基于仿真和智能算法骨骼肌超弹性本构参数的反演方法研究

从事高强度的体力工作者经常会发生肌肉软组织的损伤, 因此对骨骼肌的变形特性和应力分布的研究受到了越来越多的重视. 获取正确的本构参数对于生物软组织的力学行为的研究至关重要, 而本构参数的确定本质上是一个逆过程, 具有很大的挑战性. 本文分别采用K近邻(K-nearest neighbor, KNN)模型和支持向量机回归(support vector machine regression, SVR)模型并结合非线性有限元仿真, 提出了两种确定骨骼肌本构参数的反演方法. 首先建立了骨骼肌压缩的有限元模型, 对其压缩条件下的变形特性进行了有限元仿真, 得到了相应的变形特性及应力分布规律, 同时也建立了骨骼肌组织的名义应力和主伸长之间非线性关系的数据集. 其次, 分别利用KNN模型和SVR模型搭建了针对骨骼肌组织进行本构参数反演的机器学习智能算法, 对相应的数据集进行训练, 结合单轴压缩实验的实验数据预测了材料的本构参数. 最后, 对分别基于KNN模型和SVR模型对骨骼肌超弹性本构参数的误差结果进行了分析, 通过引入相关系数$R$和决定系数$R^{2}$对采用两种反演方法的有效性进行数值上的验证. 结果表明, 利用KNN模型和SVR模型结合有限元仿真是确定骨骼肌超弹性本构参数的有效、准确的方法, 该方法也可进一步推广到其他类型的非线性软组织的本构参数反演.      

Prediction of the softening and damage effects with permanent set in fibrous biological materials

Deformation induced softening is an inelastic phenomenon frequently accompanying mechanical response of soft biological tissues. Inelastic phenomena which occur in mechanical testing of biological tissues are very likely to be associated with alterations in the internal structure of these materials.In this study, a novel structural constitutive model is formulated to describe the inelastic effects in soft biological tissues such as Mullins type behavior, damage and permanent set as a result of residual strains after unloading. Anisotropic softening is considered by evolution of internal variables governing the anisotropic properties of the material. We consider two weight factors w_i (softening) and s_k (discontinuous damage) as internal variables characterizing the structural state of the material. Numerical simulations of several soft tissues are used to demonstrate the performance of the model in reproducing the inelastic behavior of soft biological tissues.     

Surface wrinkling of mucosa induced by volumetric growth: Theory, simulation and experiment

Mechanics of living tissues focusing on the relationships between growth, morphology and function is not only of theoretical interest but can also be useful for diagnosis of certain diseases. In this paper, we model the surface wrinkling morphology of mucosa, the moist tissue that commonly lines organs and cavities throughout the body, induced by either physiological or pathological volumetric growth. A theoretical framework of finite deformation is adopted to analyze the deformation of a cylindrical cavity covered by mucosal and submucosal layers. It is shown that compressive residual stresses induced by the confined growth of mucosa can destabilize the tissue into various surface wrinkling patterns. A linear stability analysis of the critical condition and characteristic buckling patterns indicates that the wrinkling mode is sensitive to the thicknesses of the mucosal and submucosal layers, as well as the properties of the tissues. The thinner the mucosal layer and the lower its elastic modulus, the shorter the buckling wavelength. A series of finite element simulations are performed to validate the theoretical predictions and to study local wrinkling or non-uniform patterns associated with inhomogeneous growth. Our postbuckling analysis shows that the surface pattern may evolve towards a period-doubling morphology due to continuous growth of mucosa or submucosa beyond the critical state. Finally, the theoretical predictions and numerical simulations are compared to experimental observations.     

Experimental and numerical investigations of the deformation of soft materials under tangential loading

The surface ‘tensile test’, in which tangential loads are applied through surface mounted adhesive tapes, is a viable method for the assessment of mechanical properties of soft materials, particularly biological soft materials in vivo. In the present work the deformation pattern and force–displacement relationship in the surface tensile test were experimentally investigated using surface displacement analysis (SDA) and numerically simulated using finite element modelling. The experimental and FE results showed close agreement using silicone rubber as a model material. The force–displacement relationship was found to be dependent on the tape separations. SDA measurements and FE simulation showed that the displacement and strain fields were not uniform and the distribution pattern varies with tape separation. A combined experimental–numerical approach to inversely extract material properties using multiple tests with different length scales is proposed and assessed using a model material.     

生长对超弹性球壳变形和稳定性的影响

应用连续介质力学有限变形理论建立受内压作用不可压超弹性球壳大变形问题的力学模型, 且运用基于变形梯度张量极分解的弹性体积生长理论分析生长对不可压超弹性球壳变形和稳定性的影响. 通过对球壳变形与内压关系式的数值计算得到不同生长条件下球壳的变形曲线和应力分布曲线及由生长引起的残余变形和残余应力分布. 计算结果表明生长对球壳变形特性有明显的影响, 生长影响球壳可产生不稳定变形的临界壁厚和临界内压, 且在某些情况下可改变球壳的稳定性. 生长在球壳中可产生一定的残余变形和残余应力, 对球壳中的应力分布有一定的影响. 另外当生长的程度足够大时, 即便没有外力作用,球壳仅在生长引起的残余应力作用下也可产生不稳定变形.      

Constitutive modeling of isotropic hyperelastic materials in an exponential framework using a self-contained approach

In this paper, an exponential framework for strain energy density functions of elastomers and soft biological tissues is proposed. Based on this framework and using a self-contained approach that is different from a guesswork or combination viewpoint, a set strain energy density functions in terms of the first and second strain invariants is rebuilt. Among the constructed options for strain energy density, a new exponential and mathematically justified model is examined. This model benefits from the existence of second strain invariant, simplicity, stability of parameters, and the state of being accurate. This model can capture strain softening, strain hardening and is able to differentiate between various deformation-state dependent responses of elastomers and soft tissues undergoing finite deformation. The model has two material parameters and the mathematical formulation is simple to render the possibility of numerical implementations. In order to investigate the appropriateness of the proposed model in comparison to other hyperelastic models, several experimental data for incompressible isotropic materials (elastomers) such as VHB 4905 (polyacrylate rubber), two various silicone rubbers, synthetic rubber neoprene, two different natural rubbers, b186 rubber (a carbon black-filled rubber), Yeoh vulcanizate rubber, and finally porcine liver tissue (a very soft biological tissue) are examined. The results demonstrate that the proposed model provides an acceptable prediction of the behavior of elastomers and soft tissues under large deformation for different applied loading states.     

Biaxial Mechanical Evaluation of Planar Biological Materials

A fundamental goal in constitutive modeling is to predict the mechanical behavior of a material under a generalized loading state. To achieve this goal, rigorous experimentation involving all relevant deformations is necessary to obtain both the form and material constants of a strain-energy density function. For both natural biological tissues and tissue-derived soft biomaterials, there exist many physiological, surgical, and medical device applications where rigorous constitutive models are required. Since biological tissues are generally considered incompressible, planar biaxial testing allows for a two-dimensional stress-state that can be used to characterize fully their mechanical properties. Application of biaxial testing to biological tissues initially developed as an extension of the techniques developed for the investigation of rubber elasticity [43, 57]. However, whereas for rubber-like materials the continuum scale is that of large polymer molecules, it is at the fiber-level (∼1 μm) for soft biological tissues. This is underscored by the fact that the fibers that comprise biological tissues exhibit finite nonlinear stress-strain responses and undergo large strains and rotations, which together induce complex mechanical behaviors not easily accounted for in classic constitutive models. Accounting for these behaviors by careful experimental evaluation and formulation of a constitutive model continues to be a challenging area in biomechanics. The focus of this paper is to describe a history of the application of biaxial testing techniques to soft planar tissues, their relation to relevant modern biomechanical constitutive theories, and important future trends. This revised version was published online in July 2006 with corrections to the Cover Date.     

近似不可压软材料动力分析的完全拉格朗日物质点法

物质点法(MPM)在模拟非线性动力问题时具有很好的效果,其已被广泛应用于许多大变形动力问题的分析中.然而传统的MPM在模拟不可压或近似不可压材料的动力学行为时会产生体积自锁,极大地影响模拟精度和收敛性.本文针对近似不可压软材料的大变形动力学行为,提出一种混合格式的显式完全拉格朗日物质点法(TLMPM).首先基于近似不可压软材料的体积部分应变能密度,引入关于静水压力的方程;之后将该方程与动量方程基于显式物质点法框架进行离散,并采用完全拉格朗日格式消除物质点跨网格产生的误差,提升大变形问题的模拟精度;对位移和压强场采用不同阶次的B样条插值函数并通过引入针对体积变形的重映射技术改进了算法,提升算法的准确性.此外,算法通过实施一种交错求解格式在每个时间步对位移场和压强场依次进行求解.最后,给出几个典型数值算例来验证本文所提出的混合格式TLMPM的有效性和准确性,计算结果表明该方法可以有效处理体积自锁,准确地模拟近似不可压软材料的大变形动力学行为.     

On non-physical response in models for fiber-reinforced hyperelastic materials

Soft biological tissues are sometimes composed of thin and stiff collagen fibers in a soft matrix leading to a strong anisotropy. Commonly, constitutive models for quasi-incompressible materials, as for soft biological tissues, make use of an additive split of the Helmholz free-energy into a volumetric and a deviatoric part that is applied to the matrix and fiber contribution. This split offers conceptual and numerical advantages. The purpose of this paper is to investigate a non-physical effect that arises thereof. In fact, simulations involving uniaxial stress configurations reveal volume growth at rather small stretches. Numerical methods such as the Augmented Lagrangian method might be used to suppress this behavior. An alternative approach, proposed here, solves this problem on the constitutive level.     

Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

Torsion of solid cylinders in the context of nonlinear elasticity theory has been widely investigated with application to the behavior of rubber-like materials. More recently, this problem has attracted attention in investigations of the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in nonlinear elasticity was concerned specifically with the effects of strain-stiffening on the torsional response of solid circular cylinders. The cylinders are composed of incompressible isotropic nonlinearly elastic materials that undergo severe strain-stiffening in the stress-stretch response. Here we investigate similar issues for fiber-reinforced transversely-isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect stretch induced strain-stiffening of collagen fibers on loading and have been shown to model the mechanical behavior of many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The classic Poynting effect found for rubber-like materials where torsion induces elongation of the cylinder is shown to be significantly different for the transversely-isotropic materials considered here. For sufficiently large anisotropy and under certain conditions on the amount of twist, a reverse-Poynting effect is demonstrated where the cylinder tends to shorten on twisting The results obtained here have important implications for the development of accurate torsion test protocols for determination of material properties of soft tissues.     

EFFECTIVE ELASTIC MODULUS OF BONE-LIKE HIERARCHICAL MATERIALS

A shear-lag model is used to study the mechanical properties of bone-like hierarchical materials.The relationship between the overall effective modulus and the number of hierarchy level is obtained.The result is compared with that based on the tension-shear chain model and finite element simulation,respectively.It is shown that all three models can be used to describe the mechanical behavior of the hierarchical material when the number of hierarchy levels is small.By increasing the number of hierarchy level,the shear-lag result is consistent with the finite element result.However the tension-shear chain model leads to an opposite trend.The transition point position depends on the fraction of hard phase,aspect ratio and modulus ratio of hard phase to soft phase.Further discussion is performed on the flaw tolerance size and strength of hierarchical materials based on the shear-lag analysis.     

Handedness-dependent hyperelasticity of biological soft fibers with multilayered helical structures

Collagen fibrils with multilayered helical structures widely exist in biological soft tissues, e.g., blood vessels, tendons, and ligaments. Understanding the mechanical properties of this kind of chiral materials is not only essential for evaluating the mechanical behaviors of the host tissues but also of significance for medical engineering, clinical diagnosis, and surgical operation. In this paper, a theoretical model is presented to investigate the hyperelasticity of biological soft fibers with multilayered helical structures. The effects of the initial helical angle, number and handedness of the fibers in each ply on the mechanical response of the material are examined. Our analysis reveals a switch of contact modes between two neighboring layers, which may greatly alter the overall non-linear response of the material. The Poisson׳s ratio of such a multilayered string can be greater than 0.5. The obtained results agree with relevant experiments of soft tissues. This work sheds light on the non-linear mechanics of chiral materials and may also guide the design of biomimetic materials.     

复合泡沫塑料力学行为的研究综述

复合泡沫塑料是一种重要的防护材料,它在国防工业和民用工业各部门均有许多重要的应用,对这类材料的力学行为进行研究具有重要的学术价值和应用前景.本文对复合泡沫塑料力学行为的研究文献进行了综述.首先,对复合泡沫塑料力学行为研究的早期工作进行了简介.然后,重点介绍了复合泡沫塑料力学行为研究的最新进展,其中也包括作者近期在该领域开展的一些工作;对复合泡沫塑料进行了静、动态压缩实验和细观加载实验,研究了材料的宏观变形规律和细观失效机制;在理论研究方面,探讨了复合泡沫塑料的能量吸收和缓冲特性,从宏、细观力学分析出发研究了复合泡沫塑料有关力学性能的理论预测问题;还利用计算机和通用软件对高密度复合泡沫塑料进行了有限元分析,研究了高密度复合泡沫塑料的失效行为.最后,给出对该领域研究工作的一些展望.      

A nonlinear mechanics model of bio-inspired hierarchical lattice materials consisting of horseshoe microstructures

Development of advanced synthetic materials that can mimic the mechanical properties of non-mineralized soft biological materials has important implications in a wide range of technologies. Hierarchical lattice materials constructed with horseshoe microstructures belong to this class of bio-inspired synthetic materials, where the mechanical responses can be tailored to match the nonlinear J-shaped stress–strain curves of human skins. The underlying relations between the J-shaped stress–strain curves and their microstructure geometry are essential in designing such systems for targeted applications. Here, a theoretical model of this type of hierarchical lattice material is developed by combining a finite deformation constitutive relation of the building block (i.e., horseshoe microstructure), with the analyses of equilibrium and deformation compatibility in the periodical lattices. The nonlinear J-shaped stress–strain curves and Poisson ratios predicted by this model agree very well with results of finite element analyses (FEA) and experiment. Based on this model, analytic solutions were obtained for some key mechanical quantities, e.g., elastic modulus, Poisson ratio, peak modulus, and critical strain around which the tangent modulus increases rapidly. A negative Poisson effect is revealed in the hierarchical lattice with triangular topology, as opposed to a positive Poisson effect in hierarchical lattices with Kagome and honeycomb topologies. The lattice topology is also found to have a strong influence on the stress–strain curve. For the three isotropic lattice topologies (triangular, Kagome and honeycomb), the hierarchical triangular lattice material renders the sharpest transition in the stress–strain curve and relative high stretchability, given the same porosity and arc angle of horseshoe microstructure. Furthermore, a demonstrative example illustrates the utility of the developed model in the rapid optimization of hierarchical lattice materials for reproducing the desired stress–strain curves of human skins. This study provides theoretical guidelines for future designs of soft bio-mimetic materials with hierarchical lattice constructions.     

Finite extension and torsion of fiber-reinforced non-linearly elastic circular cylinders

In the context of the theory of non-linear elasticity for rubber-like materials, the problem of finite extension and torsion of a circular bar or tube has been widely investigated. More recently, this problem has attracted considerable attention in studies on the biomechanics of soft tissues and has been applied, for example, to examine the mechanical behavior of passive papillary muscles of the heart. A recent study in non-linear elasticity was concerned specifically with the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic non-linearly elastic materials that undergo severe strain-stiffening in the stress–stretch response. For two specific material models that reflect limiting chain extensibility at the molecular level, it was shown that, in the absence of an additional axial force, a transition value γ=γ_t of the axial stretch exists such that for γ<γ_t, the stretched cylinder tends to elongate on twisting whereas for γ>γ_t, the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models for rubber such as the Mooney–Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. Here we investigate similar issues for fiber-reinforced transversely isotropic circular cylinders. We consider a class of incompressible anisotropic materials with strain-energy densities that are of logarithmic form in the anisotropic invariant. These models reflect limited fiber extensibility and in the biomechanics context model the stretch induced strain-stiffening of collagen fibers on loading. They have been shown to model the mechanical behavior of fiber-reinforced rubber and many fibrous soft biological tissues. The consideration of anisotropy leads to a more elaborate mechanical response than was found for isotropic strain-stiffening materials. The results obtained here have important implications for extension–torsion tests for fiber-reinforced materials, for example in the development of accurate extension–torsion test protocols for determination of material properties of soft tissues.     

功能铁磁材料的变形与断裂的研究进展

综述了近几十年, 特别是近十几年来铁磁材料的力磁耦合变形与断裂行为的研究概况. 传统铁磁弹性问题的研究已经有较长时间的积累, 文献中已有大量的研究结果发表. 近些年 来, 随着智能材料及结构应用与研究的兴起, 功能铁磁材料如稀土超磁致伸缩材料、铁磁相 变材料以及铁磁复合材料等的力学行为越来越受到重视, 人们在功能铁磁材料的变形与断裂 以及铁磁复合材料的有效性质等方面开展了大量的研究工作. 本文在简单介绍了经典铁磁弹 性和传统铁磁结构的力磁性能的研究背景基础上, 结合作者近年来在铁磁材料变形与断裂方 面所开展的工作, 着重评述了功能软铁磁材料在变形与断裂的实验研究,如实验设备和技术, 以及铁磁复合材料细观力学、软铁磁材料、铁磁功能材料的变形与断裂理论等方面的研究进 展, 并指出了需要进一步研究的方向.