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A strain increment $\Delta {\gamma }_j$ applied at time $t_j$ contributesLinearity lets the contributions add, and time-translation invariance makes the kernel depend only on elapsed time:In the continuous limit this is Boltzmann superposition,The material's prehistory must be included, or it must start from a specified relaxed state. Aging, yielding and large-strain changes can violate this simple superposition. An instantaneous Newtonian contribution can be written separately as ${\eta }_s\dot{\gamma }(t)$; for an ideal step it is an impulse, not an ordinary finite relaxation tail.

A linear viscoelastic fluid responds linearly to small imposed strain histories but retains memory of earlier deformation. For shear strain $\gamma (t)$ and a relaxed initial state, Boltzmann superposition givesHere $G(t)$ is the causal relaxation modulus, the stress following a unit step strain. Doubling a sufficiently small strain history doubles the stress, and responses to added histories add. Elasticity stores deformation energy; viscosity dissipates it. Linearity is an amplitude approximation, not an assertion that the material has no relaxation times.

In linear viscoelasticity, stress is a linear functional of past strain increments: scaling the deformation scales the response, and responses to small deformations add. For a time-translation-invariant material, Boltzmann superposition writes $\sigma (t)={\int }_{-\infty }^{t}G(t-s)\dot{ϵ}(s)ds$, where $G$ is a relaxation modulus. The material combines reversible storage with delayed or dissipative response; the description is restricted to small deformations about the chosen state.

The Maxwell model places an elastic spring and viscous dashpot in series. They carry the same shear stress $\sigma$, and their shear strains add:Differentiating givesHere $ϵ$ is dimensionless shear strain, $\sigma$ shear stress in pascals, $G_0$ the spring shear modulus, $\eta$ dashpot viscosity in $\mathrm{Pa}\mathrm{s}$, and $\tau$ the relaxation time. The first term describes changing elastic strain; the second is viscous flow. In this model the material flows at long times and has zero static shear modulus.