# Compliance

Hooke's law expresses the linear relation between stress and strain. It may be written either as

stress = stiffness * strain :
$\sigma _{ij}=\sum _{k,l}^{3}C_{ijkl}\varepsilon _{kl}$ or as

strain = compliance * stress :
$\varepsilon _{ij}=\sum _{k,l}^{3}S_{ijkl}\sigma _{kl}$ where the compliance tensor ${\textbf {S}}$ is a rank-4 tensor, which may be represented as a symmetric rank-2 matrix, using modified Voigt notation as:

$\{S_{\alpha \beta }\}={\begin{pmatrix}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\\\end{pmatrix}}={\begin{pmatrix}S_{1111}&S_{1122}&S_{1133}&2S_{1123}&2S_{1113}&2S_{1112}\\S_{1122}&S_{2222}&S_{2233}&2S_{2223}&2S_{2213}&2S_{2212}\\S_{1133}&S_{2233}&S_{3333}&2S_{3323}&2S_{3313}&2S_{3312}\\2S_{1123}&2S_{2223}&2S_{3323}&4S_{2323}&4S_{2313}&4S_{2312}\\2S_{1113}&2S_{2213}&2S_{3313}&4S_{2313}&4S_{1313}&4S_{1312}\\2S_{1112}&2S_{2212}&2S_{3312}&4S_{2312}&4S_{1312}&4S_{1212}\\\end{pmatrix}}$ (These modifications (the factors 2 and 4) are just conventions, which apply only to $\{S_{\alpha \beta }\}$ , not to the stiffness matrix $\{C_{\alpha \beta }\}$ . They are one reason why the matrices should be used for display only, while the underlying tensors should be used for computation.)

Compliance is an intrinsic property of the medium considered. As with stiffness, the most general anisotropic medium has 21 independent elements, as shown above.

An isotropic medium has only two independent elements:

$\{S_{\alpha \beta }^{iso}\}={\begin{pmatrix}{\frac {1}{E}}&{\frac {-\nu }{E}}&{\frac {-\nu }{E}}&0&0&0\\{\frac {-\nu }{E}}&{\frac {1}{E}}&{\frac {-\nu }{E}}&0&0&0\\{\frac {-\nu }{E}}&{\frac {-\nu }{E}}&{\frac {1}{E}}&0&0&0\\0&0&0&{\frac {1}{G}}&0&0\\0&0&0&0&{\frac {1}{G}}&0\\0&0&0&0&0&{\frac {1}{G}}\\\end{pmatrix}}$ where $E$ is Young's modulus, $G$ is shear modulus, and $\nu$ is Poisson's ratio, only two of which are independent; see Figure E-6.

A polar anisotropic ("VTI") medium has five independent elements:

$\{S_{\alpha \beta }^{plr}\}={\begin{pmatrix}S_{11}&S_{12}&S_{13}&0&0&0\\S_{12}&S_{11}&S_{13}&0&0&0\\S_{13}&S_{13}&S_{33}&0&0&0\\0&0&0&S_{44}&0&0\\0&0&0&0&S_{44}&0\\0&0&0&0&0&S_{66}\\\end{pmatrix}}={\begin{pmatrix}{\frac {1}{E_{H}}}&{\frac {-\nu _{12}}{E_{H}}}&{\frac {-\nu _{13}}{E_{V}}}&0&0&0\\{\frac {-\nu _{12}}{E_{H}}}&{\frac {1}{E_{H}}}&{\frac {-\nu _{13}}{E_{V}}}&0&0&0\\{\frac {-\nu _{13}}{E_{V}}}&{\frac {-\nu _{13}}{E_{V}}}&{\frac {1}{E_{V}}}&0&0&0\\0&0&0&{\frac {1}{G_{13}}}&0&0\\0&0&0&0&{\frac {1}{G_{13}}}&0\\0&0&0&0&0&{\frac {1}{G_{12}}}\\\end{pmatrix}}$ where $E_{V}$ is the Vertical Young's modulus, $E_{H}$ is the Horizontal Young's modulus,

$G_{13}$ is the modulus for horizontal shear stress on a horizontal plane, $G_{12}$ is the modulus for horizontal shear stress on a vertical plane,

$\nu _{12}$ is the horizontal/horizontal Poisson's ratio, and $\nu _{13}$ is the vertical/horizontal Poisson's ratio, only five of which are independent.

An orthorhombic medium has nine independent elements:

$\{S_{\alpha \beta }^{ort}\}={\begin{pmatrix}S_{11}&S_{12}&S_{13}&0&0&0\\S_{12}&S_{22}&S_{23}&0&0&0\\S_{13}&S_{23}&S_{33}&0&0&0\\0&0&0&S_{44}&0&0\\0&0&0&0&S_{55}&0\\0&0&0&0&0&S_{66}\\\end{pmatrix}}$ ## Inverse of Stiffness

Compliance is the inverse of stiffness in the sense that

$\sum _{k,l}^{3}C_{ijkl}S_{klmn}=I_{ijmn}$ where ${\textbf {I}}$ is the rank-4 symmetric identity tensor:

$I_{ijmn}={\frac {1}{2}}(\delta _{im}\delta _{jn}+\delta _{in}\delta _{jm})$ with $\delta _{ij}$ the Kronecker delta (the rank-2 identity tensor).