Viscoelastic Relaxation Modulus: Maxwell, Kelvin–Voigt & Standard Linear Solid Models

This lecture develops the relaxation modulus for the Maxwell viscoelastic model, explains why the Kelvin–Voigt model cannot represent an ideal stress-relaxation test, and introduces the Standard Linear Solid as a more realistic model of viscoelastic behavior. We begin with the Maxwell model, consisting of a spring and dashpot in series. In a stress-relaxation test, an instantaneous strain ε₀ is applied and then held constant while the stress decreases with time. The relaxation modulus is defined as Eᵣ(t) = σ(t)/ε₀. Solving the Maxwell constitutive equation for constant strain gives an exponentially decaying stress and relaxation modulus, Eᵣ(t) = E exp(-t/τ), where the characteristic relaxation time is τ = η/E. The Kelvin–Voigt model, consisting of a spring and dashpot in parallel, behaves differently. An ideal instantaneous strain would require an infinite strain rate and therefore infinite stress in the dashpot. Consequently, the Kelvin–Voigt model does not possess a finite relaxation modulus for an ideal step-strain test. The limitations of the two simple models are then compared. The Kelvin–Voigt model lacks an instantaneous elastic strain response and does not exhibit stress relaxation at constant strain. The Maxwell model permits stress relaxation but lacks time-dependent strain recovery after unloading. Neither model by itself reproduces all of the important features of a real viscoelastic material. The lecture concludes by showing how each model can be augmented to produce a Standard Linear Solid (SLS). Adding a spring in parallel with a Maxwell element, or adding a spring in series with a Kelvin–Voigt element, produces equivalent three-parameter models capable of representing both stress relaxation and time-dependent strain recovery. These are the simplest linear models that capture the essential behavior of a viscoelastic solid. Topics covered: Viscoelasticity Stress relaxation Relaxation modulus Maxwell model Kelvin–Voigt model Spring-dashpot models Characteristic relaxation time Exponential stress relaxation Eᵣ(t) = E exp(-t/τ) τ = η/E Step-strain loading Limitations of simple viscoelastic models Time-dependent strain recovery Standard Linear Solid SLS model Three-parameter viscoelastic models Linear viscoelastic constitutive models This lecture follows the development of creep compliance for the Maxwell and Kelvin–Voigt models and provides the transition from simple spring-dashpot models to more realistic linear viscoelastic constitutive models.