A microstructurally motivated anisotropic viscoelastic model for soft tissues

In this paper, a microstructurally motivated approach to take into account the anisotropic viscoelastic behaviour of soft biological tissues is proposed. The constitutive model is based on the assumption that this behaviour results from an interaction between collagen fibres and surrounding matrix constituents. Accordingly, a non–linear viscoelastic one–dimensional model for fibres and the nearby ground substance is developed. This model is then generalised to the anisotropic three–dimensional case. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An anisotropic large strain plasticity model for multifilament yarns

Important industrial rubber components like tyres, driving belts and air springs are usually reinforced by twisted polymer cords. A cord itself is a twisted structure of several multifilament yarns. Each yarn is a twisted strand consisting of a large number of polymer filaments. In this work, an anisotropic large strain plasticity model is presented which captures the yarn's filament structure and frictional behaviour. The material model is implemented in Abaqus adopting its user subroutine UMAT and validated by a simulation of a cord's twisting process and the comparison with microscope images. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fibre stability in anisotropic matrix at small strains

As acknowledged in almost all monographs on the fracture of composite materials, one of the major fracture mechanism in unidirectional fibrous composites under uniaxial compression along the reinforcing elements is the stability loss of the material structure (the structural instability). According to this mechanism, theoretical investigations of the fracture along the fibres are reduced to those of the stability loss in the material structure, and the value of external critical forces is accepted as the value of failure forces. At present, numerous theoretical investigations have been carried out in this field with the use of the three-dimensional linearized theory of stability in the framework of the piecewise-homogeneous body model. However, in all the investigations it is assumed that the matrix and the fibre material are isotropic. It is evident that in many cases it is necessary to take into account the anisotropy of the matrix material when investigating the stability loss of fibres. In view of the above, in the framework of the piecewise-homogeneous body model using the three-dimensional linearized theory of stability, the present paper considers the stability loss of the fibre in the anisotropic (transversally isotropic) matrix. The effect of the properties of the matrix material on the critical values of the external loading is examined.Submitted to the 10th International Conference of Mechanics of Composite Materials (Riga, April 20–23, 1998).Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 603–611. September–October, 1997.     

A multi-scale time-dependent constitutive model of soft collagenous tissue

This contribution presents a micro-mechanically motivated, time-dependent constitutive model of soft biological tissues. It considers seperate contributions of the matrix material, collagen fibrils, proteoglycans (PGs) as well as their interactions. It is based on the observations [6], that PG bridges facilitate sliding between fibrils. The initial overlapping lengths of the PG bridges are statistically distributed and decrease due to slippage. A linear-elastic force response of a PG bridge is assumed. Damage of the PG bridges is reversible and decays over time (cf. Gupta et al. [1]). This behaviour is taken into account by a healing model based on the evolution of the overlapping length. The damage of the PG bridges decreases the PG density and in turn increases the fibril contact, leading to fibril stretch. The strain energy function of fibrils is based on the response of single tropocollagen molecules and takes both, an entropic and an energetic regime into account. At higher strains, fibrils can additionally undergo damage, which in contrast to the PG damage is irreversible. The so obtained constitutive model is capable to predict several mechanical phenomena of soft tissues, such as non-linearity, Mullins effect, hysteresis and permanent set. Finally the model is compared against experimental data available in the literature. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational micro-macro transitions at large strains for curvilinear physical directions

Biological tissues such as those involved in the eye, heart, veins or arteries are heterogeneous on one or another spatial scale and can undergo very large elastic strains. Frequently, these tissues are characterized by shell-like structures at the macroscopic scale and the physical material directions follow curvilinear paths. We consider a homogenized macro-continuum formulated in curvilinear convective coordinates with locally attached representative micro-structures. Micro-structures attached to different macroscopic points are assumed to be rotated counterparts according to the curvilinear path of the physical material directions at the macro-scale. The solution of such multi-scale problems according to the computational homogenization scheme [1, 2, 3] would need a different RVE at each macroscopic point. The goal of this paper is to use the same initial RVE at each macroscopic point by generalizing the computational homogenization scheme to a formulation considering different physical spaces at the micro- and macro-scale. The deformation and the reference frame of the micro-structure are assumed to be coupled with the local deformation and the local reference frame at the corresponding point of the macrocontinuum. For a consistent formulation of micro-macro transitions physical reference directions are defined on both scales, where the macroscopic one follows a curvilinear path. To formulate the generalized micro-macro transitions in absolute tensor notation the operations scale-up and scale-down are introduced. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Notes on a novel finite element for anisotropy at large strains

A locking phenomenon can be observed in the case of anisotropic elasticity, due to high stiffness in preferred directions. In this contribution we propose a formulation with the goal to overcome this locking problem. We apply a Simplified Kinematic for the Anisotropic part of the free-energy by means of a constant approximation of the right Cauchy-Green tensor. For the tested boundary value problem the SKA-element performs excellent and behaves extremely robust. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Notes on a novel finite element for anisotropy at large strains

A locking phenomenon can be observed in the case of anisotropic elasticity, due to high stiffness in preferred directions. In this contribution we propose a formulation with the goal to overcome this locking problem. We apply a S implified K inematic for the A nisotropic part of the free-energy by means of a constant approximation of the right Cauchy-Green tensor. For the tested boundary value problem the SKA -element performs excellent and behaves extremely robust. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An L-like model containing very large cardinals

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with a strong form of diamond and a version of square consistent with supercompactness. This generalises a result due to the first author. There are no restrictions in our model on the structure of the class of supercompact cardinals. A. W. Apter’s research was partially supported by PSC-CUNY Grants and CUNY Collaborative Incentive Grants. J. Cummings’s research was partially supported by NSF Grant DMS-0400982.     

An anisotropic continuum model for traffic assignment in mixed transportation networks

This work deals with a two-dimensional continuum model for the problem of congested traffic assignment in an urban transportation system consisting of a set of freeways superimposed over a dense street network. The formulation leads to a system of non-linear differential equations whose unknowns are given by the travel times from arbitrary points of the network to the corresponding destinations. The governing equations are appropriately solved by means of the Finite Element Method. Then, traffic flow on every link of the network can be obtained. Numerical examples are given in order to demonstrate the efficiency of the developed model.     

An experimentally validated rod model for soft continuum robots

Soft robots are bio-inspired, highly deformable robots with the ability to interact with workpieces in a manner that complements their hard robot counterparts. To develop practical applications and reproducible designs of soft robots, new models are necessary to describe their kinematics and dynamics. In the present work, we describe experimental and numerical investigations of a popular pneumatically-actuated soft continuum arm. These works enable us to derive constitutive relations and develop a rod model for large deformations of the arm that faithfully describes its bending behavior. We show how the resulting non-classical constitutive relation can be defined either through experiments or through quasi-static finite element simulations. With the help of this relation, the resulting rod model can be used to study the dynamics of the soft robot arm in a fast and tractable manner. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters

We present a simple open-source semi-intrusive computational method to propagate uncertainties through hyperelastic models of soft tissues. The proposed method is up to two orders of magnitude faster than the standard Monte Carlo method. The material model of interest can be altered by adjusting few lines of (FEniCS) code. The method is able to (1) provide the user with statistical confidence intervals on quantities of practical interest, such as the displacement of a tumour or target site in an organ; (2) quantify the sensitivity of the response of the organ to the associated parameters of the material model. We exercise the approach on the determination of a confidence interval on the motion of a target in the brain. We also show that for the boundary conditions under consideration five parameters of the Ogden–Holzapfel-like model have negligible influence on the displacement of the target zone compared to the three most influential parameters. The benchmark problems and all associated data are made available as supplementary material.     

Thermodynamic analysis of the equations of a linear viscoelastic anisotropic solid

Constraints on the coefficients of the equations of state of anisotropic linear viscoelasticity are obtained using the positive definiteness of two quadratic forms—the potential energy and the energy dissipation rate.Mekhanika Polimerov, Vol. 3, No. 2, pp. 227–235, 1967     

Instability prevention at the modeling of large elasto-plastic strains under cyclic loading

The cyclic instability phenomenon is investigated at the modeling of large elasto-plastic strains. The possible causes of the cyclic instability and conditions ensuring cyclical stability of elasto-plastic models are analyzed for the case of large strains. Among the possible causes of the cyclic instability the following are considered: the method of strain decomposition on elastic and plastic parts; the constitutive law for the elastic deformation (hypo- and hyper-elasticity); the constitutive equation for the plastic deformation; the constitutive relation for the plastic spin; kinematic hardening law, in particular, the type of the objective rate in the generalized Prager's law. Predictions of 50 various models of the elasto-plastic material have been compared in order to find the causes of the cyclic instability. Two test problems are considered: cyclic simple shear, combined cyclic simple shear and tension-compression. Results of numerical experiments for the various material models are presented and discussed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the deformability of anisotropic viscoelastic bodies in the presence of thermodiffusion

We give the results of theoretical studies directed toward the development of model pictures of the strain process of anisotropic viscoelastic multicomponent solid solutions taking account of the processes of heat conduction and diffusion of matter. We write a complete system of equations of the model and boundary conditions. We analyze the stressed state in a viscoelastic cylinder in diffusion saturation. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 78–89.     

An anisotropic phase-field model for transversely isotropic barium titanate with bounded moduli

This contribution presents a phase-field model for transversely isotropic barium titanate, which allows for the adjustment of the full set of anisotropic material parameters. It is a direct extension of the work by Schrade et al. [1] who proposed a phase-field model for ferroelectrics in the framework of invariant theory. In the present contribution, the loss of positive definiteness is avoided by formulating energetic terms that provide upper and lower bounds for all material moduli involved. We show the characteristics of the formulation by a set of numerical examples in two and three dimensions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of the viscoelastic state of an anisotropic plate with rigid inclusions

A method for the determination of the viscoelastic state of multiconnected anisotropic plates with absolutely rigid inclusions is proposed. As an example, we consider the solution of the problem of viscoelasticity for a plate with an elliptic inclusion which, in a special case, turns into a linear one. For the case of tension of the plate, results of numerical investigations of the stress state depending on geometric parameters of the inclusion and on the time of application of load are described.     

Direct connection of rheological elements at large strains: Application to multiplicative viscoplasticity

In the field of nonlinear continuum mechanics, rheological models are often used to exemplify the structure of complex material models at large strains. For this purpose, different rheological elements are combined in series and parallel connections. Ihlemann [1] developed an innovative concept, which enables the direct connection of rheological elements within the framework of multiplicative decomposition of the deformation gradient. In the contribution at hand, this approach is applied to multiplicative viscoplasticity. Towards this end, the relations for parallel and series connections are introduced and several individual material models, i.e. the rheological elements, are defined. By analytical and numerical evaluation of the connection relations, a viscoplastic material model from the literature is reproduced. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.