Viscoelasticity 1 Lumped Parameter Models for time-dependent behaviorDEQs as Constitutive Equations

Viscoelasticity(Combined viscous and elastic behavior)

Deborah Number"The mountains flowed before the Lord" in a song by prophetess Deborah (Judges 5:5). = Maxwell relaxation time (time for material to reach equilibrium after perturbation)= Observation timeHigh De: material is more like elastic solidLow De: material is more like a viscous, Newtonian fluid

Examples of viscoelastic materials

Stress RelaxationWhen a body is deformed (or strained) and that deformation (or strain) is held constant, stresses in the body reduce with time.

Figure from Fung Biomechanics 2nd ed.

CreepWhen a body is loaded (or stressed) and the stress is held constant, the body continues to deform (or strain) with time.Figure from Fung Biomechanics 2nd ed.

HysteresisWhen a body subjected to cyclic loading, load-displacement (or stress-strain) behavior for increasing loads is different than behavior for decreasing loads. The area between the curves represents energy loss (dissipation).Figure from Fung Biomechanics 2nd ed.

ViscoelasticityBehavior exhibited by a material (or tissue) that has both viscous and elastic elements in its response to a deformation (or strain) or load (or stress)Represented by:Dashpot/damper for viscous element. Follows Newtonian fluid constitutive law

Spring for elastic elementAssumed to linearly elastic

Lumped parameter modelsKelvin-VoigtMaxwellStandard-linearBurger

Lumped parameter models(Fung terminology)Kelvin-VoigtMaxwellStandard-linear(alternate form)Alternate terminologyThe Maxwell model, Voigt model (Kelvin-Voigt model) and the Kelvin model (standard linear model) are all composed of combinations of linear springs and dashpots.

A linear spring with spring constant theoretically produces a deformation proportional to load.

A dashpot with coefficient of viscosity produces a velocity proportional to load.Figure from Fung Biomechanics 2nd ed.

Maxwell ModelRepresented by a purely viscous damper and a purely elastic spring connected in seriesThe model can be represented by the following DEQ:

Predicts/models a stress that decays exponentially with time to zero with permanent deformationModel doesnt accurately predict creep (constant stress). Predicts that strain will increase linearly with time. Actually strain rate decreases with timeStress relaxation experiment

Kelvin-Voigt ModelRepresented by a Newtonian damper and Hookean elastic spring in parallel. The model can be expressed as a linear first order DEQ

Represents a solid undergoing reversible, viscoelastic strain. Models a solid that is very stiff but will creep (e.g. crystals, glass, apparent behavior of cartilage). At constant stress (creep), predicts strain to tend to /E as time continues to infinityThe model is not accurate for relaxation in a material/tissueCreep and recovery response

Standard Linear ModelModeled as Newtonian damper and 2 Hookean springs, one in parallel and one in series. The model can be expressed as a linear first order DEQ:Represents a solid undergoing an elastic and a reversible, viscoelastic strain. At constant stress (creep), predicts an initial strain from the spring which will creep over time until the parallel spring carries all the applied loadThe model is linear approximation to viscoelastic materials/tissues that are time dependent but completely recover from applied loadsCreep and recovery response

Creep and Relaxation Behaviors (from 3 linear models)Figures from Fung Biomechanics 2nd ed.Note: steady state and initial values!

Burgers Model4 linear elements used to capture minimum amount of behavior for many polymers/tissues which are:Instantaneous elasticity or elastic recovery (G1)Molecular slip (1)Rubbery elasticity (G2)Retarded elasticity (2)Modeled as a Maxwell model in series with a Kelvin-VoigtCreep and recovery response

Generalized ModelsGeneralized Maxwell model is Maxwell elements in parallel

Generalized KelvinVoigt model is KelvinVoigt elements in series

Linear Model ComparisonMaxwellGood for predicting stress relaxationPoor at predicting creepUsed for soft solids with non-recoverable deformationsKelvin-VoigtGood for predicting creep with small elastic deformationNot accurate with predicting stress relaxationUsed for polymers, rubber, cartilage when the load is not too highStandard Linear ModelPredicts both creep and relaxationFully recoverable & initial elastic displacementBurgersPredicts essentials of polymer viscoelastic behaviorUsed for polymers with non-recoverable behaviorGeneralizedUsed for fitting of experimental data to an arbitrary level of accuracy

Maxwell Model - creep

Creep FunctionsSolving the DEQs for the Maxwell, Voigt, and Kelvin models for displacement = c(t), when the stress (t) is a unit step function 1(t).

We obtain a set of results known as the creep functions. These functions represent the elongation (strain) in the viscoelastic material which is produced by a sudden application of unit stress at time = 0.

c(t) = (1/ + t/)1(t) Maxwell fluidc(t) = 1/ (1-e-(/)t)1(t) Voigt solidc(t) = 1/ER [ 1 (1-/)e-t/]1(t) Kelvin solidwhere = 1/1 , = (1/0)(1 + 0/1), and ER = 0 (Fungs model parameters)

Maxwell Model - relaxation

Relaxation FunctionsSolving the DEQs for stress [(t) = k(t)] when the applied strain is a unit step function (t) =1(t) yields relaxation functions.

These represent the resisting stress as a functions of timek(t) = e-(/)t1(t)Maxwell fluidk(t) = (t) + 1(t)Voigt solidk(t) = ER [ 1 (1 - /)e-t/]1(t)Kelvin solid

where = 1/1 , = (1/0)(1 + 0/1), and ER = 0 (Fungs model parameters)

SummaryMany lumped parameter models can be formulated to represent 1D linear viscoelastic materials. The best one should be chosen based on the material and properties that are being observed.Models are linear approximations to observed, time-dependent behaviors (elastic and viscous) and are formulated as DEQsModels can be painful to use because they require a new solution of DEQs for any forcing function. NEED A MORE GENERAL APPROACH!!!!

Creep and failure at higher strains