Codex @codex  0

Magnetic tension is the restoring stress associated with bending magnetic field lines. In a divergence-free field, the Lorentz-force density splits into a magnetic-pressure gradient and the tension term $(\mathbf{B}\cdot \nabla )\mathbf{B}/{\mu }_0$ in SI units.

Magnetoelastic coupling links a material's magnetic state to its strain and stress. Magnetostriction describes a resulting deformation; mechanical loading can also change the magnetic state.

The total specific angular momentum transported by matter and magnetic stress isand is constant on each magnetic surface.

The energy of the residual flow and magnetic field in a maintained shear flow has source $S=-a{T}_{xy}$, where $T_{xy}=\rho v_x{v}_y-B_x{B}_y/{\mu }_0$. The kinetic and magnetic stresses can exchange energy with the imposed background. The source vanishes when their off-diagonal contributions cancel.

The magnetohydrodynamic total pressure is $\Pi =p+B^2/(2{\mu }_0)$, the sum of gas pressure and magnetic pressure. The remaining Lorentz force is magnetic tension, so this scalar does not include the entire magnetic stress.

The magnetorotational instability allows a weak magnetic field to destabilize some differentially rotating flows that are stable to axisymmetric hydrodynamic displacements. A useful picture connects two fluid elements by magnetic tension: the inner element loses angular momentum and moves inward, while the outer element gains angular momentum and moves outward. Decreasing angular velocity amplifies their separation. This makes the instability relevant to angular momentum transport in an accretion disk.

For a local, unstratified, incompressible ideal-MHD model with vertical magnetic field $B_z$, take an axisymmetric perturbation proportional to $e^{st+ikz}$. Let $v_A=|B_z|/\sqrt{{\mu }_0\rho }$ and $\kappa$ be the radial epicyclic frequency, withEliminating the horizontal velocity and magnetic perturbations givesFor a hydrodynamically stable background with ${\kappa }^2>0$, a growing root exists exactly whenIndeed, the constant term is then negative, so the quadratic in $s^2$ has one positive root. If the constant term is nonnegative, both roots are nonpositive: their sum is negative and their discriminant is ${\kappa }^4+16{\Omega }^2{k}^2{v}_A^2$. The vertical-field local limit follows from the original linear analysis.^[ref]

In a Keplerian disk, the unstable interval is $0<k^2{v}_A^2<3{\Omega }^2$. Maximizing the positive root gives $k^2{v}_A^2=15{\Omega }^2/16$ and growth rate $s_{\max }=3\Omega /4$. Strong tension stabilizes shorter wavelengths. In a finite disk the unstable wavelengths must also fit the vertical domain; magnetic diffusivity, viscosity and nonideal magnetic coupling can modify or suppress them. Thus a negative angular-velocity gradient alone is not a universal sufficient condition for every magnetized flow. Linear growth also does not by itself specify the eventual turbulent stress.

Magnetostriction is a change of a material's shape or dimensions associated with its magnetization. A magnetic field can therefore produce strain without a contact force. The resulting magnetoelastic coupling lets a ferromagnet convert changes of magnetic state into mechanical motion; a constraint can instead turn the preferred deformation into stress. James Prescott Joule published his study of iron and steel bars in 1847.^[ref]

The surprising point is that reversing the magnetization need not reverse the deformation. A material can prefer a longer axis along either $+\mathbf{m}$ or $-\mathbf{m}$: an axis has no arrow. Isotropic linear magnetostriction vanishes excludes one particular linear constitutive term, while saturated isotropic magnetostrictive strain supplies a nonzero response with a quadratic dependence on the magnetic direction. The pulsed-field optical magnetostriction experiment shows how an actual length change can be resolved, and quadratic magnetostriction and frequency doubling separates an illustrative response law from a measured material curve.

A magnetostrictive actuator uses magnetostriction to produce displacement or force. Its free motion and its blocked stress are different boundary conditions, as shown in free strain and blocked magnetostrictive stress.

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.

An effective relation between deviatoric stress and deformation rate in mantle rocks. temperature, stress, grain size, water and time scale can affect it; a constant Newtonian viscosity is a model parameter, not a universal material number.

Mark D. Zoback coauthored research using borehole breakouts to infer in-situ crustal stress. The orientation and shape of borehole-wall failure become informative through a mechanical model relating local stress concentration to the far-field stress.^[ref]

The normalization $\mu =m/A_w$ has units $\mathrm{kg}{\mathrm{m}}^{-2}$. It converts the Coriolis force into a force per water-drag area and allows comparison with wind and water stresses. Icebergs generally have much larger $\mu$ than thin ice floes, but their effective air-to-water drag-area ratio also matters. A geometric depth becomes a mass per area only after multiplication by density.

For a series Maxwell model with $\tau =\eta /k$, $E^{\prime \prime }=k\omega \tau /(1+{\omega }^2{\tau }^2)$. At fixed strain amplitude the energy dissipated per cycle is $\pi E^{\prime \prime }{\epsilon }_0^2$ and peaks at $\omega \tau =1$. At fixed stress amplitude it is instead $\pi {\sigma }_0^2/(\eta \omega )$, which has no finite-frequency maximum. Specifying the imposed variable and whether loss means energy per cycle or power is essential.

For a half-space initially at rest and a wall given a step velocity $U$, momentum conservation and the Maxwell fluid stress law give $\tau u_{tt}+u_t=(G_0\tau /\rho )u_{yy}$. The displayed decaying Laplace transform solution obeys the wall condition. Inverting on a right-half-plane Bromwich contour yields a shear signal of speed $\sqrt{G_0/\rho }$. The limit $\tau \to 0$ at fixed $G_0\tau =\eta$ gives viscous error-function penetration; $\tau \to \infty$ at fixed $G_0$ gives a sharp elastic front.

Apply the divergence theorem to ${\partial }_k(x_i{\sigma }_{jk})$ and use momentum balance ${\sigma }_{jk,k}=\rho (a_j-b_j)$ and stress symmetry to obtain the displayed mean stress identity. Applying the same theorem to the infinitesimal strain tensor gives $V{\overline{e}}_{ij}=\frac{1}{2}{\int }_S(u_i{n}_j+u_j{n}_i)dS$. The acceleration term cannot be dropped during a transient.

Mechanical strength describes the stress a material withstands before a specified limiting event, such as yielding or fracture. The event, loading mode and test conditions must be stated: tensile strength, compressive strength and yield strength measure different responses.

In a simple tension test, engineering stress is applied force divided by initial cross-sectional area, $\sigma =F/A_0$. A specimen carrying $10\mathrm{kN}$ over $100{\mathrm{mm}}^2$ experiences $100\mathrm{MPa}$ by that convention. Strength is distinct from Young's modulus, which relates tensile stress to elastic strain, and from fracture toughness, which concerns resistance to crack growth. Microstructure, defects, temperature and loading history can all influence the measured limit.

Microfluidics controls and studies small fluid volumes in confined channels. In cellular assays it can impose reproducible flow and chemical fields, but channel geometry, wall interactions and flow-induced stresses can themselves alter behaviour.

Let a convex strain energy density have derivative equal to the actual stress tensor. Under mixed displacement and traction data, any equilibrated stress ${\sigma }_0$ with the prescribed body force and boundary tractions represents the external work. For any compatible trial strain ${\epsilon }^{\prime }$, the convex supporting-plane inequality gives $W({\epsilon }^{\prime })-W(\epsilon )\ge \sigma :({\epsilon }^{\prime }-\epsilon )$. The integral of $(\sigma -{\sigma }_0):({\epsilon }^{\prime }-\epsilon )$ vanishes by integration by parts. Thus the actual strain minimizes $\int (W-{\sigma }_0:\epsilon )$.

The signed wall-turbulence length formed from stress velocity and the vertical turbulent buoyancy flux. With upward buoyancy transport positive, stable cooling has $B_0<0$ and ${\ell }_O>0$; heating from below gives ${\ell }_O<0$. The magnitude marks the transition between shear-dominated and buoyancy-dominated scaling.

The mathematical description of elastic bodies whose stress-free material state changes through growth or remodelling. A multiplicative decomposition of the deformation gradient separates the growth tensor $\mathbf{G}$ from the elastic accommodation $\mathbf{A}$. The relaxed grown configuration need not be globally compatible, so even an unloaded body can develop internal stress.