{2}}{C_{44}}\right ) \varepsilon _{11} + \underbrace{\left (C_{11}-\frac{C_{13}^{2}}{C_{33}}-\frac{C_{15}^{2}}{C_{44}}\right )}_{\text{$g_{8}$}}\varepsilon _{22} \,, $$

(90)

$$ \sigma _{12}= -\sigma _{23}\left (\frac{C_{15}}{C_{44}}\right ) - \underbrace{\left (\frac{C_{11}-C_{12}}{2}-\frac{C_{15}^{2}}{C_{44}}\right )}_{\text{$g_{9}$}}2\varepsilon _{12} \,. $$

(91)

Terms in parentheses in equations (86)–(91) correspond to various $g$; we denote them as $g_{i}$ or $g_{t}$. We notice that in case of trigonal symmetry, $g_{2}=g_{3}$, $g_{4}=g_{5}$, and $g_{6}=g_{8}$.

Appendix C: Proof of Lemma 1

Proof

Let us consider first part of Lemma 1, which states that

if $g_{2}^{\normalfont {\mathrm{ort}}}<0$ and $g_{3}^{\normalfont {\mathrm{ort}}}<0$ then $n_{1}$ and $n_{2}$ cannot be both positive.

To prove it, let us assume that $g_{2}^{\mathrm{ort}}<0$ and $g_{3}^{\mathrm{ort}}<0$. Since, according to stability conditions, $C_{33}\geq 0$, the above assumption is tantamount to $C_{13}<0$ and $C_{23}<0$. Also, assume that $n_{1}>0$ and $n_{2}>0$, which is tantamount to

$$ C_{13}C_{22}-C_{12}C_{23}>0 $$

(92)

and

$$ C_{23}C_{11}>C_{12}C_{13}\,, $$

(93)

respectively. From stability conditions, we also know that $C_{22}\geq 0$. Hence, to satisfy expression (92), $C_{12}$ must be positive. Therefore, we can rewrite inequality (92) as

$$ \frac{C_{13}C_{22}}{C_{12}}>C_{23}\,. $$

(94)

Now, let us consider inequality (93). Upon inserting some larger value in the place of $C_{23}$ the inequality will remain true. However, if we insert there the left-hand side of inequality (94), we obtain

$$ C_{11}C_{22}< C_{12}^{2}\,, $$

(95)

which is mathematically correct, but not allowed by the stability condition.

Second part of Lemma 1 states that

if $g_{2}^{\normalfont {\mathrm{ort}}}<0$ then $n_{1}>0$ together with $n_{2}<0$ are not allowed.

To prove it, let us assume that $g_{2}^{\mathrm{ort}}<0$, which is tantamount to $C_{13}<0$. Also, assume that $n_{1}>0$ and $n_{2}<0$. We get,

$$ \frac{C_{12}C_{23}}{C_{22}}< C_{13} $$

(96)

and

$$ \frac{C_{12}C_{13}}{C_{11}}>C_{23}\,. $$

(97)

Herein, we have invoked the so-called strict stability conditions (the matrix representing the elasticity tensor must be positive definite instead of positive semidefinite), which constitute the fact that $C_{11}$ and $C_{22}$ are greater than zero. We insert the left-hand side of inequality (97) in place of $C_{23}$ inside inequality (96) to again obtain

$$ C_{11}C_{22}< C_{12}^{2}\,. $$

(98)

To prove third part of Lemma 1 stating that

if $g_{3}^{\normalfont {\mathrm{ort}}}<0$ then $n_{1}<0$ together with $n_{2}>0$ are not allowed

we assume $g_{3}^{\mathrm{ort}}<0$ and consider $n_{1}<0$ and $n_{2}>0$. We obtain

$$ \frac{C_{12}C_{23}}{C_{22}}>C_{13} $$

(99)

and

$$ \frac{C_{12}C_{13}}{C_{11}}< C_{23}\,. $$

(100)

If we insert the left-hand side of inequality (99) in place of $C_{13}$ from inequality (100), which is a mathematically justified operation, we again obtain expression not allowed by stability conditions.

The last part stating that

if either $g_{2}^{\normalfont {\mathrm{ort}}}>0$ or $g_{3}^{\normalfont {\mathrm{ort}}}>0$ then $n_{1}$ and $n_{2}$ cannot be both negative,

can be proved trivially using same strategy as for the previous parts of Lemma 1. □