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Photoelasticity is the dependence of a material’s optical response on stress, used to reveal otherwise invisible mechanical patterns. A transparent material that is optically isotropic before loading can become birefringent under load: orthogonal polarizations acquire different refractive indices and therefore a relative phase retardation.^[ref]

For a thin, initially isotropic specimen in a suitable linear regime, the stress-optic law relates induced birefringence to the difference of in-plane principal stresses:Here $C$ is a material coefficient with units $\mathrm{P}{\mathrm{a}}^{-1}$, $t$ is the optical thickness, $\lambda$ the vacuum wavelength and $\delta$ the phase delay. This simple form assumes the relevant stress directions and difference do not change along the light path. A general three-dimensional specimen needs a more careful optical and mechanical model.

With ideal crossed polarizers, intensity normalized to the light after the first polarizer iswhere $\theta$ is the angle between the first polarization direction and a principal optical axis. A dark region can therefore arise from axis alignment or from an integer multiple of $2\pi$ retardation. Darkness alone does not prove that stress is zero.

For illustrative $C=3\times 10^{-12}\mathrm{P}{\mathrm{a}}^{-1}$, $t=5\mathrm{mm}$ and $\lambda =500\mathrm{nm}$, one full phase cycle corresponds toThese chosen values illustrate conversion from optical delay to a stress difference, not a calibration for every plastic. The method does not directly recover a hydrostatic stress shared equally by the two in-plane directions.

A polariscope makes the measurement visible. The Illinois teaching apparatus places a specimen between a polarizer and analyzer, applies load with a manual leadscrew, measures it with a load cell and records the pattern with a camera. Monochromatic illumination and adjustable polarization optics separate information about stress directions and differences.^[ref] Comparing patterns as load changes is a motivating experiment in connecting an optical observable to a mechanical model.

A tensor coefficient relating an increment of electrical displacement to stress, or the corresponding converse strain response to electric field, under stated constitutive conventions. Charge-per-force units $\mathrm{C}{\mathrm{N}}^{-1}$ are equivalent to displacement-per-voltage units $\mathrm{m}{\mathrm{V}}^{-1}$ for the usual $d$ coefficients.

Piezoelectricity is linear coupling between mechanical deformation and electrical response in suitable materials. The direct piezoelectric effect produces electrical polarization under applied stress; the converse piezoelectric effect produces strain under an applied electric field. The two responses are linked by the same electromechanical coupling coefficients when conventions and mechanical constraints are specified consistently.

A piezoelectric stress sensor uses the direct response to turn force or vibration into an electrical signal. A scanning probe uses the converse response to move a tip by very small distances, controlling a tunnel junction's separation. The resulting displacement can depend on drive history through piezoelectric hysteresis and can continue changing after a voltage step through piezoelectric scanner creep. These effects motivate calibration and closed-loop control.

A piezoelectric transducer converts mechanical stress into an electrical signal, or an applied electric field into mechanical deformation, using a piezoelectric material. The two directions enable both acoustic reception and transmission.

An ultrasound element is driven electrically to launch a pulse and then detects returning pressure waves through its generated charge or voltage. Mechanical resonance, electrical loading and acoustic coupling determine its bandwidth and sensitivity. The crystal response is therefore only one part of the complete instrument calibration.

These stress measures refer force or area to the reference configuration. Different index-order conventions distinguish a first Piola-Kirchhoff stress tensor from its transposed nominal stress tensor. The second Piola-Kirchhoff stress tensor has two material indices.

In a functional model, a driven piston reduces the chamber's volume. A surrounding cylinder carries the wall stress; a deformable packing or gasket closes a potential leak path around a moving or removable closure. A sample and a separate pressure sensor occupy the chamber. A transmitting liquid can load the sample on all sides; the sensor's signal passes to an external readout. This is a division of experimental functions, not a scale reconstruction of one historical machine.

For a slowly moving frictionless drive piston of effective surface area $A$, external pressure $p_{\mathrm{ext}}$ and inward applied force $F$, Newton's second law at mechanical balance givesThus $20\mathrm{kN}$ on a stipulated $20\mathrm{m}{\mathrm{m}}^2$ effective surface area corresponds ideally to $1\mathrm{GPa}$ of differential pressure. These are illustrative numbers, not apparatus specifications. The effective surface area is the one associated with the pressure difference, not necessarily an external ram's visible cross section. With resisting friction $F_f$ during inward motion, the balance becomes $p-p_{\mathrm{ext}}=(F-F_f)/A$. During outward motion the sign of the friction contribution reverses. An applied-load reading can therefore show hysteresis even when the chamber's actual pressure is well defined.

A resistance pressure gauge instead supplies an internal signal. In a generic electrical readout, measure an electric potential difference $V_e$ with a known electric current $I$ and use Ohm's law to infer electrical resistance $R=V_e/I$. This describes a possible readout principle; the lecture passage cited below identifies the sensor material and calibration landmarks without specifying this wiring.

For fixed plates $y=\pm h$, no slip, viscosity ${\mu }_0>0$, yield stress ${\tau }_0\ge 0$ and $G=-p_{,x}>0$, force balance gives shear stress $-Gy$. The central plug flow of a yield-stress fluid has half-width $y_0={\tau }_0/G$. If $y_0<h$, the velocity isIt is even in $y$; its derivative is $(-Gy+{\tau }_0\mathrm{sign}y)/{\mu }_0$ in the yielded layers and zero in the plug. If $Gh\le {\tau }_0$, there is no flow. Odd velocity is incompatible with this pressure-driven no-slip solution at nonzero flow.

In plane strain, the deformation map has $x_1=x_1(X_1,X_2)$, $x_2=x_2(X_1,X_2)$ and $x_3=X_3$. The out-of-plane stretch is one and out-of-plane shear vanishes. This kinematic condition need not make the out-of-plane stress vanish.

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.

A plastic flow rule prescribes the direction and rate of plastic straining once a yield condition is active. In a plastic-potential model its direction is the stress gradient of $g$, multiplied by a nonnegative plastic multiplier. An associated flow rule takes $g$ equal to the yield function. The stress-strain split, loading conditions and hardening evolution are needed along with this equation to close an elastoplastic model.

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.

In small-strain elastoplasticity, total infinitesimal strain tensor splits into an elastic part and a generally irreversible plastic part. The elastic constitutive equation determines stress from $e^{\mathrm{e}}$, while a yield condition and a plastic flow law determine the evolution of $e^{\mathrm{p}}$. Removing external load need not remove plastic strain, so using Hooke's law on total strain after yielding is insufficient.

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.

The polar molecular field is the functional derivative of the free energy with respect to the polar order parameter. In the positive-derivative convention, dissipative dynamics is $D_t\mathbf{p}=-\Gamma \mathbf{h}$ with positive kinetic coefficient. For a quartic local term and an isotropic gradient penalty,Some authors instead define the molecular field with a minus sign, which changes the associated stress and evolution conventions.

The pressure of fluid in pore space. On fluid-accessible contacts, it reduces effective compression and can promote frictional sliding or tensile opening at fixed total stress.

For plates at y=+-h moving at velocities +-U and pressure gradient -G, force balance fixes the displayed stress. The power-law fluid law gives $u_y=(Gh/k{)}^{1/n}\mathrm{sgn}(\alpha -y/h)|\alpha -y/h{|}^{1/n}$. Integrating from the lower no-slip wall yields $u=-U+hn(Gh/k{)}^{1/n}[|\alpha +1{|}^{1+1/n}-|\alpha -y/h{|}^{1+1/n}]/(n+1)$. The upper wall fixes alpha uniquely for k,n,G,U>0. Absolute values are essential when the shear reverses inside the gap.

A regime in which the steady material creep rate varies approximately as a power of stress, $\dot{\epsilon }\propto {\sigma }^n$, with thermally activated coefficients. In ordinary dislocation-creep fields, glide, climb and recovery give an exponent greater than the linear-stress diffusion-creep limit.

Pressure measures the compressive normal force per unit surface area. For a uniformly loaded surface of surface area $A$, the magnitude of the perpendicular force is $F_{\perp }=PA$; locally, $P=d{F}_{\perp }/dA$. Its SI unit is the pascal, $1\mathrm{Pa}=1\mathrm{N}{\mathrm{m}}^{-2}$. A force parallel to the surface contributes shear stress, rather than pressure. In a gas, the steady macroscopic force comes from many microscopic transfers of momentum; ideal gas pressure from wall collisions makes this connection explicit. ^[ref]

For a computed example, a uniform pressure of $100\mathrm{kPa}$ on a piston of surface area $10\mathrm{c}{\mathrm{m}}^2=10^{-3}{\mathrm{m}}^2$ produces a perpendicular force of $100\mathrm{N}$. If the opposite face is exposed to $80\mathrm{kPa}$, the net force due to the two pressures is $(100-80)\times 10^3\times 10^{-3}=20\mathrm{N}$, directed toward the lower-pressure side. The pressure on either face and their difference answer different questions; gauge pressure records a difference from a specified reference, often atmospheric pressure.

A scalar pressure describes a gas in local thermodynamic equilibrium with isotropic random particle velocities: every wall orientation has the same normal force per unit surface area. Without this isotropy, normal stresses can depend on direction and a pressure tensor is needed. Thus random motion alone is insufficient; a preferred direction matters.

In thermodynamics, pressure and volume are thermodynamic conjugate variables. For a simple compressible system, reversible expansion delivers work $\delta W=PdV$, and at fixed entropy $S$ and particle number $N$,where $U$ is internal energy. A large pressure therefore means both a large wall force per unit surface area and a large amount of expansion work per small increase in volume.

In Gaussian units, the gyrotropic pressure tensor adds directional stress to the magnetic tension. A transverse displacement of a uniform magnetic field obeys $\rho {\partial }_t^2{\mathbf{\xi }}_{\perp }=\mathcal{T}{\partial }_{\parallel }^2{\mathbf{\xi }}_{\perp }$. Positive $\mathcal{T}$ gives a restoring force; negative $\mathcal{T}$ gives the fluid firehose instability. This follows by combining $-\nabla \cdot \mathsf{P}$ with the divergence of the Maxwell stress tensor. In SI replace $B^2/(4\pi )$ by $B^2/{\mu }_0$.

For an inviscid interface with no surface tension, continuity of normal traction reduces to continuity of pressure. The same condition is used in a sharp Darcy law interface model that neglects capillary pressure and resolved interfacial viscous normal stress. For a general viscous interface, continuity of normal traction balances the full stress tensor; it does not by itself imply continuity of pressure.