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Concentric cylinders impose shear while torque and rotation measure a material's response. The displayed calibration holds for Newtonian laminar flow, inner-cylinder rotation and a stationary outer cylinder. In a narrow gap, the rate is approximately $\Omega R_i/(R_o-R_i)$ and inner-wall stress is $\mathcal{T}/(2\pi L{R}_i^2)$. End effects, wall slip and inertial secondary flow require control.

A viscoelastic material stores part of the deformation energy and dissipates part of it. In linear viscoelasticity, a small imposed shear strain produces a stress determined by its history, $\tau (t)={\int }_{-\infty }^{t}G(t-t^{\prime })\dot{\gamma }(t^{\prime })d{t}^{\prime }$. A shear step probes stress relaxation; a stress step probes creep compliance. A purely elastic solid has persistent stress after a held strain, while a Newtonian fluid relaxes that stress and creeps continuously under constant stress.

For a Maxwell model, a spring of shear modulus $G_0$ and a dashpot of viscosity $\eta$ are in series:For the convention $\gamma \propto e^{i\omega t}$,The storage modulus dominates at short times or high frequencies; the loss modulus dominates at long times or low frequencies. The energy dissipated per cycle per volume is $\pi G^{\prime \prime }{\gamma }_0^2$. A Kelvin-Voigt model places spring and dashpot in parallel and instead gives bounded creep $J(t)=(1-e^{-G_{0}t/\eta })/G_0$: a simple illustration of a viscoelastic solid. Real materials can require a spectrum of relaxation times rather than one element.

A small-amplitude oscillatory rheometer measures stress amplitude and phase relative to imposed strain, giving $G^{\prime }=|{\tau }_0/{\gamma }_0|\cos \delta$ and $G^{\prime \prime }=|{\tau }_0/{\gamma }_0|\sin \delta$. First sweep strain amplitude to identify the linear range; control temperature, sample ageing, wall slip and instrument inertia. Frequency-dependent phase lag alone is insufficient if these apparatus effects are unresolved.

A Couette rheometer measures the torque required to rotate one coaxial cylinder relative to another. With inner radius $R_i$, outer radius $R_o$, wetted length $L$, inner angular velocity $\Omega$ and stationary outer wall, laminar Newtonian flow hasThe shear stress at the inner wall is ${\tau }_i=\mathcal{T}/(2\pi L{R}_i^2)$. In a narrow gap $d=R_o-R_i\ll R_i$, $\dot{\gamma }\simeq \Omega R_i/d$, so $\eta \simeq \mathcal{T}d/(2\pi L\Omega R_i^3)$.

Measuring torque over a range of speeds gives a flow curve: constant viscosity identifies a Newtonian regime; decreasing or increasing apparent viscosity indicates shear thinning or thickening. A yield-stress fit needs a genuine low-rate regime and must not be confused with bearing drag or wall slip. Small oscillatory strain also probes viscoelasticity, with $\tau ={\gamma }_0[G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t]$ for imposed $\gamma ={\gamma }_0\cos \omega t$.

Control temperature, calibrate instrument torque and end effects, and check gap dependence or roughened surfaces for slip. Keep inertia small enough to avoid Taylor vortices. The exact Newtonian profile is a calibration model; a non-Newtonian material needs its own stress–rate relation.

A viscoelastic material stores part of the work of deformation and dissipates part. In the linear regime its stress depends on the strain history, not just the instantaneous strain or shear strain rate. For simple shear, a causal relaxation modulus $G(t)$ gives $\sigma (t)={\int }_{-\infty }^{t}G(t-t^{\prime })\dot{\gamma }(t^{\prime })d{t}^{\prime }$. A sudden strain then probes stress relaxation, while a constant imposed stress probes material creep; these are different experiments.

For steady sinusoidal strain use the convention $\gamma (t)={\gamma }_0\cos \omega t$ and $e^{i\omega t}$ complex amplitudes. The complex viscoelastic modulus isso $\sigma (t)={\gamma }_0[G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t]$. The storage modulus $G^{\prime }$ is the in-phase elastic component. The loss modulus $G^{\prime \prime }$ multiplies the component in phase with shear strain rate; stress leads strain by $\delta =atan2(G^{\prime \prime },G^{\prime })$. The loss tangent is $\tan \delta =G^{\prime \prime }/G^{\prime }$ when $G^{\prime }>0$.

Multiplying stress by shear strain rate and averaging gives $⟨P⟩/V=\frac{1}{2}\omega G^{\prime \prime }{\gamma }_0^2$. Hence the dissipated energy per oscillatory shear cycle isThe elastic contribution returns its work over the cycle. A passive material has nonnegative dissipation, which fixes the sign of the loss term in the chosen Fourier convention. This complex viscoelastic modulus is a response function, not the mathematical absolute value of a complex number.

A Maxwell model places a spring of shear modulus $G$ in series with a dashpot of viscosity $\eta$, so the strains add at common stress:The Maxwell fluid dynamic modulus isAt $\omega \tau \ll 1$ the response is predominantly viscous, $G^{\prime \prime }\simeq \eta \omega$ and $G^{\prime }\simeq G(\omega \tau {)}^2$. At $\omega \tau \gg 1$ it approaches the elastic plateau $G^{\prime }=G$. The two moduli cross, and $G^{\prime \prime }$ reaches its maximum, at $\omega \tau =1$. A Kelvin-Voigt model instead has the spring and dashpot in parallel, giving $G^{\ast }=G+i\eta \omega$ and no fluid-like zero-frequency terminal flow. Real materials often need a spectrum of relaxation times rather than one Maxwell time.

Oscillatory rheometry measures stress amplitude and phase while imposing a small strain. An oscillatory-shear amplitude sweep first identifies a linear regime; an oscillatory-shear frequency sweep then measures $G^{\prime }$ and $G^{\prime \prime }$. Repeat at controlled temperature, suppress evaporation, check wall slip and sample aging, and correct instrument inertia at high frequency. Frequency-dependent structure and a nonlinear response cannot be inferred reliably by fitting one spring and dashpot to one point.

Measurement of stress–deformation response using a calibrated rheometer. Geometry, temperature, sample history and possible wall slip must be controlled before interpreting a measured force or torque as a bulk constitutive response.