Extension of the Sussman-Bathe spline-based hyperelastic model to incompressible transversely isotropic materials

[EN] In this paper we extend the Sussman¿Bathe spline-based hyperelastic isotropic model to predict the behavior of transversely isotropic isochoric materials. The model is based on an uncoupled decomposition of the stored energy function and a generalization of the inversion formula used by Sussman and Bathe. The present extension introduces some approximations that, in all studied cases, do not yield relevant deviations from the experimental data. The isotropic model results in a particular case of the present formulation. Several possibilities of user-prescribed experimental data are addressed. The model is used to predict experiments of calendered rubber and of biological tissues.We acknowledge the anonymous reviewer #2 of the manuscript for providing a more elegant and accurate version of the inversion formula, Eq. (40), than the one given in the original submission. Financial support for this work has been given by the Direccion General de Proyectos de Investigacion of the Ministerio de Ciencia e Innovacion and the Ministerio de Economia y Competitividad of Spain under grants DPI2008-05423 and DPI2011-26635Latorre, M.; Montáns, FJ. (2013). Extension of the Sussman-Bathe spline-based hyperelastic model to incompressible transversely isotropic materials. Computers & Structures. 122:13-26. https://doi.org/10.1016/j.compstruc.2013.01.018132612

Stress and strain mapping tensors and general work-conjugacy in large strain continuum mechanics

[EN] In this paper we show that mapping tensors may be constructed to transform any arbitrary strain measure in any other strain measure. We present the mapping tensors for many usual strain measures in the Seth-Hill family and also for general, user-defined ones. These mapping tensors may also be used to transform their work-conjugate stress measures. These transformations are merely geometric transformations obtained from the deformation gradient and, hence, are valid regardless of any constitutive equation employed for the solid. Then, advantage of this fact may be taken in order to simplify the form of constitutive equations and their numerical implementation and thereafter, perform the proper geometric mappings to convert the results-stresses, strains and constitutive tangents- to usually employed measures and to user-selectable ones for input and output We herein provide the necessary transformations. Examples are the transformation of small strains formulations and algorithms to large deformations using logarithmic strains. (C) 2015 Elsevier Inc. All rights reserved.Partial financial support for this work has been given by grant DPI2011-26635 from the Direccion General de Proyectos de Investigation of the Ministerio de Economia y Competitividad of SpainLatorre, M.; Montáns, FJ. (2016). Stress and strain mapping tensors and general work-conjugacy in large strain continuum mechanics. Applied Mathematical Modelling. 40(5-6):3938-3950. https://doi.org/10.1016/j.apm.2015.10.04539383950405-

Fully anisotropic finite strain viscoelasticity based on a reverse multiplicative decomposition and logarithmic strains

[EN] In this paper we present a novel formulation for phenomenological anisotropic finite visco-hyperelasticity. The formulation is based on a multiplicative decomposition of the equilibrated deformation gradient into nonequilibrated elastic and viscous contributions. The proposal in this paper is a decomposition reversed respect to that from Sidoroff allowing for anisotropic viscous contributions. Independent anisotropic stored energies are employed for equilibrated and non-equilibrated parts. The formulation uses logarithmic strain measures in order to be teamed with spline-based hyperelasticity. Some examples compare the results with formulations that use the Sidoroff decomposition and also show the enhanced capabilities of the present model. (C) 2015 Elsevier Ltd. All rights reserved.Partial financial support for this work has been given by grant DPI2011-26635 from the Direccion General de Proyectos de Investigacion of the Ministerio de Economia y Competitividad of SpainLatorre, M.; Montáns, FJ. (2016). Fully anisotropic finite strain viscoelasticity based on a reverse multiplicative decomposition and logarithmic strains. Computers & Structures. 163:56-70. https://doi.org/10.1016/j.compstruc.2015.09.001567016

A new class of plastic flow evolution equations for anisotropic multiplicative elastoplasticity based on the notion of a corrector elastic strain rate

[EN] We herein present a new continuum theory for both isotropic and anisotropic elastoplasticity at large strains. The new framework has the following properties: (1) It is valid for non-moderate large strains, (2) it is valid for both elastic and plastic anisotropy, (3) its description in rate form is parallel to that of the infinitesimal formulation, (4) it is compatible with the multiplicative decomposition, (5) results in a similar framework in any stress-strain work-conjugate pair, (6) it is consistent with the principle of maximum plastic dissipation and (7) does not impose any restriction on the plastic spin, which must be given as an independent constitutive equation. Furthermore, when formulated using logarithmic strain measures in the intermediate configuration: (8) it may be easily integrated using a classical backward-Euler rule resulting in an additive update. All these properties are obtained simply by considering a plastic evolution in terms of a corrector rate of the proper elastic strain. This new continuum theory is a natural framework for elastoplasticity of both metals and soft materials and solves the (so-coined by Simo) rate issue.Financial support for this work has been given by grants DPI2011-26635 and DPI2015-69801-R from the Direction General de Proyectos de Investigation of the Ministerio de Economia y Competitividad of Spain. FJM also acknowledges the support of the Department of Mechanical and Aerospace Engineering of University of Florida during the sabbatical period in which part this work was developed and that of the Ministerio de Education, Cultura y Deporte of Spain for the financial support for the sabbatical stay under grant PRX15/00065Latorre, M.; Montáns, FJ. (2018). A new class of plastic flow evolution equations for anisotropic multiplicative elastoplasticity based on the notion of a corrector elastic strain rate. Applied Mathematical Modelling. 55:716-740. https://doi.org/10.1016/j.apm.2017.11.0037167405

Bi-modulus materials consistent with a stored energy function: Theory and numerical implementation

[EN] Many materials present different behavior in tension and compression. Within the infinitesimal isotropic theory, the widely used approach based on the Ambartsumyan theory presents only three independent constants to preserve symmetry of the elasticity tensor. The reported finite element implementation of this and similar theories are complex and often lack the convergence properties expected for a bi-linear material. In this work we address the problem through a hyperelastic approach, obtaining a simple and consistent framework which retains the four independent constants and yields the expected convergence characteristics of a bi-linear material. The Ambartsumyan model is obtained as a particular case within this framework.Partial financial support for this work has been given by grant PGC-2018-097257-B-C32 from the Ministerio de Ciencia, Innovacion y Universidades of Spain.The ADINA license for the examples has been a courtesy of ADINA R&D to UPMLatorre, M.; Montáns, FJ. (2020). Bi-modulus materials consistent with a stored energy function: Theory and numerical implementation. Computers & Structures. 229:1-19. https://doi.org/10.1016/j.compstruc.2019.10617611922

On the interpretation of the logarithmic strain tensor in an arbitrary system of representation

[EN] Logarithmic strains are increasingly used in constitutive modelling because of their advantageous properties. In this paper we study the physical interpretation of the components of the logarithmic strain tensor in any arbitrary system of representation, which is crucial in formulating meaningful constitutive models. We use the path-independence property of total logarithmic strains to propose different fictitious paths which can be interpreted as a sum of infinitesimal engineering strain tensors. We show that the angular (engineering) distortion measure is arguably not a good measure of shear and instead we propose area distortions which are an exact interpretation of the shear terms both for engineering and for logarithmic strains. This new interpretation clearly explains the maximum obtained in some constitutive models for the simple shear load case.Partial financial support for this research is given by the Direccion General de Investigacion of the Ministerio de Economia y Competitividad of Spain under Grant DPI2011-26635 of the Plan Nacional de Investigacion.Latorre, M.; Montáns, FJ. (2014). On the interpretation of the logarithmic strain tensor in an arbitrary system of representation. International Journal of Solids and Structures. 51(7-8):1507-1515. https://doi.org/10.1016/j.ijsolstr.2013.12.04115071515517-

Experimental data reduction for hyperelasticity

[EN] WYPiWYG hyperelasticity is a data-driven, model-free computational procedure for finite element analysis of soft materials. The procedure does not assume the shape of the stored energy function and does not employ material parameters, predicting accurately any smooth prescribed behavior from a complete set of experimental tests. However, fuzzy experimental data may yield useless highly oscillatory, unstable stored energy functions, and classical curvature smoothing frequently gives unsatisfactory results. Aside, the possibility of having experimental data from different specimens for the same test was not considered in previous procedures. In this work we present a novel technique based on spline regression and smoothing penalization using stability conditions. In general, this procedure reduces noisy experimental data or data from multiple specimens for ulterior determination of the stored energy. The procedure only needs the solution of a linear system of equations. Instead of classical curvature-based smoothing, we employ a novel stability-based smoothing, determining for each branch of the uniaxial stress-strain curve the most restrictive stability condition during uniaxial and equibiaxial tests. The resulting stored energy functions are smooth and stable. The procedure has little sensitivity to the number of spline segments or to the choice of the penalization parameter, which are computed automatically.Partial financial support for this work has been given by grant DPI2015-69801-R from the Direccion General de Proyectos de Investigacion of the Ministerio de Economia y Competitividad of SpainLatorre, M.; Montáns, FJ. (2020). Experimental data reduction for hyperelasticity. Computers & Structures. 232:1-16. https://doi.org/10.1016/j.compstruc.2018.02.01111623

WYPiWYG hyperelasticity without inversion formula: Application to passive ventricular myocardium

[EN] WYPiWYG hyperelasticity is a family of computational procedures for determining the stored energy density of soft materials. Instead of assuming the global analytical shape of these functions (the model), they are computed solving numerically the differential equations of a complete set of experimental tests that uniquely define the material behavior. WYPiWYG hyperelasticity traditionally uses an inversion formula to solve the differential equations, which limits the possible types of tests employed in the procedure. In this work we introduce a new method that does not need an inversion formula and that can be used with any type of tests. We apply the new procedure to determine the stored energy function of passive ventricular myocardium from five experimental simple shear tests.Partial financial support for this work has been given by grant DPI2015-69801-R from the Direccion General de Proyectos de Investigation of the Ministerio de Economia y Competitividad of Spain. FJM also acknowledges the support of the Department of Mechanical and Aerospace Engineering of University of Florida during the sabbatical period in which this paper was finished and that of Ministerio de Educacion, Cultura y Deporte of Spain for the financial support for that stay under grant PRX15/00065Latorre, M.; Montáns, FJ. (2017). WYPiWYG hyperelasticity without inversion formula: Application to passive ventricular myocardium. Computers & Structures. 185:47-58. https://doi.org/10.1016/j.compstruc.2017.03.001475818

Strain-Level Dependent Nonequilibrium Anisotropic Viscoelasticity: Application to the Abdominal Muscle

[EN] Soft connective tissues sustain large strains of viscoelastic nature. The rate-independent component is frequently modeled by means of anisotropic hyperelastic models. The rate-dependent component is usually modeled through linear rheological models or quasilinear viscoelastic (QLV) models. These viscoelastic models are unable, in general, to capture the strain-level dependency of the viscoelastic properties present in many viscoelastic tissues. In linear viscoelastic models, strain-level dependency is frequently accounted for by including the dependence of multipliers of Prony series on strains through additional evolution laws, but the determination of the material parameters is a difficult task and the obtained accuracy is usually not sufficient. In this work, we introduce a model for fully non-linear viscoelasticity in which the instantaneous and quasi-static behaviors are exactly captured and the relaxation curves are predicted to a high accuracy. The model is based on a fully nonlinear standard rheological model and does not necessitate optimization algorithms to obtain material parameters. Furthermore, in contrast to most models used in modeling the viscoelastic behavior of soft tissues, it is valid for the large deviations from thermodynamic equilibrium typically observed in soft tissuesSecretaria de Estado de Investigacion, Desarrollo e Innovacion (DPI2015-69801-R)Latorre, M.; Montáns, FJ. (2017). Strain-Level Dependent Nonequilibrium Anisotropic Viscoelasticity: Application to the Abdominal Muscle. Journal of Biomechanical Engineering. 139(10):1-9. https://doi.org/10.1115/1.4037405191391

On the tension-compression switch of the Gasser-Ogden-Holzapfel model: Analysis and a new pre-integrated proposal

[EN] Many biological soft tissues are structurally composed of a mostly isotropic matrix (elastin) and fibers (collagen). These fibers are not perfectly aligned but dispersed around some referential, preferred directions. In order to account for the dispersion of the fibers, a probability distribution is assumed. The Generalized Structure Tensor (GST) models perform a pre-integration of the distribution in order to achieve improved computational efficiency. The best known model of this kind is the Gasser-Ogden-Holzapfel (GOH) model. However, in these models no singular treatment of fibers is made. Whenever they suffer compression it is usual to consider that fibers should not contribute to the overall stiffness. At this point, a switch criterion is employed. This switch criterion is important because it changes the model predictions and may also result in unphysical stress predictions or strain ranges at which no compatible equilibrium solution is found. We perform an analysis of different tension-compression switch criteria from the literature for the GOH model and show relevant physical and computational drawbacks when using these criteria. In order to overcome these drawbacks, we make a new proposal which yields continuous stress solutions. In our proposal, pre-integrated expressions given in terms of the usual set of invariants take into account an average amount of fibers working either in tension or in compression for a given deformation gradient and fiber family. Two distinct switches naturally emerge from our procedure. Furthermore, we keep the appealing GST pre-integrated approach for any proposed stored energy, including that of the GOH model. (C) 2015 Elsevier Ltd. All rights reserved.Partial financial support for this work has been given by grant DPI2011-26635 from the Direccion General de Proyectos de Investigacion of the Ministerio de Economia y Competitividad of SpainLatorre, M.; Montáns, FJ. (2016). On the tension-compression switch of the Gasser-Ogden-Holzapfel model: Analysis and a new pre-integrated proposal. Journal of the Mechanical Behavior of Biomedical Materials. 57:175-189. https://doi.org/10.1016/j.jmbbm.2015.11.0181751895