Modeling residual stresses in arterial walls based on anisotropic growth

With the aim of obtaining a general local formulation for anisotropic growth in soft biological tissues, a model based on the multiplicative decomposition of the growth tensor is formulated. The two parts of the growth tensor are associated with the main anisotropy directions. Together with an anisotropic driving force, the model enables an effective stress reduction by including growth-induced residual stresses, which is demonstrated in a numerical example of an idealized arterial segment. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the treatment of active behaviour in continuum muscle mechanics

Two approaches of including active contractile behaviour of muscle tissue written in a continuum-mechanical formulation are presented. One approach relies on the addition of active and passive stress contributions, while the other approach is based on a multiplicative decomposition of the deformation gradient tensor. Both formulations can be stated in a thermodynamically consistent manner, each with different constraints, and both models can reproduce experimental data of passive and fully active muscle. Different behaviours are observed when comparing the active muscle models at submaximal stimulation rates. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micro-sphere-based remodelling formulation for arterial tissue including residual stresses

An essential property of soft biological tissues is the ability to adapt according to respective loading conditions, e.g. by means of fibre reorientation (remodelling). In particular with regard to arterial tissue, an externally unloaded state of the material is generally associated with residual stresses. In this contribution a three-dimensional micro-sphere-based constitutive model for anisotropic soft biological tissue is presented, which includes fibre-reorientation-related remodelling as well as residual stress-effects. As a key aspect of this contribution, time-dependent remodelling effects are incorporated by introducing evolution equations for the integration directions of the micro-sphere scheme, which thereby characterize the material's anisotropic properties. An appropriate remodelling approach for the orthotropic case is discussed, whereas the effect of residual stresses is additionally included in the model by means of a multiplicative decomposition of the deformation gradient tensor. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Oldroyd-B流体在孔隙壁面残余油上的作用力分析

为了分析计算粘弹性流体驱替残余油的微尺度力,从水动力学角度探索非牛顿流体的流变特性,选取Oldroyd-B本构方程来模拟粘弹性流体,并结合连续性方程和运动方程得到了粘弹性流体在微孔道中的流动方程,利用边界条件计算得到流动的流场,结合应力张量理论,计算出粘弹性流体作用在残余油上的法向偏应力和水平应力差,计算结果表明:沿流动方向,粘弹性流体的弹性越大,法向偏应力越大;垂直于流动方向,法向偏应力近似对称分布;随着粘弹性流体的弹性变化,水平应力差的变化趋势发生了变化,威森伯格数We越大,残余油所受的水平应力差先逐渐增加,达到峰值后降低,这种趋势更有利于残余油的变形,为下一步分析残余油的变形,并从主体上分离奠定了基础.     

Mathematical modelling of finite strain magneto-viscoelastic deformations

Large strain magneto-viscoelastic deformations in the presence of a finite magnetic field are modelled in this paper. Internal dissipation mechanisms are proposed using a multiplicative decomposition of the deformation gradient and an additive decomposition of the magnetic induction. Using thermodynamically consistent constitutive and evolution laws, numerical results showing stress relaxation and magnetic field evolution are presented to illustrate the theory. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetrization of the growth deformation and velocity gradients in residually stressed biomaterials

Some fundamental issues in the kinematic and kinetic analysis of the stress-modulated growth of residually stressed biological materials are addressed within the context of the multiplicative decomposition of deformation gradient into its elastic and growth parts. The symmetrizations of the growth part of the deformation gradient and the growth part of the velocity gradient are derived for isotropic pseudoelastic soft tissues. The significance of results in the formulation of the biomechanic constitutive theory is discussed.     

A Geometric Theory of Growth Mechanics

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. The time dependence of the metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and “shape”. We show that the time dependency of the material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange–d’Alembert principle with Rayleigh’s dissipation functions to derive the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of the deformation gradient, F=F _e F _g, into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory.     

Kinematics of finite elastoplastic deformations

In this paper we review various approaches to the decomposition of total strains into elastic and nonelastic (plastic) components in the multiplicative representation of the deformation gradient tensor. We briefly describe the kinematics of finite deformations and arbitrary plastic flows. We show that products of principal values of distortion tensors for elastic and plastic deformations define principal values of the distortion tensor for total deformations. We describe two groups of methods for decomposing deformations and their rates into elastic and nonelastic components. The methods of the first group additively decompose specially built tensors defined in a common basis (initial, current, or “intermediate”). The second group implies a certain relation connecting tensors that describe elastic and plastic deformations. We adduce an example of constructing constitutive relations for elastoplastic continuums at large deformations from thermodynamic equations.     

Symmetric tensor decomposition by alternating gradient descent

The symmetric tensor decomposition problem is a fundamental problem in many fields, which appealing for investigation. In general, greedy algorithm is used for tensor decomposition. That is, we first find the largest singular value and singular vector and subtract the corresponding component from tensor, then repeat the process. In this article, we focus on designing one effective algorithm and giving its convergence analysis. We introduce an exceedingly simple and fast algorithm for rank-one approximation of symmetric tensor decomposition. Throughout variable splitting, we solve symmetric tensor decomposition problem by minimizing a multiconvex optimization problem. We use alternating gradient descent algorithm to solve. Although we focus on symmetric tensors in this article, the method can be extended to nonsymmetric tensors in some cases. Additionally, we also give some theoretical analysis about our alternating gradient descent algorithm. We prove that alternating gradient descent algorithm converges linearly to global minimizer. We also provide numerical results to show the effectiveness of the algorithm.     

Application of a viscoplastic material model on a combined quasi-static-electromagnetic forming process

The prediction and simulation of material behavior by finite element methods has become indispensable. Furthermore, various phenomena in forming processes lead to highly differing results. In this work, we have investigated the process chain on a cross-shaped cup in cooperation between the Institute of Applied Mechanics (IFAM) of the RWTH Aachen and the Institute of Forming Technology and Lightweight Construction (IUL) of the TU Dortmund. A viscoplastic material model based on the multiplicative decomposition of the deformation gradient in the context of hyperelasticity has been used [1,2]. The finite strain constitutive model combines nonlinear kinematic and isotropic hardening and is derived in a thermodynamically consistent setting. This anisotropic viscoplastic model is based on the multiplicative decomposition of the deformation gradient in the context of hyperelasticity. The kinematic hardening component represents a continuum extension of the classical rheological model of Armstrong-Frederick kinematic hardening. The constitutive equations of the material model are integrated in an explicit manner and implemented as a user material subroutine in the commercial finite element package LS-DYNA with the electromagnetical module. The aim of the work is to show the increasing formability of the sheet by combining quasi-static deep drawing processes with high speed electromagnetic forming. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于分子动力学模拟的金属构件的弹-塑性分解方法

针对金属多晶材料构件的分子动力学（MD）模拟,本文提出了一种新的弹-塑性分解方法.文章将MD运动轨迹分解为结构变形和热振动,给出了计算结构变形的方法和近似公式;针对金属多晶材料构件的当前构型,给出了基于FCC|BCC晶胞和四原子占位的四面体单元相组合的连续变形函数及计算变形梯度的算法;利用原子-连续关联模型,发展了计算当前构型应力场和弹性张量的算法.分析了当构件承受过大载荷后在材料内部所产生的微观缺陷,并将其分类标定为位错、层错、挛晶界、晶界和空位等;对于层错和挛晶界讨论了在弹性卸载过程中应满足的刚体运动约束方程;利用极小势能原理构造了基于当前构型的弹性卸载算法,进而给出了完整的基于MD模拟的计算弹-塑性应变的算法.最后,针对单晶铜纳米线拉伸的MD模拟,计算了弹-塑性应变场,验证了本文方法的合理性.
本文提出的基于MD模拟的弹-塑性分解方法,为从微观到宏观的多尺度和多模型耦合计算提供了算法支撑.     

Efficient time integration in multiplicative inelasticity

In Shutov et al. (Comput Methods Appl Mech Eng 265:213–225, 2013), the numerical time integration of a famous large strain model of Maxwell fluid type has been considered. The underlying model is based on the multiplicative decomposition of the deformation gradient and includes a Neo-Hookean hyperelasticity relation as well as an incompressible viscous flow rule. Shutov et al. presented a time stepping algorithm for implicit time integration of the inelastic flow rule, which is based on Euler backward time discretisation, prevents error accumulation and is iteration free. In this contribution, the basic idea of the this approach is applied to more general models of multiplicative viscoelasticity. Here, extended hyperelastic relations including general functions of the first principal invariant of deformation tensors are regarded. An efficient time stepping algorithm is derived, where only one scalar equation for one scalar unknown has to be solved within every time step. The approach is applied to a specific viscoelastic model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

S—R分解定理的唯一性,存在性和客观性

对于连续体的一切物理可能的变形场，其变形梯度张量F可被分解为一个对称张量S和一个正交张量R的直和，这便是S-R分解定理．本文通过矩阵方法和张量方法证明了S-R分解定理的唯一性、存在性和客观性．     

Anisotropic finite strain hyperelasticity based on the multiplicative decomposition of the deformation gradient

In this contribution a new constitutive model for transversely isotropic materials is presented. The proposed model is based on the multiplicative decomposition of the deformation gradient into one part containing the deformation only in the direction of anisotropy and another part describing the remaining deformation. This clear assignment leads to a decoupling of the stress-state. The model is investigated analytically in view of simple tension. Moreover, an inhomogenous deformation is solved using a finite elements simulation. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)