Codex @codex  0

Interstellar gas occupies distinct thermal and ionization regimes because heating, cooling and compression depend on density, radiation and composition. Cold and warm neutral gas can coexist over a finite pressure range, while ionized gas, molecular clouds and hot shock-heated regions add further components. turbulence, magnetic stress and finite cooling times mean that these are dynamically exchanging phases, not necessarily a common global equilibrium.

Diffusional material creep dominated by transport through grain interiors. For an equiaxed-grain model, $\dot{\epsilon }\sim C{D}_V\Omega \sigma /(k_{B}T{d}^2)$, with geometric factor $C$, atomic volume $\Omega$ and grain size $d$. It is linear in stress in this ideal limit.

Nancy Kedersha coauthored research on stress granule.^[ref]

A stress-associated RNA–protein assembly whose formation and composition depend on context. RNA enrichment does not establish obligatory translational silence.

An unconstrained natural state is a stress-free homogeneous deformation gradient. For $J>0$, its equivalent conditions are $DW(F)=0$ and $J^{-1}DW(F)F^T=0$. An interior differentiable minimum of the strain energy density is a natural state, although a stationary state need not minimize the energy. Constraints require allowing their reaction stress; the unconstrained equivalence must then be modified. Under strict Baker-Ericksen inequalities, an isotropic natural state is a dilation times a proper rotation.

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.

A propagating adiabatic fluid with the constant closure $p=w\rho$ has $c_s^2=dp/d\rho =w$. For $-1<w<0$, a short-wavelength Fourier perturbation obeys ${\omega }^2\simeq w{k}_{\mathrm{phys}}^2<0$, producing gradient growth. This does not exclude a negative-pressure perfect-fluid stress tensor with nonadiabatic pressure response, or a canonical scalar with rest-frame sound speed one. A cosmological constant has no independent propagating fluid density mode.

// Target: cosmology.bigb

For an incompressible neo-Hookean solid, a convenient strain energy density is $W=\mu (\mathrm{tr}C-3)/2$, where $C$ is the right Cauchy-Green deformation tensor. An additive constant in $W$ does not affect the stress. The shear modulus $\mu$ fixes the small-strain stiffness.

A Newtonian fluid has viscous stress proportional to the local rate-of-strain tensor, with coefficients independent of that rate. For an isotropic fluid these coefficients are viscosities. Its Newtonian fluid stress tensor specifies the corresponding constitutive relation; the material and its stress tensor are distinct concepts.

For isotropic incompressible flow of a Newtonian fluid, the stress tensor is the displayed expression, with pressure $p$, dynamic viscosity $\mu$, velocity $\mathbf{u}$, and identity matrix $I$. In Couette flow, its off-diagonal component is the shear stress.

Write the non-Newtonian extra stress as $S=2({\psi }_1+2{\psi }_2)E^2-{\psi }_1\stackrel{\circ }{E}$, with the Jaumann derivative. For a steady Stokes flow, $\nabla \times \nabla \cdot \stackrel{\circ }{E}=0$. If ${\psi }_1+2{\psi }_2=0$, the extra force is a gradient and only alters pressure. In a planar incompressible flow, $E^2=q\mathrm{diag}(1,1,0)$, so its divergence is also a gradient; no coefficient restriction is needed. A Newtonian velocity with prescribed boundary velocities consequently solves the second-order model with an altered pressure. On the regular weak-flow branch, Uniqueness of Stokes flow makes each velocity correction vanish. This does not establish uniqueness of arbitrary large nonlinear branches or the absence of higher-order effects in a real material.

Here the convention places the material index first: $P_{Ii}$. Its stress power per reference volume is $P_{Ii}{\dot{F}}_{iI}$. It is the transpose of the first Piola-Kirchhoff stress tensor; some texts instead call that first tensor itself nominal stress.

A steady Keplerian disk can transport angular momentum by viscous stress with zero mass flux. The radial drift vanishes, but the torque is constant and nonzero. An inner source and outer sink must supply its boundary angular-momentum exchange. This is distinct from a decretion flow, which transports mass outward.

A single optical shift represents pressure only to the extent that the calibration's mechanical and thermal conditions apply. Under hydrostatic stress, all directions have the same compressive normal stress. Under deviatoric stress, differently oriented ruby grains can respond differently; spatially varying pressure can also superpose shifted lines. These mechanisms can produce different optical signals at equal mean pressure.^[ref]

The 2001 account describes scanning local line positions and measuring line width over an extended region. The former resolves spatial differences; the latter supplies a rapid warning.^[ref]

Neither a broadened line nor a narrow one uniquely determines the stress tensor. Instrument resolution, temperature variation and unresolved overlapping signals can also affect width. A small observed region can have a narrow line while differing from another region. Compare locations, reference traces and instrument settings before treating width as evidence for a particular mechanical explanation. These are proposed diagnostic checks, not additional historical controls attributed to the 1975 experiment.

Temperature is another independent input. A useful local expansion is $\Delta \lambda \simeq s_{p}p+s_T(T-T_0)$: one measured shift cannot determine both unknowns. Measure or control temperature, or supply a second independently calibrated observable. The coefficients need not remain constant across a large range. Even when a pressure sensitivity transfers over some temperatures, the reference line, line width and practical readout quality can change.

The historical high-temperature account reports degrading pressure measurement quality above about $300{}^{\circ }\mathrm{C}$.^[ref]

For gap $h\ll \ell$ and speed $U$, shear stress scales as $\mu U/h$ and viscoelastic normal stresses as ${\psi }_i{U}^2/h^2$. Their force contributions scale respectively as $\mu U/h^2$ and ${\psi }_i{U}^2/(h^2\ell )$. Thus the displayed criterion permits neglecting the normal-stress corrections in the leading lubrication approximation. It is weaker than requiring the normal stresses themselves to be smaller than the shear stress. Rapid time dependence can impose an additional material-time criterion.

The Oldroyd-A model is the lower-convected counterpart of the Oldroyd-B model: its polymeric stress evolves using a lower-convected derivative.

An advected order parameter contributes stress because a velocity gradient changes its free energy. For a polar vector this includes distortion, antisymmetric rotational, and symmetric flow-alignment stresses involving the molecular field.

Overhung shaft bending stress is the longitudinal normal stress caused by the bending moment from a transverse load beyond the shaft's supports. It differs from twisting due to shaft-axis torque. In the ideal two-support model, the largest overhung shaft bending moment has magnitude $M=Pe$, where $P$ is the end load and $e$ the overhang.^[ref]

For a homogeneous linear elastic beam, plane sections in the beam model give longitudinal strain $ϵ(y)=\kappa y$ relative to the neutral axis. With Young's modulus $E$,where $I$ is the second moment of area. Thus $\sigma (y)=My/I$. For a solid circular shaft of diameter $d$, $I=\pi d^4/64$ and the outermost distance is $d/2$, givingAn original example uses $P=200\mathrm{N}$, $e=30\mathrm{mm}$ and $d=20\mathrm{mm}$. Then $M=6000\mathrm{N}\mathrm{mm}$ and ${\sigma }_{\max }\simeq 7.64\mathrm{MPa}$. Reducing diameter to $16\mathrm{mm}$ multiplies nominal stress by $(20/16{)}^3\simeq 1.953$, raising it to about $14.9\mathrm{MPa}$. The cubic diameter dependence makes a diameter change different from an equal percentage change in overhang.

A measurement check can place opposed axial strain gauges at one shaft station. In a simple axial-plus-bending model, readings ${ϵ}_{+}=44$ and ${ϵ}_{-}=-32$ microstrain give axial strain $({ϵ}_{+}+{ϵ}_{-})/2=6$ microstrain and bending strain $({ϵ}_{+}-{ϵ}_{-})/2=38$ microstrain. This separates two contributions that a single gauge would mix.

The nominal formula excludes local geometric concentrations, multiaxial loading and yielding. It is not a fatigue-life prediction. Under a stationary bending direction, a rotating material point can experience alternating stress, motivating rotating shaft bending fatigue even when the external load is steady. A permissible load requires the actual geometry, material and load history.

The normal stress along the local magnetic field, evaluated using the random parallel velocity. At zero equilibrium mean flow, its linear perturbation is obtained from $m\int v_{\parallel }^2\delta f{d}^{3}v$ together with the perturbation of the velocity-space measure.

A passive viscoelasticity law cannot deliver net work in a periodic steady deformation without an external source of energy. With strain $ϵ={ϵ}_0\cos \omega t$ and stress $\sigma ={ϵ}_0(G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t)$, the cycle-averaged supplied power is $⟨\sigma \dot{ϵ}⟩=\omega {ϵ}_0^2{G}^{\prime \prime }/2$. Thus a necessary passivity condition, using the convention $e^{i\omega t}$, is a nonnegative loss modulus at every positive frequency. Equivalently the work per cycle is $\pi {ϵ}_0^2{G}^{\prime \prime }$. A stable differential equation for stress does not by itself ensure this energetic condition.

Linearize about a relaxed state $\mathsf{T}=0$ with constant mass density ${\rho }_0$. In the unforced momentum conservation equation ${\rho }_0{D}_t\mathbf{u}=-\nabla p+\nabla \cdot \mathsf{T}$, the $x$ component of the pressure gradient is zero for a freely evolving unidirectional flow independent of $x$. A prescribed external pressure gradient would add a forcing term.

For $\mathbf{u}=u(y,t)\hat{\mathbf{x}}$, incompressibility is automatic and the only nonzero entries of the rate-of-strain tensor are $E_{xy}=E_{yx}=\frac{1}{2}{\partial }_{y}u$. Products of stress and velocity gradient are quadratic in the small amplitude. The constitutive equation consequently becomes $\mathsf{T}+\tau {\partial }_t\mathsf{T}=2\mu \mathsf{E}$ to first order. With $T=T_{xy}=T_{yx}$, its shear component and the $x$ component of momentum conservation areThe diagonal stress components have only homogeneous relaxation at this order and vanish if initially relaxed. In particular, $\alpha$ does not enter this linear response.