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Hyperelasticity

Hyperelasticity

Even though experimental studies have indicated visco-elastic behavior of the myocardium [Dokos et al., 2002, Gültekin et al., 2016] a common assumption is to consider quasi-static behavior, meaning that the inertial term in (6) is negligible and static equilibrium is achieved at all points in the cardiac cycle. Therefore it is also possible to model the myocardium as a hyperelastic material,which is a type of elastic material. This means that we postulate the existence of a strain-energy density function \(\Psi:\mathrm{Lin}^+ \rightarrow \mathbb{R}^+\), and that stress is given by the relation

(10)\[\begin{align} \mathbf{P} = \frac{\partial \Psi(\mathbf{F})}{\partial \mathbf{F}}. \end{align}\]

Since stress has unit Pa, we see that the strain-energy density function is defined as energy per unit reference volume, and has units \(\frac{\text{Joule}}{m^3}\). The strain-energy density function relates the amount of energy that is stored within the material in response to a given strain. Hence, the stresses in a hyperelastic material with a given strain-energy density function, depend only on the strain, and not the path for which the material deforms. On the contrary, if the model had been visco-elastic we would expect to see hysteresis in the stress/strain curve, but this is not possible for a hyperelastic material.

Note

The second law of thermodynamics states that the total entropy production in a thermodynamic process can never be negative. Elastic materials define a special class of materials in which the entropy production is zero. Within this thermodynamic framework the strain-energy density function coincides (up to a constant) with the Helmholtz free energy density.

General requirements for the strain-energy density function

Some general requirements must hold for the strain-energy function. First of all, we require that the reference state is stress free and that the stored energy increases monotonically with the deformation. Formally this can be stated simply as

\[\begin{align*} \Psi(\mathbf{I}) = 0 \; && \text{and} &&\; \Psi(\mathbf{F}) \geq 0. \end{align*}\]

Moreover, expanding or compressing a body to zero volume would require an infinite amount of energy, i.e

\[\begin{split}\begin{align*} \Psi(\mathbf{F}) \rightarrow \infty \; && \text{as} &&\; \det \mathbf{F} \rightarrow& 0 \\ \Psi(\mathbf{F}) \rightarrow \infty \; && \text{as} &&\; \det \mathbf{F} \rightarrow& \infty \end{align*}\end{split}\]

We say that the strain energy should be objective, meaning that the stored energy in the material should be invariant with respect to change of observer. Formally we must have: given any positive symmetric rank-2 tensor \(\mathbf{C} \in \mathrm{Sym}\):

\[\begin{align} \Psi(\mathbf{C}) = \Psi(\mathbf{Q}\mathbf{C}\mathbf{Q}^T), \; \forall \mathbf{Q} \in \mathcal{G} \subseteq \mathrm{Orth}. \end{align}\]

Here \(\mathrm{Orth}\) is the group of all positive orthogonal matrices. If \(\mathcal{G} = \mathrm{Orth}\) we say that the material is isotropic, and otherwise we say that the material is anisotropic. This brings us to another important issue, which is related to the choice of coordinate-system. Having to deal with different coordinate-systems, and mapping quantities from one coordinate-system to another can results in complicated computations. Therefore it would be beneficial if we could work with quantities which do not depend on the choice of coordinate-system. Such quantities are called invariants. If the material is isotropic, the representation theorem for invariants [Wang, 1970] states that \(\Psi\) can be expressed in terms of the principle invariants of \(\mathbf{C}\), that is \(\Psi = \Psi(I_1, I_2, I_3)\). The principle invariants \(I_i, i=1,2,3\) are the coefficients in the characteristic polynomial of \(\mathbf{C}\), and is given by

(11)\[\begin{align} I_1 = \mathrm{tr} \; \mathbf{C}, && I_2 = \frac{1}{2}\left[ I_1^2 - \mathrm{tr} \;(\mathbf{C}^2)\right] && \text{and} && I_3 = \det \mathbf{C}. \end{align}\]

In the case when the material constitutes a transversely isotropic behavior, that is, the material has a preferred direction \(\mathbf{a}_0\), which in the case of the myocardium could be the direction of fiber muscle fibers, we have

\[\begin{align*} \mathcal{G} = \{ \mathbf{Q} \in \mathrm{Orth}: \mathbf{Q}(\mathbf{a}_0\otimes\mathbf{a}_0)\mathbf{Q}^T = \mathbf{a}_0\otimes\mathbf{a}_0 \}, \end{align*}\]

with \(\otimes\) being the outer product. In this case the strain-energy density function can still be expressed through invariants. However, we need to include the so called quasi-invariants, which are defined as stretches in the local microstructural coordinate-system. The transversely isotropic invariants are given by

\[\begin{align*} I_{4\mathbf{a}_0 } = \mathbf{a}_0 \cdot (\mathbf{C} \mathbf{a}_0) && \text{and} && I_5 = \mathbf{a}_0 \cdot (\mathbf{C}^2 \mathbf{a}_0). \end{align*}\]

Some of the invariants do have a physical interpretation. For instance, \(I_3\) is related to the volume ratio of material during deformation, while \(I_{4\mathbf{a}_0 } \) is related to the stretch along the direction \(\mathbf{a}_0 \). Indeed the stretch ratio in the direction \(\mathbf{a}_0\) is given by \(\lambda_{\mathbf{a}_0} = | \mathbf{F} \mathbf{a}_0 |\) and we see that \(I_{4\mathbf{a}_0 } = \mathbf{a}_0 \cdot (\mathbf{C} \mathbf{a}_0) = \mathbf{F} \mathbf{a}_0 \cdot (\mathbf{F} \mathbf{a}_0) = \lambda_{\mathbf{a}_0}^2\). For more details about invariants see e.g [Holzapfel and Ogden, 2009, Liu and others, 1982].

The theory of global existence of unique solutions for elastic problems was originally based convexity of the free energy function. An energy function \(\Psi: \mathrm{Lin}^+ \rightarrow \mathbb{R}^+\) is strictly convex if for each \(\mathbf{F} \in \mathrm{Lin}^+\) and \(\mathbf{H} \neq \mathbf{0}\) with \(\det (\mathbf{F} + (1-\lambda)\mathbf{H}) > 0\), we have

(12)\[\begin{align} \Psi(\lambda \mathbf{F} + (1-\lambda) \mathbf{H}) < \lambda \Psi(\mathbf{F}) + (1-\lambda) \Psi(\mathbf{F} + \mathbf{H}), && \lambda \in (0,1). \end{align}\]

If the response \(\mathbf{P}\) is differentiable, then condition (12) is equivalent of saying that the response is positive definite,

(13)\[\begin{align} \mathbf{H} : \frac{\partial \mathbf{P}}{\partial \mathbf{F}} : \mathbf{H} > 0, && \mathbf{F} \in \mathrm{Lin}^+, \mathbf{H} \neq \mathbf{0}. \end{align}\]

However, from a physical point of view this requirement is too strict [Ball, 1976]. A slightly weaker requirement is the strong ellipticity condition which states that (13) should hold for any \(\mathbf{H}\) of rank-one, and is analogous to say that the strain energy function is rank-one convex.

References

Bal76

John M Ball. Convexity conditions and existence theorems in nonlinear elasticity. Archive for rational mechanics and Analysis, 63(4):337–403, 1976.

DSYL02

Socrates Dokos, Bruce H Smaill, Alistair A Young, and Ian J LeGrice. Shear properties of passive ventricular myocardium. American Journal of Physiology-Heart and Circulatory Physiology, 283(6):H2650–H2659, 2002.

GultekinSH16

Osman Gültekin, Gerhard Sommer, and Gerhard A Holzapfel. An orthotropic viscoelastic model for the passive myocardium: continuum basis and numerical treatment. Computer methods in biomechanics and biomedical engineering, 19(15):1647–1664, 2016.

HO09

Gerhard A Holzapfel and Ray W Ogden. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 367(1902):3445–3475, 2009.

L+82

I Liu and others. On representations of anisotropic invariants. International Journal of Engineering Science, 20(10):1099–1109, 1982.

Wan70

C-C Wang. A new representation theorem for isotropic functions: an answer to professor gf smith's criticism of my papers on representations for isotropic functions. Archive for Rational Mechanics and Analysis, 36(3):166–197, 1970.

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