AVERAGING THIN LAYERS FOR LOW FREQUENCY BEHAVIOR

 

layercake
Figure 1 Either random or periodic thinly layered elastic materials give rise to an effective medium characterized by transverse isotropy when the layers are composed of isotropic elastic materials and the layer thickness is small compared to the wavelength.

Backus (1962) presents an elegant method of producing the effective constants for a thinly layered medium composed of either isotropic or anisotropic elastic layers. This method applies either to spatially periodic layering or to random layering, by which we mean either that the material constants change in a nonperiodic (unpredictable) manner from layer to layer (see Figure 1) or that the layer thicknesses may also be random. For simplicity, I will assume that the layers are isotropic, in which case equation (sscij) becomes

_11_22 _33 _23_31 _12 = +2& & & & & & +2& & & & & & +2& & & & & & & 2& & & & & & & 2& & & & & & & 2 e_11 e_22 e_33 e_23 e_31 e_12 .   The key idea presented by Backus is that these equations can be rearranged into a form where rapidly varying coefficients multiply slowly varying stresses or strains. By doing so, I arrive at the following equation

_11_22 -e_33 e_23e_31 _12 = 4(+)+2 & 2+2 & +2 & & & 2+2 & 4(+)+2 & +2 & & & +2 & +2 & -1+2 & & & & & & & 12 & & & & & & & 12 & & & & & & & 2 e_11 e_22 _33 _23 _31 e_12 ,   which can be averaged essentially by inspection. Equation (legtrans) can be viewed as a Legendre transform of the original equation, to a different set of dependent/independent variables. Both vectors now have components with mixed physical significance, some being stresses and some being strains. Otherwise these equations are completely equivalent to the original ones.

Performing the layer average, assuming the variation is along the z or x₃ direction, I find, using the notation of (TIss),

<_11><_22> -<e_33> <e_23><e_31> <_12> = <4(+)+2> & <2+2> & <+2> & & & <2+2> & <4(+)+2> & <+2> & & & <+2> & <+2> & -<1+2> & & & & & & & <12> & & & & & & & <12> & & & & & & & <2> e_11 e_22 _33 _23 _31 e_12 = a - f^2/c & b - f^2/c & f/c & & & b - f^2/c & a - f^2/c & f/c & & & f/c & f/c & -1/c & & & & & & 1/2l & & & & & & 1/2l & & & & & & 2m e_11 e_22 _33 _23 _31 e_12 ,       which can then be solved to yield the expressions

a = <+2>^2 <1+2>^-1 + 4<(+)+2>,  

b = <+2>^2 <1+2>^-1 + 2 <+ 2>,  

c = <1+2>^-1  

f = <+2> <1+2>^-1,  

l = <1>^-1   and

m = <>.   One very important fact that is known about these equations is that they reduce to isotropic results with a=c, b=f, and l=m, if the shear modulus is a constant, regardless of the behavior of $\lambda$.