This section reviews some tissue constitutive models which have been discussed in standard textbooks and references in order to model the biological tissue behavior. Please note that this section includes the standard formulations for some of the well established material models. The purpose of this section is to make the reader conversant with different standard material models which are used to model biological tissue. References have been mentioned throughout the write-up.
4.1 Introduction to Hyperelasticity
A hyper elastic material is a type of constitutive model for which the stress-strain relationship is established by strain energy density function. Hyperelasticity includes the materials which can experience large elastic strain that is recoverable. The material behavior for elastomers is accurately explained in many cases using hyper elastic material model. Many polymers have ingrained fibers in a matrix which are initially disoriented but once they encounter strain, these fibers are stretched and start to resist the stress. This behavior is explained with the help of Hyperelasticity on a macroscopic scale. This behavior is also very useful to explain the material behavior observed in case of muscle fibers. Hence it is essential to study some of the standard hyper elastic material models.
4.1.1. Saint Venant-Kirchhoff model
Saint Venant-Kirchhoff model is the simplest model which is used to model Hyperelasticity phenomenon. The governing equation for this material is given below:
S = λ*tr (E) + 2μE …Equation I
In this equation S denotes the second Piola Kirchhoff stress tensor and E is the green strain. λ and μ are Lame constants [Ref 53, Ref 55]. λ is the first Lame's parameter which has no physical significance while μ is the shear modulus which is the second Lame's parameter.
Given strain energy density function W, second Piola-Kirchhoff stress can be derived by computing the partial derivative with respect to Green strain.
S = ∂W / ∂E …Equation II
4.1.2 Hyper elastic models of importance
Mooney Rivelin hyper elastic model involves strain energy density function 'W' which involves two invariants of the Cauchy Green deformation tensor B1 and B2. Strain energy density function W is used to derive the stress stain relationship in this case because hyper elastic models don't exhibit a linear relationship between stress and strain quantities. This model is useful to explain certain material model behavior such as rubber.
Mooney-Rivlin model W = C1 (B1 - 3) + C2 (B2 - 3) …Equation III
Constants C1 and C2 mentioned in the above equation are determined through experimental data by means of regression analysis.
The Ogden model is another type of hyper elastic material model which uses the concept of strain energy density function to establish the stress strain governing equation. In this case, the strain energy density is a function of principal stretch ratios in this case. The stretch ratios are basically natural logarithmic functions of the strain tensors Ref 59].
Ogden model W = …Equation IV
In the above equation, are material constants which are determined from experimental data or obtained from published literature. More information on the suitability of Ogden model to model Adipose tissue behavior based on published literature in this research area is mentioned in section 5.
4.2 Stress Relaxation
Stress relaxation phenomenon is important because it underlines the behavior of material model when it is unloaded. It gives us an idea about the manner in which the stresses are relieved and variation of stress with constant strains during unloading condition. Some solids exhibit stress relaxation in a particular fashion which is non linear and closer to viscous liquids. Sometimes Viscoelastic behavior is used to explain the material model for soft tissues [Ref 48].
Figure Cyclic loading and response curves for various materials: a) Elastic material b) Viscous material c) Viscoelastic material [Web 2]
Viscoelastic material exhibits a mixed behavior of elastic solids and viscous material. In case of a simple elastic material, the stress is directly proportional to strain and obeys Hooke's law. For a viscous material, due to interlaminar shear stresses, there is resistance to the flow of material due to shear forces and stress is proportional to the strain rate. This fact is shown in figures a and b respectively. For an elastic material stress and strain curves are synchronous since they are directly proportional within elastic limit. For a viscoelastic material there is a phase difference as indicated between stress and strain curves [Ref 60]. These materials have only damping component and no stiffness component. Viscoelasticity is concerned with materials which exhibit both elastic and viscous behavior. Some of the energy stored in a viscoelastic system is recovered upon removal of the load, and there is a possibility of heat dissipation through rejection of remaining energy to the surroundings. Therefore, a phase difference occurs between loading and response curves (Figure c) [Web 2].
Since time plays an important role in the behavior of viscoelastic materials, hence viscoelastic material are considered to be having a time dependent behavior. At each point of loading, we are not only interested in computing the displacements but velocities and accelerations of the material particles as well.
4.3 Viscoelasticity
Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscoelastic material exhibit viscous behavior when subjected to shear stress hence shear stress considerations are an important parameter for these materials. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time dependent strain [Ref 60]. Although the behavior is time dependent, inertial effects are not significant and they are not considered in the dynamics.
4.3.1 Types of Viscoelasticity
Linear Viscoelasticity occurs when separate behavior is exhibited in both creep response and under loading. There are various generic equations for Viscoelasticity which relate stress with strain at a continuum level through addition of contributions due to elasticity and Viscoelasticity. Strain rate dependence is generally captured through the use of exponential functions [Ref 60]. Linear Viscoelasticity is only applicable for small deformations hence it's suitability for modeling tissue deformation depends on the type of tissue being modeled. Nonlinear Viscoelasticity is more suitable for modeling biological soft tissue materials since it can accommodate large displacements.
4.3.2 Constitutive models of linear Viscoelasticity
Simple spring and dashpot based models have been developed in order to characterize the stress strain response for viscoelastic materials and to explain the temporal dependence. These models, which include the Maxwell model, the Kelvin-Voigt model, and the Standard Linear Solid Model, are used to predict a material's response under different loading conditions. Viscoelastic behavior has elastic and viscous components modeled as linear combinations of springs and dashpots, respectively. These can be considered to be analogous to electric circuits e.g. as in resistor combinations. In an equivalent electrical circuit, stress and strain quantities are represented by analogous voltage and current quantities, since strain tends indicate a flow. The elastic modulus of a spring is analogous to a circuit's capacitance because of storage of energy and release of energy through the spring and the viscosity of a dashpot to a circuit's resistance because it is a dissipation device [Ref 53].
Formula for stress strain relationship is given by, Hooke's law. The constant E indicates elastic modulus of the material.
σ = E* ε
For a viscous material stress is related to strain rate by the equation, σ = C * dε/dt, i.e. stress is directly proportional to time dependent strain rate which indicates resistance to flow of material.
4.3.3 Kelvin Voigt model
Figure. Spring and dashpot analogy for Kelvin Voigt model
Figure. Relaxation function for Kelvin Voigt model [Ref 53]
If C and K represent the viscosity constant and elastic constant for the Kelvin Voigt model subjected to force P in both directions, then
P = Ku + C*du/dt …Equation V
T = Gγ + μ*dγ/dt …Equation VI
Where u = displacement, du/dt = strain, T = shear stress, G = Shear modulus, γ = shear strain
and μ = shear viscosity, dγ/dt = shear strain rate
The Kelvin-Voigt model, (see Figure ) has a parallel combination of spring and dashpot as shown in figure to model the viscoelastic behavior. Since they are in parallel, it accounts for both elastic and damping (viscous) behavior. It is used to explain the creep behavior of biological soft tissues.
After observing the relaxation function we can conclude that after removal of stress, material relaxation is gradual with time. During creep the material tends to flow with constant stress. Creep phenomenon is modeled as exponential function and is satisfactorily explained in most cases using this model. This model doesn't explain the relaxation characteristics correctly for real world viscoelastic solids because it's a step function (i.e. stress application is instantaneous and material response is also instantaneous) as shown in figure above, ideally the governing equation should model the phenomenon as asymptotic.
4.3.4 Maxwell model
Figure. Schematic representation of Kelvin-Voigt model
P + (C/K) dP/dt = C du/dt …Equation VII
And T = Gγ + μ dγ/dt …Equation VIII
Where T is shear stress and γ is shear strain while G is the shear modulus.
The characteristic equation of the Maxwell model is of the exponential form
i.e. G(t) = E e ^(-Et/μ) …Equation IX
Hence under this model, under constant application of strain the stress relaxation is gradual. There is no decay expected over time. One limitation of this model is that it does not predict creep accurately. Ratio μ/E is called as characteristic time or relaxation time. Stress relaxation is almost complete which suits the characteristics of viscoelastic fluids and not the solids, since residual stresses are present in solid case. In case of Creep, strain increases linearly with time and creep characteristics are not satisfactory.
Figure. Relaxation function for Maxwell model [Ref 53]
These models are suitable for viscoelastic solids and liquids at some particular temperature and stress conditions. However biological soft tissue requires a sophisticated multilayered material model that we are going to explain in detail in section 5.
