Displacements, Stresses, and strains

This section recalls the relationships between the displacement vector, the strain matrix and the stress matrix in the framework of the linear theory of elasticity. If the displacements along axis $x_i$ are infinitesimal ($u_i$ where $i$ may be 1, 2 or 3), the strain matrix is defined by

$$\varepsilon_{ij}=\frac{1}{2}\left( \frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i} \right)$$ (3.6)

The stress matrix is linked to the strain matrix by the means of the Hooke tensor $c_{ijkl}$ (81 components reducing to 21 due to symmetries). Using the summation rule for replicated indices inside a product, the stress tensor can be written as

$$\sigma_{ij}=c_{ijkl}\varepsilon_{kl}=\sum_{k=1}^3\sum_{l=1}^3 c_{ijkl}\varepsilon_{kl}$$ (3.7)

In the case of isotropic medium, the 21 independent components reduce to the two Lamé moduli, $\lambda$ and $\mu$ , and equation (3.7) is now

$$\sigma_{ij}=\lambda \delta_{ij} (\delta_{kl} \varepsilon_{kl})+ \mu (\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\varepsilon_{kl}$$ (3.8)

where $\delta_{ij}$ is the Kronecker symbol ( $\delta_{ij}=1$ if $i=j$ or 0 if $i\ne j$ ).

In the absence of volumetric forces, the equation of motion is a differential equation of displacements and stresses.

$$\rho \frac{d^2 u_i}{dt^2}=\sum_{j=1}^3\frac{\partial \sigma_{ji}}{\partial x_j}$$ (3.9)

where $\rho$ is the density. For clarity, in the next sections, numerical indices $i$ are replaced by indices $x$ , $y$ , and $z$ , and $x_i$ are replaced by $x$ , $y$ , and $z$ .