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Cold working is permanent shaping by plastic deformation at a temperature low enough that substantial concurrent recrystallization does not erase the deformed microstructure. The relevant comparison is with the material's thermal response, rather than with a universal room-temperature boundary. Drawing wire and rolling sheet are examples of imposing large controlled shape changes.

In a crystalline metal, cold working changes the dislocation population and can create preferred crystal orientations. Dislocation interactions can increase resistance to further deformation through work hardening, while uneven deformation can leave residual stress. A practical comparison measures the stress-strain curve before and after a fixed amount of shaping, then unloads to identify the permanent strain. Subsequent heating can modify the stored microstructure through recovery or recrystallization, so processing history is part of specifying a specimen rather than a detail recoverable from its chemical composition alone.

Forest hardening is resistance to dislocation glide caused by dislocations that cut across a glide plane. An advancing dislocation must negotiate their intersections, form or break junctions, or bow between obstacles. The resulting obstacle network is called a forest because it obstructs motion across the plane, even though it is made of line defects rather than solid inclusions.

The mechanism connects dislocation density with work hardening: a shorter spacing between effective obstacles generally demands greater applied shear stress. It also explains why different slip systems need not evolve independently; slip on one system changes the obstacles encountered by another, contributing to latent hardening. Comparing a specimen before and after controlled plastic deformation tests the increase in flow resistance, while microstructural observation tests whether an assumed obstacle mechanism is plausible. Not every change of yield stress is forest hardening; other microstructural changes can also strengthen a material.

A junction produced when suitable partial dislocations on intersecting FCC slip planes react to form a low-energy sessile stair-rod segment. It obstructs further slip and can promote work hardening. Frank energetic favorability alone is not a complete calculation of junction geometry or mobility.

Plastic flow involving more than one active slip system. Interactions between dislocations on intersecting systems can give substantially faster work hardening than easy single slip.

Necking is localization of tensile deformation into a narrowing part of a specimen. A thinner cross-section carries greater local stress at the same axial force, encouraging further thinning. Work hardening opposes this feedback by increasing the stress required for additional plastic deformation.

For a uniformly extending specimen whose plastic flow approximately conserves volume, write $A=A_0{e}^{-\epsilon }$ with logarithmic length change $\epsilon =\log (L/L_0)$. Its load is $F=\sigma A$, where $\sigma$ is axial Cauchy stress tensor. ThenThe maximum-load condition is $d\sigma /d\epsilon =\sigma$. This ideal estimate assumes uniform uniaxial deformation up to onset and neglects rate effects and damage. In tensile testing, recording both axial force and local specimen diameter distinguishes the start of localization from the later fracture; the maximum engineering stress need not be the maximum local Cauchy stress tensor.

The in-plane equilibrium equations are ${\tau }_{xx,x}+{\tau }_{xy,y}=0$ and ${\tau }_{xy,x}+{\tau }_{yy,y}=0$. On a simply connected patch, each divergence-free row has a streamfunction: write ${\tau }_{xx}={\varphi }_y$, ${\tau }_{xy}=-{\varphi }_x=-{\psi }_y$, ${\tau }_{yy}={\psi }_x$. Symmetry gives ${\varphi }_x={\psi }_y$, so another potential exists with $A_y=\varphi$, $A_x=\psi$. HenceIf two such Airy stress functions give the same stress, their difference has zero Hessian and is therefore an arbitrary affine function $a+bx+cy$ on a connected domain. This is the full nonuniqueness for a fixed stress. The existence argument is local, or global on a simply connected domain; domains that are multiply connected regions require the associated force and moment period conditions for a single-valued global potential. Equilibrium alone should not silently erase that topological qualification.

For plane strain put $s=e_{xx}+e_{yy}$. The constitutive equation gives ${\tau }_{xx}+{\tau }_{yy}=2(\lambda +\mu )s$. ConsequentlyThe displacement Saint-Venant compatibility equations $e_{xx,yy}+e_{yy,xx}-2e_{xy,xy}=0$ then becomesFor stable elastic moduli the coefficient is nonzero, so $\boxed{{\Delta }^{2}A=0}$. If $D$ is bounded and smooth and $A={\partial }_{n}A=0$ on its boundary, integrating twice by parts gives $0={\int }_{D}A{\Delta }^{2}AdV={\int }_D(\Delta A{)}^{2}dV$. Thus $A$ is harmonic with zero boundary value and $\boxed{A=0}$ by uniqueness of the Dirichlet problem. The normal-derivative condition is what removes the extra fourth-order boundary terms.

For the pressurized annulus, substituting the radial constitutive equations into equilibrium gives $u^{\prime \prime }+u^{\prime }/r-u/r^2=0$, so $u=Cr+D/r$. The stresses areUsing gauge stresses relative to ambient, the boundaries are ${\tau }_{rr}(a)=-P$, ${\tau }_{rr}(b)=0$. ThereforeandThe maximum occurs at the bore. Once the chosen yield criterion is reached, Hooke's law cannot describe the total strain alone: write total strain as elastic plus plastic strain and supplement the elastic stress law with a yield condition, a plastic flow rule and any required work hardening law. A plastic region starts near the bore and must be matched to the remaining elastic region. The exact conversion between a principal-stress difference and the quoted yield stress depends on whether the material criterion is Tresca or von Mises. This is the elastic stress concentration in a pressurized thick cylinder.

Use the two partials on each plane as obtained above. The four pairwise sums, in units of $a/6$, areThe shortest resultant gives the largest line-energy reduction in the $G{b}^2$ approximation:The initial summed squared magnitude is $a^2/3$ and the final value is $a^2/18$, a sixfold reduction in that part of the energy estimate.

The line of intersection of the two original planes is parallel toIts dot product with the resultant Burgers vector is zero, so the junction is edge in character. A plane containing both line and Burgers vector has normal $[100]$, rather than an FCC close-packed $\{111\}$ glide normal. The resulting stair-rod dislocation is therefore sessile for ordinary glide and can form a Lomer-Cottrell lock. It obstructs further dislocation motion and promotes work hardening.

This energetic comparison assumes comparable line lengths and elastic prefactors. Geometry, line character and stacking-fault geometry affect a complete junction-energy calculation, but the small resultant identifies the favorable candidate requested by the vector model.

The alloy shear yield stress can be roughly $70/0.78\simeq 90$ times the stated pure-aluminium critical resolved shear stress. Strengthening raises the stress required for dislocation glide rather than requiring homogeneous shearing of a perfect crystal.

These mechanisms can coexist. The numerical comparison is an order-of-magnitude comparison of shear strengths; it is not a conversion of the alloy's shear yield stress directly into a tensile yield stress.

A second slip system begins to operate as the tensile-axis rotation makes its resolved shear stress comparable to that on the primary system. Dislocations on intersecting planes interact much more strongly, giving the transition from easy single slip to rapid work hardening. The standard geometric estimate neglects differences between the evolving critical stresses of the two systems.

The primary direction is $\mathbf{s}\parallel [101]$. Single-slip shear carries the line joining the specimen ends, expressed in crystal coordinates, alongThe initial fundamental sector has $x<y<z$. Its first reached boundary is $x=y$, at $t=1$, giving $\mathbf{r}=[225]$. At that orientation $(\overline{1}11)[101]$ and $(1\overline{1}1)[011]$ have equal Schmid factors. Since the component perpendicular to the primary slip direction is unchanged by that shear, the length ratio follows directly:ThereforeThe corresponding logarithmic strain is $\ln (L/L_0)=0.2260$. This is an ideal equal-critical-stress geometric onset estimate. Real secondary slip can start earlier or later because of latent hardening, nonuniform deformation and grip constraints; the problem supplies no hardening law that would determine that shift.^[ref]

At small load the deformation is elastic: ${\epsilon }_{\parallel }=\sigma /E$, and, using isotropic elasticity with $E=2G(1+\nu )$, $\nu =0.30$ gives lateral contraction ${\epsilon }_{\perp }=-0.30{\epsilon }_{\parallel }$. At maximum load,For the specified slip system, the tensile direction is proportional to $(-2,-3,5)$, the slip direction to $(1,-1,1)$ and the plane normal to $(1,0,-1)$. Hence the Schmid factor isSchmid's law predicts glide when this reaches the critical resolved shear stress. Before that, unloading reverses the strain; after activation, glide produces permanent strain and may generate work hardening and additional slip. The supplied data do not fix the yield load. Neither the adjective “ultra-strong” nor $E/G$ supplies a critical shear stress or an absolute modulus. Other slip systems must also be checked before claiming which activates first. In particular, the specimen remains elastic through the maximum load if every relevant system stays below its activation threshold.

Irradiation softening occurs when irradiation removes pre-existing strengthening obstacles faster than it creates new ones. It is particularly plausible in an initially hardened alloy, rather than an initially soft annealed reference.

In a material strengthened by cold working, the starting dislocation density is high. Irradiation-generated point defects can assist dislocation climb, rearrangement and annihilation, reducing the contribution of work hardening. A schematic estimate iswhere $\mu$ is the shear modulus, $b$ the magnitude of the Burgers vector, ${\rho }_d$ the dislocation density, and the last two terms represent precipitate and irradiation-obstacle strengthening. It is an illustrative decomposition, not an exact additive law for every alloy. A sufficiently large decrease in the pre-existing contributions can exceed the increase in $\Delta {\sigma }_{\mathrm{irr}}$.

In a material strengthened by precipitation hardening, ballistic mixing can dissolve or disorder the original obstacles. Radiation-enhanced diffusion can change their composition or accelerate coarsening; increasing obstacle spacing can reduce strength. In an ordered alloy, irradiation-induced loss of order can also remove a strengthening contribution. None of these outcomes is inevitable: irradiation can instead create new strengthening precipitates.

The required conditions therefore include an appropriate initial microstructure, enough displacement dose to modify it, and a temperature and displacement dose rate at which obstacle removal or recovery can compete with new defect accumulation. Compare with an unirradiated specimen held for the same time at the same temperature to separate ordinary thermal aging from an irradiation effect.

A drop in yield strength or hardness need not restore ductility: cavities, helium embrittlement or localization of dislocation glide can coexist with softening. Conversely, an initially annealed steel may show strong irradiation hardening under the same exposure conditions.

Perfect plasticity idealizes a material whose yield surface does not evolve as plastic deformation accumulates. In one-dimensional elastic-perfectly plastic loading, $\sigma =E(\epsilon -{\epsilon }^p)$ while $|\sigma |\le {\sigma }_y$; continued plastic extension can occur at the fixed yield stress ${\sigma }_y$. Unloading can still be elastic, so perfect plasticity does not mean that the elastic response is absent.

This model isolates permanent flow from work hardening and is useful in limiting-load calculations. A rigid-perfectly plastic model makes the additional approximation of neglecting elastic strain, as in classical slip-line field theory. A real specimen may harden, soften, damage or show rate dependence, so the constant threshold is an idealization rather than a universal property of metals. It is also a local material law: decreasing cross-sectional area can change the measured axial force even when the local flow stress is constant.

Plastic deformation is the part of a body's deformation that remains after the applied stress has been removed. Bend a metal wire gently and it springs back through elastic deformation; bend it far enough and its unloaded shape changes. In small-strain plasticity in continuum mechanics, the total strain is separated as $\mathbf{\epsilon }={\mathbf{\epsilon }}^e+{\mathbf{\epsilon }}^p$, where the superscripts denote elastic and plastic parts.

In a crystal, dislocation glide can produce plastic deformation without all the chemical bonds across a plane breaking simultaneously. Repeated plastic deformation can increase the required yield stress through work hardening. A loading-and-unloading tensile testing experiment distinguishes the recoverable extension from the residual extension; the area of a closed stress versus strain curve measures net supplied mechanical work per unit initial volume in the small-strain description. It equals dissipation when the material's internal state also returns to its starting value. This distinction is essential when shaping a component permanently while keeping its service loads within the elastic range.

Plasticity in continuum mechanics models permanent deformation after loading passes a material's yield criterion. It supplements the balances of continuum mechanics with an evolution law for plastic strain, and often with internal variables representing work hardening. This separates the recoverable elastic response from changes that remain after unloading.

In one-dimensional small-strain loading, $\epsilon =\sigma /E+{\epsilon }^p$. Unloading without further plastic deformation follows slope $E$, the Young's modulus, and leaves residual strain ${\epsilon }^p$ when $\sigma$ returns to zero. A tensile testing experiment with repeated unloading makes this distinction visible. A yield surface generalizes the threshold to multiaxial stress; a flow rule determines the direction of further plastic strain. The threshold and flow rule are separate ingredients, so a yield criterion alone cannot predict a component's permanent shape.

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.