Reformulation of the virtual fields method using the variation of elastic energy for parameter identification of ${\textbf {QR}}$ decomposition-based hyperelastic models

Abstract

${\textbf {QR}}$ decomposition-based constitutive relations for hyperelastic materials have attracted great attention from the community of solid mechanics, as hyperelastic models in terms of the distortion tensor $\varvec{\widetilde{F}}$ have obvious physical meanings. However, there are few works systematically discussing the material parameter identification for ${\textbf {QR}}$ decomposition-based hyperelastic models. In this work, we reformulate the virtual fields method by considering the internal virtual work as the variation of elastic energy caused by virtual displacements. This approach (together with the ${\textbf {QR}}$ decompositions) is more concise and easier to be implemented when compared with the conventional approach, which requires specific stresses, such as Cauchy stress, first or second Piola–Kirchhoff stress, and conjugate virtual strains to calculate the internal virtual work. To validate the reformulated virtual fields method, we derive the Mooney–Rivlin model under the ${\textbf {QR}}$ framework, and then identify its material parameters for incompressible silicone specimens under biaxial tensile tests. The results indicate that the proposed virtual fields method works very well for ${\textbf {QR}}$ decomposition-based models.

Appendices

Appendix A Derivation of IVW based on Promma and Grediac’s approach

Assuming that the material was incompressible and a plane-stress condition, we can have the deformation gradient tensor expressed as

$$\begin{aligned} {\textbf {F}} = \begin{bmatrix} F_{11} &{}\quad F_{12} &{}\quad 0 \\ F_{21} &{}\quad F_{22} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \frac{1}{D}\\ \end{bmatrix} \end{aligned}$$

(A.1)

with $D = F_{11}F_{22}-F_{12}F_{21}$. Therefore, we have

$$\begin{aligned} {\textbf {F}}^{-T} = \frac{1}{D}\begin{bmatrix} F_{22} &{}\quad -F_{21} &{}\quad 0 \\ -F_{12} &{}\quad F_{11} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad D^2\\ \end{bmatrix} \end{aligned}$$

(A.2)

and

$$\begin{aligned} {\textbf {C}} = {\textbf {F}}^{T}{} {\textbf {F}} = \begin{bmatrix} F_{11}^2 + F_{21}^2 &{}\quad F_{11}F_{12}+F_{21}F_{22} &{}\quad 0 \\ F_{11}F_{12}+F_{21}F_{22} &{}\quad F_{12}^2+F_{22}^2 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \frac{1}{D^2}\\ \end{bmatrix} \end{aligned}$$

(A.3)

According to Eq. 32, Cauchy stress components of Mooney–Rivlin model are

$$\begin{aligned} \sigma _{11}&= 2C_{10}\left( F_{11}^2+F_{12}^2-\frac{1}{D^2}\right) + 2C_{01}\left( D^2 - \frac{F_{21}^2}{D^2}-\frac{F_{22}^2}{D^2}\right) \nonumber \\ \sigma _{22}&= 2C_{10}\left( F_{21}^2+F_{22}^2-\frac{1}{D^2}\right) + 2C_{01}\left( D^2 - \frac{F_{11}^2}{D^2}-\frac{F_{12}^2}{D^2}\right) \nonumber \\ \sigma _{12}&= 2C_{10}(F_{12}F_{22}+F_{11}F_{21})+2C_{01} \frac{1}{D^2}\left( F_{12}F_{22}+F_{11}F_{21}\right) \end{aligned}$$

(A.4)

The first Piola–Kirchhoff stress $\textbf{P}$ is derived from Cauchy stress tensor $\varvec{\sigma }$ using the following expression:

$$\begin{aligned} \textbf{P} = J\sigma \textbf{F}^{-T} \end{aligned}$$

(A.5)

with $J=1$ for incompressible materials. We obtain the first Piola–Kirchhoff stress whose components are expressed as below

$$\begin{aligned} P_{11}&=2C_{10}\left( -\frac{F_{22}}{D^3}+F_{11}\right) +2C_{01}\left( -\frac{F_{22}}{D^3}(F_{11}^2+F_{21}^2 +F_{12}^2+F_{22}^2)+F_{22}D+\frac{F_{11}}{D^2}\right) \nonumber \\ P_{12}&=2C_{10}\left( \frac{F_{21}}{D^3}+F_{12}\right) +2C_{01}\left( \frac{F_{21}}{D^3}(F_{11}^2+F_{21}^2+F_{12}^2 +F_{22}^2)-F_{21}D+\frac{F_{12}}{D^2}\right) \nonumber \\ P_{21}&=2C_{10}\left( \frac{F_{12}}{D^3}+F_{21}\right) +2C_{01}\left( \frac{F_{12}}{D^3}(F_{11}^2+F_{21}^2+F_{12}^2 +F_{22}^2)-F_{12}D+\frac{F_{21}}{D^2}\right) \nonumber \\ P_{22}&=2C_{10}\left( -\frac{F_{11}}{D^3}+F_{22}\right) +2C_{01}\left( -\frac{F_{11}}{D^3}(F_{11}^2+F_{21}^2+F_{12}^2 +F_{22}^2)+F_{11}D+\frac{F_{22}}{D^2}\right) \end{aligned}$$

(A.6)

The derivative of virtual displacement² is written as follows:

$$\begin{aligned} \frac{\partial {{\textbf {U}}^{\textbf {*}}}}{\partial {\textbf {X}}} = \frac{\partial \varvec{\delta U}}{\partial {\textbf {X}}}=\delta \frac{\partial {\textbf {U}}}{\partial {\textbf {X}}} \end{aligned}$$

(A.7)

The virtual displacement field $\delta {\textbf {u}}$ is independent of the actual displacement field ${\textbf {u}}$ and can be expressed in terms of spatial coordinates or material coordinates. Therefore, we have $\delta {\textbf {U}} = \delta {\textbf {u}}$

$$\begin{aligned} \delta \frac{\partial {\textbf {U}}}{\partial {\textbf {X}}} = \delta \frac{\partial {\textbf {u}}}{\partial {\textbf {X}}} = \delta ({\textbf {F}}-{\textbf {I}}) = \delta {\textbf {F}} \end{aligned}$$

(A.8)

Therefore, the IVW for a single mesh element within the body of the specimen is written as follows:

$$\begin{aligned} {\textbf {P}}: \frac{\partial {{\textbf {U}}^{\textbf {*}}}}{\partial {\textbf {X}}} = {\textbf {P}}: \delta {\textbf {F}} = P_{11}\delta F_{11}+ P_{12}\delta F_{12} + P_{21}\delta F_{21}+ P_{22}\delta F_{22} \end{aligned}$$

(A.9)

After plugging the components of the first Piola–Kirchhoff stress (Eq. A.6) into the above equation, we can find the formulation of IVW for hyperelastic materials based on the conventional VFM proposed by Promma and Grediac [20] is equivalent to the formulation of IVW in VEE-VFM (Eq. 13).

Appendix B Detailed expression of $\delta \varvec{\widetilde{F}}$

$$\begin{aligned} \delta \widetilde{F}_{11}=\frac{\partial \widetilde{F}_{11}}{\partial F_{11}}\delta F_{11} + \frac{\partial \widetilde{F}_{11}}{\partial {F}_{21}}\delta {F}_{21} \end{aligned}$$

(B.1)

with

$$\begin{aligned}{} & {} \frac{\partial \widetilde{F}_{11}}{\partial F_{11}} = \frac{F_{11}}{\sqrt{F_{11}^2+F_{21}^2}}, \quad \frac{\partial \widetilde{F}_{11}}{\partial F_{21}} = \frac{F_{21}}{\sqrt{F_{11}^2+F_{21}^2}} \end{aligned}$$

(B.2)

$$\begin{aligned}{} & {} \delta \widetilde{F}_{12}=\frac{\partial \widetilde{F}_{12}}{\partial F_{11}}\delta F_{11} + \frac{\partial \widetilde{F}_{12}}{\partial {F}_{12}}\delta {F}_{12} + \frac{\partial \widetilde{F}_{12}}{\partial {F}_{21}}\delta {F}_{21} + \frac{\partial \widetilde{F}_{12}}{\partial {F}_{22}}\delta {F}_{22} \end{aligned}$$

(B.3)

with

$$\begin{aligned} \frac{\partial \widetilde{F}_{12}}{\partial F_{11}}&= \frac{F_{21}(F_{12}F_{21}-F_{11}F_{22})}{(F_{11}^2+F_{21}^2)\sqrt{F_{11}^2+F_{21}^2}} \nonumber \\ \frac{\partial \widetilde{F}_{12}}{\partial F_{21}}&= \frac{F_{11}(F_{11}F_{22}-F_{12}F_{21})}{(F_{11}^2+F_{21}^2)\sqrt{F_{11}^2+F_{21}^2}} \nonumber \\ \frac{\partial \widetilde{F}_{12}}{\partial F_{12}}&= \frac{F_{11}}{\sqrt{F_{11}^2+F_{21}^2}}\nonumber \\ \frac{\partial \widetilde{F}_{12}}{\partial F_{22}}&= \frac{F_{21}}{\sqrt{F_{11}^2+F_{21}^2}} \end{aligned}$$

(B.4)

$$\begin{aligned} \delta \widetilde{F}_{22}&=\frac{\partial \widetilde{F}_{22}}{\partial F_{11}}\delta F_{11} + \frac{\partial \widetilde{F}_{22}}{\partial {F}_{12}}\delta {F}_{12} + \frac{\partial \widetilde{F}_{22}}{\partial {F}_{21}}\delta {F}_{21} + \frac{\partial \widetilde{F}_{22}}{\partial {F}_{22}}\delta {F}_{22} \end{aligned}$$

(B.5)

with

$$\begin{aligned} \frac{\partial \widetilde{F}_{22}}{\partial F_{11}}&= \frac{F_{22}(F_{11}F_{22}-F_{12}F_{21})(F_{11}^2+F_{21}^2)-F_{11}(F_{11}F_{22}-F_{12}F_{21})^2}{(F_{11}^2+F_{21}^2)^2\sqrt{(F_{11}F_{22}-F_{12}F_{21})^2}}\sqrt{F_{11}^2+F_{21}^2} \nonumber \\ \frac{\partial \widetilde{F}_{22}}{\partial F_{21}}&= \frac{F_{12}(F_{12}F_{21}-F_{11}F_{22})(F_{11}^2+F_{21}^2)-F_{21}(F_{11}F_{22}-F_{12}F_{21})^2}{(F_{11}^2+F_{21}^2)^2\sqrt{(F_{11}F_{22}-F_{12}F_{21})^2}}\sqrt{F_{11}^2+F_{21}^2} \nonumber \\ \frac{\partial \widetilde{F}_{22}}{\partial F_{12}}&= \frac{F_{21}(F_{12}F_{21}-F_{11}F_{22})}{\sqrt{F_{11}^2+F_{21}^2}\sqrt{(F_{11}F_{22}-F_{12}F_{21})^2}}\nonumber \\ \frac{\partial \widetilde{F}_{22}}{\partial F_{22}}&= \frac{F_{11}(F_{11}F_{22}-F_{12}F_{21})}{\sqrt{F_{11}^2+F_{21}^2}\sqrt{(F_{11}F_{22}-F_{12}F_{21})^2}} \end{aligned}$$

(B.6)

$$\begin{aligned} \delta \widetilde{F}_{33}&=\frac{\partial \widetilde{F}_{22}}{\partial F_{11}}\delta F_{11} + \frac{\partial \widetilde{F}_{33}}{\partial {F}_{12}}\delta {F}_{12} + \frac{\partial \widetilde{F}_{33}}{\partial {F}_{21}}\delta {F}_{21} + \frac{\partial \widetilde{F}_{33}}{\partial {F}_{22}}\delta {F}_{22} \end{aligned}$$

(B.7)

with

$$\begin{aligned} \frac{\partial \widetilde{F}_{33}}{\partial F_{11}}&= -\frac{F_{33}\sqrt{(F_{11}F_{22} -F_{12}F_{21})^2}}{(F_{11}F_{22}-F_{12}F_{21})^3} \nonumber \\ \frac{\partial \widetilde{F}_{33}}{\partial F_{21}}&= -\frac{F_{12}\sqrt{(F_{11}F_{22} -F_{12}F_{21})^2}}{(F_{11}F_{22}-F_{12}F_{21})^3} \nonumber \\ \frac{\partial \widetilde{F}_{33}}{\partial F_{12}}&= -\frac{F_{21}\sqrt{(F_{11}F_{22} -F_{12}F_{21})^2}}{(F_{11}F_{22}-F_{12}F_{21})^3} \nonumber \\ \frac{\partial \widetilde{F}_{33}}{\partial F_{22}}&= -\frac{F_{11}\sqrt{(F_{11}F_{22} -F_{12}F_{21})^2}}{(F_{11}F_{22}-F_{12}F_{21})^3} \end{aligned}$$

(B.8)

Fig. 8

The change of the ratio between the current and initial volume of the specimens with the stretch ratio (averaged from three specimens) under uniaxial tension

Appendix C Incompressibility justification

Uniaxial tensile tests were performed on rectangular silicone elastomer. Three specimens were tested up to 60% strain under a strain rate of 1%/s. Ncorr, a Matlab-based digital image correlation software, was used for strain measurement across the surface of the specimens. A rectangular ROI which was far away from the clamps was selected for data analysis. The strain within this region was uniformly distributed. Details about the testing protocol can be found in this paper [28]. The stretch ratio along the stretching direction ($\lambda _x$) and stretch ratio across the stretching direction ($\lambda _y$) were extracted and averaged over the selected ROI. Since specimens were under uniaxial tension, we assumed that $\lambda _y = \lambda _z$. Therefore, the ratio of the current volume over the initial volume was calculated as $\lambda _x*{\lambda _y}^2$. Figure 8 shows that the ratio between the current and initial volumes slightly increases with the stretch ratio. The maximum change of the volume occurred in the last deformation step, which was about 0.46%. This small difference can be considered negligible. Therefore, the material can be assumed to be incompressible.

Appendix D Parameter identification with VEE-VFM for Mooney–Rivlin model under polar decomposition

After extracting the full-field displacement data and forming triangular mesh elements based on nodal positions, the deformation gradient and virtual strain can be calculated respectively. The integral of the IVW for a hyperelastic model under polar decomposition in Eq. 12 can be approximated by a discrete sum as follows

$$\begin{aligned} IVW = \sum _{i=1}^{n_e} \delta W \left( \textbf{x}(\textbf{X}\right) , \textbf{U}^*) \cdot A_i t_i \end{aligned}$$

(D.1)

where $n_e$ represents the total number of mesh elements; and $A_i$ and $t_i$ represent the area and the thickness of a specific element under the reference configuration, which do not depend on time; W is the strain energy density function of the hyperelastic material. Note that $\delta W$ for Mooney–Rivlin model under the polar decomposition is expressed as $C_{10}\delta I_1 + C_{01}\delta I_2$. $\delta I_1$ and $\delta I_2$ can be calculated based on Eq. 15.

For each deformation step k, we can construct one virtual field ${\textbf {U}}^{*}_k$. According to the principle of virtual work, we have

$$\begin{aligned} \sum _{i=1}^{n_e} \delta W({\textbf {U}}^{*}_k) \cdot A_it_i = \delta W_{ext}({\textbf {U}}^{*}_k) \end{aligned}$$

(D.2)

where $\sum _{i=1}^{n_e} \delta W({\textbf {U}}^{*}_k) \cdot A_it_i$ represents the sum of the IVW of all the mesh elements, and $\delta W_{ext}({\textbf {U}}^{*}_k)$ represents the EVW. In the case of Mooney–Rivlin model, $\delta W({\textbf {U}}^{*}_k)$ is expressed as

$$\begin{aligned} \delta W({\textbf {U}}^{*}_k) = C_{10} \cdot \delta I_1({\textbf {U}}^{*}_k) + C_{01} \cdot \delta I_2({\textbf {U}}^{*}_k) \end{aligned}$$

(D.3)

Therefore, we can have

$$\begin{aligned} \sum _{i=1}^{n_e} ( C_{10} \cdot \delta I_1({\textbf {U}}^{*}_k) + C_{01} \cdot \delta I_2({\textbf {U}}^{*}_k) ) \cdot A_it_i = \delta W_{ext}({\textbf {U}}^*) \end{aligned}$$

(D.4)

Note that we constructed three virtual fields in this paper. The rest of the procedure to identify material parameters is the same as that described in Sect. 3.3.