Viscoelastic Response and Relaxation

Consider a one-dimensional elastic spring and a Newtonian dashpot connected in series. Here their variables are stress and strain, so the spring stiffness parameter is a modulus in pascals rather than an ordinary spring constant in newtons per metre. Both elements transmit the same stress, while their strains add. The spring obeys a linear stress–strain law and the dashpot obeys a linear stress–strain-rate law. The dashpot viscosity consequently has units of pascal seconds.

Differentiate the sum of strains and replace each element’s strain rate using its own law. The result is the Maxwell differential constitutive equation. It specifies the relation between a material point’s stress history and strain history; it is not Newton’s equation for an attached moving mass. The characteristic relaxation time is viscosity divided by modulus. Positive values of both parameters permit elastic storage and positive viscous dissipation. A single pair of constants is an approximation over a chosen strain, temperature, and time range, not a universal description of every polymer.

Apply an ideal instantaneous strain step and then keep the strain fixed. The dashpot cannot acquire a finite instantaneous strain under finite stress, so the initial strain is carried by the spring. After the step, total strain rate is zero, and the constitutive equation becomes a first-order decay equation for stress. Stress falls exponentially toward zero because strain transfers from the spring into the dashpot. The relaxation modulus is stress divided by the imposed strain; it has units of pascals and is not a dimensionless fraction.

Now impose constant stress instead. After the spring’s initial extension, the dashpot continues to extend at a constant rate. This Maxwell creep strain grows without bound for an indefinitely sustained nonzero stress. That is appropriate for a simple viscoelastic fluid model, but many viscoelastic solids retain a nonzero equilibrium stiffness. A parallel equilibrium spring added to a Maxwell branch creates a standard linear solid and a nonzero long-time relaxation plateau. Do not fit a decaying-to-zero model to such a plateau and interpret the discrepancy as mere measurement noise.

A relaxation time only becomes informative relative to an observation or loading time. Watch a Maxwell material for a duration much shorter than its relaxation time and little stress has relaxed. Wait many relaxation times and the stress has nearly disappeared under fixed strain. Temperature and molecular structure can change the relevant parameters substantially, and real materials often need a spectrum of relaxation times. Linear viscoelasticity also permits superposition of small loading histories, but that does not justify extending the same constants through yielding or very large strains.

Use the oscillator lab to investigate an analogy with an explicit boundary. At fixed natural frequency, predict that increasing damping ratio from 0.2 to 0.5 will make the underdamped envelope decay faster, then compare the histories. The lab solves a second-order unit-mass oscillator released from unit displacement and zero velocity. Its position can overshoot and oscillate, whereas Maxwell stress under the specified strain step decays monotonically. The damping-ratio slider is not a viscosity control, and its natural-frequency slider is not a modulus measurement. The shared lesson is that a decay needs a governing equation and a time scale, not that the models are interchangeable.