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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.

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 torsion stress function for a simply connected bar. The only nonzero strain components are $e_{xz}=(w_x-\Omega y)/2$ and $e_{yz}=(w_y+\Omega x)/2$. ThusThe axial equilibrium equation is ${\partial }_x{\tau }_{xz}+{\partial }_y{\tau }_{yz}=0$. Since the section is simply connected, the planar divergence-free stress field has a stream function:Equality of the mixed warping derivatives gives $\boxed{\Delta \varphi =-2\mu \Omega }$. On the lateral boundary the axial traction is ${\varphi }_y{n}_x-{\varphi }_x{n}_y$, the tangential derivative of $\varphi$. Traction-free walls therefore make $\varphi$ constant on the connected boundary, and its arbitrary additive constant can be chosen so $\boxed{\varphi =0\text{ on }\partial D}$.

The axial torque isThe divergence theorem removes the boundary term because $\varphi =0$.

Tresca's yield criterion requires the maximum shear stress, half the largest principal-stress difference, not to exceed the material's shear yield stress ${\tau }_Y$. The stress eigenvalues here are $0,\pm \sqrt{{\tau }_{xz}^2+{\tau }_{yz}^2}$, so the maximum shear is $|\nabla \varphi |$ andIf yield is instead quoted as a uniaxial tensile stress, that parameter is $2{\tau }_Y$; the present convention is the source's shear parameter.

For the ellipse, the proposed quadratic vanishes on its boundary. Its Laplacian fixesThe corresponding warping is $w=\Omega (b^2-a^2)xy/(a^2+b^2)+\text{constant}$. Elliptic polar coordinates $x=ar\cos \vartheta$, $y=br\sin \vartheta$ give ${\int }_D\varphi =C\pi ab/2$, henceTo locate first yield, maximize $|\nabla \varphi |=2|C|\sqrt{x^2/a^4+y^2/b^4}$. It increases radially to the boundary. For $a>b$, the maximum there occurs at $(0,\pm b)$ and equals $2|C|/b$. ThereforeThe positive-twist value is the requested expression. The ends of the minor axis yield first, not the ends of the longer axis. This is elliptical-bar torsion and first yield.

In the completely elastic exterior, radial elastic compatibility makes $q+2t$ constant. The far-field stress vanishes, so $q+2t=0$ and $t=-q/2$. Radial force balance then gives $q^{\prime }+3q/r=0$, whence $q=C/r^3$. The pressure boundary condition is $q(a)=-P$: positive applied pressure is compressive. ThereforeFor completeness, the corresponding radial displacement field is $u=P{a}^3/(4\mu r^2)$, which gives these components through Hooke's law and decays at infinity.

The Tresca yield criterion becomesFor $P\ge 0$ this is largest at the cavity wall and decreases strictly with radius when $P>0$. The first yield point is therefore at $r=a$, where $3P/4={\tau }_Y$. HenceThe normalization matters: ${\tau }_Y$ in this paper is a shear yield stress, rather than the uniaxial yield strength.

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.

The hypothesis has two distinct components: the tissue changes its material response, and that change causally affects collective cell migration. A moving tissue may generate different stresses without becoming mechanically different. Conversely, a changed viscosity or yield stress need not generate directed movement without a driving force. The experiment should distinguish force production, material response and geometry.

First obtain a time-resolved three-dimensional map of the embryo using minimally perturbing live-cell imaging. Label nuclei and membranes, register tissue motion, and measure cell speeds, directional persistence, neighbour exchanges, strain rates and the timing of morphogenesis. Track cells relative to their neighbours as well as to the microscope: bulk translation alone does not show internal rearrangement. Compare equivalent developmental stages and anatomical regions, and quantify cell proliferation, death and differentiation so that changes in tissue composition are not mistaken for migration.

Next apply a calibrated mechanical perturbation. An embedded deformable droplet with known interfacial properties can report anisotropic stresses; magnetic actuation can impose a controlled deformation and reveal recovery, viscoelasticity and a possible yield stress. A controlled force step followed by relaxation probes different timescales. Laser ablation of a loaded junction provides a complementary recoil measurement, but recoil velocity depends on both pre-existing tension and dissipation: it is not an absolute tension measurement without a mechanical model. Probe different directions because tissue response may be anisotropic.

An informative precedent combined magnetically actuated droplets and cell tracking in the extending zebrafish body axis. Posterior tissue was more fluid-like than the more mature anterior region. Perturbations of N-cadherin and contractility helped connect adhesion and rearrangements to mechanical integrity and elongation.^[ref] This motivates a specific mechanism to test in the new embryo; it does not establish that all migrating tissues undergo the same transition.

For causality, change the proposed material property locally and acutely, before migration changes. For example, use optogenetics to alter contractility in a defined region, or a reversible adhesion perturbation, with unilluminated and light-only controls. Measure the material response again rather than assuming the manipulation made the tissue stiffer. Then test whether the spatial pattern and timescale of migration change as predicted. Restore the perturbed component or reverse the manipulation to seek rescue. Mosaic interventions help distinguish changes in migrating cells from changes in their surroundings, although altered signal transduction remains a potential confounder.

Finally fit a mechanical model to measured stresses, geometry and material parameters, then predict a held-out perturbation rather than merely fitting the original flow. Replicate across embryos, randomize treatment, analyse trajectories blind to treatment, and use the embryo as the independent biological replicate. Evidence is strongest when a measured material change precedes movement, targeted intervention changes both in the predicted way, and rescue restores both without simply restoring tissue viability.

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.

Where the stress in a yield-stress fluid remains below the yield stress, the shear rate vanishes and the material moves as an undeformed plug bounded by yielded layers.

For a Bingham layer on a stationary cylinder of radius $a$ with a shear-free cylindrical surface at $b$, an axial pressure gradient $-G$ produces ${\tau }_{rz}=G(b^2/r-r)/2$. Flow starts when $G(b^2-a^2)/(2a)$ exceeds the yield stress.

A slump test for yield stress probes a material by allowing a molded sample to deform under its own weight. Fill a known mold on a level support, remove the mold, and measure the decrease in height or the spread of the final deposit. Gravity supplies the driving stress; resistance associated with the yield stress can arrest the motion before the sample becomes a thin layer.

The measured distance is not itself a stress. Inferring yield stress requires a calibrated relation or a mechanical model using the initial geometry, final shape and density. The final slump alone generally cannot identify both yield stress and dynamic viscosity: a time-resolved measurement provides additional information. A modified concrete test, for example, records the time required for a specified fall as well as the final slump.^[ref] The procedure is useful as a simple field measurement, with interpretation dependent on sample history and geometry.

The Tresca yield criterion bounds the largest resolved shear stress, expressed through the principal stresses. In spherical symmetry, the two tangential principal stresses coincide, so $f=|{\tau }_{rr}-{\tau }_{\theta \theta }|/2$. Here ${\tau }_Y$ is a shear yield stress; the corresponding uniaxial yield strength is $2{\tau }_Y$.

The Von Mises yield criterion depends only on the second invariant of deviatoric stress. With shear yield stress $k$, its surface is ${\sigma }^{\prime }:{\sigma }^{\prime }=2k^2$; the uniaxial yield stress is $\sqrt{3}k$. An associated flow rule gives plastic strain rate proportional to ${\sigma }^{\prime }$, up to a positive normalization of the plastic multiplier.

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.

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-stress fluid can sustain a finite shear stress without steady flow, but flows when the applied loading exceeds its yield stress. The ideal Bingham plastic treats the unyielded region as rigid and gives, after yield,Here $\dot{\gamma }$ is the shear rate, ${\tau }_y$ the yield stress and ${\eta }_p$ the post-yield dynamic viscosity. Real materials can undergo small elastic deformations below the flow threshold; an ideal rigid branch does not describe every part of their response.

A pressure-driven experiment makes the spatial distinction visible. Where the local shear stress is below threshold, a plug flow of a yield-stress fluid can move without internal shearing, while adjacent yielded layers deform. If every point remains below threshold and the walls have no wall slip, the ideal material does not flow. Slowly evolving microstructure and thixotropy make the apparent threshold depend on preparation, waiting time and the loading protocol.