$$ \newcommand{\partd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\partdd}[2]{\frac{\partial^{2} #1}{\partial {#2}^{2}}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\Int}{\int\limits} \newcommand{\D}{\displaystyle} \newcommand{\ie}{\textit{i.e. }} \newcommand{\dA}{\; \mbox{dA}} \newcommand{\dz}{\; \mbox{dz}} \newcommand{\tr}{\mathrm{tr}} \renewcommand{\eqref}[1]{Eq.~(\ref{#1})} \newcommand{\reqs}[2]{\req{#1} and \reqand{#2}} \newcommand{\rthreeeqs}[3]{Eqs.~(\ref{#1}), (\ref{#2}), and (\ref{#3})} $$
Cardiovascular biomechanics

 

 

 

4 Elasticity

A material is (Cauchy) elastic if the stresses are functions of the deformations and position only: $$ \begin{equation} \boldsymbol{T} = \boldsymbol{T(E,r)} \tag{4.1} \end{equation} $$ An equation, like (4.1), which expresses the relation between the stresses and strains for a material is normally referred to as a constitutive equation or a material equation. The relation in equation (4.1) may be linear or nonlinear, and is expressed by some material parameters which defines a given material. The constitutive equation is referred to as homogeneous if the elastic properties are the same in every particle or position. For a homogeneous material there is no explicit dependency of the position and equation (4.1) reduces to: $$ \begin{equation} \boldsymbol{T} = \boldsymbol{T(E)} \tag{4.2} \end{equation} $$

Figure 25: Coaxial stresses and strains.

If the material properties are the same in every direction through a particle, the material is isotropic. For an isotropic material the principal directions for the stress and strain tensors are co-axial, i.e., the principal stresses and principal strains have the same orientation. To see this, consider a box element with edges oriented parallel to the principal stress directions. The box element is illustrated in figure 25, in both an undeformed state (corresponding to the unloaded situation) and a deformed state. The deformations are highly exaggerated for illustrative purposes. The diagonal planes \( P_1 \) and \( P_2 \) will be equally stretched due to the symmetry in the stresses and the isotropic elastic properties. Consequently, the normal angels betwen the edges of the box element will be maintained after deformation,and concequently the principal directions for the stresses and strains will be co-axial.

Finally, the constitutive model is referred to as linear elastic if the stress is a linear function of strain. For the linear case the six elements of the stress tensor \( T_{ij} \), are linear functions of the six elements of the strain tensor \( E_{ij} \). For a fully anisotropic material, this corresponds to \( 6\times 6 = 36 \) material parameters, which normally are denoted elasticities or stiffnesses. These elasticities are constants for a homogenous material.

4.0.1 Fundamental properties of elastic materials

In this section some prominent features of elastic materials (meaning materials which stress-strain relation is represented by equation (4.1))are noted.

The first feature of reversibility follows due to the assumption that the stress is a function of the strain only, and not of the rate of strain (or stress), the stress (or strain) level, or the stress (or strain) history. Many materials (in particular biomaterials) exhibits such effects, and may consequently not be denoted elastic. Reversibility means that strain curves (which may be both linear and nonlinear as seen in figures 26 and 27) are identical during loading and unloading, and independent of stress/strain levels and history.

Figure 26: Uniaxial behaviour of linear elastic materials.

Figure 27: Uniaxial behaviour of linear elastic materials elasticity

An elastic material is also non-dissipative, as no net work is performed from an external actor for a closed loading-unloading loop (see figures 26 and 27), and thus the deformation energy may be recovered upon unloading.