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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.

Contact mechanics studies how touching bodies deform, transmit forces and separate. Even a ball that appears to touch a flat surface at one point develops a finite contact patch: local elastic deformation spreads the load into a nonuniform pressure distribution. Hertzian contact is a useful ideal model when the surfaces are smooth, the contact patch is small and adhesion and friction can be neglected.

The contact patch is part of the solution, rather than a prescribed loaded area. In nonadhesive contact the nonpenetration constraint prevents overlap, and the normal contact force can push but cannot pull; unilateral contact complementarity joins those requirements. Pressing a spherical probe into a specimen while recording force and indentation tests the contact law and can determine a reduced elastic modulus. Surface roughness, plastic deformation and adhesion require additional modeling, especially when interpreting a measured pull-off force.

Crashworthiness is the ability of a structure to manage a collision through controlled deformation and energy dissipation while limiting transmitted loads and intrusion into the space it protects. A crushable member can use plastic deformation to spread the loss of kinetic energy over a useful stopping distance. High static strength alone does not specify this response: the complete force-displacement curve and the available deformation stroke matter.

For an original one-dimensional model, a $100\mathrm{kg}$ carriage approaches a stationary crush member at $4\mathrm{m}{\mathrm{s}}^{-1}$. Neglect rebound, friction elsewhere and external work during crushing. The member must absorbA constant resisting force over $s=0.20\mathrm{m}$ is $4000\mathrm{N}$; over $0.40\mathrm{m}$ it is $2000\mathrm{N}$. A triangular force-displacement curve rising from zero over the shorter stroke needs a peak of $8000\mathrm{N}$ to absorb the same energy. Thus equal absorbed energy does not imply equal peak load. Doubling the approach speed quadruples the required energy. The long-stroke design is useful only if that stroke is physically available without unacceptable intrusion.

A physical test must record the load history as well as the final crushed shape. One primary example used two vertical drop tests at $12.8\mathrm{m}{\mathrm{s}}^{-1}$ with an energy-absorbing aircraft-seat prototype, a test dummy and acceleration measurements, using crash pulses informed by twelve light-aircraft crash tests. This tests a dynamic load path rather than merely a material's static strength.^[ref] The carriage calculation above is an idealized energy budget, not a reconstruction of those tests.

The dislocation core is the small region near a dislocation line where the smooth linear-elastic continuum description fails. For a screw dislocation, the far-field displacement gradient scales as the Burgers-vector magnitude divided by radial distance, so the predicted elastic stress diverges on the axis. Atomic structure, nonlinear response or local plastic deformation replaces that singular prediction in the core.

Egon Orowan developed the dislocation account of plastic deformation in crystals. Motion of a dislocation permits successive local rearrangements instead of simultaneous sliding of an entire atomic plane, helping explain why measured strengths can be much lower than ideal-crystal estimates. The National Academy memoir traces his 1934 contribution and later work on deformation and failure.^[ref]

The boundary of the loading range within which a material recovers its original configuration on unloading. Elasticity need not be linear; beyond the elastic range, permanent plastic deformation or damage may remain.

Erich Schmid studied the plastic deformation of single crystals, connecting applied loading to the resolved shear stress acting on a slip system. Schmid's law expresses the geometric projection that underlies this connection. The University of Vienna's account of his scientific work describes his research on crystal deformation and his jointly authored treatment of crystal plasticity.^[ref]

Finite elasticity describes reversible deformations without assuming that stretches or rotations are small. A material point initially at $\mathbf{X}$ moves to $\mathbf{x}=\mathbf{\chi }(\mathbf{X})$, with deformation gradient $F_{ij}=\partial x_i/\partial X_j$. A useful measure of local stretching is the Green-Lagrange strain tensor,For a pure rotation, $\mathbf{F}$ is an orthogonal matrix and $\mathbf{E}=0$: rotating a specimen does not stretch it. This property is essential when, for example, a thin elastic rod bends through a large angle.

Both geometry and the constitutive equation matter. Current area and volume can differ substantially from their values in the reference configuration, and different stress tensors refer to different areas. A hyperelastic material supplies the response through a strain energy density. Rubber stretching and large bending motivate this framework even when the specimen returns to its original shape on unloading; large reversible deformation does not imply plastic deformation.

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.

Frank Nabarro developed the theory of dislocations and their role in plastic deformation. His work connected microscopic defects to macroscopic resistance to deformation, and his monograph Theory of Crystal Dislocations helped organize the subject. A scientific obituary by a colleague describes both the theoretical contributions and his long influence on materials science.^[ref]

Hertzian contact is the small-contact solution for smooth bodies obeying linear elasticity, with friction and adhesion neglected. For a sphere of radius $R$ pressed into a flat surface by indentation $\delta$, the contact radius $a$ and compressive normal contact force $F$ satisfyThe reduced elastic modulus $E^{\ast }$ includes both bodies. The contact pressure is largest at the centre and falls to zero at the edge:These formulas describe an ideal contact patch, rather than a uniformly loaded disk.^[ref]

The increasing contact area explains the nonlinear load law even though the material itself obeys linear elasticity. In a probe experiment, $F^{2/3}$ should be proportional to $\delta$ in the Hertzian regime. Systematic departures can signal adhesion, plastic deformation, a thin specimen or a contact patch too large for the assumed local geometry.

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.

For antiplane displacement $u=(0,0,w)$ the only nonzero strain components are $e_{xz}=w_x/2$ and $e_{yz}=w_y/2$. Commutation of mixed partial derivatives gives the necessary local compatibility conditionThe supplied stress is symmetric, independent of $z$ and trace-free. Its first two equilibrium equations are automatic; its third givesaway from the origin. Antiplane shear has zero trace, so isotropic Hooke's law reduces to $\boxed{e=\tau /(2\mu )}$. HenceLocally this integrates to $w=\arg (x+iy)+C$, represented on a suitable branch by $\arctan (y/x)+C$. The strain condition is satisfied at every point of the punctured plane, but the displacement cannot be made globally single-valued there:A gradient of a globally single-valued smooth displacement would instead integrate to zero around any closed curve. Local compatibility is therefore not sufficient on this multiply connected region.

A physical construction is a Volterra screw dislocation: cut the cylinder along a radial half-plane ending on its axis, shift one cut face relative to the other parallel to the axis, and rejoin. The axial mismatch is the Burgers vector. In the given normalization its magnitude is $2\pi$; with a dimensional mismatch $b$ one uses $w=b\arg (x+iy)/(2\pi )$. An intact pristine solid cannot reach this state by a smooth single-valued elastic deformation alone. The axis is a dislocation core where the continuum elastic field is singular and must be regularized by microscopic structure or plastic deformation. On the punctured domain the differential condition holds locally; globally its nonzero circulation detects the defect. In a distributional description, that incompatibility is concentrated on the axis. This is local strain compatibility versus screw-dislocation circulation.

A molecule is an electrically neutral bound assembly of more than one atom. It can contain identical atoms, as in ${\mathrm{O}}_2$, or different atoms, as in ${\mathrm{H}}_{2}O$. Discrete molecules are not a universal description of matter: an extended ionic crystal does not consist of separate neutral molecules corresponding to each empirical formula unit. The IUPAC definition also specifies binding sufficient to support a vibrational state.^[ref]

Elasticity is a body's tendency to recover its original configuration when the deforming force is removed. Within a linear regime, Hooke's law makes stress proportional to strain, but elasticity does not require exact linearity. The elastic limit matters: permanent plastic deformation is a different response.

Inertia is the property expressed by Newton's first law: without a resultant external force, a body's velocity remains constant in an inertial frame of reference. Mass measures translational inertia. It is not a force acting against motion, and a moving body does not require a continuing forward force merely to keep moving.

A force describes an interaction that changes momentum. In Newtonian mechanics,A force has magnitude and direction; its point of application also matters for the torque and hence the body's rotational response.

Specify what “onset” means. I would distinguish the first local, perhaps short-lived subduction event from persistent, self-sustaining recycling in a connected global network of tectonic plates. Modern plate tectonics combines relatively rigid moving lithosphere, localized boundary deformation, seafloor spreading and sustained convergence. A sinking drip or a single imposed descending sheet is not sufficient evidence of that global regime. Different processes may become established at different times, so a transition interval may be a better hypothesis than an instantaneous worldwide switch.

Construct competing physical models. Solve conservation of mass, momentum and energy for evolving mantle convection and lithosphere, including temperature- and stress-dependent rheology, brittle yielding, plastic deformation, melting, compositional buoyancy and relevant phase changes. Use geodynamic modelling to compare a stagnant lid, episodic overturn, drip-dominated recycling and a sustained mobile lid. In a simplified incompressible creeping-flow framework,Here $\mathbf{u}$ is velocity, $p$ pressure, $\eta$ effective viscosity, $\dot{\mathbf{ϵ}}$ the strain-rate tensor, $\rho$ density, $\mathbf{g}$ gravitational acceleration, $c_p$ specific heat capacity, $k$ thermal conductivity, $H$ radiogenic power per unit mass, and $\Phi$ dissipative heating per unit volume. These equations are a framework: compressibility, latent heat and chemical transport may require additional terms. A simple yield stress law ${\tau }_y=C+\mu (\rho gz-P_f)$ represents cohesion $C$, friction coefficient $\mu$ and pore pressure $P_f$, illustrating why hydration and interface weakening can change the outcome.

Vary the early-planet controls rather than assuming the answer. Explore plausible initial mantle potential temperature, radiogenic heating, crust thickness and composition, water inventory, grain-size evolution, surface temperature, and boundary strength. Hotter mantle lowers viscosity but can produce thicker buoyant basaltic crust; it does not automatically make sustained subduction easier. A cold strong plate can transmit slab pull yet resist bending and initiation. Water weakening of rocks, melt-induced weakening, inherited faults and mechanically weak sediments can change this balance. Test proposed triggers such as mantle plume impingement or gravitational instability without presupposing an established global plate-force system.

Useful diagnostics include the Rayleigh number,with thermal expansion coefficient $\alpha$, layer thickness $D$, thermal diffusivity $\kappa$ and temperature contrast $\Delta T$, together with surface mobility, the localization of deformation, plate longevity, melt production and material recycling. A large Rayleigh number establishes vigorous convection under suitable conditions, not necessarily a mobile lid. A convincing model should sustain one-sided subduction, connected boundary evolution and seafloor spreading after any artificial trigger is removed. Compare initiation thresholds with continuation thresholds: hysteresis can allow an established regime to persist under conditions in which it would not spontaneously start.

As an example of the type of experiment, three-dimensional models have tested mantle plume-induced subduction initiation, finding sensitivity to plate buoyancy, magmatic weakening and hydration of the interface. The scientific use of such results here is to identify a feasible mechanism and its predicted signatures, not to adopt their success as a direct measurement of an ancient onset date.^[ref]

Connect a model regime to a date through independent evidence. Couple the experiments to candidate planetary cooling histories, then predict observable pressure-temperature-time paths, magmatic compositions, deformation patterns and recycling rates for each regime. If independently dated rocks constrain a thermal or compositional state, ask which regimes can reproduce that state and the observed geological associations. Repeat over uncertain initial conditions and constitutive parameters, check numerical resolution, and compare two- and three-dimensional results where geometry matters. Models that prescribe a descending slab, fix plate velocity, or insert a weak boundary may be excellent forward models of a working subduction zone but do not demonstrate spontaneous initiation.

The result is a conditional statement: persistent plate tectonics is physically feasible under a range of conditions, and dated geological observations may favor that range at a particular time. Uncertain cooling histories, several feasible regimes and uncertain rheology prevent numerical feasibility alone from dating onset. The strongest constraint comes when the same parameter range explains multiple independently dated observations and alternative regimes fail those tests.

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.

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.

The secant modulus at a specified point of a stress-strain curve is the slope of the line from the declared unloaded origin to that point:It depends on the chosen point when the curve is nonlinear. Mississippi State University's solid-mechanics laboratory material defines it geometrically in this way.^[ref] Specify both the strain measure and the origin when reporting a secant modulus.

For an original branch $\sigma =E_0\epsilon +B{\epsilon }^2$, with $E_0=2\mathrm{GPa}$ and $B=100\mathrm{GPa}$,At ${\epsilon }_{\ast }=0.02$, stress is $80\mathrm{MPa}$, the secant modulus is $4\mathrm{GPa}$ and the tangent modulus is $6\mathrm{GPa}$. The first slope relates accumulated stress and strain; the second predicts a small further increment. In linear elasticity through the unloaded origin they both equal Young's modulus.

The relation is especially clear for a smooth curve with $\sigma (0)=0$. The fundamental theorem of calculus givesThus the secant modulus averages the local tangent slopes over that interval. A chord modulus instead joins two specified states, with slope $({\sigma }_2-{\sigma }_1)/({\epsilon }_2-{\epsilon }_1)$; its first state need not be the unloaded origin.

For a tensile measurement using initial cross-sectional area $A_0$ and local gauge length $L_g$, engineering stress and engineering strain givewhere extension is measured from the appropriate unloaded state. An original $F=800\mathrm{N}$, $A_0=10\mathrm{m}{\mathrm{m}}^2$, $L_g=50\mathrm{mm}$ and $\Delta L_g=1\mathrm{mm}$ yields $E_{\sec }=4000\mathrm{N}\mathrm{m}{\mathrm{m}}^{-2}=4\mathrm{GPa}$. A local extensometer avoids attributing all machine compliance to specimen extension. Retain the loading history, temperature and target stress or strain: a secant from an arbitrary preload and an unloading slope are different measurements, particularly after plastic deformation.