Softening in random networks of non-identical beams

Random fiber networks are assemblies of elastic elements connected in random configurations. They are used as models for a broad range of fibrous materials including biopolymer gels and synthetic nonwovens. Although the mechanics of networks made from the same type of fibers has been studied extensively, the behavior of composite systems of fibers with different properties has received less attention. In this work we numerically and theoretically study random networks of beams and springs of different mechanical properties. We observe that the overall network stiffness decreases on average as the variability of fiber stiffness increases, at constant mean fiber stiffness. Numerical results and analytical arguments show that for small variabilities in fiber stiffness the amount of network softening scales linearly with the variance of the fiber stiffness distribution. This result holds for any beam structure and is expected to apply to a broad range of materials including cellular solids.     

Elastic–plastic behavior of non-woven fibrous mats

Electrospinning is a novel method for creating non-woven polymer mats that have high surface area and high porosity. These attributes make them ideal candidates for multifunctional composites. Understanding the mechanical properties as a function of fiber properties and mat microstructure can aid in designing these composites. Further, a constitutive model which captures the membrane stress–strain behavior as a function of fiber properties and the geometry of the fibrous network would be a powerful design tool. Here, mats electrospun from amorphous polyamide are used as a model system. The elastic–plastic behavior of single fibers are obtained in tensile tests. Uniaxial monotonic and cyclic tensile tests are conducted on non-woven mats. The mat exhibits elastic–plastic stress–strain behavior. The transverse strain behavior provides important complementary data, showing a negligible initial Poisson's ratio followed by a transverse:axial strain ratio greater than ?1:1 after an axial strain of 0.02. A triangulated framework has been developed to emulate the fibrous network structure of the mat. The micromechanically based model incorporates the elastic–plastic behavior of single fibers into a macroscopic membrane model of the mat. This representative volume element based model is shown to capture the uniaxial elastic–plastic response of the mat under monotonic and cyclic loading. The initial modulus and yield stress of the mat are governed by the fiber properties, the network geometry, and the network density. The transverse strain behavior is linked to discrete deformation mechanisms of the fibrous mat structure including fiber alignment, fiber bending, and network consolidation. The model is further validated in comparison to experiments under different constrained axial loading conditions and found to capture the constraint effect on stiffness, yield, post-yield hardening, and post-yield transverse strain behavior. Due to the direct connection between microstructure and macroscopic behavior, this model should be extendable to other electrospun systems and other two dimensional random fibrous networks.     

Continuous versus discrete (invariant) representations of fibrous structure for modeling non-linear anisotropic soft tissue behavior

The non-linear anisotropic mechanical response of soft tissue is largely dependent on the structure of the underlying collagen network. Collagen structure has been successfully quantified for various tissue types in terms of a locally defined fiber orientation distribution function. The continuous distribution function derived from structural data can be directly incorporated into an integral representation of the strain energy function for modeling tissue behavior. Alternatively, non-integral (often invariant-based) strain energy functions have been developed in which the collagen network structure is approximated using a discrete set of fiber classes. The advantage of such an approach is increased computational efficiency since the values of the strain energy and its derivatives (e.g. stress) can be evaluated without numerical integration. However, because of the structural simplifications such models are presumably unable to predict mechanical data as accurately as the models which incorporate a continuous orientation distribution function. In this work the ability of discrete versus continuous fiber models to capture the non-linear anisotropic response of soft tissue is critically analyzed. Both unimodal and bimodal fiber distributions are considered. A general formulation has been developed in terms of an arbitrary fiber strain energy function, such that the analysis can be performed for any suitable fiber material model. For tissue structures in which a discrete representation is suitable, techniques are presented for establishing the range of loading conditions in which model accuracy is not significantly compromised, thus justifying the use of an invariant-based modeling approach.     

Visco-electro-elastic models of fiber-distributed active tissues

We present a constitutive model for stochastically distributed fiber reinforced visco-active tissues, where the behavior of the reinforcement depends on the relative orientation of the electric field. Following our previous works, for the passive behaviors we adopt a second order approximation of the strain energy density associated to the parameters of the fiber distribution. Consistently, we also assume that the active behavior accounts for the stochastic distribution of the fibers. The ensuing mechanical quantities result to be dependent on two average structure tensors. We introduce an extended Helmholtz free energy density characterized by the inclusion of a directional active potential, dependent on a stochastic anisotropic permittivity tensor. The permittivity tensor is expanded in Taylor series up to the second order, allowing to obtain an approximated active potential with the same structure of the passive Helmholtz free energy density. In particular, the explicit expression of active stress and stiffness are dependent on the two average structure tensors that characterize the passive response. Anisotropy follows from the fiber distribution and inherits its stochastic nature through statistics parameters. The active fiber distributed model is extended here to viscous materials by including the contribution of a dual dissipation potential in the variational formulation of the constitutive updates. Additionally, we present a computational example of application of the electro-viscous-mechanical material model by simulating peristaltic contractions on a portion of human intestine.     

Effect of Network Topology on Relative Permeability

We consider the role of topology on drainage relative permeabilities derived from network models. We describe the topological properties of rock networks derived from a suite of tomographic images of Fontainbleau sandstone (Lindquist et al., 2000, J. Geophys. Res. 105B, 21508). All rock networks display a broad distribution of coordination number and the presence of long-range topological bonds. We show the importance of accurately reproducing sample topology when deriving relative permeability curves from the model networks. Comparisons between the relative permeability curves for the rock networks and those computed on a regular cubic lattice with identical geometric characteristics (pore and throat size distributions) show poor agreement. Relative permeabilities computed on regular lattices and on diluted lattices with a similar average coordination number to the rock networks also display poor agreement. We find that relative permeability curves computed on stochastic networks which honour the full coordination number distribution of the rock networks produce reasonable agreement with the rock networks. We show that random and regular lattices with the same coordination number distribution produce similar relative permeabilities and that the introduction of longer-range topological bonds has only a small effect. We show that relative permeabilities for networks exhibiting pore–throat size correlations and sizes up to the core-scale still exhibit a significant dependence on network topology. The results show the importance of incorporating realistic 3D topologies in network models for predicting multiphase flow properties.     

Pore-Level Modeling of Gas and Condensate Flow in Two- and Three-Dimensional Pore Networks: Pore Size Distribution Effects on the Relative Permeability of Gas and Condensate

We present a mechanistic model of retrograde condensation processes in two- and three-dimensional capillary tube networks under gravitational forces. Condensate filling-emptying cycles in pore segments and gas connection–isolation cycles are included. With the pore-level distribution of gas and condensate in hand, we determine their corresponding relative permeabilities. Details of pore space and displacement are subsumed in pore conductances. Solving for the pressure field in each phase, we find a single effective conductance for each phase as a function of condensate saturation. Along with the effective conductance for the saturated network, the relative permeability for each phase is calculated. Our model porous media are two- and three-dimensional regular networks of pore segments with distributed size and square cross-section. With a Monte Carlo sampling we find the optimum network size to avoid size effects and then we investigate the effect of network dimensionality and pore size distribution on the relative permeabilities of gas and condensate.     

Growing actin networks regulated by obstacle size and shape

Growing actin networks provide the driving force for the motility of cells and intracellular pathogens. Based on the molecular-level processes of actin polymerization, branching, capping, and depolymerization, we have devel-oped a modeling framework to simulate the stochastic and cooperative behaviors of growing actin networks in pro-pelling obstacles, with an emphasis on the size and shape effects on work capacity and filament orientation in the grow-ing process. Our results show that the characteristic size of obstacles changes the protrusion power per unit length, with-out influencing the orientation distribution of actin filaments in growing networks. In contrast, the geometry of obstacles has a profound effect on filament patterning, which influ-ences the orientation of filaments differently when the drag coefficient of environment is small, intermediate, or large. We also discuss the role of various parameters, such as the aspect ratio of obstacles, branching rate, and capping rate, in affecting the protrusion power of network growth.     

Pore Connectivity Between Organic and Inorganic Matter in Shale: Network Modeling of Mercury Capillary Pressure

Each of the two major mineral components found in shale samples—organic matter (OM) and inorganic matter (iOM)—has a distinct pore system revealed by scanning electron microscope images, low-pressure nitrogen adsorption, and high-pressure mercury injection tests. Although a vast amount of research has been conducted to detect and measure pore sizes in OM and iOM separately, the connectivity of the pores in these two components remains unclear. In permeability models, pore connectivity between OM and iOM components plays an important role in studying and predicting fluid flow. We studied pore connectivity between OM and iOM by developing pore-network models to mimic the composite nature of distributed OM patches in shale. Input parameters to generate network models were porosity, pore- and throat-size distribution, and total organic content. Mercury injection and capillary-pressure curves were then simulated through generated network models using percolation theory. To study the effects of pore connectivity between OM and iOM, we changed the size and locale of OM patches in the generated network models. Simulation results showed that the locale of OM affects mercury saturation (location and numbers of invaded pores) at given applied pressures. To study the effect of pore-size overlap between OM and iOM pores, we simulated mercury injection for two groups of constructed pore networks: non-overlapping and overlapping. In non-overlapping cases, first all iOM pores were invaded with mercury; then, only OM pores at very high pressure were invaded. In overlapping cases, OM and iOM pores can be invaded simultaneously because some of the pores have similar sizes in both components. The simulated capillary-pressure curves show distinct behavior in the non-overlapping and overlapping cases. Non-overlapping capillary-pressure curves show a sudden increase when OM pores are invaded, whereas overlapping capillary-pressure curves are smoother. Results of this work increase understanding of the connectivity of pores from measured capillary-pressure curves for further implementation in permeability-predictive models.     

Strain gradient plasticity effects in whisker-reinforced metals

A metal reinforced by fibers in the micron range is studied using the strain gradient plasticity theory of Fleck and Hutchinson (J. Mech. Phys. Solids 49 (2001) 2245). Cell-model analyses are used to study the influence of the material length parameters numerically, for both a single parameter version and the multiparameter theory, and significant differences between the predictions of the two models are reported. It is shown that modeling fiber elasticity is important when using the present theories. A significant stiffening effect when compared to conventional models is predicted, which is a result of a significant decrease in the level of plastic strain. Moreover, it is shown that the relative stiffening effect increases with fiber volume fraction. The higher-order nature of the theories allows for different higher-order boundary conditions at the fiber-matrix interface, and these boundary conditions are found to be of importance. Furthermore, the influence of the material length parameters on the stresses along the interface between the fiber and the matrix material is discussed, as well as the stresses within the elastic fiber which are of importance for fiber breakage.     

On the coupling between macroscopic material degradation and interfiber bond fracture in an idealized fiber network

The numerical analysis performed here, using a finite element network model, provides a number of important results regarding the evolution of micro fractures in planar random fiber networks where the only active microscopic fracture mechanism is bond fracture. The fibers are randomly distributed in the network meaning that the network is considered having in-plane isotropic properties on the macroscopic scale. The network is loaded so that, in an average sense, homogenous macroscopic stress and strain fields are present.Several conclusions are drawn. It is found that the development of macroscopic material degradation follows an exponential two-parameter law, consisting of an onset parameter and a fracture rate parameter, justifying a previous theory derived by the authors. The fracture rate parameter is linearly related to the inverse of the bond density above a certain density limit (percolation) and increases with increasing slenderness ratio of the fibers when keeping the bond density at a constant level. The strain energies stored in interfiber bonds are exponentially distributed over the whole network. The numerical analysis reveals that there is a linear relation between the ratio of fractured and initial number of loaded bonds, and the network’s macroscopic material stiffness normalized with its pristine stiffness, confirming earlier findings based on experimental observations. At localization the analyzed theory looses its validity because the fracture process is no longer randomly distributed over the whole network. Localization coincides with location of peak load in force–displacement tensile tests.     

Comparative study of different discrete element models and evaluation of equivalent micromechanical parameters

Comparative studies of different discrete element models of a rock-type material are presented. The discrete element formulation employs spherical particles with the cohesive interaction model combining linear elastic behaviour with brittle failure. Numerical studies consisted in simulation of the uniaxial compression test. Two cylindrical specimens with particle size distributions yielding different degree of heterogeneity have been used. Macroscopic response produced by different discrete element models has been compared. The main difference between the compared models consists in the evaluation of micromechanical constitutive parameters. Two approaches are compared. In the first approach, the contact stiffness and strength parameters depend on the local particle size, while in the second approach, global uniform contact parameters are assumed for all the contacting pairs in function of average geometric measures characterizing the particle assembly. The size dependent contact parameters are calculated as functions of geometric parameters characterizing each contacting particle pair. As geometric scaling parameters, the arithmetic and harmonic means, as well as the minimum of the radii of two contacting particles are considered. Two different models with size dependent contact parameters are formulated. The performance of these models is compared with that of the discrete element model with global uniform contact parameters. Equivalence between the models with size dependent and uniform contact parameters has been checked. In search of this equivalence, different methods of evaluation of global uniform parameters have been studied. The contact stiffness has been evaluated in terms of the average radius of the particle assembly or in terms of the averages of the arithmetic and harmonic means of the contact pair radii, the geometric parameters used in the evaluation of the contact stiffness in the size-dependent models. The uniform contact strengths have been determined as functions of the averages of radii squares, squares of arithmetic radii means or squares of minimum radii of the contacting pairs.For the more homogenous specimen, the models with local size dependent parameters and models with global uniform parameters give similar response. The models with uniform parameters evaluated according to the averages of the geometric parameters used in the evaluation of local parameters ensure better agreement with the respective models with size-dependent parameters than the models with uniform parameters evaluated according to the particle radii. Simulations using the more heterogenous specimen reveal differences between the considered models. There are significant differences in stress–strain curves as well as in the failure pattern. The models with local size-dependent parameters are more sensitive to the change of heterogeneity than the model with global uniform parameters.     

一种新形式的钢纤维混凝土冲击动态本构关系及材料参数的确定

采用?75 mm大口径SHPB系统进行了钢纤维体积率为0%、0.75%、1.5%三种混凝土材料动态性能实验,得出了不同钢纤维含量、不同应变率下的材料应力-应变关系曲线,实验结果表明:随着纤维含量及应变率的增加,钢纤维混凝土材料的峰值应变、峰值应力都随之提高,并在峰值应力之后出现应力的应变软化现象。以此实验结果为基础,提出了一种依赖于应变和应变率相关函数的新型非线性黏塑性动态本构关系,并通过对实验曲线的三步逐次最小二乘优选模拟,得到了相应的材料参数。结果表明,该本构关系对实验数据的模拟效果较好。     

A cohesive model based fragmentation analysis: effects of strain rate and initial defects distribution

A one-dimensional fragmentation analysis that incorporates elastic wave propagation and a cohesive failure process is presented. An irreversible cohesive law models the internal crack nucleation and opening process, and the elastodynamic states of the intact material are calculated using the method of characteristics. Both the average fragment size and the fragment size distribution are obtained. The fragmentation of a model ceramic system is simulated over a wide range of strain rates, and the calculated results are compared to existing theoretical, numerical and experimental results. In the high strain-rate regime, the calculated average fragment size is smaller than that predicted by energy models, but at quasistatic rates the calculated average size is larger than that estimated by such models. The intrinsic mechanisms leading to these deviations are discussed. The fragment size distributions exhibit similarity under all strain-rate range. The effect of the distribution of internal defects on the fragmentation and fragment size distribution is also investigated using this methodology.     

含共晶界面陶瓷复合材料的损伤应变场分析

研究了含共晶界面陶瓷复合材料的损伤应变场及其尺度效应。根据含共晶界面复合陶瓷的细观结构特性,利用含共晶界面陶瓷复合材料中三相胞元内的应力场分布规律,得出棒状共晶体内的无损应变场分布规律。针对棒状共晶体内存在损伤的现象,通过引入损伤变量,利用三相模型法得到了棒状共晶体内存在损伤时的应变场分布规律;根据应变和纤维状夹杂直径之间的关系,分析了棒状共晶体内的损伤应变场及其尺度效应。结果表明,含共晶界面陶瓷复合材料内三相胞元中基体、界面相和纤维夹杂内的损伤应变场对纤维夹杂直径具有明显的尺度效应。     

Evolution of bond fractures in a randomly distributed fiber network

Fracture in a planar randomly ordered fiber network subjected to approximately homogenous macroscopic stress and strain field is considered. A theory describing material degradation on a macroscopic scale is derived via Griffith’s energy balance for an internal fractured area in the network assuming the active fracture process on the microscopic level is fiber–fiber bond breakage. Attention is confined to a purely mechanical theory assuming isothermal processes and the theory relies on equations commonly used in theories of statistical physics. In the theory, a bond breaking driving force is stated to be equal to the elastic strain energy density of a non-fractured network. A debond fraction can be coupled to a linearly decrease of the network’s macroscopic stiffness. The rate of the fracture processes is determined by the network’s inherent properties (bond and fiber density, bond strength, etc.). During the loading process, until onset of localization, the bond breaks occur at randomly distributed locations spread over the fiber network and the theory estimate material degradation on a macroscopic level. When localization takes place, the fracture process changes from a two-dimensional randomly distributed process to a one-dimensional process and other theories have to be included to describe post-localization behavior. An approximately in-plane isotropic low-density paper is used in tensile experiments while monitoring acoustic emission activity to evaluate the theory. The experimentally obtained results support the theory surprisingly well.     

A novel deterministic hybrid complex network model created by inner–outer iteration

Complex networks are ubiquitous in real-life systems. Most previous models of complex networks are stochastic. In order to decrease the randomness and make it more direct to gain a visual understanding on complex network evolving mechanism, we present a deterministic algorithm that generates a hybrid network model with the characteristics of small-world networks and random networks. In this model, the network growth is determined by triangle inner and outer node and edge iterations. We analyze the main topological properties by both theoretical predictions and numerical simulations. The results show that our growing model has a low average path length, a high clustering and an exponential degree distribution.     

基于规则网络的碳酸盐岩多尺度网络模型构建方法研究

碳酸盐岩油藏非均质性强,孔隙大小变化可达好几个数量级,描述碳酸盐岩油藏多尺度孔隙特征具有重要意义.本文首先基于三维规则网络模型建立了不同物理尺寸的溶洞网络、大孔隙网络和微孔隙网络;然后提出一种耦合算法,以溶洞网络为基础,通过添加适当比例的大孔隙和微孔隙,构建出碳酸盐岩多尺度网络模型;最后对比分析了各网络模型的几何性质、拓扑性质和绝对渗透率.结果表明,碳酸盐岩多尺度网络模型能够同时描述不同尺度孔隙的几何和拓扑特征;且相比各单一尺度的孔隙网络模型,多尺度网络模型有着较高的绝对渗透率,这是由于各尺度孔隙之间的相互连通极大地提高了网络的整体连通性和流动能力,为碳酸盐岩油藏微观渗流模拟提供了重要的研究平台.