Alternatively, viscoelastic behavior can be quantified with creep and stress relaxation experiments. The output signal is in phase with the input signal for a purely elastic material and out of phase for a viscoelastic material. Often, a sinusoidal input (stress or strain) is applied to tissue, and an output signal (the corresponding strain or stress) is measured. To measure viscoelasticity, the strain rate-, frequency-, or time-dependent mechanical behavior of a material must be measured. Since viscoelastic material behavior is also time-dependent, the loading-unloading stress-strain behavior also depends on strain rate ( Figure 1d). Due to this energy dissipation, loading and unloading behaviors are not identical and loading moduli can be determined separately from unloading moduli. The energy lost during a loading cycle is equal to the hysteresis area between the loading and unloading curves. In contrast, viscoelastic materials dissipate energy upon deformation, which can be observed through hysteresis in the stress-strain curve ( Figure 1c). Linear and nonlinear elastic materials do not dissipate energy after deformation or exhibit time-dependent behavior therefore, stress-strain behavior of these materials is not different between loading and unloading. (d) Typical hysteresis loops of nonlinear viscoelastic cardiovascular tissue exhibiting strain rate dependence. (c) Typical stress-strain curve of nonlinear viscoelastic cardiovascular tissue exhibiting energy dissipation resulting in distinct nonlinear loading and unloading curves. Low-strain and high-strain behavior can be quantified by fitting elastic moduli to those regions of the stress-strain curve. (b) Stress-strain curve for a typical elastic material displaying nonlinear behavior. (a) Stress-strain curve for a linear elastic material. ![]() In arteries, for example, it is often convenient to quantify the behavior in low and high strain regions separately and to calculate low and high strain moduli (E low and E high), respectively, which is discussed in Section 2.3. In this case, elastic moduli can be defined at any point along the stress-strain curve. Materials that are nonlinearly elastic respond differently to different levels of strain and remain time-independent ( Figure 1b). ![]() A common metric of material elasticity is the elastic modulus, E, which is the slope of the stress-strain curve. For example, linear elastic materials subjected to an applied load exhibit a time-independent stress that is linearly proportional to strain ( Figure 1a). Elasticity, viscosity, and viscoelasticity can be quantified from mechanical testing techniques that relate the dynamics of a tissue’s deformation to an applied load. Cardiovascular tissues are viscoelastic, exhibiting behaviors that combine features of elastic solids and viscous fluids.
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