Tensile fracture behavior of heterogeneous materials based on fractal geometry

Many of heterogeneous structural materials, like concrete, have different behavior under tensile stresses in comparison to their behavior under compressive stresses. The aim of this paper is to interpret behavior of such materials subjected to tensile stresses, by using newly introduced concept of fractal geometry. In the first part of this paper, tensile behavior of granular composites has been studied by using fractal geometry. It is shown that the fractality of the cross section in this kind of composites can be used to interpret the size effect on tensile strength. In fact, this work is a modification with innovations on the previous studies on fractal based size effect.This hypothesis that the fracture surfaces of quasi-brittle materials are fractals has been verified by several investigations. Accordingly, in the other part of this paper, softening process in heterogeneous materials is studied. Resulting from presented approach, a new softening curve for quasi-brittle materials is proposed. This new softening curve is denominated “Quasi-fractal softening curve” and is consisted of two parts, a linear portion in beginning part and an exponential portion in rest of the curve. This makes it very compatible to the pre-existing softening curves.     

Three-dimensional fractal analysis of concrete fracture at the meso-level

The fracture process in concrete-like materials cannot be properly modelled in an Euclidean framework, due to its complex morphology at the micro- and meso-level. The inherent flaws interact through a multi-scale process, leading to self-affine fracture surfaces. Moreover, the self-organized network of microcracks displays fractal properties prior to the formation of the final fracture surface. At the same time, due to the presence of pores and voids, the stress-carrying cross section is a rarefied fractal domain, even from the beginning of the loading process. A new experimental equipment has been developed which allows the entire fracture surface, or any plane cross section, to be digitised and analysed. This represents an important progress with respect to the study of one-dimensional profiles. In this paper, the three-dimensional algorithms for evaluating the fractal dimension of invasive surfaces and lacunar sections are described. The invasive fractal character of the fracture surfaces is confirmed. Moreover, the lacunar fractal character of the stress-carrying cross sections, a priori assumed by Carpinteri [A. Carpinteri, Mechanics of Materials 18 (1994) 259–266], is now proven experimentally.     

A friction contact stiffness model of fractal geometry in forced response analysis of a shrouded blade

To investigate the nonlinear vibration behavior of a shrouded blade with friction dynamic contact interface, a friction contact stiffness model is proposed to describe the friction force at different rough interfaces and different normal loads. In the proposed model, the friction contact interface is discretized to a series of friction contact pairs and each of them can experience stick, slip, or separate states. Fractal geometry is used to simulate the topography of contact surfaces. The contact stiffness is calculated using the Hertz contact theory and fractal geometry, which is related to contact interfaces parameters including normal load, roughness, Young??s modulus, and Poisson??s ratio. The trajectory tracking method is used to predict the friction force and it is not necessary to judge the transition condition among stick, slip, and separate states. It is suitable for complicated periodic motion of the contact interfaces. The forced response of a real shrouded blade is predicted using the proposed model and the multi-harmonic balance method. The effect of surface roughness, initial normal load, and contact area on the forced response of a shrouded blade is studied. It is shown that contact stiffness increases with normal load and fractal dimension. The resonant amplitude is sensitive to the initial normal load and contact surface roughness. The response can be influenced by the contact area, which is an important parameter for blade designers.     

DEM analysis of the effect of joint geometry on the shear behavior of rocks

In order to comprehensively investigate the effect of different joint geometries on the shear behavior of rocks, the Distinct Element Method (DEM) was utilized with a new bond contact model. A series of direct shear tests on coplanar and non-coplanar jointed rocks was simulated using the PFC2D software, which incorporates our bond contact model. Both coplanar jointed rocks with different joint persistence and non-coplanar ones with different joint inclinations were simulated and investigated numerically. The numerical results were compared and discussed with relevant laboratory tests as well as some reported numerical works. The results show that for coplanar jointed rocks, the peak shear stress decreases nonlinearly with the joint persistence, and the failure process can be divided into four stages: elastic shearing phase, crack propagation, failure of rock bridges, and residual phase. For non-coplanar jointed rocks, as the absolute value of the inclination angle of the rock joints increases, its shear strength increases, changing the failure patterns and the length of new fractures between existing cracks. When the absolute value increases from 15° to 30°, the average shear capacity increases the most as 39%, while the shear capacity increases the least as 2.9% when the absolute value changes from 45° to 60°. There is a good consistency of the failure patterns obtained from experiments and numerical tests. All these demonstrate that the DEM can be further applied to rock mechanics and practical rock engineering with confidence in the future.     

On the numerical solution of structures with fractal geometry: The FE approach

The scope of the present paper is to present the numerical aspects of the theory developed in [1]. The fractal geometry of structure(s) is approximated either through the IFS (iterated function system) method or through the FI (fractal interpolation) method. These approximations of the fractal through classical curves and surfaces are combined with the FEM in order to get numerical results for important technical problems, which cannot be satisfactorily treated by other methods.

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Sommario Lo scopo del lavoro é quello di discutere gli aspetti numerici della teoria sviluppata in [1]. La geometria frattale della/e struttura é approssimata sia attraverso il metodo IFS (iterated function system) che il metodo FI (fractal interpolation). Queste approssimazioni frattali, attraverso curve e superfici classiche, sono combinate con il metodo degli elementi finiti, onde potere ottenere risultati numerici per importanti problemi tecnici che non potrebbero venire soddisfacentemente affrontati con altri metodi.

     

Time- and frequency-domain analysis using lumped mass global collocation

Quite recently, a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures was proposed (Archive of Applied Mechanics: DOI: ). This paper extends the latter methodology on several levels, in both the time and frequency domain. Firstly the formulation is updated so that it can also deal with rods of variable cross section. Then, the fully populated mass matrices of the previous formulation are properly replaced by lumped masses, thus saving still more computer effort. Subsequently, a new general formulation for the transient response analysis is proposed. Finally, a novel procedure for the coupling of two neighboring collinear rods is presented. The theory is supported by six test cases concerning elastic rods of constant and variable cross sections. Among these, transient analysis refers to the response of a single rod due to a Heaviside-type loading as well as to the impact between two collinear rods of different cross sections.     

Static analysis of two-dimensional elastic structures using global collocation

Based on previous findings concerning the numerical solution of one-dimensional elastodynamical problems [Provatidis in Arch Appl Mech 78(4):241–250, 2008] this paper extends the methodology to the static analysis of two-dimensional problems in quadrilateral domains. This target is achieved by replacing the Galerkin/Ritz procedure involved in Lagrangian (or Gordon–Coons) type finite elements by a global collocation scheme. In brief, the boundary conditions are fulfilled at all boundary nodes, while the governing equation is fulfilled at internal points. The theory is supported by four test cases concerning rectangular and curvilinear structures under plane-stress or plane-strain conditions, where the convergence rate is successfully compared with that of conventional bilinear finite elements with the same mesh density.     

Free vibration analysis of elastic rods using global collocation

This paper investigates the performance of a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures when either basis or shape functions are used to approximate the displacement field. Although the methodology is generally applicable, numerical results are presented only for rods in which one-dimensional basis functions in the form of a power series, as well as equivalent Lagrange, Bernstein or Chebyshev polynomials are used. The new feature of the proposed methodology is that it can deal with any type of boundary conditions; therefore, the cases of two Dirichlet as well as one Dirichlet and one Neumann condition were successfully treated. The basic finding of this work is that all these polynomials lead to results identical to those obtained by the power series expansion; therefore, the solution depends on the position of the collocation points only.     

Stress analysis of a layered elastic solid in contact with a rough surface exhibiting fractal behavior

A contact stress analysis is presented for a layered elastic half-space in contact with a rough surface exhibiting self-affine (fractal) behavior. Relationships for the mean contact pressure versus representative strain and the real half-contact width versus elastic properties of the layer and the substrate, asperity radius, layer thickness, and truncated half-contact width were derived from finite element simulations of a layered medium compressed elastically by a rigid cylindrical asperity. These relationships were incorporated in a numerical algorithm that was used to obtain the contact pressure distributions and stresses generated by the asperity contacts formed at the interface of the layered medium and the fractal surface. Analytical solutions illustrate the significance of the elastic material properties, layer thickness, and surface topography (roughness) on global parameters such as normal load and real contact area. Results for the contact pressure distribution and the surface and subsurface stresses provide insight into the initiation of yielding and the tendency for cracking in the layered medium. It is shown that cracking at the surface and the layer/substrate interface is more likely to occur in the case of a stiff layer, whereas surface cracking is more prominent for a relatively compliant layer.     

A slip-line plasticity analysis of sliding friction of rough surfaces exhibiting self-affine (fractal) behavior

A plasticity analysis of sliding friction of rough (fractal) surfaces sliding against smooth surfaces was developed based on a slip-line model of a rigid spherical asperity (wear particle) plowing and cutting through a soft semi-infinite medium. Solutions of the fraction of fully plastic asperity microcontacts responsible for the evolution of friction and energy dissipation were obtained in terms of the total normal load (global interference), interfacial adhesion characteristics, topography (fractal) parameters of the hard surface, and elastic–plastic material properties of the soft surface. This was accomplished by incorporating the slip-line model of a single microcontact into a friction analysis of sliding surfaces demonstrating multi-scale roughness. Numerical results provide insight into the effects of global interference (normal load), fractal parameters (surface roughness) of the hard surface, interfacial shear strength (adhesion), and material properties of the soft surface on plastic deformation at the microcontact level, global coefficient of friction, and frictional energy dissipated during sliding.     

Nonlinear dynamic analysis of parabolic leaf springs using ANCF geometry and data acquisition

Nonlinear Dynamics - The parabolic leaf spring is widely used in modern vehicle suspension systems because it has many desirable features, such as weak interleaf friction and light weight. In this...     

Spatio-temporal correlation analysis of turbulent flows using global and single-point measurements

A method to extract whole-field spatio-temporal correlations by combining global and single-point measurement techniques of different time resolutions is proposed. For fluid mechanics applications, the emphasis is on the combination of low repetition rate particle image velocimetry (PIV) results with experimental data obtained at largely higher sampling frequencies. The experimental feasibility of the procedure is established from results obtained in the wake of a cylinder, using PIV and constant temperature hot wire anemometry (CTA). The method is then applied to examine the shear layer in the core of a round subsonic jet using PIV and laser Doppler velocimetry (LDV). The accuracy of the cross-correlation functions is compared to the auto- and cross-correlation functions obtained from series of LDV and CTA measurements.     

混凝土拉伸断裂的细观数值分析

根据混凝土试件拉伸和三点弯曲的物理模型,用梁-颗粒模型BPM 2D(B eam-Particle M ode l)模拟了混凝土拉伸和三点弯曲试件微裂纹的萌生、扩展直至试件宏观破坏的全过程。在梁-颗粒模型中用三种类型梁单元形成混凝土细观数值模型,每种类型梁单元的力学性质均按韦伯(W e ibu ll)分布随机赋值以模拟混凝土细观结构的非均匀性。数值模拟结果给出了混凝土拉伸应力-应变曲线和三点弯曲载荷-位移曲线,以及混凝土试件破坏过程最大应力分布图和裂纹扩展图。数值模拟结果显示混凝土破坏过程实际上就是微裂纹萌生、扩展、贯通,直到宏观裂纹产生导致混凝土失稳断裂的过程。通过对数值模拟结果的分析,揭示出混凝土在拉伸条件下裂纹尖端的拉应力集中是裂纹扩展的动力,混凝土组成材料力学性质的非均匀性是造成裂纹扩展路径曲折的重要原因。     

Spatial behavior in the electromagnetic theory of microstretch elasticity

This paper is concerned with the electromagnetic theory of microstretch elasticity. First, the initial boundary value problem is formulated in the framework of the linear dynamic theory of microstretch magnetoelectroelastic solids. Then, the spatial behavior of solutions is studied in both bounded and unbounded regions. The obtained result gives an exact idea of the domain of influence, in the sense that for each fixed time in a given interval, the entire activity vanishes at distanced from the support of the given data greater than a time-dependent threshold value. The study of spatial behavior is completed by an exponential decay estimate inside the domain of influence. As a by product a uniqueness result holding for both bounded and unbounded bodies is derived. Finally, the effect of a concentrated microstretch body force is studied.     

Experimental analysis of liquid–gas interface at low Weber number: interface length and fractal dimension

The present paper reports an experimental investigation on atomizing liquid flows produced by simplified cavity nozzles. The Weber number being kept low, the sprays produced by these injectors depend on the liquid flow characteristics only, and more precisely, on the non-axial kinetic energy and of the turbulent kinetic energy at the nozzle exit. The investigation reported here concentrates on the characterization of liquid flows during atomization by measuring the spatial variation of the local interface length and of the local interface fractal dimension. Both parameters were found representative of the physics of atomization process: they depend on the characteristics of the flow issuing from the nozzle and they are related to the subsequent drop size distribution. The local interface length is representative of the amount of liquid–gas interface surface area, and is a function of both the non-axial and the turbulent kinetic energies at the nozzle exit. The fractal dimension is representative of the tortuosity of the liquid–gas interface and, as expected, is mainly related to the turbulent kinetic energy at the nozzle exit. As far as the drop size distribution is concerned, it is found that the local interface length at the instant of break-up determines a representative drop diameter of some kind, whereas the fractal dimension at the same instant controls the dispersion of the distribution.     

基于分形理论的煤层气非达西流动分析

在考虑了煤层的分形特征和启动压力梯度影响的基础上,建立了无限大煤层中气体低速非达西流动的数学模型,并求得了量纲为一的井底压力的Laplace空间解析解,并根据数值求解结果绘制了典型的井底压力动态曲线。研究结果表明:在定产量生产的情况下,分形维数和量纲为一的启动压力梯度对早期井底压力动态无显著影响;在生产的中后期,由于受二者的影响,压力导数曲线上的径向流水平直线段消失;量纲为一的井储系数的影响主要表现在续流阶段,而吸附系数则主要影响煤基质向裂隙扩散的过渡阶段。     

Fracture analysis of concrete using singular fractal functions with lattice beam network and confirmation with acoustic emission study

In the present study singular fractal functions (SFF) were used to generate stress-strain plots for quasi-brittle material like concrete and cement mortar and subsequently stress-strain plot of cement mortar obtained using SFF was used for modeling fracture process in concrete. The fracture surface of concrete is rough and irregular. The fracture surface of concrete is affected by the concrete’s microstructure that is influenced by water cement ratio, grade of cement and type of aggregate [1], [2], [3] and [4]. Also the macrostructural properties such as the size and shape of the specimen, the initial notch length and the rate of loading contribute to the shape of the fracture surface of concrete. It is known that concrete is a heterogeneous and quasi-brittle material containing micro-defects and its mechanical properties strongly relate to the presence of micro-pores and micro-cracks in concrete [1], [2], [3] and [4]. The damage in concrete is believed to be mainly due to initiation and development of micro-defects with irregularity and fractal characteristics. However, repeated observations at various magnifications also reveal a variety of additional structures that fall between the ‘micro’ and the ‘macro’ and have not yet been described satisfactorily in a systematic manner [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [15], [16] and [17]. The concept of singular fractal functions by Mosolov was used to generate stress-strain plot of cement concrete, cement mortar and subsequently the stress-strain plot of cement mortar was used in two-dimensional lattice model [28]. A two-dimensional lattice model was used to study concrete fracture by considering softening of matrix (cement mortar). The results obtained from simulations with lattice model show softening behavior of concrete and fairly agrees with the experimental results. The number of fractured elements are compared with the acoustic emission (AE) hits. The trend in the cumulative fractured beam elements in the lattice fracture simulation reasonably reflected the trend in the recorded AE measurements. In other words, the pattern in which AE hits were distributed around the notch has the same trend as that of the fractured elements around the notch which is in support of lattice model.