Quasi Linear ViscoelasticityEdit

Quasi Linear Viscoelasticity (QLV) is a constitutive framework used to describe the time-dependent, rate-sensitive behavior of soft tissues and similar polymers. It combines a nonlinear instantaneous elastic response with a linear viscoelastic relaxation function to capture how tissue stress evolves under a history of strain. The approach aims to be both physically interpretable and computationally tractable, enabling researchers to fit experimental data with a modest number of parameters. The theory owes much to the biomechanics tradition, particularly the work of Y. C. Fung and collaborators, who sought to extend linear viscoelastic concepts to the nonlinear strains typical of living tissues while preserving a clear causal structure.

QLV has become standard in biomechanics and biomedical engineering, where it informs models of Arteries, Ligaments, Tendons, and other connective tissues. Its appeal lies in a separable structure: the stress response can be viewed as a convolution of a time-dependent relaxation function with an instantaneous elastic response derived from a suitable energy function. This separation makes it relatively straightforward to interpret material behavior, calibrate parameters from experiments, and implement the model in finite element analyses. Despite its popularity, QLV is not the only path forward, and its assumptions are the subject of ongoing discussion in the literature.

Theory and formulation

Historical background

The quasi-linear framework emerged from attempts to extend concepts of Viscoelasticity to nonlinear regimes encountered in soft tissues. The core idea is to retain a linear time-dependent relaxation mechanism while allowing the instantaneous, strain-driven response to be nonlinear. Foundational work in this area is associated with Y. C. Fung and coworkers, who highlighted the practical benefits of a model that could capture essential rate effects without resorting to fully nonlinear time-dependent rheology.

Mathematical formulation

In broad terms, the QLV constitutive relation expresses the stress S(t) as a history-dependent integral: S(t) = ∫_0^t G(t − s) d[S^e(ε(s))]

Here: - G(t) is the reduced relaxation function (with G(0) = 1 in many formulations), capturing how stress decays under a unit instantaneous strain. - S^e(ε) is the instantaneous elastic response, obtained from a hyperelastic energy function W(ε) such that S^e is the derivative of W with respect to the strain ε. - ε(s) is the strain history, and the integral embodies a time-domain convolution between the history of a nonlinear elastic response and a linear viscoelastic relaxation kernel.

The practical choice of strain measures (e.g., Green-Lagrange strain for large deformations) and the specific form of W(ε) determine the precise expression for S^e(ε). In many implementations, G(t) is represented by a Prony series or similar, which provides a compact, parameter-efficient way to describe relaxation over multiple time scales. For numerical work, this framework lends itself to straightforward incorporation into Finite element method simulations and related computational approaches.

Practical considerations

The instantaneous elastic part S^e(ε) is commonly derived from a hyperelastic energy function, which may be chosen to reflect tissue anisotropy or preferred symmetry (e.g., isotropic or transversely isotropic forms).
The relaxation function G(t) is typically fit from stress-relaxation experiments, ramp-and-hold tests, or cyclic loading data.
Calibration often uses a two-step procedure: first determine the elastic parameters from instantaneous tests, then fit the relaxation parameters to time-dependent data.
The model’s simplicity facilitates interpretation and comparison across tissues, loading conditions, and species, but at the cost of potential limitations under certain loading regimes.

Applications

Biomedical tissues

QLV is widely applied to soft tissues where viscoelastic time dependence is important but large, fully nonlinear time-dependent models would be unwieldy. Typical targets include: - Arteries and vascular walls, where wall mechanics exhibit rate dependence and time-delayed responses. - Ligaments and Tendons, where collagenous networks produce nonlinear elastic responses coupled with relaxation phenomena. - Skin and other connective tissues, which show quasi-linear relaxation behavior over relevant physiological strains. - Intervertebral discs and other dense connective tissues, where rate effects influence load sharing and deformation.

Engineering and materials science

Beyond biology, QLV has been used to model polymers and elastomeric composites that display nonlinear instantaneous stiffening or softening with time-dependent relaxation, while keeping a manageable set of time- and strain-dependent parameters.

Calibration and data

Experimental tests

Stress-relaxation tests, where a prescribed strain is held and the resulting stress decay is recorded, are central to extracting G(t).
Ramp-and-hold and creep tests provide complementary information about both the instantaneous elastic response and the relaxation function.
Cyclic loading experiments help assess the model’s ability to capture hysteresis and rate dependence.

Parameter estimation

Elastic parameters are typically obtained from instantaneous loading data, while relaxation parameters come from time-dependent tests.
Prony-series representations of G(t) enable efficient fitting with a small number of exponential terms, balancing accuracy with interpretability.
Model validation often involves predicting responses to novel loading protocols and comparing with experimental measurements.

Limitations and alternatives

Assumptions of separability

A central limitation of QLV is the assumption that time dependence acts through a linear relaxation kernel that is separable from the nonlinear instantaneous response. In some tissues and loading regimes, this separability may not hold, leading to inaccuracies in predicted stresses or energy dissipation.

Large-strain and anisotropy

QLV often relies on a hyperelastic, sometimes isotropic, instantaneous response. While extensions to anisotropy (e.g., fiber-reinforced models) exist, capturing strong directional dependence and large-strain complications can require more complex constitutive formulations or entirely different frameworks.

Alternative models

Nonlinear viscoelastic models that treat the time dependence and nonlinear elasticity without strict separability.
Internal-variable formulations that introduce additional state variables capturing material history beyond a single relaxation function.
Fractional viscoelastic models, which use fractional derivatives to describe a continuum of relaxation times more compactly.
Fully empirical or data-driven approaches that bypass explicit constitutive laws in favor of machine-learning-based surrogates. These alternatives are often motivated by improvements in accuracy for complex loading, better representation of history effects like Mullins-type damage, or improved fit to large-deformation data.