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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.

Work hardening is an increase in resistance to further plastic deformation caused by earlier deformation. A useful macroscopic test loads a metal beyond yield, unloads it and reloads in the same direction: further plastic flow then requires a larger stress. In a constitutive equation, this history is represented by evolving internal variables rather than by the current elastic strain alone.^[ref]

In a crystal, multiplication and interaction of dislocations can make later dislocation glide more difficult. Forest hardening and latent hardening describe particular aspects of that obstacle formation. Increased resistance to plastic flow is not the same as a large change in Young's modulus: the elastic unloading slope and the load required to restart permanent deformation measure different properties. Work hardening can delay necking by compensating for the shrinking loaded area.

A wrench represents a system of loads on a rigid body by its resultant force and resultant torque about a stated reference point. In the moment-first convention, writeThe ${\mathbf{C}}_j$ are applied force couples. The six entries package two different physical units: newtons for force and newton-metres for torque. The primary robotics text and its author video use this representation to transform measured forces and moments between reference frames.^[ref]

An original example has a force $(0,10,0)\mathrm{N}$ applied at $(0.30,0,0)\mathrm{m}$, plus a free couple $(0,0,2)\mathrm{N}\mathrm{m}$. Its resultant torque about the origin is $(0,0,5)\mathrm{N}\mathrm{m}$. Shift the reference point to $P=(0.10,0,0)\mathrm{m}$ without rotating the axes. ThenThe physical loading has not changed, but its torque coordinates depend on the reference point. A rotation of axes additionally rotates the vector components.

The dual velocity object is a rigid-body twist, ordered as $(\mathbf{\omega },{\mathbf{v}}_O)$. Their pairing is mechanical power:Since ${\mathbf{v}}_P={\mathbf{v}}_O+\mathbf{\omega }\times {\mathbf{r}}_{OP}$, substituting the shifted torque cancels the extra scalar triple product and preserves power. This explains why changing the force-sensor frame must transform torque and force together.

A nonzero resultant force does not always replace the whole wrench by one force alone. Define wrench pitchThen ${\mathbf{r}}_0\times \mathbf{F}={\mathbf{\tau }}_O-h\mathbf{F}$. A force along the central line through ${\mathbf{r}}_0$ leaves the parallel residual couple $h\mathbf{F}$. For $\mathbf{F}=(0,0,10)\mathrm{N}$ and ${\mathbf{\tau }}_O=(2,0,3)\mathrm{N}\mathrm{m}$, ${\mathbf{r}}_0=(0,0.20,0)\mathrm{m}$ and $h=0.30\mathrm{m}$. The remaining axial couple is $(0,0,3)\mathrm{N}\mathrm{m}$. If the resultant force is zero, a nonzero pure couple must instead be retained directly.

A proposed check uses a calibrated force-torque sensor and known lever arms to compare the reported six components at two reference points. Agreement tests the transformation and loading model; it does not establish that two externally equivalent loads produce identical internal stress in a deformable part.

A yield criterion specifies the admissible elastic stress region, possibly depending on internal hardening variables $q$. Plastic loading occurs on its boundary. The Tresca yield criterion uses maximum principal-stress difference, while the Von Mises yield criterion uses a quadratic deviatoric-stress invariant. Thus a numerical stress difference must be compared to a yield stress using the convention of the selected criterion.

A conservative material-screening upper bound obtained by requiring wall penetration before fast fracture even at a stated yield-limited stress. Substituting a thickness-dependent pressure stress instead yields a different fixed-pressure inequality; the two comparisons must not be conflated.

The yield stress is the stress threshold beyond which a material sustains irreversible flow under the specified loading conditions. Time-dependent materials require an operational measurement protocol; a single elastic modulus does not determine their yield stress.

A yield surface separates yielded material, where stress exceeds the yield criterion and deformation occurs, from an unyielded plug where the deformation rate vanishes.

Young's modulus is the ratio of axial stress to axial strain in uniaxial linear elasticity. For an isotropic plate, the bending rigidity is $E{t}^3/[12(1-{\nu }^2)]$, with thickness $t$ and Poisson's ratio $\nu$.

A tensile modulus experiment asks how much a specimen stretches under an axial force, while allowing its sides to contract freely. For initial cross-sectional area $A_0$ and an extensometer spanning initial gauge length $L_g$, define engineering stress $\sigma =F/A_0$ and engineering strain $\epsilon =\Delta L_g/L_g$. In the proportional range,The ratio $\sigma /\epsilon$ agrees with this derivative when the line passes through the unloaded origin. A fitted offset should not be mistaken for a material property. Young's modulus has units of pressure: $1\mathrm{Pa}=1\mathrm{N}{\mathrm{m}}^{-2}$. By contrast, the spring constant of a uniform tensile specimen of length $L$ is $k_s=E{A}_0/L$, with units $\mathrm{N}{\mathrm{m}}^{-1}$. Two rods made of the same material can therefore have very different spring constants while sharing Young's modulus.

The normalization removes a useful geometrical ambiguity: doubling $L$ doubles the extension at a given force, whereas doubling $A_0$ halves it. For a round specimen $A_0=\pi d^2/4$; for a rectangular one $A_0=wt$, using initial diameter $d$, width $w$ and thickness $t$. Diameter error matters twice at first order: $\delta A_0/A_0\simeq 2\delta d/d$. Specify material direction, temperature and loading rate when comparing Young's modulus. Lateral restraint changes the stress state and can prevent this experiment from measuring the uniaxial modulus.

The name commemorates Thomas Young, whose formulation of a measure of elasticity is recorded by the Royal Society. This attribution alone does not establish priority for every modern definition or experimental method.^[ref]