A generalized strain approach to anisotropic elasticity

This work proposes a generalized Lagrangian strain function $$f_\alpha$$ f α (that depends on modified stretches) and a volumetric strain function $$g_\alpha$$ g α (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions $$f_\alpha$$ f α and $$g_\alpha$$ g α to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which is attractive, in aiding experiment design and the construction of specific forms of the strain energy. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy function. The resulting constitutive equations can be easily converted, to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. Implementation of the constitutive equations in Finite Element software is discussed. The suggested crude specific strain function forms are able to fit the theory well with experimental data and managed to predict several sets of experimental data.

A Generalized Strain Approach to Anisotropic Elasticity

This work proposes a generalized Lagrangian strain function fα (that depends on modified stretches) and a volumetric strain function gα (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions fα and gα to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which is attractive, in aiding experiment design and the construction of specific forms of the strain energy. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy function. The resulting constitutive equations can be easily converted, to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. Implementation of the constitutive equations in Finite Element software is discussed. The suggested crude specific strain function forms are able to fit the theory well with experimental data and managed to predict several sets of experimental data.

Proposal and validation of polyconvex strain-energy function for biological soft tissues

BACKGROUND: Mechanical simulations for biological tissues are effective technology for development of medical equipment, because it can be used to evaluate mechanical influences on the tissues. For such simulations, mechanical properties of biological tissues are required. For most biological soft tissues, stress tends to increase monotonically as strain increases. OBJECTIVE: Proposal of a strain-energy function that can guarantee monotonically increasing trend of biological soft tissue stress-strain relationships and applicability confirmation of the proposed function for biological soft tissues. METHOD: Based on convexity of invariants, a polyconvex strain-energy function that can reproduce monotonically increasing trend was derived. In addition, to confirm its applicability, curve-fitting of the function to stress-strain relationships of several biological soft tissues was performed. RESULTS: A function depending on the first invariant alone was derived. The derived function does not provide such inappropriate negative stress in the tensile region provided by several conventional strain-energy functions. CONCLUSIONS: The derived function can reproduce the monotonically increasing trend and is proposed as an appropriate function for biological soft tissues. In addition, as is well-known for functions depending the first invariant alone, uniaxial-compression and equibiaxial-tension of several biological soft tissues can be approximated by curve-fitting to uniaxial-tension alone using the proposed function.

Specialized Strain Energy Functions for Modeling the Contribution of the Collagen Network (Waniso) to the Deformation of Soft Tissues

A popular framework in continuum mechanics modeling of soft tissues is the use of an additive split of the total strain energy function (W) into the contribution of the isotropic matrix (Wiso) and the anisotropic collagen fiber networks (Waniso): W = Wiso + Waniso. This paper presents specialized strain energy functions for the Waniso part of this additive split, in the form of Waniso(I4) or Waniso(I4, I6) for one or two fiber families, respectively, accounting for the deformation and contribution of the collagen fibers’ network. The models have their origins in the statistical mechanics treatment of chains network based on a non-Gaussian, a Gaussian, and a modified Gaussian approach. The models are applied to extant experimental stress-stretch data, across multi-scales from a single collagen molecule to the network ensemble, demonstrating an excellent agreement. Due to the direct physical structural basis of the model parameters and therefore their objectivity and uniqueness, these models are proposed as advantageous options next to the existing phenomenological continuum-based strain energy functions in the literature. In addition, and while not exploited in this paper, since the model parameters are inherent structural properties of the collagen molecular chains, they may be established a priori via imaging or molecular techniques. Therefore, the proposed models allow the important possibility of precluding the need for destructive mechanical tests and calibration a posteriori, instead of paving the way for predicting the mechanical behavior of the collagen network from pre-established structural parameters. These features render the proposed models as attractive choices for application in continuum-based modeling of collagenous soft tissues.

Numerical and Experimental Investigation of Oil Palm Shell Reinforced Rubber Composites

This paper presents a pioneering effort to ascertain the suitability of hyperelastic modelling in simulating the stress–strain response of oil palm shell reinforced rubber (ROPS) composites. ROPS composites with different oil palm shell contents (0%, 5%, 10% and 20% by volume) were cast in the laboratory for the experimental investigation. ROPS specimens with circular, square, hexagon, and octagon shapes (loading surface) were considered to evaluate the accuracy of finite element simulation considering the shape effect of composites. Strain-controlled (compressive) tests with ε ≈ 50% at 0.8 Hz frequency were conducted in the laboratory and the test data obtained was used as input to simulate material coefficients corresponding to the strain energy functions chosen. Five different strain energy functions were selected and utilized for the hyperelastic modelling in this study using finite element approach. The shape effect was then used to ascertain any variation in the simulation outcomes and to discuss the effect of shape on the behaviour of ROPS composites in comparison to existing literature. The numerical predictions using the Yeoh model (error ≤ 2.7% for circular shaped ROPS) were found to perform best in comparison with the experimental results, thus a more stable and suitable hyperelastic model to this end. The Marlow (error ≤ 4.6% for circular shaped ROPS) and Arruda Boyce (error ≤ 4.7% for circular shaped ROPS) models were amongst the next alternatives to perform better. Even with the other shapes considered in this study, Yeoh, followed by the Marlow function, were more appropriate models. The shape effect was then studied with particular emphasis on comparing and assessing them with that observed in the literature. To this end, adopting the Yeoh function in the finite element model is the ideal approach to estimate the stress–strain response of ROPS composites.

Finite Bending and Straightening of Hyperelastic Materials: Analytical Solution and FEM

In this research, an incompressible, isotropic, nonlinear elastic rectangular block and a circular cylindrical sector are studied under bending and straightening moments, respectively. Analytical approaches are presented on implementing of the left Cauchy–Green tensor and Cauchy stresses. In addition, finite element analysis of both problems is carried out using UHYPER user-defined subroutine in ABAQUS to verify the analytical methods. Four different invariant-based strain energy functions, including neo-Hookean, Mooney–Rivlin, Arruda–Boyce, and recently proposed polynomial Exp-Exp models, are examined, and the results are compared. Material parameters of silicon rubber for the strain energy functions are identified by applying an optimization procedure. Finite element method results confirmed the analytical approach with great compatibility. Results showed that the length of the unbent beam does not affect the stress. Likewise, the initial angle of curved structure does not affect the unbending moment and stresses. Moreover, the Exp-Exp model had a slightly different result rather than other strain energies, which means that this model is more conservative than its counterparts. Furthermore, the Exp-Exp strain energy function is calibrated for tissue-like phantom and is compared with experimental data.

A CONSTITUTIVE MODEL FOR BOTH LOW AND HIGH STRAIN NONLINEARITIES IN HIGHLY FILLED ELASTOMERS AND IMPLEMENTATION WITH USER-DEFINED MATERIAL SUBROUTINES IN ABAQUS

ABSTRACT Strain energy functions (SEFs) are used to model the hyperelastic behavior of rubberlike materials. In tension, the stress–strain response of these materials often exhibits three characteristics: (i) a decreasing modulus at low strains (<20%), (ii) a constant modulus at intermediate strains, and (iii) an increasing modulus at high strains (>200%). Fitting an SEF that works in each regime is challenging when multiple or nonhomogeneous deformation modes are considered. The difficulty increases with highly filled elastomers because the small strain nonlinearity increases and finite-extensibility occurs at lower strains. One can compromise by fitting an SEF to a limited range of strain, but this is not always appropriate. For example, rubber seals in oilfield packers can exhibit low global strains but high localized strains. The Davies–De–Thomas (DDT) SEF is a good candidate for modeling such materials. Additional improvements will be shown by combining concepts from the DDT and Yeoh SEFs to construct a more versatile SEF. The SEF is implemented with user-defined material subroutines in Abaqus/Standard (UHYPER) and Abaqus/Explicit (VUMAT) for a three-dimensional general strain problem, and an approach to overcome a mathematically indeterminate stress condition in the unstrained state is derived. The complete UHYPER and VUMAT subroutines are also presented.

Molecular chain networks and strain energy functions in rubber elasticity

A relatively simple molecular–statistical model for rubber elasticity, similar to the Wang–Guth and the Arruda–Boyce (AB) models, is presented. Discussion of some approximate inverse Langevin functions leads to the selection of a new one as the most attractive balance between accuracy and simplicity. Use of this approximation with the AB model, in particular, yields the logarithmic strain energy introduced by Gent that exhibits limiting chain extensibility. A rather unusual facet of the relationship between the three strain energies is that the new one is the mean of the other two. This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.