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.

For an objective smooth strain energy density $W(QF)=W(F)$, differentiation along $Q=e^{tK}$ shows $DW(F)F^T$ is symmetric. Hence the associated Cauchy stress tensor is symmetric. At a stress-free reference $DW(I)=0$, differentiation of this identity shows $D^{2}W(I)[H]$ is symmetric for every $H$. In components this gives output minor symmetry. The Hessian has pair-interchange symmetry, which then gives the input minor symmetry and the displayed annihilation of skew increments. The stress-free condition matters: at a prestressed reference the derivative includes the extra term $DW(I)H^T$.

For a continuously differentiable energy on the connected set of positive-determinant matrices, material frame indifference differentiates to the displayed covariance of the first Piola-Kirchhoff stress tensor. Conversely that covariance makes $W(QF)-W(F)$ independent of $F$. Call the constant $c(Q)$. Composition gives $c(Q_1{Q}_2)=c(Q_1)+c(Q_2)$. Continuity and compactness of the rotation group make $c$ bounded, while $c(Q^n)=nc(Q)$ forces every $c(Q)$ to vanish. Thus no additive rotation-dependent energy ambiguity remains. Homogeneity means the energy has no explicit dependence on the material point; it is separate from this observer-independence property.

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.

Resolution of a stress tensor onto differently oriented planes. For two compression-positive principal stresses, the resolved normal stress and shear stress trace a Mohr circle as the plane normal rotates, with double the physical rotation angle.

The load carried by the solid skeleton or contacts after allowing for pore fluid pressure. In compression-positive form an elementary law is ${\mathbf{\sigma }}^{\prime }={\mathbf{\sigma }}^c-\alpha P\mathbf{I}$; the contact approximation often takes $\alpha =1$. Fixed-total-stress pressure changes translate a Mohr circle without changing its radius.

A geometric representation of resolved normal stress and shear stress. For two principal stresses, its centre is $C=({\sigma }_1+{\sigma }_3)/2$ and radius $R=({\sigma }_1-{\sigma }_3)/2$. A physical plane-normal rotation by $\theta$ corresponds to $2\theta$ on the circle, with the sense depending on the shear-sign convention. Named for Christian Otto Mohr.

The locus of critical combinations of normal stress and shear stress. Contact of a Mohr circle with this boundary identifies an ideal failure threshold; it does not by itself describe post-failure dynamics.

A frictional shear-strength approximation $|\tau |=c+{\sigma }_n^{\prime }\tan \phi$, with cohesion $c$ and internal friction angle $\phi$. Effective compression increases shear resistance. A separate tensile cutoff is generally needed at low confinement. The names refer to Christian Otto Mohr and Charles-Augustin de Coulomb.

For a circular tunnel of radius $a$ in a plane-strain elastic medium, isotropic far compression $p_{\infty }$ and wall bracing pressure $P$ give ${\tau }_{rr}=-p_{\infty }-(P-p_{\infty })a^2/r^2$ and ${\tau }_{\theta \theta }=-p_{\infty }+(P-p_{\infty })a^2/r^2$. The Mohr circle center is $-p_{\infty }$ and its radius is $|P-p_{\infty }|a^2/r^2$. Its maximum is at the wall, proving that first yield occurs there. Cohesionless frictional tangency requires this radius to equal $p_{\infty }\sin \varphi$. Lowering $P$ from hydrostatic confinement yields the displayed threshold and stress ratio ${\tau }_{\theta \theta }/{\tau }_{rr}=(1+\sin \varphi )/(1-\sin \varphi )$. Increasing bracing instead reaches $P=p_{\infty }(1+\sin \varphi )$ with the reciprocal ratio. The two cases must not be confused when identifying the first loading threshold.

The angle whose tangent gives the slope of a linear frictional failure envelope. It controls both the confinement dependence of strength and the optimum new shear-fracture orientation.

The shear-strength intercept $c$ at zero effective normal stress in a specified frictional failure envelope. It is a model parameter and is not automatically the tensile strength.

The longitudinal normal stress in an elongated body. For uniform loading it is the axial resultant force divided by the relevant cross-sectional area; Cauchy stress tensor components use current area. Positive values conventionally indicate tension and negative values compression.

A stress remaining after the applied loads have been removed. Spatially nonuniform permanent strain can leave a self-equilibrated residual stress field, even though every boundary traction vanishes.

Engineering stress is axial force divided by the initial cross-sectional area. The initial area remains the denominator throughout a tensile modulus experiment, even if the specimen later changes shape.

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.

Cauchy's stress theorem states that local contact traction depends linearly on the oriented unit normal vector: $t_i(\mathbf{n})={\sigma }_{ij}{n}_j$, with ${\sigma }_{ij}={\mathbf{e}}_i\cdot \mathbf{t}({\mathbf{e}}_j)$. It follows by balancing forces on a shrinking tetrahedron with three coordinate faces and one oblique face. Contact forces scale as the face area, while bounded body force and inertial contributions scale as its volume; dividing by the oblique face area and shrinking eliminates the latter. The coordinate-face projected areas are the oblique face area times $n_j$, giving the displayed linear map. Symmetry of the Cauchy stress tensor requires the additional balance of angular momentum in a continuum without independent couples.

The contact traction on an oriented material cut is assumed to depend locally on the point, time and unit normal, with no independent dependence on the cut's curvature or global geometry. Together with local momentum balance and regularity sufficient for shrinking-volume arguments, this implies linear normal dependence through a stress tensor. Referential traction is measured per reference area and yields the first Piola-Kirchhoff stress tensor; spatial traction is measured per current area and yields the Cauchy stress tensor. These are different representations and the nominal tensor is not generally symmetric.

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.

A plane whose unit normal vector is an eigenvector of the stress tensor. Its traction has no tangential component, so the resolved shear stress vanishes. This is distinct from a principal plane in geometrical optics.

The difference between greatest and least principal stresses. It is twice the radius of the largest Mohr circle.

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.

Elastic strain is the reversible part of strain that determines stress through the elastic constitutive equation. In a small-strain model with plastic and thermal effects, subtract the plastic strain and thermal strain from total strain to obtain it. A stress-free heated body can therefore have nonzero total strain but zero elastic strain; after plastic loading, unloading can likewise remove elastic strain while leaving permanent deformation.

The time rate of strain; a scalar may describe an idealized deformation component, while general deformation uses a strain-rate tensor.

The first-order relative change of volume for a small displacement field. It is the trace of the infinitesimal strain tensor. For a radial displacement field $u(r){\mathbf{e}}_r$ in spherical coordinates, $e=u^{\prime }+2u/r$.

The fractional change in the length of a material line in a specified direction. For infinitesimal deformation with unit direction $\mathbf{n}$, it is $\mathbf{n}\cdot \mathbf{\epsilon }\mathbf{n}$, where $\mathbf{\epsilon }$ is the small strain tensor.

Engineering shear strain is the first-order change of the right angle between material lines initially directed along two perpendicular axes. For a displacement $\mathbf{u}$, ${\gamma }_{xy}={\partial }_y{u}_x+{\partial }_x{u}_y=2{\epsilon }_{xy}$, where ${\epsilon }_{xy}$ is the tensor strain component. Slender Euler-Bernoulli beam equation models neglect the transverse shear strain contribution to deflection.

The time derivative of engineering shear strain. A Newtonian simple-shear stress is $\eta \dot{\gamma }$. This scalar is twice the off-diagonal symmetric rate-of-strain-tensor component.

The infinitesimal strain tensor $\mathrm{diag}(-e,e)$ compresses one principal direction and stretches the perpendicular one by the same amount. Rotating the coordinate axes through $45^{\circ }$ turns it into a tensor with zero normal strains and off-diagonal entries of magnitude $e$. The engineering shear strain is twice the tensor shear component, so its shear angle has magnitude $\gamma =2e$. Likewise principal stresses $(-S,S)$ give zero normal stress and shear stress of magnitude $S$ on those rotated faces. The shear modulus law $S=G\gamma$ therefore gives $S=2Ge$.

Engineering strain is the change in length divided by that segment's initial length. For an extensometer, use its own initial gauge length and extension.

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.

The deviatoric strain is the trace-free part of the infinitesimal strain tensor. It removes the isotropic volume change. The decomposition is orthogonal for the Frobenius inner product, because a trace-free matrix has zero Frobenius inner product with $I$. ConsequentlyFor $\epsilon =\frac{1}{2}(\nabla v+(\nabla v{)}^{\mathsf{T}})$, its trace is $\mathrm{div}v$.