Stress and Balance Laws¶
This chapter organizes the definitions and interrelationships of stress tensors, together with the equilibrium equations and stress symmetry derived from conservation of mass, linear momentum, and angular momentum. The three stress tensors—Cauchy stress \(\boldsymbol{\sigma}\), first Piola-Kirchhoff stress \(\boldsymbol{P}\), and second Piola-Kirchhoff stress \(\boldsymbol{S}\)—are introduced according to which configuration is used for the reference surface and force vector. See Motion, Deformation, and Strain for definitions of configuration, motion, and deformation gradient, and List of Physical-Quantity Symbols for the symbol list.
Cauchy Stress Tensor¶
Let \(\boldsymbol{x}\) be the force acting on an infinitesimal surface at a point \(d\Gamma\) in the current configuration (area \(\boldsymbol{n}\), outward unit normal \(d\boldsymbol{f}\)), and define the traction vector \(\boldsymbol{t}\) by the force per unit area, \(d\boldsymbol{f} = \boldsymbol{t}\, d\Gamma\). \(\boldsymbol{t}\) depends on both position \(\boldsymbol{x}\) and normal \(\boldsymbol{n}\). Applying conservation of momentum to an infinitesimal tetrahedron shows that \(\boldsymbol{t}\) is linear in \(\boldsymbol{n}\), so there exists a second-order tensor \(\boldsymbol{\sigma}(\boldsymbol{x}, t)\) such that
for any \(\boldsymbol{n}\) (Cauchy's stress theorem). This tensor \(\boldsymbol{\sigma}\) is called the Cauchy stress tensor, and is also called true stress because it represents force per unit area measured with respect to the geometry of the current configuration. The component \(\sigma_{ij}\) represents the \(x_j\) component of force per unit area acting on an infinitesimal surface normal to coordinate axis \(x_i\) in the current configuration.
As shown later from conservation of angular momentum, the Cauchy stress is a symmetric tensor:
In three dimensions it therefore has six independent components (\(\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}, \sigma_{31}\)). Vectorization of stress in Voigt notation assumes this symmetry (see Tensor Notation and Mathematical Foundations). Cauchy stress is used for stress output in FrontISTR geometrically nonlinear analysis with the Updated Lagrange method and in small-deformation analysis.
First and Second Piola-Kirchhoff Stresses¶
For finite deformation it is often convenient to express stress with reference to surfaces and vectors in the reference configuration, so two Piola-Kirchhoff stress tensors are introduced. Let the area of an infinitesimal surface in the reference configuration be \(d\Gamma_0\) and its outward unit normal be \(\boldsymbol{N}\).
The first Piola-Kirchhoff stress (first PK stress, nominal stress) \(\boldsymbol{P}\) is defined as the stress tensor obtained when the force \(d\boldsymbol{f}\) in the current configuration is applied to an infinitesimal surface in the reference configuration:
Using Nanson's formula \(\boldsymbol{n}\, d\Gamma = J \boldsymbol{F}^{-T} \boldsymbol{N}\, d\Gamma_0\) together with Cauchy's stress theorem gives the relation to Cauchy stress
The first PK stress is generally nonsymmetric.
The second Piola-Kirchhoff stress (second PK stress) \(\boldsymbol{S}\) is defined as the stress tensor obtained by pulling the current-configuration force \(d\boldsymbol{f}\) back to the reference configuration with \(\boldsymbol{F}^{-1}\) and applying it to an infinitesimal surface in the reference configuration:
Both the force vector and the surface on which it acts are expressed in reference-configuration quantities, making it invariant under rigid-body rotation and symmetric. Its relation to the first PK stress and the transformation from Cauchy stress are
The second PK stress is derived from a hyperelastic strain-energy function \(W(\boldsymbol{C})\) as \(\boldsymbol{S} = 2\,\partial W / \partial \boldsymbol{C}\) and, in the Total Lagrange method, is used as the work-conjugate pair \(\boldsymbol{E}\) with Green-Lagrange strain \((\boldsymbol{S}, \boldsymbol{E})\).
The reference configurations and symmetry properties are summarized below. In the small-deformation limit (\(\boldsymbol{F} \to \boldsymbol{I}\), \(J \to 1\)), the three stresses coincide.
| Stress tensor | Reference surface | Force vector | Symmetry | Application |
|---|---|---|---|---|
| Cauchy stress \(\boldsymbol{\sigma}\) | Current configuration \(d\Gamma, \boldsymbol{n}\) | Current configuration \(d\boldsymbol{f}\) | Symmetric | Updated Lagrange method / small-deformation analysis |
| First PK stress \(\boldsymbol{P}\) | Reference configuration \(d\Gamma_0, \boldsymbol{N}\) | Current configuration \(d\boldsymbol{f}\) | Generally nonsymmetric | Equilibrium equations in the reference configuration |
| Second PK stress \(\boldsymbol{S}\) | Reference configuration \(d\Gamma_0, \boldsymbol{N}\) | Reference configuration \(\boldsymbol{F}^{-1} d\boldsymbol{f}\) | Symmetric | Total Lagrange method / hyperelasticity |
Conservation of Mass and Momentum and the Equilibrium Equations¶
Let \(\rho\) be the mass density in the current configuration and \(\rho_0\) the mass density in the reference configuration. The conservation-of-mass law \(\int_{\Omega} \rho\, dv = \int_{\Omega_0} \rho_0\, dV\), together with the volume-element transformation \(dv = J\, dV\), reduces to the local form
This is the local conservation-of-mass relation.
Let \(\boldsymbol{g}\) be the body force per unit mass and \(\boldsymbol{a}\) the acceleration. Applying Cauchy's stress theorem and Gauss's divergence theorem to conservation of linear momentum (Euler's first law of motion) gives the local equilibrium equation (equation of motion) in the current configuration:
Rewriting the integral in the reference configuration using Nanson's formula and \(\boldsymbol{P} = J \boldsymbol{\sigma} \boldsymbol{F}^{-T}\) gives the local form in the reference configuration:
The two forms are equivalent through the stress-transformation rule and \(\rho_0 = J\rho\). Neglecting the inertial term gives
which is the static equilibrium equation and forms the starting point for FrontISTR static analysis (linear and nonlinear).
Conservation of Angular Momentum and Stress Symmetry¶
Combining conservation of angular momentum (Euler's second law of motion) with the equilibrium equation from conservation of linear momentum yields the symmetry of the Cauchy stress tensor:
Thus it has six independent components in three dimensions. Taking the transpose of both sides of the transformation \(\boldsymbol{\sigma} = J^{-1} \boldsymbol{F} \boldsymbol{S} \boldsymbol{F}^T\) and using the nonsingularity of \(\boldsymbol{F}\) shows that the second PK stress is also symmetric:
and therefore also has six independent components. By contrast, the first PK stress \(\boldsymbol{P} = \boldsymbol{F} \boldsymbol{S}\) is generally not symmetric; only the relation \(\boldsymbol{P} \boldsymbol{F}^T = \boldsymbol{F} \boldsymbol{P}^T\) holds (that is, \(\boldsymbol{P} \boldsymbol{F}^T\) is symmetric), and it has nine independent components.
These symmetries are the basis for expressing stress as a six-component vector in Voigt notation. See Tensor Notation and Mathematical Foundations and Linear Elasticity for Voigt-notation conventions and construction of material matrices.