Constitutive equation

A constitutive equation relates continuum fields such as stress, strain, and the rate-of-strain tensor to characterize a material beyond universal balance laws.

A constitutive equation is frame indifferent when a superposed rigid spatial rotation changes its outputs by their tensor transformation rules. For an elastic Cauchy stress tensor, this requires $\sigma (QF)=Q\sigma (F)Q^T$. It differs from material isotropy, which rotates reference material directions.

An objective spatial second-rank tensor transforms by the displayed rule under every rigid observer change $x^{\ast }=c(t)+Q(t)x$. This permits time-dependent rotations, not merely constant rotations. Cauchy stress tensor is objective because contact traction and surface normal both rotate by Q. The symmetric rate-of-strain tensor is objective because the additional observer spin in the velocity gradient is antisymmetric.

Stress is force transmitted per unit area inside a continuum. The Cauchy stress tensor maps a surface normal to the traction acting across that surface.

A stress measure $T$ is work-conjugate to a strain measure $E$ when their contraction gives the mechanical power per specified volume for every admissible deformation rate. The reference-volume power is $P_{Ii}{\dot{F}}_{iI}=\tau :D$, with Kirchhoff stress tensor $\tau$ and rate-of-strain tensor $D$. A strain constraint can leave the conjugate stress undetermined by a reaction component that does no admissible work; uniqueness should therefore not be assumed on a restricted set of rates.

Use the material-first nominal stress $P_{Ii}$ and $F=RU$. The symmetric Biot stress conjugate to the Biot strain tensor is $T^{(1)}=\mathrm{sym}(PR)$. To prove it, set $M=P^T$ and differentiate $F=RU$. Conservation of angular momentum makes $M{F}^T$ symmetric, so the contraction of $R^{T}M$ with $R^T\dot{R}U$ is zero: it is a contraction of a symmetric tensor with a skew tensor after cyclically moving $U$. Thus $M:\dot{F}=\mathrm{sym}(R^{T}M):\dot{U}=T^{(1)}:{\dot{E}}^{(1)}$. No assumption that $PR$ itself is symmetric is needed.

With the right Cauchy-Green deformation tensor $C=F^{T}F$ and the second Piola-Kirchhoff stress tensor $S$, this mixed material stress is generally nonsymmetric. In the material-first nominal stress tensor convention, $PF=SC=M^T$. Thus replacing $P{F}^{-T}$ by $PF$ changes the stress measure. The relation follows from $\mathcal{P}=FS$, without any assumption of material isotropy.

The Kirchhoff tensor is the Cauchy stress tensor multiplied by the local volume ratio. Its stress power is per reference volume. Pushing forward the second Piola-Kirchhoff stress tensor gives the displayed formula.

These stress measures refer force or area to the reference configuration. Different index-order conventions distinguish a first Piola-Kirchhoff stress tensor from its transposed nominal stress tensor. The second Piola-Kirchhoff stress tensor has two material indices.

The second tensor has two material indices and is symmetric when the Cauchy stress tensor is symmetric. The displayed relation uses the material-first nominal stress tensor convention. In particular, $PF$ is generally not this tensor. These conventions agree with A. F. Bower's definitions of stress measures.

Here the convention places the material index first: $P_{Ii}$. Its stress power per reference volume is $P_{Ii}{\dot{F}}_{iI}$. It is the transpose of the first Piola-Kirchhoff stress tensor; some texts instead call that first tensor itself nominal stress.

The stress tensor maps a surface normal to the traction exerted across that surface. Its divergence is the force density in local momentum balance. For an incompressible Newtonian fluid it combines pressure with twice the viscosity times the symmetric strain-rate tensor.

The component of traction normal to a surface with unit normal $n$. For an incompressible Newtonian fluid, $\sigma =-pI+2\mu e$, so ${\sigma }_{nn}=-p+2\mu n\cdot en$. A free-interface normal-stress jump includes the curvature force from surface tension.

The principal stresses are the eigenvalues of the symmetric Cauchy stress tensor. In their orthonormal eigenbasis, each traction is normal to its corresponding principal plane.

The deviatoric part of a three-dimensional stress tensor is ${\mathbf{\sigma }}_{\mathrm{dev}}=\mathbf{\sigma }-\mathrm{tr}(\mathbf{\sigma })\mathbf{I}/3$. It is traceless. Adding an isotropic pressure does not change a normal-stress difference.

Traction is the force per unit area transmitted across an oriented surface. For unit normal vector $\mathbf{n}$, the stress tensor gives $\mathbf{t}=\mathbf{\sigma }\mathbf{n}$. Its normal component is normal stress, and its tangential component is shear stress.

The hydrodynamic torque on a body about a reference point is $T={\int }_{S}r\times (\sigma \cdot n)dS$, with $r$ measured from that point and $n$ pointing from the body into the fluid. The stress includes pressure and viscous contributions, and $\sigma \cdot n$ is the fluid traction on the body. Changing the reference point by $d$ changes the torque by $-d\times F$, where $F={\int }_S\sigma \cdot ndS$ is the fluid force. A torque-free body has zero resultant torque, including any external torque.

Strain measures local deformation relative to a reference configuration. In small-deformation theory it is the symmetric part of the displacement gradient.

The infinitesimal strain tensor is the symmetric part of a displacement gradient tensor. It is the linear approximation to a finite strain measure for small displacement gradients. An infinitesimal rigid rotation contributes no strain, so the elastic energy depends on the symmetric part rather than the entire gradient.