On the strong discontinuity approach in finite deformation settings

Taking the strong discontinuity approach as a framework for modelling displacement discontinuities and strain localization phenomena, this work extends previous results in infinitesimal strain settings to finite deformation scenarios. By means of the strong discontinuity analysis, and taking isotropic damage models as target continuum (stress–strain) constitutive equation, projected discrete (tractions–displacement jumps) constitutive models are derived, together with the strong discontinuity conditions that restrict the stress states at the discontinuous regime. A variable bandwidth model, to automatically induce those strong discontinuity conditions, and a discontinuous bifurcation procedure, to determine the initiation and propagation of the discontinuity, are briefly sketched. The large strain counterpart of a non‐symmetric finite element with embedded discontinuities, frequently considered in the strong discontinuity approach for infinitesimal strains, is then presented. Finally, some numerical experiments display the theoretical issues, and emphasize the role of the large strain kinematics in the obtained results. Copyright © 2003 John Wiley & Sons, Ltd.     

Initial response of a micro-polar hypoplastic material under plane shearing

The behavior of an infinite strip of a micro-polar hypoplastic material located between two parallel plates under plane shearing is investigated. The evolution equation of the stress tensor and the couple-stress tensor is described using tensor-valued functions, which are nonlinear and positively homogeneous of first order in the rate of deformation and the rate of curvature. For the initial response of the sheared layer an analytical solution is derived and discussed for different micro-polar boundary conditions at the bottom and top surfaces of the layer. It is shown that polar quantities appear within the shear layer from the beginning of shearing with the exception of zero couple stresses prescribed at the boundaries.     

From continuum mechanics to fracture mechanics: the strong discontinuity approach

The paper deals with the strong discontinuity approach and shows the links with the decohesive fracture mechanics provided by that approach. On the basis of 1D continuum damage models it is shown that, by introducing some few ingredients like the strong discontinuity kinematics, discrete constitutive models (traction vs. displacement jumps) are automatically induced. For the general 2D-3D cases it is shown that the weak discontinuity concept is an additional ingredient, necessary in order to fulfill the strong discontinuity conditions, which allows to establish additional links with the fracture process zone concept. Also classical fracture mechanics properties as the fracture energy are related to the continuum model properties in a straightforward manner.     

A variational formulation for finite elasticity with independent rotation and Biot-axial fields

The fundamentals of the geometrically nonlinear mechanics of the three-dimensional elastic continuum are derived, starting from a general variational framework established for the polar model and passing through a constitutive definition of the non-polar medium itself. A constrained variational setting follows, having as unknown vector fields the displacement, the rotation vector and the axial of the Biot stress. It embraces both the rotational equilibrium and the characterization of the rotation as Euler-Lagrange equations. These conditions can then be satisfied in a weak sense within discrete approximations. It is also shown that the classical approach of the non-polar continuum can be accomodated as a particular case of the present formulation. A consistent linearization is then proposed and a simple solid finite element developed to test the computational viability of the formulation. A few examples assess the capability of the element to represent large three-dimensional rotations. Communicated by S. N. Atluri, 2 August 1996     

Initial response of a micro-polar hypoplastic material under plane shearing

The behavior of an infinite strip of a micro-polar hypoplastic material located between two parallel plates under plane shearing is investigated. The evolution equation of the stress tensor and the couple-stress tensor is described using tensor-valued functions, which are nonlinear and positively homogeneous of first order in the rate of deformation and the rate of curvature. For the initial response of the sheared layer an analytical solution is derived and discussed for different micro-polar boundary conditions at the bottom and top surfaces of the layer. It is shown that polar quantities appear within the shear layer from the beginning of shearing with the exception of zero couple stresses prescribed at the boundaries.     

模糊参数平面连续体结构的拓扑优化设计

研究了具有模糊参数的连续体结构在模糊载荷作用下的拓扑优化设计问题。利用信息熵将模糊变量转换为随机变量,构建了随机载荷作用下的随机参数的连续体结构的拓扑优化设计数学模型,以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,满足单元应力可靠性为约束条件,利用分布函数法对应力可靠性约束进行了等价显式化处理。基于随机因子法,利用代数综合法导出了应力响应的数字特征的计算表达式。采用双方向渐进结构优化(BESO)方法求解。通过两个算例验证了该文模型及求解方法的合理性和有效性。     

Combined continuum damage‐embedded discontinuity model for explicit dynamic fracture analyses of quasi‐brittle materials

In this paper, a novel constitutive model combining continuum damage with embedded discontinuity is developed for explicit dynamic analyses of quasi‐brittle failure phenomena. The model is capable of describing the rate‐dependent behavior in dynamics and the three phases in failure of quasi‐brittle materials. The first phase is always linear elastic, followed by the second phase corresponding to fracture‐process zone creation, represented with rate‐dependent continuum damage with isotropic hardening formulated by utilizing consistency approach. The third and final phase, involving nonlinear softening, is formulated by using an embedded displacement discontinuity model with constant displacement jumps both in normal and tangential directions. The proposed model is capable of describing the rate‐dependent ductile to brittle transition typical of cohesive materials (e.g., rocks and ice). The model is implemented in the finite element setting by using the CST elements. The displacement jump vector is solved for implicitly at the local (finite element) level along with a viscoplastic return mapping algorithm, whereas the global equations of motion are solved with explicit time‐stepping scheme. The model performance is illustrated by several numerical simulations, including both material point and structural tests. The final validation example concerns the dynamic Brazilian disc test on rock material under plane stress assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

Impact of temperature and concentration dispersion on the physiology of blood nanofluid: links to atherosclerosis

A theoretical model of blood-silver nanofluid flow through an \(\omega \)-shaped tapered stenotic artery in the presence of a catheter is understood in detail. The rheology of the blood is considered as that of a micro-polar fluid. Suspension of the silver nano-particles in the micro-polar fluid is proposed to investigate the temperature and concentration dispersion from the immersed temperature-sensitive drug-coated nano-particles. The effects of velocity discontinuity at the arterial wall in the stenotic and non-stenotic regions are considered. The outer surface of the catheter is layered with the temperature-sensitive, drug-coated nano-particles. The resulting mathematical formulation involving the coupled non-linear momentum, temperature and concentration equations is solved using the Homotopy Perturbation Method. The efficiency and convergence of the method to the modelled equations are discussed in detail. The consequent effects of the fluid and geometric parameters on pressure drop, flow rate, impedance and wall shear stress of the fluid flow are computed. It is noticed that the high volume fraction of the nano-particles in the blood results in high flow velocity, contributing to secondary flow regions, thus resulting in higher shear stress. Such high volume fraction of the nano-particle may lead to the pathological disorder called aneurysm. This physical model has an important application of drug delivery in biomedical and pharmaceutical industry to prevent obstructions in arteries. Further, the results obtained could be very useful in the manufacturing of related artificial devices.     

Continuum model of two-dimensional crystal lattice of metamaterials

The continuum models of two-dimensional crystal lattice of metamaterial are investigated in this paper. The equivalent classical continuum requires the introduction of frequency-dependent mass density that becomes negative or infinite near the resonance frequency. In order to avoid the frequency-dependent mass density and make the dispersive characteristic of the elastic waves propagating in the equivalent continuum approximating that of lattice wave in two-dimensional crystal lattice of metamaterial, three kinds of continuum models, namely, the multiple displacements continuum model, the strain gradient continuum model and the nonlocal strain gradient continuum model, are investigated. It is found that the nonlocal gradient continuum model may better represent the dispersive relation and the bandgap characteristics of the metamaterial by the appropriate selection of nonlocal parameters.     

Enhanced continua and discrete lattices for modelling granular assemblies

This article discusses the derivation of continuum models that can be used for modelling the inhomogeneous mechanical behaviour of granular assemblies. These so-called kinematically enhanced models are of the strain-gradient type and of the strain-gradient micro-polar type, and are derived by means of homogenizing the micro-structural interactions between discrete particles. By analysis of the body wave dispersion curves, the enhanced continuum models are compared to corresponding discrete lattice models. Accordingly, it can be examined up to which deformation level the continuum models are able to accurately describe the discrete particle behaviour. Further, the boundary conditions for the enhanced continuum models are formulated, and their stability is considered. It is demonstrated how to use the body wave dispersion relations for the assessment of stability.     

层状材料的高阶连续统模型

本文研究了由两种各向同性线性弹性材料组成的层状复合材料。同时利用位移在上下层面处的值及各阶导数对层内位移展开,并引用层面处应力矢量为参变量,使得层面处的应力和位移连续条件自动满足。然后利用Hamilton原理,按无约束条件变分,得到界面上的Euler方程,由此得到高阶连续统模型的场方程。考虑了场方程的长波近似和短波近似,所得结果和现存理论符合得很好。      

Discrete modelling results of a direct shear test for granular materials versus FE results

The intention of this paper is to present a comparison of the results of discrete element and finite element simulations of a simple shear test for medium dense cohesionless sand. Such a comparison may provide useful information on the limitations and possible advantages of micro-polar continuum models for granular media as compared with discrete element models. To simulate the discrete nature of sand at the micro-level during shearing, the 3D discrete open-source model YADE developed at Grenoble University was used. Contact moments at spheres were assumed to capture the influence of force eccentricities due to grain roughness. Attention was paid to some micro-structural events (such as vortices, force chains, vortex structures, local void ratio fluctuations) appearing in a shear zone and kinetic, elastic and dissipated energies in granular specimen. The results of the discrete element simulations were compared with the corresponding finite element (FE) solutions based on a micro-polar hypoplastic constitutive model for granular material. A satisfactory agreement between discrete and FE results was achieved. Advantages and disadvantages of both approaches are outlined.     

Kinetics of scattered fracture in structural metals under elastoplastic deformation

The paper presents a procedure whereby the damage accumulation kinetics in structural materials, such as steel 45, stainless steel 12Kh18N10T, aluminum alloy D16T, and titanium alloy VT22, under elastoplastic deformation is studied based on variation parameters of elastic modulus and resistivity. For complex stress conditions, a continuum model for damage accumulation is proposed which relates the damage parameter to the intensity of accumulated plastic strains. The data calculated by the proposed continuum model are compared to the experimental findings of the investigation of the damage accumulation kinetics for some structural metals. __________ Translated from Problemy Prochnosti, No. 3, pp. 23–34, May–June, 2007.