Stress ($\mathbf{\sigma }$)

Stress describes how internal forces are transmitted across imagined surfaces within a material. Dividing a small contact force by the cut's area gives the traction, a vector that depends on surface orientation. The Cauchy stress tensor $\sigma$ collects that dependence through $\mathbf{t}(\mathbf{n})=\sigma \mathbf{n}$, where $\mathbf{n}$ is the oriented unit normal vector.

The part of traction perpendicular to a cut is normal stress; its tangential part is shear stress. For uniform axial loading, stress is $F/A$, measured in pascals, but a general three-dimensional state needs the full tensor. A fluid at rest has $\sigma =-pI$ in the tension-positive convention, whereas a stretched bar has a preferred loading direction. Changing the orientation of the imagined cut changes its traction without changing the physical loading. This is why one scalar pressure cannot represent every solid stress state.

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.

Residual stress is stress present after the external loading that produced it has been removed. Different regions can acquire incompatible preferred shapes through nonuniform plastic deformation or thermal expansion. Because the regions must still fit together, elastic deformation balances those mismatches and leaves an internal stress field.

A free specimen can therefore have zero resultant external force while containing regions in tension and compression. For example, cooling different parts at different rates can leave different permanent strains, even after their temperatures become equal. Cutting the specimen releases some constraints, and the resulting shape change can reveal the stored stress. In a simple traction-free static body with no body force, the residual stress must satisfy internal force balance and zero surface traction. It affects later loading because the applied stress adds to a pre-existing field rather than starting from zero everywhere.

Engineering stress is axial force divided by the initial cross-sectional area $A_0$:It uses a fixed geometric reference even as a specimen stretches. The axial Cauchy stress tensor, by contrast, is $F/A$ for a uniform current cross-section $A$. During extension with decreasing area, the two quantities diverge.

For uniform deformation at approximately constant volume, $AL=A_0{L}_0$. With engineering strain $e=(L-L_0)/L_0$, the relation is ${\sigma }_{\mathrm{Cauchy}}=(1+e){\sigma }_{\mathrm{eng}}$. It assumes a uniform axial stress state and fails as a local conversion after pronounced necking unless the local geometry is measured. In a tensile modulus experiment, engineering stress versus engineering strain gives the initial Young's modulus. Its later maximum identifies a maximum load per original area, rather than a universal local failure stress.

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.

The first Piola-Kirchhoff stress tensor expresses the current force transmitted through a material surface per unit reference area. In the spatial-first index convention used here,where $F$ is the deformation gradient, $J=\det F$, $\sigma$ the Cauchy stress tensor, and $\mathbf{N},\mathbf{n}$ the reference and current unit normal vectors. Nanson's formula supplies the transformation of oriented surface area.

Its first index refers to current force direction and its second to reference surface orientation, so it is generally nonsymmetric even when $\sigma$ is symmetric. The mechanical power per reference volume is $\mathcal{P}:\dot{F}$. In a uniform uniaxial test the relevant component is the axial force divided by original area, connecting this tensor to engineering stress. The material-first nominal stress tensor in this repository is $P={\mathcal{P}}^{\mathsf{T}}$; keeping this convention explicit prevents misplaced transposes.

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 differential stress is the difference between the greatest and least principal stresses, $\Delta \sigma ={\sigma }_{\max }-{\sigma }_{\min }\ge 0$. It measures the spread of the principal loading, rather than the isotropic part of the stress. Adding the same pressure contribution to every principal stress leaves it unchanged.

For a symmetric three-dimensional stress tensor, the largest possible resolved shear stress is $\Delta \sigma /2$. In a Mohr circle representation this is the radius of the outer circle. A uniaxial load with principal values $(S,0,0)$ therefore has differential stress $|S|$, while hydrostatic loading has zero differential stress even when the pressure is large. This distinction is useful in yield criterion: a threshold based on maximum shear stress depends on principal-stress differences, rather than simply on the magnitude of the largest compression.

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.