Stress, strain, and energy in continuum physics

Deformation in continuum physics

The deformation of coordinates

$x^I$ gives the referential coordinates, $r^i$ gives the deformed coordinates

$x^I \xrightarrow{\text{deform}~F} r^i$

The deformed coordinates $r^i$ is related to the undeformed coordinates $x^I$ by the deformation matrix $\mathbf{F}$ . The components of $\mathbf{F}$ are given by

$F_I^i = \frac{\partial r^i}{\partial x^I}.$

The deformation of volume

The Jacobian $J$ , or what actually more commonly referred to as (volumic) “strain”, $\alpha$ , is the determinant of the deformation matrix

$J = \det \mathbf{F}.$

The deformation of surface

$\mathbf{n}^0\,\mathrm{d}\Sigma^0 \xrightarrow{\text{deform } F} \mathbf{n}^t\,\mathrm{d}\Sigma^t$

Notations

The uppercase subscripts $\square_{IJKL}$ denote quantities in referential coordinates, whereas the lowercase subscripts $\square_{ijkl}$ denote quantities in the deformed coordinates.

Stress

Definitions of stress

Lagrangian description of the Cauchy stress $\mathbf{T}^\mathrm{L}$

$\mathrm{d}\mathbf{f}^\mathrm{L} = \hat{\mathbf{n}}^t \mathrm{d}\Sigma^t \, \mathbf{T}^\mathrm{L} \quad \text{ or } \quad \mathrm{d}f^\mathrm{L}_I = n^t_j \, \mathrm{d}\Sigma^t \, T^\mathrm{L}_{jI}$

Eulerian description of the Cauchy stress $\mathbf{T}^\mathrm{E}$

$\mathrm{d}\mathbf{f}^\mathrm{E} = \hat{\mathbf{n}}^t \mathrm{d}\Sigma^t \, \mathbf{T}^\mathrm{E} \quad \text{ or } \quad \mathrm{d}f^\mathrm{E}_i = n^t_j \, \mathrm{d}\Sigma^t \, T^\mathrm{E}_{ji}$

First Piola-Kirchhoff stress $\mathbf{T}^\mathrm{PK}$

$\mathrm{d}\mathbf{f}^\mathrm{E} = \hat{\mathbf{n}}^0 \mathrm{d}\Sigma^0 \, \mathbf{T}^\mathrm{PK} \quad \text{ or } \quad \mathrm{d}f^\mathrm{E}_i = n^0_J \, \mathrm{d}\Sigma^0 \, T^\mathrm{PK}_{Ji}$

Second Piola-Kirchhoff stress $\mathbf{T}^\mathrm{SK}$

$\mathrm{d}\mathbf{f}^\mathrm{L} = \hat{\mathbf{n}}^0 \mathrm{d}\Sigma^0 \, \mathbf{T}^\mathrm{SK} \quad \text{ or } \quad \mathrm{d}f^\mathrm{L}_I = n^0_J \, \mathrm{d}\Sigma^0 \, T^\mathrm{SK}_{JI}$

Relations between the three definitions of stress

Because of the deformation relations, the relations between these definitions can be given.

Comparison with Birch (1947)

In Birch (1947), Eq. (10) is actually Cauchy stress with this respect, because $\rho = J^{-1}\rho_0$ , $\partial a_p / \partial x_r = F_{iI}$

$T_{rs} = \rho\frac{\partial\phi}{\partial \eta_{pq}} \frac{\partial a_p}{\partial x_r} \frac{\partial a_q}{\partial x_s} \quad \to \quad T_{ij}^\mathrm{L} = J^{-1} \rho_0\frac{\partial\phi}{\partial E_{IJ}} F^I_i F^J_j \quad \text{or} \quad \mathbf{T}^\mathrm{L} = J^{-1} \mathbf{F}\mathbf{T}^\mathrm{SK}\mathbf{F}^\mathrm{T}$

And Dahlen & Tromp (1998) usually focuses on $\mathbf{T}^\mathrm{SK}$ and $\rho_0\partial\phi/\partial E_{IJ}$ .

Reference

Birch, F., 1947. Finite Elastic Strain of Cubic Crystals. Phys. Rev. 71, 809–824. https://doi.org/10.1103/PhysRev.71.809
Dahlen, F.A., Tromp, J., 1998. Theoretical Global Seismology. Princeton University Press.