# Isotropic hyperelasticity in principal stretches: explicit elasticity tensors and numerical implementation

## Abstract

Elasticity tensors for isotropic hyperelasticity in principal stretches are formulated and implemented for the Finite Element Method. Hyperelastic constitutive models defined by this strain measure are known to accurately model the response of rubber, and similar materials. These models may not be available in the library of a Finite Element Analysis software, but a numerical implementation of the constitutive model may be provided by a programmed subroutine. The implementation proposed here is robust and accurate, with straightforward user input. It is presented in multiple configurations with novel features, including efficient definition of isochoric stress and elasticity coefficients, symmetric dyadic products of the principal directions, and development of a stable and accurate algorithm for equal and similar principal stretches. The proposed implementation is validated, for unique, equal and similar principal stretches. Further validation in the Finite Element Method demonstrates the developed implementation requires lower computational effort compared with an alternative, well-known implementation.

## Keywords

Hyperelasticity Principal stretches Finite Element Method Numerical implementation Elasticity tensors## 1 Introduction

In the Finite Element Method (FEM), a constitutive model is defined to describe the response for each component. The isothermal, quasi-static and rate-independent behaviour of rubber, and materials of a similar phenomenological description, is described by a hyperelastic constitutive model. When more complex behaviour is considered (viscoelasticity, plasticity or damage), the hyperelastic response may represent the equilibrium elastic behaviour [29, 40].

Hyperelastic constitutive models typically define a Helmholtz free energy function, dependent only on the state of strain. Hence, they are often referred to as strain energy (density) functions. These may be categorised as either micro-mechanical or phenomenological [15, 30, 42], depending on whether or not their parameters are physically defined. Otherwise, they may be categorised by their strain measure [21]. Hyperelastic constitutive models defined in terms of principal stretches are of particular interest, as this class of model has been shown to capably predict the elastic response of rubber [10, 18, 23, 25, 33, 35, 46]. However, a definitive numerical implementation with explicit validation is not known from literature.

The implementation of a constitutive model within an implicit FEM requires the definition of a stress tensor and its consistent tangent moduli, in a form that is dependent on the solver. The consistent tangent moduli for hyperelastic constitutive models are defined by an elasticity tensor and any additional terms required for objectivity [14]. For isotropic hyperelasticity in principal stretches, various forms of the elasticity tensor are known from literature. This ambiguity is due to the nature of the principal stretches. The squared principal stretches and associated principal directions are found from the Cauchy-Green deformation tensors, where they are equivalent to eigenvalues and eigenvectors respectively. The original derivation [3, 4, 34] of the elasticity tensor relies on the explicit calculation of these eigenvalues and eigenvectors. However, the seminal implementation in the FEM developed by Simo and Taylor [41] avoided explicit calculation of the eigenvectors, due to the required computational effort, and developed alternative elasticity tensors. Though Simo [39] later stated that explicit calculation of eigenvectors by a Jacobi method should be preferred. This is due to the known numerical instabilities of the alternative elasticity tensors and eigenvector replacements [12, 16, 39]. Despite this, direct methods based on Simo and Taylor [41] have found continued use [5, 7, 8, 16, 18, 23, 26, 36].

In this paper an efficient implementation of isotropic hyperelasticity in principal stretches is proposed and validated. Fortran programs are created to investigate the numerical implementation in reference and current configurations. The commercial FEM software Abaqus is used to further investigate the numerical implementation. Abaqus enables the implementation of user defined constitutive models through UMAT subroutines, written in Fortran 77 standard. The stress and elasticity tensors are derived by explicit calculation of the principal stretches and associated principal directions. These are computed using an efficient Jacobi method algorithm from Kopp [22] (provided open-source). The stress and elasticity tensors are first derived in the reference configuration in terms of the material 2nd Piola-Kirchhoff stress tensor and associated elasticity tensor. These tensors are transformed into the current configuration in terms of the spatial Kirchhoff stress and the elasticity tensor is defined in terms of the Oldroyd rate of the Kirchhoff stress. Transformations to the Cauchy stress and the Jaumann rate of the Cauchy stress are defined, as these are the stress and elasticity tensors required by Abaqus. It is assumed throughout that the constitutive model is defined by an isochoric-volumetric split [9]. The developed expressions are therefore given in terms of isochoric principal stretches and the volumetric contributions are defined for completeness.

The developed numerical implementation has some novel features: the stress and elasticity coefficients are efficiently implemented, symmetric dyadic products of the principal directions are utilised, and the numerical instabilities associated with equal and similar principal stretches are resolved by derived approximations with L’Hôpital’s rule. The implementation is validated by evaluating the stress and elasticity tensors for constitutive models typically described in terms of Cauchy-Green invariants. By expressing these constitutive models in terms of principal invariants, the presented implementation can be compared to the well-established implementation of Cauchy-Green invariants, which are not subject to numerical instabilities for deformations with equal and similar principal stretches. This form of validation is not otherwise known from literature.

## 2 Isotropic hyperelasticity in principal stretches

In this section, the stress and elasticity tensors for isotropic hyperelasticity in principal stretches are defined. The complete derivations are omitted but referenced throughout. The material tensors are defined with respect to the reference configuration, then transformed to their spatial equivalent form in the current configuration by a push-forward operation. Some additional aspects of the developed numerical implementation are also detailed, including the treatment of numerically similar and equal eigenvalues. Firstly, the kinematics of hyperelasticity are defined.

### 2.1 Kinematical description of hyperelasticity

In three-dimensional real coordinate space \({\mathbb {R}}^3\), a reference configuration \({\varvec{\Omega }}_0\) and a current configuration \(\varvec{\Omega }\) are defined for a body of interest. A material point \(\mathbf X \), in the reference configuration \(\mathbf X {\in }\varvec{{{\Omega }}_0}\), is mapped to its current position \(\mathbf x \), in the current configuration \(\mathbf x {\in }{\Omega }\), by \(\varvec{\upchi }\), where \(\varvec{\upchi }\ {:}{{\Omega }}_0\rightarrow \, {\mathbb {R}}^3\) and therefore \(\mathbf x = \varvec{\upchi }\left( \mathbf X \right) \). The motion \(\varvec{\upchi }\) describes the both the deformation and the rigid body motions (translation and rotation). To remove translation, the two-point deformation gradient tensor, defined by \(\mathbf F = {\partial \varvec{\upchi }}/{\partial \mathbf X }\) is used. The volume ratio \(J = {V}/{V_0}\) is calculated by its determinant \(J = \mathrm{det}\left( \mathbf F \right) \).

### 2.2 Definition of stress tensors

*Material stress tensor*In the reference configuration, the 2nd Piola-Kirchhoff stress tensor \(\mathbf S \) defines the material stress. For a hyperelastic constitutive model this is calculated as the derivative of the energy \(\psi \) with respect to the right Cauchy-Green deformation tensor \(\mathbf C \)

*Spatial stress tensors*The spatial stress tensor is first defined in terms of the Kirchhoff stress, \(\varvec{\tau }\). This stress tensor is related to the 2nd Piola-Kirchhoff stress tensor by

### 2.3 Definition of elasticity tensors

*Material Elasticity Tensor*The material elasticity tensor is defined using the additive split of isochoric and volumetric contributions as

*Spatial elasticity tensors*The spatial elasticity tensor may be defined in various forms due to differences in the stress and strain tensors and the choice of objective rate. The spatial elasticity tensor is first defined in terms of the Oldroyd rate of the Kirchhoff stress, \(\mathrm {c}\). This is connected to the material elasticity tensor, in index notation, by

### 2.4 Aspects of numerical implementation

With the stress and elasticity tensors defined in reference and current configurations, these terms may be implemented in numerical methods. Some additional considerations are now discussed regarding particular aspects of the developed numerical implementation for any hyperelastic constitutive model defined in terms of principal stretches. The first consideration is the simplified implementation of the isochoric stress and elasticity coefficients \({\beta }_a\) and \({\gamma }_{ab}\). The next is the use of symmetric dyadic products of the principal directions, which enables Voigt notation and reduces the computational effort. The final consideration is the required algorithm for detection of numerical similarity in the squared principal stretches and employment of the L’Hôpital’s rule approximations to avoid numerical instability. These features can be understood in more depth by examining the developed programs and subroutines available in the referenced dataset [6].

*Implementation of isochoric stress and elasticity coefficients* In the proposed implementation, the required user input is minimised. In other implementations [18, 23, 41], the stress coefficients \({\beta }_a\) are required. Here it is noted from (12) that the use of a generic expression for \(\frac{\partial W}{\partial {\overline{\lambda }}_a}\) may be defined by considering a specified principal stretch e.g. \(\frac{\partial W}{\partial {\overline{\lambda }}_1}\). As, due to isotropy, this derivative is symbolically equivalent to \(\frac{\partial W}{\partial {\overline{\lambda }}_2}\) and \(\frac{\partial W}{\partial {\overline{\lambda }}_3}\), a general expression is therefore defined to compute all three derivatives. The stress coefficients \({\beta }_a\) are then computed using these derivatives, as defined in (12).

*Symmetric dyadic products of principal directions* For all variations of the elasticity tensors defined in this paper, the stress and deformation tensors used in their derivation are symmetric. For computational efficiency, symmetric 2nd-order tensors may be defined using Voigt notation. This notation allows 2nd-order tensors to be represented by six component matrices, and hence 4th-order tensors are defined by six by six matrices. In the presented numerical implementation, the convention used by Abaqus is followed where the integers 1 to 6 represent the 1,1; 2,2; 3,3; 1,2; 1,3; and 2,3 components, respectively.

The stress tensors are defined in terms of the stress coefficients and the dyadic products of the relevant principal directions, \(\mathbf{n }_{a}\otimes \mathbf{n }_{a}\) or \(\mathbf{N }_{a}\otimes \mathbf{N }_{a}\). These dyadic products are inherently symmetric since they are constructed from the same vectors, as \(\mathbf a \otimes \mathbf b = {\left( \mathbf a \otimes \mathbf b \right) }^{\mathrm{T}}\) if \(\mathbf a = \mathbf b \). These can therefore be represented in Voigt notation. However, the definition of the elasticity tensors, (22) and (31), contains several non-symmetric 2nd-order tensors, for example, in (22), \(\mathbf{N }_{{a}}\otimes \mathbf{N }_{{b}}\) where \({a}{\ne } {b}\) and by the nature of eigenvectors \(\mathbf{N }_{{a}}{\ne }\mathbf{N }_{{b}}\) for \({a}{\ne } {b}\). In this form, these cannot be represented in Voigt notation and nor can their 4th-order dyadic products, e.g. \(\mathbf{N }_{{a}}\otimes \mathbf{N }_{{b}}\otimes \mathbf{N }_{{a}}\otimes \mathbf{N }_{{b}}\). A modification is therefore required.

*Equal and similar principal stretches* When two or three of the squared principal stretches (eigenvalues of the Cauchy-Green deformation tensors) are equal, the elasticity tensors result in a divide by zero error. In a numerical method, the finite floating-point precision limit of the computation means that the solution also encounters numerical inaccuracy when the eigenvalues are numerically similar. With equal eigenvalues, the application of L’Hôpital’s rule provides an exact alternative solution, though numerically similar eigenvalues require additional consideration. It is therefore necessary to find an approximate numerical tolerance at which the use of L’Hôpital’s rule gives an approximation more accurate than the original function. There is also a requirement for an algorithm to compare the similarity of the eigenvalues.

*tol*is the magnitude of the numerical tolerance.

Proposed algorithm for robust computation of the spatial elasticity tensor

An equivalent algorithm is used in the numerical implementation of the isochoric material elasticity tensor defined in (22). Both can be found in the dataset [6]. The algorithm prevents divide by zero errors provided that a suitable tolerance value is selected. Optimisation of the tolerance value is investigated in Sect. 3.3.

## 3 Numerical validation

This numerical investigation is divided into three studies, all of which use Fortran programs. These programs allow computation of the stress and elasticity tensors for a user defined deformation gradient with a chosen constitutive model and definition of its parameters and derivatives. The error is then computed by using an equivalent Cauchy-Green invariant implementation. The proposed implementation is first validated for unique eigenvalues, then compared for two and three equal eigenvalues. The tolerance value is then optimised using similar eigenvalues. The validated and optimised implementation is then used to create UMAT user subroutines to investigate performance in the FEM using Abaqus in Sect. 4.

The Fortran programs and UMAT user subroutines used in these investigations, as well as templates for a generic implementation of isochoric-volumetric constitutive models, are provided in the dataset [6]. Both Cauchy-Green invariant and principal stretch variations are included.

### 3.1 Unique eigenvalues

*I*to represent the indices 1 to 6, as defined previously. The 2nd Piola-Kirchhoff stress tensors \({\varvec{\mathrm {S}}}^{\overline{\varvec{\lambda }}}\) and \({\varvec{\mathrm {S}}}^{{\overline{\varvec{I}}}_{\varvec{12}}}\) represent the components computed in terms of principal stretches and Cauchy-Green invariants respectively. The same method is applied for calculating the error of the spatial Kirchhoff and Cauchy stress tensors, \(E_{\varvec{\uptau }}\) and \(E_{\varvec{\upsigma }}\) respectively.

*J*, they are omitted from the comparisons.

Error of stress tensors: material (2nd Piola-Kirchhoff), spatial (Kirchhoff), and spatial (Cauchy)

\(E_\mathbf{S }\) | \(E_{\varvec{\uptau }}\) | \(E_{\varvec{\upsigma }}\) | |
---|---|---|---|

\(\mathbf{F }_{{1}}\) | 1.306E-15 | 3.249E-15 | 3.248E-15 |

\(\mathbf{F }_{{2}}\) | 3.085E-15 | 1.145E-15 | 1.000E-15 |

Error of elasticity tensors: material, spatial (Oldroyd), and spatial (Jaumann)

\(E_{\mathbb {C}}\) | \(E_{\mathrm {c}}\) | \(E_{{\mathrm {c}}^{J}}\) | |
---|---|---|---|

\(\mathbf{F }_{{1}}\) | 1.525E-15 | 5.867E-16 | 1.449E-15 |

\(\mathbf{F }_{{2}}\) | 1.642E-15 | 1.243E-15 | 1.589E-15 |

### 3.2 Equal eigenvalues

With equal squared principal stretches, the algorithm of Table 1 is required to avoid undefined divide by zero errors for all variations of the elasticity tensors. As the eigenvectors are identical, any tolerance value *N* is acceptable for which \(1+N>1\) is true. The computation of the stress tensor is not affected by equal or similar eigenvalues and is therefore not considered henceforth. The eigenvalue and eigenvector computation would be a cause of concern in these scenarios. However, the Jacobi method algorithm is known to be stable and accurate [22].

Error of elasticity tensors with two equal eigenvalues

\(E_{\mathbb {C}}\) | \(E_{\mathrm {c}}\) | \(E_{{\mathrm {c}}^{J}}\) | |
---|---|---|---|

\(\mathbf{F }_{{3}}\) | 1.436E-16 | 5.928E-15 | 7.398E-16 |

\(\mathbf{F }_{{4}}\) | 1.395E-15 | 5.478E-15 | 3.083E-15 |

Error of elasticity tensors with three equal eigenvalues

\(E_{\mathbb {C}}\) | \(E_{\mathrm {c}}\) | \(E_{{\mathrm {c}}^{J}}\) | |
---|---|---|---|

\(\mathbf{F }_{{5}}\) | 5.617E-16 | 3.378E-16 | 3.144E-16 |

\(\mathbf{F }_{{6}}\) | 7.411E-16 | 4.798E-16 | 4.577E-16 |

### 3.3 Eigenvalue similarity tolerance

The algorithm presented in Table 1 is designed such that the tolerance value corresponds to the perturbation magnitude. The optimal tolerance magnitude is therefore approximated as \({10}^{-6}\), the results for which are also plotted in Fig. 1. For the four deformation gradients, assumed to be the limits of physical deformations, and for all configurations, the elasticity tensor is computed accurately, with a maximum error of less than \({10}^{-10}\).

## 4 Finite Element Method investigation

### 4.1 Inhomogeneous tension test

The first benchmark simulation is a 3D inhomogeneous tension test of a rubber specimen. The cuboidal specimen has a 1 mm by 1 mm cross-section and is 1.5 mm long. One of the 1 mm by 1 mm faces of the specimen is fully fixed and a displacement load of 1 mm is applied to the opposing face in the longitudinal direction in a minimum of 10 increments using automatic incrementation. The body is meshed by hybrid, higher-order tetrahedral elements (C3D10H) to further ensure that the principal directions are computed appropriately. The model is solved for five levels of mesh refinement, giving additional insight into any differences in computational effort. Incompressibility is assumed such that only the isochoric constitutive model requires consideration.

In the incremental solution of the inhomogeneous tensile test, the built-in model, the presented implementation and the Cauchy-Green invariant implementation achieve convergence in the specified minimum of 10 increments for all levels of mesh refinement, with the same number of total iterations. However, the Simo and Taylor [41] implementation achieves convergence in 10 increments for only the simplest mesh, where it requires additional iterations the solution of the first increment. For an increasingly refined mesh, the Simo and Taylor [41] implementation requires additional increments due to the numerical instability in the early increments of the solution. The difference between the implementations is present only in their elasticity tensors, however, and so the converged simulation results are identical for all tested outputs at each level of mesh refinement across the four implementations.

The von Mises stress results are shown for the third mesh with 5617 elements in Fig. 2 for solved models using the built-in model and the proposed implementation. For this mesh, the maximum residual forces during convergence have a maximum difference of \(\Delta F\le {10}^{-7}\ N\) between the presented implementation’s model and the built-in model. There are therefore no observable differences between the solved models for all increments.

### 4.2 Combined tension-torsion test

The combined tension torsion-test reveals similar findings to the previous simulated model. The minimum number of increments are required for the built-in model, the Cauchy-Green invariant implementation and the presented implementation for all levels of mesh refinement, each with the same number of total iterations. The implementation of Simo and Taylor [41] requires additional increments for all meshes and a higher number of total iterations. As in the previous example, the physical results are seemingly identical in the converged solutions. The maximum principal stress results are shown for the third mesh with 5376 elements in Fig. 4, showing seemingly identical results for the built-in and the proposed implementation models.

The computational effort is compared using solve times relative to the built-in model. The results are shown for the five levels of mesh refinement in Fig. 5 with the same labelling as before. It is confirmed that increasing mesh density leads to a smaller difference in solve times for all implementations. However, the developed implementation remains significantly more computationally efficient compared with that based on Simo and Taylor [41] due to its more stable form requiring fewer total increments and iterations. It does have a higher computation time compared with the Cauchy-Green invariant implementation and the latter implementation should be preferred for a constitutive model that is expressible in terms of the Cauchy-Green invariants.

### 4.3 Discussion

The numerical validation and Finite Element Method investigation results show that the developed implementation is accurate and of satisfactory computational efficiency. Although it is noted that the proposed implementation is not optimised for computational efficiency. The priority of the developed programs and user subroutines is the ease of user implementation. If all implementations were optimised for computational efficiency then the results may differ.

In the case of equal and similar eigenvalues, it is shown that the approximated terms using L’Hôpital’s rule are accurate and valid. It is apparent that in the case of similar eigenvalues that the algorithm and tolerance value could be optimised and improved. However, this would result in imperceptible changes to the accuracy of the Finite Element result and is not considered to be necessary. The developed algorithm and the approximate tolerance of \({10}^{-6}\) is found to produce converging solutions with a maximum error in the order of \({10}^{-11}\).

The spatial elasticity tensor is given in two forms convenient to the derivation. However, it also exists in several other forms due to the requirement of fulfilling objectivity, which is not inherent as in the reference configuration [38]. The developed spatial elasticity tensors are both known to be objective. However, a possible concern is that they may not satisfy work-conjugacy [1, 2, 17, 32, 47] due to a missing volumetric term in both of the objective rates of the elasticity tensors used in this implementation. The error is therefore insignificant for near and full incompressibility. If the compressive behaviour is not negligible, the modification outlined in Bažant et al. [2] should be applied.

## 5 Conclusions

A Finite Element implementation of hyperelasticity in principal stretches has been proposed. The implementation involves simple user input while attaining numerical robustness and accuracy. The user input is unchanged in the definition of constitutive models in the reference and current configurations. The user is required only to define the derivatives of the isochoric and volumetric strain energy functions with respect to the isochoric principal stretches and the volume ratio. The isochoric principal stretches and associated principal directions are computed explicitly using an iterative Jacobi method. Explicit computation requires additional computational effort, compared with a direct method, but is shown to require less total computational effort due to its improved numerical stability and accuracy. Compared with a built-in model and a Cauchy-Green invariant implementation, the computational effort is satisfactory, which is in part due to the novel use of symmetric dyadic products of the principal directions. These enable Voigt notation to be used throughout, reducing the overall number of computations.

The proposed implementation was validated by numerical investigations of hyperelastic constitutive models that are typically defined in terms of the Cauchy-Green invariants. Evaluations were made between the proposed implementation and a conventional Cauchy-Green invariant implementation. The evaluations show that the proposed implementation is accurate, stable and of similar computational effort. These evaluations were also beneficent in developing and validating the proposed algorithm for applying approximations derived from L’Hôpital’s rule when principal stretches are numerically equal or similar, as to avoid numerical instability and divide by zero errors. The FEM implementation in Abaqus, through use of a UMAT subroutine, shows that the method produces the desired numerical results, with acceptable solve times. This demonstrates that the proposed implementation is suitable for implementing isotropic hyperelastic constitutive models in terms of principal stretches for the FEM.

## Notes

### Compliance with ethical standards

### Data statement

Programs and subroutines created during this research are made openly available from the University of Strathclyde “Pure” data archive at DOI 10.15129/b1cc7acc-a170-479e-8b26-b74395352b26.

## References

- 1.Bažant ZP (1971) A correlation study of formulations of incremental deformation and stability of continuous bodies. J Appl Mech 38(4):919. https://doi.org/10.1115/1.3408976 CrossRefzbMATHGoogle Scholar
- 2.Bažant ZP, Gattu M, Vorel J (2012) Work conjugacy error in commercial finite-element codes: its magnitude and how to compensate for it. Proc R Soc A Math Phys Eng Sci 468(2146):3047–3058. https://doi.org/10.1098/rspa.2012.0167 MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Chadwick P, Ogden RW (1971a) A theorem of tensor calculus and its application to isotropic elasticity. Arch Ration Mech Anal 44(1):54–68. https://doi.org/10.1007/BF00250828 MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Chadwick P, Ogden RW (1971b) On the definition of elastic moduli. Arch Ration Mech Anal 44(1):41–53. https://doi.org/10.1007/BF00250827 MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Chen YC, Dui G (2004) The derivative of isotropic tensor functions, elastic moduli and stress rate: I. eigenvalue formulation. Math Mech Solids 9(5):493–511. https://doi.org/10.1177/1081286504038672 MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Connolly SJ (2019) Isotropic hyperelasticity in principal stretches: Fortran programs and subroutines. dataset. University of Strathclyde. https://doi.org/10.15129/b1cc7acc-a170-479e-8b26-b74395352b26
- 7.de Souza Neto EA (2004) On general isotropic tensor functions of one tensor. Int J Numer Methods Eng 61(6):880–895. https://doi.org/10.1002/nme.1094 MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Dui G, Wang Z, Ren Q (2007) Explicit formulations of tangent stiffness tensors for isotropic materials. Int J Numer Methods Eng 69(4):665–675. https://doi.org/10.1002/nme.1776 MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Flory PJ (1961) Thermodynamic relations for high elastic materials. Trans Faraday Soc 57:829–838. https://doi.org/10.1039/tf9615700829 MathSciNetCrossRefGoogle Scholar
- 10.Gendy AS, Saleeb AF (2000) Nonlinear material parameter estimation for characterizing hyper elastic large strain models. Comput Mech 25(1):66–77. https://doi.org/10.1007/s004660050016 CrossRefzbMATHGoogle Scholar
- 11.Gent AN (1996) A new constitutive relation for rubber. Rubber Chem Technol 69(1):59–61. https://doi.org/10.5254/1.3538357 MathSciNetCrossRefGoogle Scholar
- 12.Govindjee S (2004) Numerical issues in finite elasticity and viscoelasticity. In: Saccomandi G, Ogden RW (eds) Mechanics and thermomechanics of rubberlike solids. Springer, Vienna, pp 187–232. https://doi.org/10.1007/978-3-7091-2540-3 CrossRefGoogle Scholar
- 13.Hartmann S (2003) Computational aspects of the symmetric eigenvalue problem of second order tensors. Technische Mechanik 23(2–4):283–294Google Scholar
- 14.Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, Hoboken. https://doi.org/10.1023/A:1020843529530 zbMATHGoogle Scholar
- 15.Hossain M, Steinmann P (2013) More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study. J Mech Behav Mater 22(1–2):27–50. https://doi.org/10.1515/jmbm-2012-0007 Google Scholar
- 16.Jeremić B, Cheng Z (2005) Significance of equal principal stretches in computational hyperelasticity. Commun Numer Methods Eng 21(9):477–486. https://doi.org/10.1002/cnm.760 CrossRefzbMATHGoogle Scholar
- 17.Ji W, Waas AM, Bažant ZP (2013) On the importance of work-conjugacy and objective stress rates in finite deformation incremental finite element analysis. J Appl Mech 80(4):041024. https://doi.org/10.1115/1.4007828 CrossRefGoogle Scholar
- 18.Kaliske M, Heinrich G (1999) An extended tube-model for rubber elasticity: statistical-mechanical theory and finite element implementation. Rubber Chem Technol 72(4):602–632. https://doi.org/10.5254/1.3538822 CrossRefGoogle Scholar
- 19.Kaliske M, Rothert H (1997) On the finite element implementation of rubber like materials at finite strains. Eng Comput 14(2):216–232. https://doi.org/10.1108/02644409710166190 CrossRefzbMATHGoogle Scholar
- 20.Kato T (1995) Perturbation theory for linear operators, vol 132. Springer, Berlin. https://doi.org/10.1007/978-3-642-66282-9 zbMATHGoogle Scholar
- 21.Khiêm VN, Itskov M (2016) Analytical network-averaging of the tube model: rubber elasticity. J Mech Phys Solids 95:254–269. https://doi.org/10.1016/j.jmps.2016.05.030 MathSciNetCrossRefGoogle Scholar
- 22.Kopp J (2008) Efficient numerical diagonalization of hermitian 3 x 3 matrices. Int J Mod Phys C 19(03):523–548. https://doi.org/10.1142/S0129183108012303 MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Le Saux V, Marco Y, Bles G, Calloch S, Moyne S, Plessis S, Charrier P (2011) Identification of constitutive model for rubber elasticity from micro-indentation tests on natural rubber and validation by macroscopic tests. Mech Mater 43(12):775–786. https://doi.org/10.1016/j.mechmat.2011.08.015 CrossRefGoogle Scholar
- 24.Liu CH, Hofstetter G, Mang HA (1994) 3D finite element analysis of rubber-like materials at finite strains. Eng Comput 11(2):111–128. https://doi.org/10.1108/02644409410799236 CrossRefGoogle Scholar
- 25.Marckmann G, Verron E (2006) Comparison of hyperelastic models for rubber-like materials. Rubber Chem Technol 79(5):835–858. https://doi.org/10.5254/1.3547969 CrossRefGoogle Scholar
- 26.Miehe C (1993) Computation of isotropic tensor functions. Commun Numer Methods Eng 9(11):889–896. https://doi.org/10.1002/cnm.1640091105 CrossRefzbMATHGoogle Scholar
- 27.Miehe C (1994) Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int J Numer Methods Eng 37(12):1981–2004. https://doi.org/10.1002/nme.1620371202 MathSciNetCrossRefzbMATHGoogle Scholar
- 28.Miehe C, Göktepe S (2005) A micro-macro approach to rubber-like materials. Part II: the micro-sphere model of finite rubber viscoelasticity. J Mech Phys Solids 53(10):2231–2258. https://doi.org/10.1016/j.jmps.2005.04.006 MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Miehe C, Keck J (2000) Superimposed finite elastic-viscoelastic-plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation. J Mech Phys Solids 48(2):323–365. https://doi.org/10.1016/S0022-5096(99)00017-4 CrossRefzbMATHGoogle Scholar
- 30.Miehe C, Göktepe S, Lulei F (2004) A micro-macro approach to rubber-like materials-Part I: the non-affine micro-sphere model of rubber elasticity. J Mech Phys Solids 52(11):2617–2660. https://doi.org/10.1016/j.jmps.2004.03.011 MathSciNetCrossRefzbMATHGoogle Scholar
- 31.Mooney M (1940) A theory of large elastic deformation. J Appl Phys 11(9):582–592. https://doi.org/10.1063/1.1712836 CrossRefzbMATHGoogle Scholar
- 32.Nedjar B, Baaser H, Martin RJ, Neff P (2018) A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes. Comput Mech 62(4):635–654. https://doi.org/10.1007/s00466-017-1518-9 MathSciNetCrossRefzbMATHGoogle Scholar
- 33.Ogden RW (1972) Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc A Math Phys Eng Sci 326(1567):565–584. https://doi.org/10.1098/rspa.1972.0026 CrossRefzbMATHGoogle Scholar
- 34.Ogden RW (1997) Non-linear elastic deformations. Dover Publications, New York. https://doi.org/10.1016/0955-7997(84)90049-3 Google Scholar
- 35.Ogden RW, Saccomandi G, Sgura I (2004) Fitting hyperelastic models to experimental data. Comput Mech 34(6):484–502. https://doi.org/10.1007/s00466-004-0593-y CrossRefzbMATHGoogle Scholar
- 36.Peyraut F, Feng ZQ, He QC, Labed N (2009) Robust numerical analysis of homogeneous and non-homogeneous deformations. Appl Numer Math 59(7):1499–1514. https://doi.org/10.1016/j.apnum.2008.10.002 MathSciNetCrossRefzbMATHGoogle Scholar
- 37.Rivlin RS (1948) Large elastic deformations of isotropic materials. I. fundamental concepts. Philos Trans R Soc A Math Phys Eng Sci 240(822):459–490. https://doi.org/10.1098/rsta.1948.0002 MathSciNetCrossRefzbMATHGoogle Scholar
- 38.Simo JC (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Methods Appl Mech Eng 60(2):153–173. https://doi.org/10.1016/0045-7825(87)90107-1 CrossRefzbMATHGoogle Scholar
- 39.Simo JC (1998) Numerical analysis and simulation of plasticity. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, numerical methods for solids (Part 3), Elsevier Science B.V., pp 183–499, https://doi.org/10.1016/S1570-8659(98)80009-4
- 40.Simo JC, Ortiz M (1985) A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comput Methods Appl Mech Eng 49(2):221–245. https://doi.org/10.1016/0045-7825(85)90061-1 CrossRefzbMATHGoogle Scholar
- 41.Simo JC, Taylor RL (1991) Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms. Comput Methods Appl Mech Eng 85(3):273–310. https://doi.org/10.1016/0045-7825(91)90100-K MathSciNetCrossRefzbMATHGoogle Scholar
- 42.Steinmann P, Hossain M, Possart G (2012) Hyperelastic models for rubber-like materials: consistent tangent operators and suitability for Treloar’s data. Arch Appl Mech 82(9):1183–1217. https://doi.org/10.1007/s00419-012-0610-z CrossRefzbMATHGoogle Scholar
- 43.Tanaka M, Fujikawa M, Balzani D, Schröder J (2014) Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis. Comput Methods Appl Mech Eng 269:454–470. https://doi.org/10.1016/j.cma.2013.11.005 MathSciNetCrossRefzbMATHGoogle Scholar
- 44.Tanaka M, Sasagawa T, Omote R, Fujikawa M, Balzani D, Schröder J (2015) A highly accurate 1st- and 2nd-order differentiation scheme for hyperelastic material models based on hyper-dual numbers. Comput Methods Appl Mech Eng 283:22–45. https://doi.org/10.1016/j.cma.2014.08.020 MathSciNetCrossRefzbMATHGoogle Scholar
- 45.Treloar LRG (1944) Stress–strain data for vulcanized rubber under various types of deformation. Rubber Chem Technol 17(4):813–825. https://doi.org/10.5254/1.3546701 CrossRefGoogle Scholar
- 46.Valanis KC, Landel RF (1967) The strain-energy function of a hyperelastic material in terms of the extension ratios. J Appl Phys 38(7):2997–3002. https://doi.org/10.1063/1.1710039 CrossRefGoogle Scholar
- 47.Vorel J, Bažant ZP (2014) Review of energy conservation errors in finite element softwares caused by using energy-inconsistent objective stress rates. Adv Eng Softw 72:3–7. https://doi.org/10.1016/j.advengsoft.2013.06.005 CrossRefGoogle Scholar
- 48.Yeoh OH (1990) Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber Chem Technol. https://doi.org/10.5254/1.3538289 Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.