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Use a rheometer, for example a cone-and-plate rheometer or a parallel-plate rheometer, to impose a small deformation and measure the required torque, or impose a torque and measure the deformation. The apparatus geometry converts these into shear stress and strain. First check that reducing the imposed amplitude does not change the normalized response; this identifies the linear regime and avoids yielding or structural rearrangement.

Three complementary protocols characterize the material's memory. In a stress relaxation experiment impose a small step ${ϵ}_{0}H(t)$ and hold the deformation fixed. The output is the relaxation modulus $G(t)=\sigma (t)/{ϵ}_0$, revealing how stored stress decays and whether an elastic plateau remains. In a creep compliance measurement impose a small stress step ${\sigma }_{0}H(t)$ and record $J(t)=ϵ(t)/{\sigma }_0$. Removing the load also distinguishes recoverable deformation from permanent flow.

In small-amplitude oscillatory shear, impose sinusoidal strain over a range of frequencies and measure both stress amplitude and phase. With $ϵ={ϵ}_0\cos \omega t$, fitThe outputs are the storage modulus $G^{\prime }$ and loss modulus $G^{\prime \prime }$, or equivalently amplitude ratio and loss tangent $\tan \delta =G^{\prime \prime }/G^{\prime }$. These measure elastic storage and dissipative response, respectively. The step experiments give time-domain information and the oscillatory experiment gives the corresponding frequency-domain response. All comparisons must use the same deformation convention and controlled temperature.

The dynamic modulus, or complex viscoelastic modulus, is the ratio of complex stress amplitude to complex strain amplitude in the periodic steady response:This defines the sign convention; for passive response the loss modulus is nonnegative at positive frequency. A Newtonian fluid obeys $\sigma =\eta \dot{ϵ}$, while a Hooke's law solid obeys $\sigma =G_0ϵ$. HenceThe first is purely dissipative and the second purely elastic. Here $ϵ$ denotes the scalar deformation used in the problem, with any shear-versus-extension geometric factors already included in its modulus and viscosity.

Parallel elements have the same strain, while their stresses add. The spring supplies $G_0ϵ$ and the viscous element supplies ${\eta }_0\dot{ϵ}$, so the Kelvin-Voigt model has constitutive equationFor the oscillatory deformation this givesPhysical stress and strain are the real parts of these expressions. Thus the storage modulus is $G_0$, independent of frequency, and the loss modulus is $\omega {\eta }_0$. The equation is the quasistatic material law; an inertial equation for an attached mass would require additional mechanical data.

For elements in series the stress is common and the strains add. If ${\widehat{ϵ}}_K$ and ${\widehat{ϵ}}_1$ are the Kelvin element and added spring amplitudes, thenThe compliances therefore add, yieldingThis is a standard linear solid model. DefineWriting $x=\omega \tau$, its storage modulus and loss modulus areThus $G^{\prime }$ increases from the low-frequency plateau $G_L$ to the high-frequency plateau $G_H$. The loss modulus has low-frequency slope $+1$ and high-frequency slope $-1$ on a log-log plot, with maximum $\Delta G/2$ at $\omega =1/\tau$. At that frequency $G^{\prime }=(G_L+G_H)/2$. There need not be a crossing of the two curves for arbitrary spring constants.

A Kelvin-Voigt model has no finite instantaneous elastic compliance: the viscous element prevents a sudden deformation under a finite stress step. Its creep compliance isConversely an ideal strain step requires an impulsive stress,After the impulse the held-strain response is simply constant. There is no finite stress-decay interval, and the loss modulus grows without bound as $\omega \to \infty$. This fails to represent a material with both a finite instantaneous stiffness and delayed stress relaxation.

The added series spring resolves this particular defect. At the instant of loading the Kelvin element cannot move but the added spring can. The standard linear solid model consequently hasandIts instantaneous stiffness is $G_H=G_1$, its long-time stiffness is $G_L$, and it exhibits genuine stress relaxation as well as delayed creep. The creep retardation time ${\eta }_0/G_0$ differs from the stress-relaxation time $\tau ={\eta }_0/(G_0+G_1)$. These properties also follow directly from its constitutive equation,It remains an idealized solid with one relaxation mode: it cannot describe unrestricted long-time viscous flow, nonlinear response, or a broad spectrum of relaxation times.

By definition, the relaxation modulus is the stress generated by a unit step of strain from an initially relaxed state. Set $ϵ(t)=H(t)$, where $H$ is the Heaviside step function, and write $\sigma (t)=G(t)$. Integrating the linear ordinary differential equation across the step shows that neither $G$ nor $G^{\prime }$ can jump: a jump would produce a Dirac delta function or its derivative, absent from the forcing. Hence $G(0^{+})=G^{\prime }(0^{+})=0$.

For $t>0$, a particular solution is $1$. The characteristic equation factors asso $G=1+A{e}^{-t/5}+B{e}^{-t/3}$. The initial conditions give $A+B=-1$ and $A/5+B/3=0$, whenceThe time and modulus units are those implicit in the printed coefficients. A Laplace transform check gives $\hat{G}(s)=1/[s(5s+1)(3s+1)]$.

The original PDF really does put undifferentiated strain on the right-hand side. The resulting response rises from zero to one, with $G^{\prime }(t)=\frac{1}{2}(e^{-t/5}-e^{-t/3})>0$, rather than exhibiting ordinary stress relaxation. There is a stronger physical problem if this is intended as a passive material law: with the convention $e^{i\omega t}$ the stress-to-strain amplitude ratio isThe loss modulus would give negative cycle-averaged supplied power $⟨\sigma \dot{ϵ}⟩=\omega G^{\prime \prime }|{ϵ}_0{|}^2/2$, contrary to passivity in linear viscoelasticity. The boxed expression is therefore the formal relaxation modulus of the equation as printed, not a claim that this equation describes an ordinary passive viscoelasticity model. No derivative of the input has been silently inserted.

For a standard linear solid model with $0<G_R<G_U$, put $x=\omega \tau$. Its storage modulus is $G^{\prime }=G_R+(G_U-G_R)x^2/(1+x^2)$ and its loss modulus is $G^{\prime \prime }=(G_U-G_R)x/(1+x^2)$. The latter peaks at $x=1$, whereas its loss tangent isDifferentiation puts the loss-tangent peak at $x=\sqrt{G_R/G_U}$, with height $(G_U-G_R)/(2\sqrt{G_R{G}_U})$. A loss-modulus peak and a loss-tangent peak therefore need not occur at the same frequency. For weak relaxation, $G_U/G_R$ is close to one and their frequencies are approximately equal.

Rheology studies how matter deforms and flows under applied stress, including the dependence on loading history and observation time. Its central task is to connect a measured stress to strain, strain rate and relevant internal state through a constitutive equation. A Newtonian fluid has an instantaneous proportional response to strain rate; an elastic solid responds to strain; a viscoelastic material retains information about previous deformation.

These differences can be exposed by changing the experiment rather than only changing the load. Holding stress fixed measures material creep; holding strain fixed measures stress relaxation; small-amplitude oscillatory shear separates storage modulus from loss modulus. A rheometer supplies controlled motion and measures the resulting force or torque. Such measurements help predict whether a paste will remain in place, how much pressure is needed to drive a fluid flow, and whether a material will recover after loading.

A standard linear solid model has a finite instantaneous stiffness, a nonzero long-time stiffness and one exponential stress-relaxation mode. A Kelvin-Voigt model with spring $G_0$ and viscosity $\eta$ in series with spring $G_1$ givesThese expressions follow by adding series strains under common stress. Its storage modulus is $G_L+(G_H-G_L)x^2/(1+x^2)$ and its loss modulus is $(G_H-G_L)x/(1+x^2)$ for $x=\omega \tau$. Its relaxation modulus is $G(t)=G_L+(G_H-G_L)e^{-t/\tau }$, while its creep compliance is $J(t)=G_1^{-1}+G_0^{-1}(1-e^{-G_{0}t/\eta })$. Thus creep retardation and stress-relaxation times differ. This solid has bounded creep and cannot represent terminal fluid flow.

An experimental physicist who coauthored a method for inferring viscoelasticity from the motion of embedded probe particles. With David A. Weitz, he connected dynamic light scattering data to the storage modulus and loss modulus. The method makes Brownian motion informative: the particles' fluctuations encode how their surroundings store and dissipate mechanical energy.^[ref]

Viscoelasticity combines recoverable elastic response with dissipative and time-dependent response. Two specimens may react similarly to a rapid pull but behave differently when the same load is maintained. In a material creep test, constant stress produces evolving strain; in a stress relaxation test, fixed strain produces evolving stress.

The Maxwell model makes the distinction explicit by placing an elastic spring and viscous element in series. In simple shear strain it obeysAfter a small step to fixed ${\gamma }_0$, its shear stress decays as $\tau (t)=G{\gamma }_0{e}^{-t/\lambda }$ with viscoelastic relaxation time $\lambda =\eta /G$. Under constant shear stress ${\tau }_0$, it instead continues to deform, with $\gamma (t)={\tau }_0/G+{\tau }_{0}t/\eta$. These are predictions of this particular model: a Kelvin-Voigt model has bounded creep and different short-time behavior. Small-amplitude oscillatory shear measures the same competition through storage modulus and loss modulus, rather than assuming that one constant dynamic viscosity characterizes the material.