Hyperelastic materials are the special class of materials which tends to respond elastically when they are subjected to very large strains. They shows both a nonlinear material behavior as well as large shape changes. They posses certain characteristics:
All the polymers tends to show the hyperelastic behaviour such as elastomers, rubbers, sponges and other soft flexible materials.
Hyperelastic materials are mostly used where high flexibility on a long run is required under large loads. The typical examples of there uses are as elastomeric pads in brigdes, rail pads, car door seal, car tyres etc.
In finite element analysis, hyperelasticity theory is used to represent the non-linear response of hyperelastic materials at large strains. Hyperlasticity is popular due to it’s ease of use in finite element models. Normally stress-strain curve data from experiments is used to find the constants of theoretical models to fit the material response. There are some good available choices of hyperelasticity models which are available in SimScale platform including:
The stress-strain relation for hyperelastic materials is normally calculated through strain energy density function. The formulation of which is shown below:
Suppose a solid is subjected to a displacement which can be represented by ui(xk). Than, the deformation gradient F and it’s Jacobian J (the total change in volume at point) can be represeted by:
$$\mathbf{F} = F_{ij} = \delta_{ij} + \frac {\partial u_i}{\partial x_j}\ , J = det(\mathbf{F})\quad$$
For simplicity,
$${\mathbf{\bar F}} = J^{\frac {-1}{3}} \mathbf{F}$$
From this, we have a left Cauchy-Green deformation tensor,
\(\bar B\) = \(\bar F\bar F\)T Bij = FikFjk
Invariants of B are represented as:
$$\bar I_1 = trace \ \mathbf{\bar B} = \bar B_{kk} \\
\bar I_2 = \frac {1}{2} (\bar I_1^2 – trace \ (\mathbf{\bar B}.\mathbf{\bar B)}) = \frac {1}{2} (\bar I_1^2 – \bar B_{ik}\bar B_{ki})\\
\bar I_3 = det(\mathbf{B}) = J^2$$
The invariant \(\bar I_3\) is a volumetric invariant (zero for incompressible case).
Furthermore,
$$\delta {\bar I_1} = 2\mathbf{\bar B}\ :\ \delta \mathbf{\epsilon_v}\\
\delta {\bar I_2} = 2(\bar I_2\mathbf{\bar B} – \mathbf{\bar B}.\mathbf{\bar B})\ :\ \delta \mathbf{\epsilon_v}\\
J = J \delta \mathbf{\epsilon_v}\\ (where,\ \delta \mathbf{\epsilon_v}\ is\ a\ virtual\ deviatoric\ strain\ rate)$$
Variation of the strain energy density \(U = U(\bar I_1,\bar I_2,\bar J)\) for isotropic and compressible materials can be represented as,
$$\delta U = \frac {\partial U}{\partial \bar I_1} \delta \bar I_1 + \frac {\partial U}{\partial \bar I_2} \delta \bar I_2 + \frac {\partial U}{\partial \bar J} \delta \bar J$$
Using \(\bar I_1\), \(\bar I_2\) and J as defined above,
$$\delta U = 2 \left[\left(\frac {\partial U}{\partial \bar I_1} + \bar I_1 \frac {\partial U}{\partial \bar I_2}\right)\mathbf{\bar B}
– \frac {\partial U}{\partial \bar I_2} \mathbf{\bar B}.\mathbf{\bar B}\right]\ :\ \mathbf{\epsilon_v} + J\frac {\partial U}{\partial \bar J} \delta \epsilon^v \\
(where,\ \delta \epsilon^v\ is\ a\ virtual\ volumetric\ strain\ rate)$$
This variation of strain energy density is used to deduce the stress-strain law in case of hyperelastic materials, Therefore one can write:
$$\mathbf{\sigma} = 2\ DEV\ \left[\left(\frac {\partial U}{\partial \bar I_1} + \bar I_1 \frac {\partial U}{\partial \bar I_2}\right)\mathbf{\bar B}
– \frac {\partial U}{\partial \bar I_2} \mathbf{\bar B}.\mathbf{\bar B}\right] + J\frac {\partial U}{\partial \bar J} \mathbf{I}$$
For the incompressible cases (e.g. rubber), the last term containing J is neglected.
Before giving the appropriate material parameters to define a specific hyperelastic materials, one should know the strain energy density forms of the hyperelasticity models. In this section, we have mentioned the strain energy density forms of all the available hyperelasticity models on SimScale platform (already mentioned in Hyperelasticity).
The basic strain energy density form of reduced polynomial is represented as:
$$U = \displaystyle\sum_{i=1}^{N} C_{i0}(\bar I_1 – 3)^i + \displaystyle\sum_{i=1}^{N} \frac {1}{D_i}(J – 1)^{2i}$$
where, Ci0(i = 1, 2, 3) are the constants that needs to be deduced in order to get the appropriate material behaviour. For the compressibility of a material, one have to give the value of Di(i = 1, 2, 3). As a first approximation, one can use a basic formulation for elastic case:
$$D_i = \frac {3(1 – 2 \nu)}{G_o (1 + \nu)}\quad with,\ G_o = 2 C_{10}$$
If material is incompressible, than one can take the value of ν nearest to 5.0 i.e. ν = 0.499.
For N = 1 in above formulation, one obtains the most basic form known as Neo Hookean, represented as:
$$U = C_{10}(\bar I_1 – 3) + \frac {1}{D_1}(J – 1)^2$$
For N = 2 in above formulation, one obtains:
$$U = C_{10}(\bar I_1 – 3) + C_{20}(\bar I_1 – 3)^2 + \frac {1}{D_1}(J – 1)^2 + \frac {1}{D_2}(J – 1)^4$$
For N = 3 in above formulation, one obtains Yeoh, represented as:
$$U = C_{10}(\bar I_1 – 3) + C_{20}(\bar I_1 – 3)^2 + C_{30}(\bar I_1 – 3)^3 + \frac {1}{D_1}(J – 1)^2 + \frac {1}{D_2}(J – 1)^4 + \frac {1}{D_3}(J – 1)^6$$
The basic strain energy density form of polynomial is represented as:
$$U = \displaystyle\sum_{i+j=1}^{N}C_{ij}(\bar I_1 – 3)^i(\bar I_2 – 3)^j + \displaystyle\sum_{i=1}^{N} \frac {1}{D_i}(J – 1)^{2i}$$
As a first approximation for Di, same eq. eq2 can be used with Go = 2(C10 + C01).
For N = 1 in above formulation, one obtains the enhancement to Neo Hookean form known as Mooney Rivlin, represented as:
$$U = C_{10}(\bar I_1 – 3) + C_{01}(\bar I_2 – 3) + \frac {1}{D_1}(J – 1)^2$$
For N = 2 in above formulation, one obtains:
$$U = C_{10}(\bar I_1 – 3) + C_{01}(\bar I_2 – 3) + C_{20}(\bar I_1 – 3)^2 + C_{02}(\bar I_2 – 3)^2 + C_{11}(\bar I_1 – 3)(\bar I_2 – 3)
+ \frac {1}{D_1}(J – 1)^2 + \frac {1}{D_2}(J – 1)^4$$
Strain energy density form for Signorini is represented as:
$$U = C_{10}(\bar I_1 – 3) + C_{01}(\bar I_2 – 3) + C_{20}(\bar I_1 – 3)^2 + \frac {1}{D_1}(J – 1)^2$$
Compared to the previous forms, Ogden is represented by the principal stretches \(\bar \lambda_i (i = 1,2,3)\) compared to the deviatoric strain invariants \(\bar I_i (i = 1,2,3)\). The basic strain energy density form of Ogden is represented as:
$$U = \displaystyle\sum_{i=1}^{N} \frac {2 \mu_i}{\alpha_{i}^{2}}(\bar \lambda_{1}^{\alpha_i} + \bar \lambda_{2}^{\alpha_i} + \bar \lambda_{3}^{\alpha_i} – 3)
+ \displaystyle\sum_{i=1}^{N} \frac {1}{D_i}(J – 1)^{2i}$$
where, μi and αi for (i = 1, 2, 3) are the constants that needs to be deduced in order to get the appropriate material behaviour. For the compressibility of a material, eq. eq2 can be used as a first approximation for Di with \(G_o = \frac {1}{2} \displaystyle\sum_{i=1}^{N} \mu_i \alpha_i\)
The strain energy density of Arruda-Boyce has the following form:
$$U = \mu \displaystyle\sum_{i=1}^{5} \frac {C_i}{\lambda_{m}^{2i-2}} \left(\bar I_{1}^{i} – 3 \right) + \frac {1}{D} \left(\frac {J^2 – 1}{2} – lnJ \right)$$
where, \(C_1 = \frac {1}{2}\), \(C_2 = \frac {1}{20}\), \(C_3 = \frac {11}{1050}\), \(C_4 = \frac {19}{7000}\) and \(C_5 = \frac {519}{673750}\).
For the compressibility of a material, eq. eq2 can be used as a first approximation for Di with \(G_o = \mu (1 + \frac {3}{5 \lambda_{m}^{2}} + \frac {99}{175 \lambda_{m}^{4}} + \frac {513}{875 \lambda_{m}^{6}} + \frac {42039}{67375 \lambda_{m}^{8}})\)
In experiments, the response of a material for a specific load is normally deduced by the stress-strain relationship. As an example, a basic stress-strain curve of an elastomer for Uniaxial (Tension), Biaxial and Shear load (provided by Axel Product, Inc.) is shown below:
In order to fit the material properties to the experimental stress-strain curve, one has to take in to account eq. eq1 above. This equation shows the stress as a function of invariants containing principal stretches (1 + “strain values”) and strain energy density (of any of the above model). Using this equation, one can calculate the constants of the undertaken model through numerical scheme to get an accurate fit to the stress-strain material curve. In most of the cases, the accuracy of the fit increases with the increasing order of the model.
As an example, let us consider a stress-strain relation of an incompressible uniaxial case for a general polynomial model:
The principal stretches of uniaxial case are:
$$\lambda_1 = \lambda, \quad \lambda_2 = \lambda_3 = \lambda^\frac{-1}{2} \\
(where, \lambda = 1 + \epsilon)$$
and the deviatoric strain invariants are:
$$\bar I_1 = \lambda^2 + 2 \lambda^{-1}, \quad \bar I_2 = \lambda^{-2} + 2 \lambda$$
From principle of virtual work, it follows that:
δU = σδλ
Than stress can be represented as:
$$\sigma = \frac {\partial U}{\partial \lambda} = \frac {\partial U}{\partial \bar I_1} \frac {\partial \bar I_1}{\partial \lambda}
+ \frac {\partial U}{\partial \bar I_2} \frac {\partial \bar I_2}{\partial \lambda}\\
= 2(\lambda – \lambda^{-2}) \left(\frac {\partial U}{\partial \bar I_1} + \frac {\partial U}{\partial \bar I_2} \right)$$
Therefore, for the general polynomial case:
$$\sigma = 2(\lambda – \lambda^{-2}) \displaystyle\sum_{i+j=1}^{N}[iC_{ij}(\bar I_1 – 3)^i(\bar I_2 – 3)^j + jC_{ij}(\bar I_1 – 3)^i(\bar I_2 – 3)^j]$$
By using the experimental stress-strain data in the above equation, one can calculate the material constants for a specific order of polynomial through a numerical scheme e.g. Least square method.
Once the material constants of a specific model are obtained for a good fit to experimental data, one can input these constants to the specified material model in SimScale under material properties. In case a polynomial of order 2 is used, the following can be interpreted:
Same cases are applied to all other models.
