Definition of the Problem
A solid-like body, or part of it, denoted ℬ, is considered in a reference configuration \(\kappa _{R}(\mathscr{B})\). Under the application of loads, it is assumed that the solid has a nonlinear elastic response which is governed by a finite number \(N\) of material properties (defining local constitutive equations and their possible spatial variations).
These are denoted as a vector \(\boldsymbol{\beta }=(\beta _{1},\beta _{2},\dots ,\beta _{q},\dots , \beta _{n},\dots ,\beta _{N})\).
At time \(t\), after the application of traction \({\mathbf{h}}(t)\) on a part of the boundary of ℬ, denoted by \(\Gamma _{h}\), ℬ undergoes a motion described by a mapping \(\chi _{\boldsymbol{\beta }}\), depending on material properties \(\boldsymbol{\beta }\), from a reference configuration \(\kappa _{R}(\mathscr{B})\) to a current configuration \(\kappa _{\boldsymbol{\beta }}(\mathscr{B})\), such as
$$ \textbf{x} = \chi _{\boldsymbol{\beta }}(\textbf{X},t) = \textbf{X} + \textbf{u}(\boldsymbol{\beta },X,t)~, $$
(1)
where \(\textbf{X}\) and \(\textbf{x}\) are position vectors relative to reference and current configurations and \(\textbf{u}\) is the displacement vector field. In the following, we introduce the deformation gradient
$$ \textbf{F} = \frac{\partial \chi _{\boldsymbol{\beta }}(\textbf{X},t)}{\partial \textbf{X}}= \textbf{I}+ \frac{\partial \textbf{u}(\boldsymbol{\beta },\textbf{X},t)}{\partial \textbf{X}}, $$
(2)
the right Cauchy–Green tensor
$$ \textbf{C}=\textbf{F}^{T} \textbf{F}, $$
(3)
and the Jacobian
$$ J=\textrm{det}(\textbf{F}). $$
(4)
We also introduce the isochoric right Cauchy–Green tensor
$$ \hat{\textbf{C}}=\hat{\textbf{F}}^{T} \hat{\textbf{F}}, $$
(5)
where we denote the isochoric part of the deformation gradient as
$$ \hat{\textbf{F}}=J^{-1/3}\textbf{F}. $$
(6)
We assume that an experimental measurement of the displacement field \(\textbf{u}(\textbf{X},t)\) is available across the solid at time \(t\). The measured displacement field \(\tilde{\textbf{u}}(t)\) may be different than the actual displacement \(\textbf{u}(\textbf{X},t)\) because of measurement noise.
The inverse / identification problem consists of finding the values of the approximate parameters \(\tilde{\boldsymbol{\beta }}\) that minimizes the absolute error between \(\textbf{u}(\tilde{\boldsymbol{\beta }},\textbf{X},t)\) and \(\tilde{\textbf{u}}(\textbf{X},t)\).
Constitutive and Equilibrium Equations
Equilibrium equations in \(\kappa _{\boldsymbol{\beta }}(\mathscr{B})\) must be satisfied by the Cauchy stress \(\textbf{T}\), which may be written, in absence of accelerations and body forces, as
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \boldsymbol{\nabla }_{\kappa _{\boldsymbol{\beta }}}.\textbf{T} & = & \textbf{0} & \mathrm{on} & \chi _{\boldsymbol{\beta }}(\mathscr{B},t), \\ \textbf{T}.{\mathbf{n}} & = & {\mathbf{h}}(t) & \mathrm{on} & \chi _{ \boldsymbol{\beta }}(\Gamma _{h},t). \end{array} $$
(7)
In Eq. (7), the equilibrium on the boundaries is only written where known tractions are applied as we will use these equilibrium equations to solve the identification problem and we do not want to introduce supplemental unknowns such as the reaction tractions on the boundaries where Dirichlet boundary conditions would be applied.
In this paper, we focus on compressible hyperelastic materials. It is assumed that their strain energy density function \(\Phi \) can be written in an uncoupled form as
$$ \Phi = U(J,\boldsymbol{\beta }) + \hat{W}(\hat{\textbf{C}}, \boldsymbol{\beta }), $$
(8)
where \(U(J,\boldsymbol{\beta })\) describes the volumetric response and \(\hat{W}(\hat{\textbf{C}},\boldsymbol{\beta })\) describes the deviatoric response.
The second Piola-Kirchhoff stress \(\textbf{S}\) can be written as
$$ \textbf{S}=2 \frac{\mathrm{d}\Phi }{\mathrm{d}\textbf{C}} = \frac{\mathrm{d}U}{\mathrm{d}J} J \textbf{C}^{-1} + 2 J^{-2/3} ~ \textbf{Dev} \left ( \frac{\partial \hat{W}}{\partial \hat{\textbf{C}}} \right ), $$
(9)
where
$$ \textbf{Dev} \left ( \frac{\partial \hat{W}}{\partial \hat{\textbf{C}}} \right ) = \left ( \frac{\partial \hat{W}}{\partial \hat{\textbf{C}}} \right ) - \frac{1}{3}\left [ \left ( \frac{\partial \hat{W}}{\partial \hat{\textbf{C}}} \right ): \hat{\textbf{C}} \right ]\hat{\textbf{C}}^{-1}. $$
(10)
In the following, we introduce \(\hat{\textbf{S}}=2\frac{\partial \hat{W}}{\partial \hat{\textbf{C}}}\). The Cauchy stress \(\textbf{T}\) can be written as
$$ \textbf{T}= J^{-1}\textbf{F}^{T}\textbf{S}\textbf{F} = \frac{\mathrm{d}U}{\mathrm{d}J}\textbf{I}+ \textbf{dev} \left ( \hat{\textbf{T}} \right ), $$
(11)
where \(\hat{\textbf{T}}=J^{-1} \hat{\textbf{F}}\hat{\textbf{S}} \hat{\textbf{F}}^{T}\), and
$$ \textbf{dev} \left ( \hat{\textbf{T}} \right ) = \hat{\textbf{T}} - \frac{1}{3}\textrm{Tr}\left ( \hat{\textbf{T}} \right )\textbf{I}~. $$
(12)
Moreover, we introduce
$$ \textbf{S}_{,\beta _{q}} = \frac{\partial \textbf{S} }{\partial \beta _{q}} = \frac{\mathrm{d} U_{,\beta _{q}}}{\mathrm{d}J}J \textbf{C}^{-1} + J^{-2/3} \textbf{Dev} \left ( \hat{\textbf{S}}_{,\beta _{q}} \right ), $$
(13)
and
$$ \textbf{T}_{,\beta _{q}} = \frac{\partial \textbf{T} }{\partial \beta _{q}} = \frac{\mathrm{d} U_{,\beta _{q}}}{\mathrm{d}J}\textbf{I}+ J^{-1} \textbf{dev} \left ( \hat{\textbf{F}}~\hat{\textbf{S}}_{,\beta _{q}}~ \hat{\textbf{F}}^{T} \right ), $$
(14)
where
$$ \hat{\textbf{S}}_{,\beta _{q}} = \frac{\partial \hat{\textbf{S}}}{\partial \beta _{q}} = 2 \frac{\partial \frac{\partial \hat{W}}{\partial \hat{\textbf{C}}}}{\partial \beta _{q}} = 2 \frac{\partial \frac{\partial \hat{W}}{\partial \beta _{q}}}{\partial \hat{\textbf{C}}} = 2 \frac{\partial \hat{W}_{,\beta _{q}}}{\partial \hat{\textbf{C}}}, $$
(15)
$$ \hat{W}_{,\beta _{q}} = \frac{\partial \hat{W}}{\partial \beta _{q}} ~~~~ \textrm{and}~~~~ U_{,\beta _{q}} = \frac{\partial U}{\partial \beta _{q}}. $$
(16)
Definition of an Intermediate Configuration
Decomposition of the Deformation Gradient
Let us start with a set of parameters \(\boldsymbol{\beta }^{o}\) which corresponds to an initial estimation of the unknown constitutive parameters based on existing literature on the same tissue for instance.
Unstressed “intermediate configuration” were traditionally used in the literature for the multiplicative split of the deformation gradient in finite strain plasticity [38]. We define here a stressed intermediate configuration \(\kappa _{\boldsymbol{\beta }^{o}}(\mathscr{B})\) for which the solid with the vector of material properties \(\boldsymbol{\beta }^{o}\) is at equilibrium (Fig. 1). We emphasize that this is a different use of the “intermediate configuration” terminology than in finite strain plasticity [38].
Then, let the position in the intermediate (stressed) configuration be denoted by \(\textbf{x}^{o} = \chi _{\boldsymbol{\beta }^{o}}(\textbf{X},t)\), corresponding to displacement \(\textbf{u}^{o}(\textbf{X},t) = \textbf{x}^{o} - \textbf{X}\).
We assume that a small displacement \(\boldsymbol{\delta }\textbf{x}^{o}\) superimposed upon the large deformation \(\textbf{u}^{o}\), yields the current position \(\textbf{x}\) at time \(t\) for which the solid with a vector of material properties \(\boldsymbol{\beta }=\boldsymbol{\beta }^{o} + \boldsymbol{\delta }\boldsymbol{\beta }^{o}\) is at equilibrium. We assume that \(\boldsymbol{\delta }\boldsymbol{\beta }^{o}\) is a small variation of \(\boldsymbol{\beta }^{o}\) such as
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c} ||\boldsymbol{\delta }\textbf{x}^{o}|| & \ll & ||\textbf{u}^{o} ||, \\ ||\boldsymbol{\delta }\boldsymbol{\beta }^{o}|| & \ll & || \boldsymbol{\beta }^{o}||. \end{array} $$
(17)
The small displacement \(\boldsymbol{\delta }\textbf{x}^{o}\) corresponds to the deformation between the intermediate configuration (at which the solid with \(\boldsymbol{\beta }^{o}\) material properties is at equilibrium) and the current configuration (at which the solid with \(\boldsymbol{\beta }=\boldsymbol{\beta }^{o} + \boldsymbol{\delta }\boldsymbol{\beta }^{o}\) material properties is at equilibrium).
The current position can thus be written as
$$ \textbf{x} = \textbf{x}^{o} + \boldsymbol{\delta }\textbf{x}^{o}. $$
(18)
The deformation gradient associated with mappings from the reference to the intermediate is thus given by
$$ \textbf{F}^{o} = \frac{\partial \chi _{\boldsymbol{\beta }^{o}}(\textbf{X},t)}{\partial \textbf{X}}~. $$
(19)
The deformation gradient representing a mapping from the intermediate configuration to the current configuration (corresponding to the variations of the configuration for a variation of material properties \(\boldsymbol{\delta }\boldsymbol{\beta }^{o}\)) may be expressed as \(\textbf{I} + \boldsymbol{\delta }\textbf{F}^{o}\) where
$$ \boldsymbol{\delta }\textbf{F}^{o} = \frac{\partial \boldsymbol{\delta }\textbf{x}^{o}}{\partial \textbf{x}^{o}}~. $$
(20)
Then, gradients of the successive motions are obtained with the chain rule, such as
$$ \textbf{F}= \frac{\partial \textbf{x}}{\partial \textbf{x}^{o}} \frac{\partial \textbf{x}^{o}}{\partial \textbf{X}} = \frac{\partial \textbf{x}}{\partial \textbf{x}^{o}} \textbf{F}^{o} = \left [ \frac{\partial (\textbf{x}^{o}+\boldsymbol{\delta }\textbf{x}^{o})}{\partial \textbf{x}^{o}} \right ] \textbf{F}^{o} = \frac{\partial \textbf{x}^{o}}{\partial \textbf{x}^{o}}\textbf{F}^{o} + \frac{\partial \boldsymbol{\delta }\textbf{x}^{o}}{\partial \textbf{x}^{o}} \textbf{F}^{o} = \textbf{F}^{o} + \boldsymbol{\delta }\textbf{F}^{o} ~ \textbf{F}^{o} ~. $$
(21)
Then, the identification problem can be formulated as, find the values of \(\boldsymbol{\delta }\boldsymbol{\beta }^{o}\) that minimizes the error between \(\boldsymbol{\delta }\textbf{x}^{o}\) and \(\tilde{\textbf{u}}-\textbf{u}^{o}\).
Cauchy Stress Tensor
Using Eq. (9) and Eq. (11) specifically with \(\textbf{F}^{o}\) deformation gradient, the second Piola-Kirchhoff stress \(\textbf{S}^{o}\) and Cauchy stress \(\textbf{T}^{o}\) in the intermediate configuration may be written respectively
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c} \textbf{S}^{o} & = & \displaystyle \frac{\mathrm{d}U^{o}}{\mathrm{d}J} J^{o} \left (\textbf{C}^{o}\right )^{-1} + 2 \left (J^{o}\right )^{-2/3} \textbf{Dev} \left ( \frac{\partial \hat{W}^{o}}{\partial \hat{\textbf{C}}} \right )~, \\ \textbf{T}^{o} & = & \displaystyle \frac{\mathrm{d}U^{o}}{\mathrm{d}J} \textbf{I}+ 2\left (J^{o}\right )^{-1} \textbf{dev} \left ( \hat{\textbf{F}^{o}} \frac{\partial \hat{W}^{o}}{\partial \hat{\textbf{C}}} \hat{\textbf{F}^{o}}^{T} \right )~, \end{array} $$
(22)
where \(U^{o}=U(J^{o},\boldsymbol{\beta }^{o})\) and \(\hat{W}^{o}=\hat{W}(\hat{\textbf{F}^{o}},\boldsymbol{\beta }^{o})\).
Note that the Cauchy stress \(\textbf{T}^{o}\) must satisfy the equilibrium equations on the \(\kappa _{\boldsymbol{\beta }^{o}}\) configuration (further denoted \(\kappa _{o}\)) which may be rewritten
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \boldsymbol{\nabla }.\textbf{T}^{o} & = & \textbf{0} & \mathrm{on} & \chi _{\boldsymbol{\beta }^{o}}(\mathscr{B},t)~, \\ \textbf{T}^{o}.{\mathbf{n}} & = & {\mathbf{h}}(t) & \mathrm{on} & \chi _{ \boldsymbol{\beta }^{o}}(\Gamma _{h},t)~. \end{array} $$
(23)
The second Piola-Kirchhoff stress \(\textbf{S}\) can be related to \(\textbf{S}^{o}\) using a Taylor expansion of first order. For that we substitute into Eq. (9), \(\boldsymbol{\beta }=\boldsymbol{\beta }^{o}+\boldsymbol{\delta }\boldsymbol{\beta }^{o}\) and \(\textbf{C}=\textbf{C}^{o}+\boldsymbol{\delta }\textbf{C}^{o}\), which yields
$$ \mathbf{S} \simeq \textbf{S}^{o} + \boldsymbol{\delta }\textbf{S}^{o} \simeq \textbf{S}^{o} + \frac{\partial \textbf{S}}{\partial \textbf{C}} : \boldsymbol{\delta }\textbf{C}^{o} + \sum _{q=1}^{N} \delta \beta _{q}^{o}~\textbf{S}^{o}_{, \beta _{q}} ~, $$
(24)
where
$$ \textbf{S}^{o}_{,\beta _{q}} = \frac{\mathrm{d}U_{,\beta _{q}}^{o}}{\mathrm{d}J}J^{o} \left ( \textbf{C}^{o}\right )^{-1} + 2 \left (J^{o}\right )^{-2/3} \textbf{Dev} \left ( \frac{\partial \hat{W}_{,\beta _{q}}^{o}}{\partial \hat{\textbf{C}}} \right ) ~. $$
(25)
Neglecting the second order terms in the deformations from the intermediate to the current configuration, it may be written
$$ \textstyle\begin{array}{l@{\quad }c@{\quad }l} \boldsymbol{\delta }\textbf{C}^{o} & = & \left (\left ( \boldsymbol{\delta }\textbf{F}^{o}+\textbf{I} \right )\textbf{F}^{o} \right )^{T} \left (\left (\boldsymbol{\delta }\textbf{F}^{o}+\textbf{I} \right )\textbf{F}^{o} \right ) -\textbf{F}^{oT} \textbf{F}^{o} \\ & = & {\textbf{F}^{o}}^{T} \left ( \left (\boldsymbol{\delta }\textbf{F}^{o}+ \textbf{I} \right )^{T} \left (\boldsymbol{\delta }\textbf{F}^{o}+ \textbf{I} \right ) - \textbf{I} \right ) \textbf{F}^{o} \\ & = & {\textbf{F}^{o}}^{T} \left (\boldsymbol{\delta }\textbf{F}^{o}+ \left (\boldsymbol{\delta }\textbf{F}^{o} \right )^{T} + \left ( \boldsymbol{\delta }\textbf{F}^{o} \right )^{T} \boldsymbol{\delta }\textbf{F}^{o} \right ) \textbf{F}^{o} \\ & \simeq & 2 {\textbf{F}^{o}}^{T} \boldsymbol{\delta }\textbf{E}^{o} ~ \textbf{F}^{o}~, \end{array} $$
(26)
where \(\boldsymbol{\delta }\textbf{E}^{o}=\frac{1}{2}\left (\boldsymbol{\delta }\textbf{F}^{o}+\left (\boldsymbol{\delta }\textbf{F}^{o} \right )^{T} + \left (\boldsymbol{\delta }\textbf{F}^{o} \right )^{T} \boldsymbol{\delta }\textbf{F}^{o} \right )\) signifies the infinitesimal strain induced by a slight variation of material properties when \(\boldsymbol{\delta }\textbf{F}^{o}\) is small.
Finally,
$$ \boldsymbol{\delta }\textbf{S}^{o} = \pmb{\mathbb{K}}^{o} : \left ( { \textbf{F}^{o}}^{T} \boldsymbol{\delta }\textbf{E}^{o} ~\textbf{F}^{o} \right ) + \sum _{q=1}^{N} \delta \beta _{q}^{o}~\textbf{S}_{,\beta _{q}}^{o} ~, $$
(27)
where
$$ \pmb{\mathbb{K}}^{o} = 2 \frac{\partial \textbf{S}^{o}}{\partial \textbf{C}}. $$
(28)
Next, we can derive the Cauchy stress \(\textbf{T}\) for the current configuration from \(\textbf{T}^{o}\) as
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }l} \mathbf{T} & = & \textrm{det}(\textbf{F}^{o}+\boldsymbol{\delta }\textbf{F}^{o})^{-1}\left (J^{o}\right )^{-1}\left (\textbf{I} + \boldsymbol{\delta }\textbf{F}^{o} \right ) \textbf{F}^{o} \left ( \textbf{S}^{o}+ \boldsymbol{\delta }\textbf{S}^{o} \right ) \left ( \textbf{F}^{o}\right )^{T} \left (\textbf{I} + \boldsymbol{\delta }\textbf{F}^{o} \right )^{T} \\ & \simeq & \mathbf{T}^{o} + \boldsymbol{\delta }\textbf{T}^{o}~, \end{array} $$
(29)
where
$$ \textstyle\begin{array}{l} \boldsymbol{\delta }\textbf{T}^{o} = - \textrm{Tr}(\boldsymbol{\delta }\textbf{E}^{o}) \mathbf{T}^{o} + \boldsymbol{\delta }\textbf{F}^{o} \mathbf{T}^{o} + \mathbf{T}^{o} \left (\boldsymbol{\delta }\textbf{F}^{o} \right )^{T} \\ + \displaystyle \sum _{n=1}^{N} \delta \beta _{n}~ \textbf{T}_{,\beta _{n}}^{o} + \left (J^{o}\right )^{-1} \textbf{F}^{o} \left [ \pmb{\mathbb{K}}^{o} : \left ( {\textbf{F}^{o}}^{T} \boldsymbol{\delta }\textbf{F}^{o}~ \textbf{F}^{o} \right ) \right ] \left (\textbf{F}^{o}\right )^{T}~, \end{array} $$
(30)
and
$$ \textbf{T}_{,\beta _{q}}^{o} = \frac{\mathrm{d} U_{,\beta _{q}}^{o}}{\mathrm{d}J} \textbf{I}+ 2\left (J^{o} \right )^{-1} \textbf{dev} \left ( \hat{\textbf{F}^{o}} \frac{\partial \hat{W}_{,\beta _{q}}^{o}}{\partial \hat{\textbf{C}}} \hat{\textbf{F}^{o}}^{T} \right )~. $$
(31)
Equation 30 may be rewritten by introducing a 4\(^{\text{th}}\) order tensor \(\pmb{\mathbb{L}}^{o}\) such as
$$ \boldsymbol{\delta }\textbf{T}^{o} = \pmb{\mathbb{L}}^{o}: \boldsymbol{\delta }\textbf{F}^{o} + \displaystyle \sum _{q=1}^{N} \delta \beta _{q}~ \textbf{T}_{,\beta _{q}}^{o}~. $$
(32)
The components of the \(\pmb{\mathbb{L}}^{o}\) tensor may be written such as
$$ \mathbb{L}^{o}_{ijkl}=\frac{1}{2}(\mathscr{C}^{o}_{ijkl}+\mathscr{D}^{o}_{ijkl}+ \mathscr{C}^{o}_{ijlk}-\mathscr{D}^{o}_{ijlk})~, $$
(33)
where, the summation convention being adopted for the repeated indices,
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }l} \mathscr{C}^{o}_{ijkl} & = & - \delta _{kl} T^{o}_{ij} + \delta _{ik}T^{o}_{lj} + T^{o}_{ik}\delta _{jl} + \left (J^{o}\right )^{-1} F^{o}_{iA}F^{o}_{jB}F^{o}_{kC}F^{o}_{lD} {\mathbb{K}}^{o}_{ABCD}~, \\ \mathscr{D}^{o}_{ijkl} & = & \delta _{ik}T^{o}_{lj} + T^{o}_{ik} \delta _{jl}~. \end{array} $$
(34)
Equilibrium must be satisfied in the current configuration. Given that Eq. (23) is satisfied by \(\textbf{T}^{o}\) on the intermediate configuration, and assuming that the intermediate configuration is infinitesimally close to the current configuration (owing to \(\boldsymbol{\delta }\boldsymbol{\beta }^{o}/\boldsymbol{\beta }^{o} \ll 1\)), we can assume that the following equations are satisfied by \(\boldsymbol{\delta }\textbf{T}^{o}\),
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \boldsymbol{\nabla }_{\kappa _{o}}.\boldsymbol{\delta }\textbf{T}^{o} & = & \textbf{0} & \mathrm{on} & \chi _{\boldsymbol{\beta }^{o}}(\mathscr{B},t) ~, \\ \boldsymbol{\delta }\textbf{T}^{o}.{\mathbf{n}} & = & \textbf{0} & \mathrm{on} & \chi _{\boldsymbol{\beta }^{o}}(\Gamma _{h},t)~. \end{array} $$
(35)
The equilibrium equations may be rewritten such as
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \displaystyle \sum _{q=1}^{N} \delta \beta _{q}~ \boldsymbol{\nabla }. \textbf{T}_{,\beta _{q}}^{o} & = & -\boldsymbol{\nabla }_{\kappa _{o}}. \left (\pmb{\mathbb{L}}^{o}: \boldsymbol{\delta }\textbf{F}^{o} \right ) & ~~~~\mathrm{on} & \chi _{\boldsymbol{\beta }^{o}}(\mathscr{B},t) ~, \\ \displaystyle \sum _{q=1}^{N} \delta \beta _{q}~\textbf{T}_{,\beta _{n}}^{o}.{ \mathbf{n}} & = & -\left (\pmb{\mathbb{L}}^{o}: \boldsymbol{\delta }\textbf{F}^{o} \right ).{\mathbf{n}} & ~~~~\mathrm{on} & \chi _{ \boldsymbol{\beta }^{o}}(\Gamma _{h},t)~. \end{array} $$
(36)
In the following, for the sake of simplification, the following abuse of notation will be used for the gradient and divergence in the \(\kappa _{o}\) configuration: \(\boldsymbol{\nabla }\equiv \boldsymbol{\nabla }_{\kappa _{o}}\).
The Virtual Fields Method
\(\boldsymbol{\delta }\textbf{T}^{o}\) should satisfy the equilibrium equations (Eq. (36)) written just above. These equations can be written in their weak form such as
$$ \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B},t)} \boldsymbol{\delta }\textbf{T}^{o} : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{o(n)} dV^{o} = 0~, $$
(37)
where \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\) is a virtual displacement field which equals zero on \(\chi _{\boldsymbol{\beta }^{o}}(\partial \mathscr{B} \backslash \Gamma _{h})\) (boundary where traction are not applied). In \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\), the index \((n)\) indicates that at least \(N\) virtual fields are necessary to establish a system of \(N\) equations of the \(N\) unknown material properties \(\beta _{n}\).
As full-field measurements are available, \(\boldsymbol{\delta }\textbf{F}^{o}\) and \(\boldsymbol{\delta }\textbf{E}^{o}\) from Eq. (30) can be replaced by their measures denoted respectively \(\tilde{\textbf{H}}\) and \(\tilde{\boldsymbol{\epsilon }}\), such as
$$ \tilde{\textbf{H}}=\boldsymbol{\nabla }(\tilde{\textbf{u}}-\textbf{u}^{o})~,~ \tilde{\boldsymbol{\epsilon }}=\frac{1}{2}(\tilde{\textbf{H}}+ \tilde{\textbf{H}}^{T})~, $$
(38)
yielding
$$ \textstyle\begin{array}{l} \displaystyle \sum _{q=1}^{N} \delta \beta _{q}^{o}~\displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \textbf{T}_{,\beta _{q}}^{o} : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{o(n)} dV^{o} \\ = - \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \left ( - \textrm{Tr}(\tilde{\boldsymbol{\epsilon }}) \mathbf{T}^{o} + \tilde{\textbf{H}}\mathbf{T}^{o} + \mathbf{T}^{o} \tilde{\textbf{H}}^{T} \right ): \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{o(n)} dV^{o} \\ - \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \left ( \left (J^{o}\right )^{-1} \textbf{F}^{o} \left [ \pmb{\mathbb{K}}^{o} : \left ( {\textbf{F}^{o}}^{T} \tilde{\boldsymbol{\epsilon }} \textbf{F}^{o} \right ) \right ] \left ( \textbf{F}^{o}\right )^{T} \right ) : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{o(n)} dV^{o} ~. \end{array} $$
(39)
Equation 39 may be rewritten by introducing the \(\pmb{\mathbb{L}}^{o}\) tensor such as
$$ \textstyle\begin{array}{l} \displaystyle \sum _{q=1}^{N} \delta \beta _{q}^{o}~\displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \textbf{T}_{,\beta _{q}}^{o} : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{o(n)} dV^{o} \\ = - \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \left ( \pmb{\mathbb{L}}^{o}: \tilde{\textbf{H}} \right ): \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{o(n)} dV^{o} \\ = - \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \tilde{\textbf{H}} : \left ( \boldsymbol{\pmb{\mathbb{L}}^{o}}^{T}: \nabla \boldsymbol{\delta }\mathfrak{u}^{o(n)} \right ) dV^{o} ~. \end{array} $$
(40)
The previous equation can be written \(N\) times with \(N\) virtual fields \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\). The obtained system of equations may be written such as
In the next section we provide a methodology to choose the set of \(N\) virtual fields.
Derivation of the Virtual Fields for Parameter Identification
In the previous subsection, we showed that the unknown constitutive parameters can be identified by solving a linear system of equations. To establish this system of equations, one needs to define \(N\) virtual fields denoted \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\), such that \([\mathbf{A}^{o}]\) is invertible. The invertibility is ensured by relating each virtual field \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\) to the sensitivity of the Cauchy stress to each unknown parameter, denoted as \(\textbf{T}_{,\beta _{n}}^{o}\) in Eq. (31) and introduced in Eq. (30), since it is assumed that the different \(\textbf{T}_{,\beta _{n}}^{o}\) constitute a set of \(N\) linearly independent tensorial functions.
Therefore, a possible choice of \(N\) linearly independent virtual fields could simply be: \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}=\boldsymbol{\nabla }.\textbf{T}_{, \beta _{n}}^{o}\). However, the virtual field must equal zero on \(\chi _{\boldsymbol{\beta }^{o}}(\partial \mathscr{B} \backslash \Gamma _{h})\) and remain continuous. To meet these requirements and benefit from the linear independence between the \(\textbf{T}_{,\beta _{n}}^{o}\) fields, we defined \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\) as the vectorial fields satisfying
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \boldsymbol{\nabla }.\left ({\pmb{\mathbb{L}}^{o}}^{T}: \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{o(n)} \right ) & = & \boldsymbol{\nabla }.\left (\textbf{T}_{,\beta _{n}}^{o}\right ) & \mathrm{on} & \chi _{\boldsymbol{\beta }^{o}}(\mathscr{B},t), \\ \left ({\pmb{\mathbb{L}}^{o}}^{T}:\boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{o(n)} \right ).\textbf{n} & = & \left ( \textbf{T}_{,\beta _{n}}^{o}\right ) .\textbf{n}~~ & \mathrm{on} & \chi _{\boldsymbol{\beta }^{o}}(\Gamma _{h}), \\ \boldsymbol{\delta }\mathfrak{u}^{o(n)} & = & \textbf{0} & \mathrm{on} & \chi _{\boldsymbol{\beta }^{o}}(\partial \mathscr{B} \backslash \Gamma _{h}). \end{array} $$
(42)
The virtual fields \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\) are eventually obtained by solving the linear elastic problems defined in Eq. (42) using the finite-element method (Sect. 2.6). Let us introduce the \(\mathfrak{L}\) linear operator such as \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}=\mathfrak{L}(\textbf{T}_{, \beta _{n}}^{o})\) is the solution of Eq. (42).
Moreover using the integration by parts, we can write
$$ \textstyle\begin{array}{l} \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \tilde{\textbf{H}} : \left ( {\pmb{\mathbb{L}}^{o}}^{T} : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{n} \right ) dV^{o} = \displaystyle \oint _{\chi _{\boldsymbol{\beta }^{o}}(\Gamma _{h})} ( \tilde{\textbf{u}}-\textbf{u}^{o}) . \left ( {\pmb{\mathbb{L}}^{o}}^{T} : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{n} \right ). \textbf{n}~dS^{o} \\ - \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} ( \tilde{\textbf{u}}-\textbf{u}^{o}) . \boldsymbol{\nabla }. \left ( { \pmb{\mathbb{L}}^{o}}^{T} : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{n} \right ) dV^{o}~, \\ \\ \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \tilde{\textbf{H}} : \left ( {\pmb{\mathbb{L}}^{o}}^{T} : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{n} \right ) dV^{o} = \displaystyle \oint _{\chi _{\boldsymbol{\beta }^{o}}(\Gamma _{h})} ( \tilde{\textbf{u}}-\textbf{u}^{o}) . \left ( \textbf{T}_{,\beta _{n}}^{o} \right ).\textbf{n}~dS^{o} \\ - \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} ( \tilde{\textbf{u}}-\textbf{u}^{o}) . \boldsymbol{\nabla }.\left ( \textbf{T}_{,\beta _{n}}^{o}\right ) dV^{o}~, \\ \\ \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \tilde{\textbf{H}} : \left ( {\pmb{\mathbb{L}}^{o}}^{T} : \boldsymbol{\nabla }\boldsymbol{\delta }\mathfrak{u}^{n} \right ) dV^{o} = \displaystyle \int _{\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B})} \tilde{\textbf{H}} : \textbf{T}_{,\beta _{n}}^{o} dV^{o}. \end{array} $$
(43)
Then we can build the system of equations of Eq. (41) by replacing each \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\) by their expression coming from Eq. (42), and use also the simplifications derived in Eq. (43), yielding
(44)
This system of equations can be rewritten
$$ [\mathbf{A}^{o}] \{ \boldsymbol{\delta }\boldsymbol{\beta }^{o} \} = \{ \mathbf{b}^{o} \} ~. $$
(45)
Then, the unknown constitutive parameters can be obtained as
$$ \boldsymbol{\beta }= \boldsymbol{\beta }^{o} + [\mathbf{A}^{o}]^{-1}\{ \mathbf{b}^{o} \} ~. $$
(46)
Finite-Element Implementation
The numerical implementation of the proposed method requires the use of finite-element analyses at two different levels:
-
1.
find the intermediate configuration and the deformation gradient \(\textbf{F}^{o}\) by solving the partial differential equations (PDEs) of the forward problem (Eq. (23)) with a set of parameters \(\boldsymbol{\beta }^{o}\),
-
2.
compute integrals in Eq. (41) or Eq. (44) to identify the unknown constitutive parameters.
In this section, we propose a possible numerical implementation. As standard in finite-elements [39–42]), the computational domain ℬ is discretised into a finite number of non-overlapping elements \(e \in \mathbb{E}\) such that
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \mathscr{B} & \approx & \mathscr{B}^{h} & = & \bigcup \limits _{e \in \mathbb{E}}\mathscr{B}^{e}~, \\ \chi _{\boldsymbol{\beta }^{o}}(\mathscr{B}) & \approx & \chi _{ \boldsymbol{\beta }^{o}}(\mathscr{B}^{h}) & = & \bigcup \limits _{e \in \mathbb{E}} \chi _{\boldsymbol{\beta }^{o}}(\mathscr{B}^{e}) ~.\end{array} $$
(47)
The fields are discretised using the following standard vectorial shape functions \(\pmb{\mathscr{N}}(\mathbf{X})\) as
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c} \mathbf{u}^{o}(\mathbf{X}) & = & \displaystyle \sum _{a=1}^{N_{w}} {u}^{o}_{a} \pmb{\mathscr{N}}^{a}(\mathbf{X})~, \\ \tilde{\mathbf{u}}(\mathbf{X}) & = & \displaystyle \sum _{a=1}^{N_{w}} \tilde{{u}}_{a} \pmb{\mathscr{N}}^{a}(\mathbf{X})~, \\ \boldsymbol{\delta }\mathfrak{u}^{o(n)}(\mathbf{X}) & = & \displaystyle \sum _{a=1}^{N_{w}} \delta {u}^{o(n)}_{a} \pmb{\mathscr{N}}^{a}(\mathbf{X})~, \end{array} $$
(48)
where \({u}^{o}_{a}\) are nodal variables for the \(\textbf{u}^{o}\) field, \(\tilde{{u}}_{a}\) are nodal variables for the \(\textbf{u}^{o}\) field and \(\delta {u}^{o(n)}_{a}\) are nodal variables for the \(\boldsymbol{\delta }\mathfrak{u}^{o(n)}\) field.
A standard finite-element discretisation enables the following discrete form for the components of Eq. (44),
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }l} \{\textbf{b}^{o}\}_{n} & = & \displaystyle \int _{\chi _{ \boldsymbol{\beta }^{o}}(\mathscr{B})} \tilde{\textbf{H}} : \textbf{T}_{, \beta _{n}}^{o} dV^{o} \\ & & \\ & = & \displaystyle \sum _{a=1}^{N_{w}} \left (\tilde{{u}}_{a}-{u}_{a}^{o} \right ) {\mathscr{F}}^{o(n)}_{a} \\ & & \\ & = & \left ( \pmb{\mathscr{F}}^{o(n)} \right )^{T} \left ( \tilde{\textbf{u}}-\textbf{u}^{o}\right )~, \end{array} $$
(49)
where \(\pmb{\mathscr{F}}^{o(n)}\) is obtained such as
$$ {\mathscr{F}}^{o(n)}_{a} = \displaystyle \int _{\chi _{ \boldsymbol{\beta }^{o}}(\mathscr{B})} \boldsymbol{\nabla }\pmb{\mathscr{N}}(\textbf{x}^{o}) : \textbf{T}_{,\beta _{n}}^{o} dV^{o}~, $$
(50)
and
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }l} [\textbf{A}^{o}]_{qn} & = & \displaystyle \int _{\chi _{ \boldsymbol{\beta }^{o}}(\mathscr{B})} \textbf{T}_{,\beta _{q}}^{o} : \boldsymbol{\nabla }\mathfrak{L}(\textbf{T}_{,\beta _{n}}^{o}) dV^{o} \\ & & \\ & = & \displaystyle \sum _{a=1}^{N_{w}}\sum _{b=1}^{N_{w}} { \mathscr{F}}^{o(n)}_{a} [{\mathscr{K}}^{o}]^{-1}_{ab} {\mathscr{F}}^{o(q)}_{b} \\ & & \\ & = & \left ( \pmb{\mathscr{F}}^{o(n)} \right )^{T} [{ \pmb{\mathscr{K}}^{o}}]^{-1} \pmb{\mathscr{F}}^{o(q)}~, \end{array} $$
(51)
where \(\pmb{\mathscr{K}}^{o}\), which is the stiffness matrix needed to solve the elastic problem of Eq. (42) to derive the virtual fields, and which satisfies \(\pmb{\mathscr{K}}^{o} \boldsymbol{\delta }\mathfrak{u}^{o(n)} = \pmb{\mathscr{F}}^{o(n)}\), is obtained such as
$$ {\mathscr{K}}^{o}_{ab} = \int _{\chi _{\boldsymbol{\beta }^{o}}( \mathscr{B})} \left (\boldsymbol{\nabla }\pmb{\mathscr{N}}(\textbf{x}^{o}) \right ):\left ({\pmb{\mathbb{L}}^{o}}^{T}(\textbf{x}^{o})\right ): \left (\boldsymbol{\nabla }\pmb{\mathscr{N}}(\textbf{x}^{o})\right )~dV^{o} ~. $$
(52)
Finally, the expression of \([\textbf{A}^{o}]\) and \(\{\textbf{b}^{o}\}\) with this finite-element discretisation are
$$ \{\textbf{b}^{o}\}_{n} =- \left (\pmb{\mathscr{F}}^{o(n)}\right )^{T} \left (\tilde{\textbf{u}}-\textbf{u}^{o}\right )~, $$
(53)
and
$$ [\textbf{A}^{o}]_{qn} = \left (\pmb{\mathscr{F}}^{o(n)}\right )^{T} \left [\pmb{\mathscr{K}}^{o}\right ]^{-1} \pmb{\mathscr{F}}^{o(q)}~. $$
(54)
Convergence of Parameter Identification
Based on the details discussed in the previous subsection, it is possible to derive \(\boldsymbol{\delta }\boldsymbol{\beta }^{o}\) from the choice of an initial guess of \(\boldsymbol{\beta }^{o}\) by solving Eq. (45). The question that remains to be solved is how to choose \(\boldsymbol{\beta }^{o}\). In practice, an initial estimation of the unknown constitutive parameters based on existing literature on the same tissue can be used to initiate the resolution. However, this does not guarantee the criterion in Eq. (17) is satisfied. Then, a non infinitesimal deviation between the intermediate configuration and the reference configuration would cause Eq. (36) to not be satisfied on \(\chi _{\boldsymbol{\beta }^{o}}(\mathscr{B},t)\). Therefore, the concept of “intermediate configuration” has to be iterative. From the choice of a first set of parameters \(\boldsymbol{\beta }^{o}\), an intermediate configuration can be found by solving the forward problem in Eq. (23). Evaluating \([\textbf{A}^{o}]\) and \(\{\textbf{b}^{o}\}\) (from equations 53 and 54) using the obtained \(\pmb{\mathbb{L}^{o}}\) and \(\textbf{T}_{,\beta _{n}}^{o}\) expressions (from equations 33 and 31) provides an update for \(\boldsymbol{\beta }^{o}\). The process is repeated until the deviation between \(\mathbf{u}^{o}\) and \(\tilde{\mathbf{u}}\) becomes small enough (Fig. 2).
The convergence criterion of the inverse algorithm is that the relative difference between the current estimation of material properties \(\boldsymbol{\beta }^{o}\) and its update \([\mathbf{A}^{o}]^{-1}\{ \mathbf{b}^{o} \}\) is less than the tolerance delta
$$ \frac{\left \| [\mathbf{A}^{o}]^{-1}\{ \mathbf{b}^{o} \} \right \| }{\left \| \boldsymbol{\beta }^{o} \right \| } < 10^{-6}~. $$
(55)
Although deriving analytically the convergence rate is not possible, we verified numerically in the following results that a quadratic convergence was obtained for cases where existence and uniqueness of the solution are guaranteed.
Examples of Hyperelastic Constitutive Models
In the following sections, we applied the VFM method and algorithm in Fig. 2 to determine the parameters of 3 hyperelastic strain energy potentials commonly used to describe collagenous tissues.
Neo-Hookean Model
For the Neo-Hookean constitutive model, the strain energy density function is
$$ \Phi = \frac{1}{2} \left [ \mu (\hat{I}_{1} - 3) + \kappa ( \mathrm{ln} J)^{2} \right ] , $$
(56)
where \(\hat{I}_{1}=\textrm{Tr}\left ( \hat{\textbf{C}} \right )=J^{-2/3} \textrm{Tr}\left ( \textbf{C} \right )\).
The strain energy density function depends linearly on two unknown constitutive parameters denoted \(\mu \) and \(\kappa \). Then in an identification problem, the vector of unknown parameters would be: \(\beta =(\mu ,\kappa )\).
The second Piola-Kirchhoff stress is written
$$ \textbf{S} = \kappa (\mathrm{ln} J) \mathbf{C}^{-1} + \mu \left ( J^{-2/3} \mathbf{I} -\frac{1}{3}\hat{I}_{1} \mathbf{C}^{-1} \right ) . $$
(57)
The associated Cauchy stress is written as
$$ \textbf{T} = J^{-1} \left [ \kappa (\mathrm{ln} J) \mathbf{I} + \mu ( \hat{\mathbf{B}} -\frac{1}{3}\hat{I}_{1} \mathbf{I} ) \right ] . $$
(58)
The sensitivity of the second Piola-Kirchhoff stress to each parameter is written
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }l} \textbf{S}_{,\beta _{1}} & = & \frac{\partial \textbf{S} }{\partial \beta _{1}} & = & J^{-2/3} \mathbf{I} -\frac{1}{3}\hat{I}_{1} \mathbf{C}^{-1}, \\ \textbf{S}_{,\beta _{2}} & = & \frac{\partial \textbf{S} }{\partial \beta _{2}} & = & (\mathrm{ln} J) \mathbf{C}^{-1}. \end{array} $$
(59)
The sensitivity of the Cauchy stress to each parameter is written
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }l} \textbf{T}_{,\beta _{1}} & = & \frac{\partial \textbf{T} }{\partial \beta _{1}} & = & J^{-1} ( \hat{\mathbf{B}} -\frac{1}{3}\hat{I}_{1} \mathbf{I} ) , \\ \textbf{T}_{,\beta _{2}} & = & \frac{\partial \textbf{T} }{\partial \beta _{2}} & = & J^{-1} ( \mathrm{ln} J) \mathbf{I} . \end{array} $$
(60)
Therefore,
$$ \mathbf{T}^{o} = \mu ^{o} \mathbf{T}_{,\beta _{1}}^{o} + \kappa ^{o} \mathbf{T}_{,\beta _{2}}^{o} , $$
(61)
with
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }l} \textbf{T}_{,\beta _{1}}^{o} & = & (J^{o})^{-1} (\hat{\mathbf{B}}^{o} - \frac{1}{3}\hat{I}_{1}^{o} \mathbf{I} ) , \\ \textbf{T}_{,\beta _{2}}^{o} & = & (J^{o})^{-1} [\mathrm{ln} (J^{o}) ] \mathbf{I} . \end{array} $$
(62)
Moreover, we have
$$ \pmb{\pmb{\mathbb{K}}}^{o} = \mu ^{o} \pmb{\mathbb{K}}_{,\beta _{1}}^{o} + \kappa ^{o} \pmb{\mathbb{K}}_{,\beta _{2}}^{o} , $$
(63)
with
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }l} \pmb{\mathbb{K}}_{,\beta _{1}}^{o} & = & -\frac{2}{3} J^{-2/3} \left ( \mathbf{I} \otimes (\mathbf{C}^{o})^{-1} + (\mathbf{C}^{o})^{-1} \otimes \mathbf{I} \right ) + \frac{2}{9} \hat{I}_{1} (\mathbf{C}^{o})^{-1} \otimes (\mathbf{C}^{o})^{-1} \\ & & + \frac{2}{3} \hat{I}_{1} (\mathbf{C}^{o})^{-1} \odot (\mathbf{C}^{o})^{-1} , \\ & & \\ \pmb{\mathbb{K}}_{,\beta _{2}}^{o} & = & (J^{o})^{-1} (\mathbf{C}^{o})^{-1} \otimes (\mathbf{C}^{o})^{-1} - (\mathrm{ln} J^{o}) (\mathbf{C}^{o})^{-1} \odot (\mathbf{C}^{o})^{-1} , \end{array} $$
(64)
where \(\mathbf{C}^{-1} \odot \mathbf{C}^{-1}=- \frac{\partial \mathbf{C}^{-1}}{\partial \mathbf{C}}\) and ⊗ represents the dyadic multiplication symbol.
Mooney-Rivlin Model
For the Mooney-Rivlin constitutive model, the strain energy density function is
$$ \Phi = \frac{1}{2} \left [ \mu (\hat{I}_{1} - 3) + \alpha (\hat{I}_{2} - 3) + \frac{1}{2} \kappa (\mathrm{ln} J)^{2} \right ] , $$
(65)
where \(\hat{I}_{2}=J^{-4/3} I_{2} = J^{-4/3}\textrm{Tr}\left ( \hat{\textbf{C}} \right )=J^{-2/3}\textrm{Tr}\left ( \textbf{C} \right )\).
The strain energy density function depends linearly on three unknown constitutive parameters denoted \(\mu \), \(\alpha \) and \(\kappa \). Then in an identification problem, the vector of unknown parameters would be: \(\beta =(\mu , \alpha , \kappa )\).
The second Piola-Kirchhoff stress is written
$$ \textbf{S} = \kappa (\mathrm{ln} J) \mathbf{C}^{-1} + \mu \left ( J^{-2/3} \mathbf{I} -\frac{1}{3}\hat{I}_{1} \mathbf{C}^{-1} \right ) + \alpha \left ( J^{-2/3} \hat{I}_{1} \mathbf{I} -\frac{2}{3}\hat{I}_{2} \mathbf{C}^{-1} - J^{-4/3} \mathbf{C} \right ) . $$
(66)
The associated Cauchy stress is written
$$ \textbf{T} = J^{-1} \left [ \kappa (\mathrm{ln} J) \mathbf{I} + \mu ( \hat{\mathbf{B}} -\frac{1}{3}\hat{I}_{1} \mathbf{I} ) + \alpha ( \hat{I}_{1} \hat{\mathbf{B}} -\frac{2}{3}\hat{I}_{2} \mathbf{I} - \hat{\mathbf{B}}^{2} ) \right ] . $$
(67)
The sensitivity of the second Piola-Kirchhoff stress to each parameter is written
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }l} \textbf{S}_{,\beta _{1}} & = & \frac{\partial \textbf{S} }{\partial \beta _{1}} & = & J^{-2/3} \mathbf{I} -\frac{1}{3}\hat{I}_{1} \mathbf{C}^{-1} , \\ \textbf{S}_{,\beta _{2}} & = & \frac{\partial \textbf{S} }{\partial \beta _{2}} & = & J^{-2/3} \hat{I}_{1} \mathbf{I} -\frac{2}{3}\hat{I}_{2} \mathbf{C}^{-1} - J^{-4/3} \mathbf{C} , \\ \textbf{S}_{,\beta _{3}} & = & \frac{\partial \textbf{S} }{\partial \beta _{3}} & = & (\mathrm{ln} J) \mathbf{C}^{-1} . \end{array} $$
(68)
The sensitivity of the Cauchy stress to each parameter is written
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }l} \textbf{T}_{,\beta _{1}} & = & \frac{\partial \textbf{T} }{\partial \beta _{1}} & = & J^{-1} ( \hat{\mathbf{B}} -\frac{1}{3}\hat{I}_{1} \mathbf{I} ) , \\ \textbf{T}_{,\beta _{2}} & = & \frac{\partial \textbf{T} }{\partial \beta _{2}} & = & J^{-1} ( \hat{I}_{1} \hat{\mathbf{B}} -\frac{2}{3}\hat{I}_{2} \mathbf{I} - \hat{\mathbf{B}}^{2} ) , \\ \textbf{T}_{,\beta _{3}} & = & \frac{\partial \textbf{T} }{\partial \beta _{3}} & = & J^{-1} ( \mathrm{ln} J) \mathbf{I} . \end{array} $$
(69)
Therefore,
$$ \mathbf{T}^{o} = \mu ^{o} \mathbf{T}_{,\beta _{1}}^{o} + \alpha ^{o} \mathbf{T}_{,\beta _{2}}^{o} + \kappa ^{o} \mathbf{T}_{,\beta _{3}}^{o} , $$
(70)
with
$$ \textstyle\begin{array}{c@{\quad }c@{\quad }l} \textbf{T}_{,\beta _{1}}^{o} & = & (J^{o})^{-1} \left [ \hat{\mathbf{B}}^{o} -\frac{1}{3}\hat{I}_{1}^{o} \mathbf{I} \right ] , \\ \textbf{T}_{,\beta _{2}}^{o} & = & (J^{o})^{-1} \left [ \hat{I}_{1}^{o} \hat{\mathbf{B}}^{o} -\frac{2}{3}\hat{I}_{2}^{o} \mathbf{I} - ( \hat{\mathbf{B}}^{o})^{2} \right ] , \\ \textbf{T}_{,\beta _{3}}^{o} & = & (J^{o})^{-1} [\mathrm{ln} (J^{o}) ] \mathbf{I} . \end{array} $$
(71)
Moreover, we have
$$ \pmb{\mathbb{K}}^{o} = \mu ^{o} \pmb{\mathbb{K}}_{,\beta _{1}}^{o} + \alpha ^{o} \pmb{\mathbb{K}}_{,\beta _{2}}^{o} + \kappa ^{o} \pmb{\mathbb{K}}_{,\beta _{3}}^{o} , $$
(72)
with
$$ \textstyle\begin{array}{rcl} \pmb{\mathbb{K}}_{,\beta _{1}}^{o} & = & -\frac{2}{3} J^{-2/3} \left ( \mathbf{I} \otimes (\mathbf{C}^{o})^{-1} + (\mathbf{C}^{o})^{-1} \otimes \mathbf{I} \right ) + \frac{2}{9} \hat{I}_{1} (\mathbf{C}^{o})^{-1} \otimes (\mathbf{C}^{o})^{-1} \\ & & + \frac{2}{3} \hat{I}_{1} (\mathbf{C}^{o})^{-1} \odot (\mathbf{C}^{o})^{-1} , \\ \pmb{\mathbb{K}}_{,\beta _{2}}^{o} & = & 2 J^{-4/3} \left ( \mathbf{I} \otimes \mathbf{I} - \mathbb{I} \right ) -\frac{4}{3} \hat{I}_{1} J^{-2/3} \left ( \mathbf{I} \otimes (\mathbf{C}^{o})^{-1} + (\mathbf{C}^{o})^{-1} \otimes \mathbf{I} \right ) \\ & & + \frac{8}{9} \hat{I}_{2} (\mathbf{C}^{o})^{-1} \otimes ( \mathbf{C}^{o})^{-1} + \frac{4}{3} \hat{I}_{2} (\mathbf{C}^{o})^{-1} \odot (\mathbf{C}^{o})^{-1} \\ & & + \frac{4}{3} J^{-4/3} \left [ \mathbf{C}^{o} \otimes (\mathbf{C}^{o})^{-1} + (\mathbf{C}^{o})^{-1} \otimes \mathbf{C}^{o} \right ] , \\ \pmb{\mathbb{K}}_{,\beta _{3}}^{o} & = & (J^{o})^{-1} (\mathbf{C}^{o})^{-1} \otimes (\mathbf{C}^{o})^{-1} - (\mathrm{ln} J^{o}) (\mathbf{C}^{o})^{-1} \odot (\mathbf{C}^{o})^{-1} , \end{array} $$
(73)
where \(\mathbb{I}\) is the fourth order identity tensor.
Veronda-Westmann Model
In the two previous constitutive models, the strain energy density function depends linearly on the material properties. In this subsection, the Veronda-Westmann model [43] which involves an exponential function of a material property introduced. The strain energy density function is written such as
$$ \Phi = \frac{c_{1}}{2} \left ( e^{c_{2}(\hat{I}_{1} - 3)} -1 \right ) - \frac{c_{1} c_{2}}{2} (\hat{I}_{2} - 3) + \frac{1}{2} \kappa ( \mathrm{ln} J)^{2} , $$
(74)
where \(c_{1}\), \(c_{2}\) and \(\kappa \) are material properties. Note that \(c_{2}\) is a parameter controlling the nonlinear behavior of the Veronda-Westmann solid, and \(\mu = c_{1} c_{2}\) is the shear modulus controlling the linear behavior of the Veronda-Westmann solid. In the following, we rewrite the model such as
$$ \Phi = \frac{\mu }{2c_{2}} \left ( e^{c_{2}(\hat{I}_{1} - 3)} -1 \right ) -\frac{\mu }{2} (\hat{I}_{2} - 3) + \frac{1}{2} \kappa ( \mathrm{ln} J)^{2} . $$
(75)
The strain energy density function depends linearly on two unknown constitutive parameters denoted \(\mu \) and \(\kappa \), and non linearly on the unknown parameter \(c_{2}\). In the identification problem, the vector of unknown parameters would be written: \(\beta =(\mu ,c_{2},\kappa )\).
The second Piola-Kirchhoff stress is written
(76)
The associated Cauchy stress is written
$$ \textbf{T} = J^{-1} \left [ \kappa (\mathrm{ln} J) \mathbf{I} + \mu \left ( (\hat{\mathbf{B}} -\frac{1}{3}\hat{I}_{1} \mathbf{I} )e^{c_{2}( \hat{I}_{1} - 3)} + ( \hat{I}_{1} \hat{\mathbf{B}} -\frac{2}{3} \hat{I}_{2} \mathbf{I} - \hat{\mathbf{B}}^{2} ) \right ) \right ] . $$
(77)
The sensitivity of the second Piola-Kirchhoff stress to each parameter is written
$$ \textstyle\begin{array}{rcl} \textbf{S}_{,\beta _{1}} & = & \frac{\partial \textbf{S} }{\partial \beta _{1}} = \left ( J^{-2/3} \mathbf{I} -\frac{1}{3}\hat{I}_{1} \mathbf{C}^{-1} \right )e^{c_{2}( \hat{I}_{1} - 3)} \\ & & + \left ( J^{-2/3} \hat{I}_{1} \mathbf{I} -\frac{2}{3}\hat{I}_{2} \mathbf{C}^{-1} - J^{-4/3} \mathbf{C} \right ) , \\ \textbf{S}_{,\beta _{2}} & = & \frac{\partial \textbf{S} }{\partial \beta _{2}} = \mu (\hat{I}_{1} - 3)\left ( J^{-2/3} \mathbf{I} -\frac{1}{3}\hat{I}_{1} \mathbf{C}^{-1} \right )e^{c_{2}(\hat{I}_{1} - 3)} , \\ \textbf{S}_{,\beta _{3}} & = & \frac{\partial \textbf{S} }{\partial \beta _{3}} = (\mathrm{ln} J) \mathbf{C}^{-1} . \end{array} $$
(78)
The sensitivity of the Cauchy stress to each parameter is written
$$ \textstyle\begin{array}{rcl} \textbf{T}_{,\beta _{1}} & = & \frac{\partial \textbf{T} }{\partial \beta _{1}} = J^{-1} \left [ ( \hat{\mathbf{B}} -\frac{1}{3}\hat{I}_{1} \mathbf{I} )e^{c_{2}(\hat{I}_{1} - 3)} + ( \hat{I}_{1} \hat{\mathbf{B}} -\frac{2}{3}\hat{I}_{2} \mathbf{I} - \hat{\mathbf{B}}^{2} ) \right ] , \\ \textbf{T}_{,\beta _{2}} & = & \frac{\partial \textbf{T} }{\partial \beta _{2}} = J^{-1} \mu ( \hat{I}_{1} - 3) (\hat{\mathbf{B}} -\frac{1}{3}\hat{I}_{1} \mathbf{I} )e^{c_{2}( \hat{I}_{1} - 3)} , \\ \textbf{T}_{,\beta _{3}} & = & \frac{\partial \textbf{T} }{\partial \beta _{3}} = J^{-1} ( \mathrm{ln} J) \mathbf{I} . \end{array} $$
(79)
Then we approximate \(\mathbf{T}^{o}\) such as
$$ \mathbf{T}^{o} = \mu ^{o} \mathbf{T}_{,\beta _{1}}^{o} + c_{2}^{o} \mathbf{T}_{,\beta _{2}}^{o} + \kappa ^{o} \mathbf{T}_{,\beta _{3}}^{o} , $$
(80)
with
$$ \textstyle\begin{array}{rcl} \textbf{T}_{,\beta _{1}}^{o} & = & (J^{o})^{-1} \left [ ( \hat{\mathbf{B}}^{o} -\frac{1}{3}\hat{I}_{1}^{o} \mathbf{I} )e^{c_{2}^{o}( \hat{I}_{1}^{o} - 3)} + ( \hat{I}_{1}^{o} \hat{\mathbf{B}}^{o} - \frac{2}{3}\hat{I}_{2}^{o} \mathbf{I} - (\hat{\mathbf{B}}^{o})^{2} ) \right ] , \\ \textbf{T}_{,\beta _{2}}^{o} & = & (J^{o})^{-1} \mu ^{o} (\hat{I}_{1}^{o} - 3) (\hat{\mathbf{B}}^{o} -\frac{1}{3}\hat{I}_{1}^{o} \mathbf{I} )e^{c_{2}^{o}( \hat{I}_{1}^{o} - 3)} , \\ \textbf{T}_{,\beta _{3}}^{o} & = & (J^{o})^{-1} [\mathrm{ln} (J^{o}) ] \mathbf{I} . \end{array} $$
(81)
Moreover, we also approximate
$$ \pmb{\mathbb{K}}^{o} = \mu ^{o} \pmb{\mathbb{K}}_{,\beta _{1}}^{o} + c_{2}^{o} \pmb{\mathbb{K}}_{,\beta _{2}}^{o} + \kappa ^{o} \pmb{\mathbb{K}}_{, \beta _{3}}^{o} ~, $$
(82)
with
$$ \textstyle\begin{array}{rcl} \pmb{\mathbb{K}}_{,\beta _{1}}^{o} & = & \left [ -\frac{2}{3} J^{-2/3} \left ( \mathbf{I} \otimes (\mathbf{C}^{o})^{-1} + (\mathbf{C}^{o})^{-1} \otimes \mathbf{I} \right ) + \frac{2}{9} \hat{I}_{1} (\mathbf{C}^{o})^{-1} \otimes (\mathbf{C}^{o})^{-1} \right . \\ & & + \left . \frac{2}{3} \hat{I}_{1} (\mathbf{C}^{o})^{-1} \odot ( \mathbf{C}^{o})^{-1} \right ] e^{c_{2}^{o}(\hat{I}_{1}^{o} - 3)} + 2 J^{-4/3} \left ( \mathbf{I} \otimes \mathbf{I} - \mathbb{I} \right ) \\ & & -\frac{4}{3} \hat{I}_{1} J^{-2/3} \left ( \mathbf{I} \otimes ( \mathbf{C}^{o})^{-1} + (\mathbf{C}^{o})^{-1} \otimes \mathbf{I} \right ) + \frac{8}{9} \hat{I}_{2} (\mathbf{C}^{o})^{-1} \otimes ( \mathbf{C}^{o})^{-1} \\ & & + \frac{4}{3} \hat{I}_{2} (\mathbf{C}^{o})^{-1} \odot (\mathbf{C}^{o})^{-1} + \frac{4}{3} J^{-4/3} \left [ \mathbf{C}^{o} \otimes (\mathbf{C}^{o})^{-1} + (\mathbf{C}^{o})^{-1} \otimes \mathbf{C}^{o} \right ] \\ & & + 2 c_{2}^{o} \left [ J^{-4/3} \mathbf{I} \otimes \mathbf{I} - \frac{1}{3} J^{-2/3} \hat{I}_{1}^{o} \left ( (\mathbf{C}^{o})^{-1} \otimes \mathbf{I} + \mathbf{I} \otimes (\mathbf{C}^{o})^{-1} \right ) \right . \\ & & \left . + \frac{1}{9} (\hat{I}_{1}^{o})^{2} (\mathbf{C}^{o})^{-1} \otimes (\mathbf{C}^{o})^{-1} \right ] e^{c_{2}^{o}(\hat{I}_{1}^{o} - 3)} , \\ \pmb{\mathbb{K}}_{,\beta _{2}}^{o} & = & \mu ^{o}(\hat{I}_{1}^{o} - 3) e^{c_{2}^{o}(\hat{I}_{1}^{o} - 3)} \left [ -\frac{2}{3} J^{-2/3} \left ( \mathbf{I} \otimes (\mathbf{C}^{o})^{-1} + (\mathbf{C}^{o})^{-1} \otimes \mathbf{I} \right ) \right . \\ & & \left . + \frac{2}{9} \hat{I}_{1} (\mathbf{C}^{o})^{-1} \otimes ( \mathbf{C}^{o})^{-1} + \frac{2}{3} \hat{I}_{1} (\mathbf{C}^{o})^{-1} \odot (\mathbf{C}^{o})^{-1} \right ] \\ & & + 2 \mu ^{o} \left (1 + c_{2}^{o}(\hat{I}_{1}^{o} - 3) \right ) e^{c_{2}^{o}( \hat{I}_{1}^{o} - 3)} \left [ J^{-4/3} \mathbf{I} \otimes \mathbf{I} \right . \\ & & \left . -\frac{1}{3} J^{-2/3} \hat{I}_{1}^{o} \left ( (\mathbf{C}^{o})^{-1} \otimes \mathbf{I} + \mathbf{I} \otimes (\mathbf{C}^{o})^{-1} \right ) + \frac{1}{9} (\hat{I}_{1}^{o})^{2} (\mathbf{C}^{o})^{-1} \otimes ( \mathbf{C}^{o})^{-1} \right ] , \\ \pmb{\mathbb{K}}_{,\beta _{3}}^{o} & = & (J^{o})^{-1} (\mathbf{C}^{o})^{-1} \otimes (\mathbf{C}^{o})^{-1} - (\mathrm{ln} J^{o}) (\mathbf{C}^{o})^{-1} \odot (\mathbf{C}^{o})^{-1} . \end{array} $$
(83)
Case Study for Verification
Using the previous formulas derived for the Neo-Hookean, the Mooney-Rivlin and the Veronda-Westmann models, it is straightforward to implement the novel VFM method and apply it to solve identification problems based on full-field data. Note also that the approach also extends naturally to more complex strain energy density functions involving exponential functions with invariants \(I_{4}\) and \(I_{6}\) [28] for instance. In the next section we show results for the 3 constitutive models, which were previously introduced, and discuss the convergence and the feasibility of the proposed approach in the case of compression and tension tests onto 2 simple geometries (Fig. 3):
-
1.
Compression is applied on a cube of 1 cm edge which is fully fixed on its bottom face (Fig. 3a). For the sake of verification of the approach, the cube is simply discretized uniformly by \(4 \times 4 \times 4\) elements.
-
2.
We also consider another geometric model as presented in (Fig. 3b) where we apply tension on the top face and fix the bottom face.
In both cases, full-field displacement measurements were simulated by solving finite-element models using the open source software FEBio [44]. For each case, the full-field displacement measurements were simulated with a set of target material properties, named the target, and the proposed VFM approach was employed to recover the target for the Neo-Hookean, the Mooney-Rivlin and the Veronda-Westmann model, using different initialization values to assess the convergence of the method.
Case Study for Validation
The proposed VFM method will be used in this case study to determine the material properties of the LC, a connective tissue structure in the ONH of great interest to researchers studying development and progression of glaucoma [45]. The LC in humans is approximately 1.5 mm - 2.0 mm in diameter and 450 microns thick and is located in the back of the eye. The mechanics of the LC is thought [46] to play an important role in mediating the progressive vision loss associated with glaucoma, a degenerative disease that is a leading cause of blindness worldwide [47]. As shown in Fig. 4, the LC is situated beneath the prelaminar neural tissue (PLNT). We further split the LC into an anterior region (ALC), 250 \({\mu }\)m posterior to the anterior posterior surface, and a posterior region (PLC), which includes the remainder of the imageable volume of the LC.
Spectral domain OCT imaging (Heildelberg Engineering) was applied to acquire 24 radial scans centered about the ONH of the left eye of a glaucoma patient. The OCT imaging was performed at Johns Hopkins University’s Wilmer Eye Institute in the Glaucoma Center of Excellence, and was approved by the appropriate Institutional Review Board. The following structural features were marked in the 24 radial scans to segment the tissue structures of the ONH as described in Midgett et al. [48]: Bruch’s membrane opening, the anterior boundary of the PLNT, the anterior LC surface, and the boundary of the imageable volume below the anterior LC surface. The manual marking were imported into Cubit (Coreform, Orem, UT, USA) to construct surface geometries using closed splines. The PLNT, ALC, and PLC volumes were defined by extruding a cylinder from Bruch’s membrane posteriorly to intersect the anterior PLNT surface and the anterior LC surface (for the PLNT), the anterior ALC surface and a surface positioned 250 microns posterior to the anterior ALC surface (for the ALC), and that posterior surface and a surface marking the end of the imageable volume of the LC (see Fig. 4a). The final solid volume was meshed in Cubit linear 4-node tetrahedral elements and exported into FEBio, where the linear elements were converted to 10-node quadratic tetrahedral elements [49] to avoid mesh locking and improve accuracy. The compressible Neo-Hookean constitutive model (Eq. (56)) was chosen for all three materials.
To validate the present VFM method, we simulated a displacement field by solving the forward problem with target parameter values, which are reported in Table 1. In the forward problem, zero displacement boundary conditions were applied to the posterior and lateral surfaces. To account for the 10 mmHg pressure decrease, a pressure boundary condition was applied to the ONH surface, with a magnitude of −10 mmHg. The induced displacement is shown in Fig. 4c.
Table 1 Results obtained with the novel VFM for identifying compressibility modulus values of the 3 separate regions in the ONH using simulated data Eventually the proposed VFM approach was employed to process the simulated displacement fields and recover the target compressibility modulus values for each separate material.