A pseudo active kinematic constraint for a biological living soft tissue: An effect of the collagen network

Recent studies in mammalian hearts show that left ventricular wall thickening is an important mechanism for systolic ejection, and that during contraction the cardiac muscle develops significant stresses in the muscular cross-fiber direction. We suggested that the collagen network surrounding the muscular fibers could account for these mechanical behaviors. To test this hypothesis we develop a model for large deformation response of active, incompressible, nonlinear elastic and transversely isotropic living soft tissue (such as cardiac or arteries tissues) in which we include a coupling effect between the connective tissue and the muscular fibers. Then, a three-dimensional finite element formulation including this internal pseudo-active kinematic constraint is derived. Analytical and finite element solutions are in a very good agreement. The numerical results show this wall thickening effect with an order of magnitude compatible with the experimental observations.     

Extremal loading of soft fibrous tissues: multi-scale mechanics and constitutive modeling

In the current contribution, we present a multi-scale constitutive model capturing macroscopic inelastic effects (like stress softening and permanent set) in soft tissues under cyclic loading. Soft biological tissues can be described as a biological composite material. The extracellular matrix is hereby reinforced by collagen fibers which themself are an assembly of collagen fibrils embedded in a proteoglycan (PG) rich matrix. Micro-damage induced by cyclic loading is treated by an interaction scenario between the fibrils and the PGs. At the low strain regime PGs promote sliding between fibrils [1] which leads to the yielding of statistical distributed overlapping segments. The breakage of the PG-bridges is defined by a decreasing PG-density. Due to the accumulated damage of the PG connections at high tissue strains, the strains at the fibril level increases. This finally drives the over-stretching of the fibrils, which is associated with a permanent rupture of the hydrogen bonds inside of the tropocollagen molecules [2]. The so obtained model is in line with recent experimental findings [1, 2] and was additionally validated against experimental data available in literature. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Determination of Damping in Metal Matrix Composites

Damping in metal matrix composites is mainly caused by inelastic matrix deformation induced by the great difference in the mechanical properties of the single constituents of the materials. In this study, the finite-element method in combination with a highly accurate material model is employed to examine the effects of both the fiber volume fraction and the external loading amplitude on the energy dissipation process in an Al/SiC composite under a cyclic mechanical load.     

Modeling of deformation and failure in rubber-toughened polymers – effect of distributed crazing

A continuum mechanical model for rubber-toughened polymers undergoing inelastic deformation solely by distributed crazing is introduced. Scaling relations with regard to microstructural parameters are derived analytically from a simple unit cell model. The constitutive model is calibrated from experimental data for a commercial ABS material and well captures various aspects of its deformation and failure behaviour. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient evaluation of the material response of tissues reinforced by statistically oriented fibres

For several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e. the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: (a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; (b) a Taylor expansion of the fibre constitutive function in the structure tensor; (c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument. Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e. different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation.     

人体动脉瘤生成与破裂的力学分析

在大变形超弹性理论框架下研究了内压、轴向拉伸和扭转联合作用下人体动脉壁的力学响应,应用结构不稳定性理论对动脉瘤生成的可能性进行了解释,应用材料强度理论对动脉瘤破裂的可能性进行了分析．考虑动脉壁中残余应力和平滑肌主动作用的影响,用纤维加强各向异性不可压超弹性复合材料两层厚壁圆筒模型来模拟动脉壁的力学特性．给出了正常和几种非正常状态下动脉壁的变形曲线和应力分布．变形和稳定性分析结果表明该文模型可以模拟正常状态下动脉壁的均匀变形,还可以模拟在动脉壁中弹性蛋白纤维和胶原蛋白纤维强度降低的非正常状态下动脉瘤生成的可能性及动脉瘤的增长．应力和强度分析结果表明该文模型可以模拟当动脉瘤中的最大应力超过管壁的强度时动脉瘤破裂的可能性．     

Experimental and numerical study of the relaxation behavior of facial soft tissue

Challenges in computational simulation of the mechanical behavior of soft tissues and organs for clinical applications are related to the reliability of the models with respect to the anatomy, the mechanical interactions between different tissues, and the non linear (time dependent) force deformation characteristics of soft biological materials. In this paper a 3D finite element model of the face and neck, which has applications in surgical devices optimization and surgery planning, is presented. Bones, muscles, skin, fat, and superficial muscoloaponeurotic system (SMAS) were reconstructed from magnetic resonance images and their shape, constraints and interactions have been modeled according to anatomical, plastic and reconstructive surgery literature. Non linear time dependent constitutive equations are implemented in the numerical model, based on the Rubin-Bodner model. For the present calculations a simplified hyperelastic formulation has been used. The corresponding model parameters were selected according to previous work with mechanical measurements ex vivo on facial soft tissue. For determination of model parameters, in particular the ones corresponding to the time dependent behavior, an instrument for measuring the relaxation behavior of the face tissue in vivo was developed. The experimental set-up is described and results are presented for tests performed on different locations of the face (jaw, mid-face, parotid regions) and neck. The measured “long term” reaction force of the facial soft tissue is compared to numerical results. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Response of Kidney Tissue to Dynamical Loading

In shock-wave lithotripsy – a medical procedure to fragment kidney stones – the patient is subjected to hypersonic waves focused at the kidney stone. Although this procedure is widely applied, the physics behind this medical treatment, in particular the question of how the injuries of the surrounding kidney tissue arise, is still under investigation. Here we contribute to the solution of this problem with large scale numerical simulations of a human kidney under shock-wave loading. For this purpose we developed a complex constitutive model of the bio-mechanical kidney system. Assuming a multiplicative decomposition of the deformation gradient and adopting an internal variable formulation for the inelastic deformation the model is able to handle large deformations, time-effects, rate-sensitivity and material damage. By finite element simulations we study the shock-wave propagation into the kidney tissue and analyze the resulting stress states. Unknown material parameters are adapted and special attention is paid on the bubble expansion within the soft tissue. The numerical simulations predict localized damage in the human kidney within the focal region of the shock waves. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of the deformation characteristics and heat generation in pneumatic aircraft tires Part II. Thermal modeling

A numerical procedure to determine the temperature rise in aircraft tires under free rolling conditions is presented in this article. Energy dissipation from cyclic inelastic deformation is considered the main heat generation source. This modeling considers the deformation process of the tire to be a steady-state problem, where all concurrent cycles are assumed to be the same as the first. The inelastic energy is determined by imposing a phase lag between the strain and the stress fields. The phase lag is assumed to be frequency independent in the range of interest, in keeping with the experimental observations in aircraft tire materials. It is further assumed that the inelastic energy is completely converted into volumetric heat input for a transient thermal conduction analysis. A conduction model is described and results are compared against thermocouple data recorded by Clark and Dodge [1].     

Mathematical analysis of a thermo-visco-plastic model with Bodner–Partom constitutive equations

We study a thermo-mechanical model, where the mechanical model of inelastic deformation due to S.R. Bodner and Y. Partom is coupled with a heat equation. The main result is the existence and uniqueness of the solution to the thermo-visco-plastic model.     

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)     

A Microscopic model for a cell-seeded material

The aim of the present paper is to account for the growth of fiber which is observed in a cell-seeded material stimulated in a bioreactor. For this purpose, the change of mass is considered in the balance laws, and the deformation energy is assumed to be a function of varying mass and the Helmholtz free-energy. Fiber growth at the microscopic level causes a macroscopic change of the material's mechanical properties. The study is a first approach towards a micromechanical model accounting for remodelling in cartilage replacement materials. In so doing, constitutive equations for renewable soft tissues are proposed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micro-sphere-based remodelling formulation for anisotropic biological tissues

Soft biological tissues possess a pronounced composite-type multi-scale structure together with strongly anisotropic mechanical properties. A fibre-like network structure is characteristic for this kind of materials. If the tissue is exposed to mechanical loading, the initially possibly unstructured collagen fibre network tends to reorient with the local dominant stretch direction – it adapts according to the particular loading conditions. In general, biological tissues exhibit changes in mass, also denoted as growth, and internal structure, which is commonly referred to as remodelling. In this regard, an anisotropic micromechanically motivated model that incorporates such time-dependent remodelling effects will be discussed in this contribution. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An anisotropic viscoelastic soft tissue model at large strains

Soft biological tissues represent complex inhomogeneous, and as a rule multiphase materials subjected to large strains under in vivo mechanical conditions. Apart from a number of other structural-related features they are characterized by a ratedependent material behavior which is attributed to fluid-solid interactions as well as intrinsic viscoelastic properties of the solid matrix. The authors propose to model rate-dependent phenomena of the solid phase of soft biological tissues within the context of a thermodynamically consistent phenomenological material approach resulting from an overstress concept. Due to the presence of directed fibrous constituents soft tissues should be considered as anisotropic materials. Therefore, the viscous overstress model has been completed by a transversely isotropic approach. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Simulation of Anisotropy Directions in Soft Tissues and in Particular in Skin

Orientation of collagen fibers and their spatial distribution predefine macroscopic mechanical properties of the soft tissue and in particular its anisotropy directions. In this contribution, we apply two different procedures to automatically generate these directions for a 3D FE-model. The first procedure is based on an analogy with a heat conduction problem. Accordingly, a thermal flux under certain temperature boundary conditions is calculated by the same FE model and is further utilized for the definition of the anisotropy directions. The numerical result shows good agreement with Langer's lines data in human skin. Within the second procedure, the fiber vector field is calculated by the Laplacian smoothing method based on the user defined fiber direction sketches. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Cellular articular cartilage replacement tissues: Modeling and analysis

Long-term studies reveal that mechanical stimulation causes growth and remodeling phenomena within biological tissues. The main aim of this research is to fully understand and control these phenomena. For accomplishing that, two steps are considered: first, we determine a suitable numerical model based on different approaches by a comparative study using experimental validations, and second, investigate the mechanical properties of the tissue specimens after a remodeling process. We start with the first step by choosing a convenient model that mimics the biotissue for running the numerical simulations in the second step. There are different models available that determine the mechanical properties of soft replacement tissues seeded with human chondrocytes in modern medical applications. It is our objective to achieve a common methodology of theory and experiments that allows the determination of the mechanical properties of the native material. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A thermodynamical consistent model for modeling of ionic electroactive polymers (EAPs) within the framework of the Theory of Porous Media

Ionic electroactive polymers are widely used in many engineering fields. These kind of materials can be stimulated to change their shape and size, see [1]. Since, the material under consideration has a complex multiphasic microstructure, such multiphasic materials are best described by a continuum mechanical approach. Thus, the presented model is based on the Theory of Porous Media (TPM), cf. [2]. In this contribution, we consider the Ionic Polymer Metal Composites (IPMCs). Stimulating by an electrical voltage, a structural deformation will be caused. Responsible for this deformation are the mobile ions. The focus of the presented model is to capture this material behavior, e.g. the distribution of the mobile cations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relationship of the ultrastructure and mechanical properties in tendinous collagen,a highly ordered macromolecular composite

A relationship has been found between the stress and deformation in tendon and, from an analysis of the data obtained, it was established that there is a major stress-bearing fiber, the diameter of which increases with increasing age of the animal. The relationship of the results obtained with the structure and properties of tendon as well as with the aging process is stressed. Models of the ultrastructure of collagen fibers in rat tail tendon or in other soft connective tissue were prepared by sealing rigid fibers in a considerably softer elastic matrix. Loss of fiber resistance was achieved by contraction of this matrix. These synthetic composite systems exactly reflect a number of the features of the collagen ultra-structure. Models were also used for studying stresses in the matrix about the fibers as this necessary information could not be obtained by a direct study of the biological tissues. Examination of the reaction between tropocollagen and the connective tissue polysaccharides in the process of collagen fiber formation shows that a mechanism exists through which the fibers may lose their resistance also in vivo.