Consideration of the yarn–yarn interactions in meso/macro discrete model of fabric. Part I: Single yarn behaviour

A mesoscopic discrete model of dry fabric has been developed, based on the yarn–yarn interactions occurring at the yarns crossing points. The fabric yarns, described initially by a Fourier series development, are discretized into elastic straight bars represented by stretching springs and connected at frictionless hinges by rotational springs. The motion of each node is described by a lateral displacement and a rotation. The expression of the reaction force exerted by the transverse yarns at the contact points is assessed, from which the work of the reaction forces is established. The equilibrium shape of the yarn is obtained as the minimum of its total potential energy, accounting for the work of the reaction forces due to the transverse yarns. Simulations of a traction curve of a single yarn are performed, that evidence the effect of the yarn interactions. The two principal deformation mechanisms, the variation of undulation and the yarn stretching, are separately analysed.     

Equivalent properties of monolayer fabric from mesoscopic modelling strategies

A general and systematic approach for the development of mesostructurally-based continuum model of woven fabrics has been elaborated, relating the fabric behavior at the macroscopic continuum scale to the response and geometry of the fabric’s mesostructure (geometrical configuration of the weave and the yarn properties). Mesoscopic discrete models of dry fabric have been developed based on a discretization of the yarn geometry, accounting for the yarn–yarn interactions at the yarns crossing points. The yarns are modeled within a unit cell consisting of the repetitive fabric pattern as curved planar beams submitted to the reaction forces of the transverse yarns at discrete crossover points. Those reaction forces are expressed in semi-analytical form versus the yarn geometry and mechanical properties for general armour from beam theory. The equilibrium shape of the woven fabric is obtained by minimizing its total potential energy, accounting for the work of the reaction forces due to the transverse yarns. The absolute minimum of the structure’s total potential energy is achieved by a classical genetic algorithm. Simulation results show that plain weave presents a nonlinear response in the early deformation stage due to the crimp change, whereas twill shows a quasi linear response due to yarn extension being the dominant deformation mechanism. Plain weave fabric overall exhibits an orthotropic constitutive law, as biaxial simulations show. The transverse behavior of plain weave fabric is presently evaluated in terms of Poisson’s ratio, based on virtual simulations at the mesoscopic scale of analysis. Simulation results show that Poisson’s ratio first increases towards a maximum due to the rapid shrinkage of the sample in the transverse direction, and decreases thereafter when the crimp changes become limited by the reaction forces of the transverse yarns. The influence of the mechanical properties of both warp and weft on Poisson’s coefficient is assessed. The predictions of the mesoscopic models regarding the impact of yarn geometry and mechanical properties on the overall behavior provide a guideline for the design of woven fabrics.     

Un modèle discret du couplage entre les fils dans une structure tisséeA discrete model for the coupling between the yarns in a woven fabric

The coupling between yarns in a piece of fabric has been analysed at the mesoscopic scale, in terms of its impact on the macroscopic unidirectional behaviour. Starting from a discrete model of a woven structure associated to a variational formulation of the equilibrium of the structure, the coupling between both yarns is introduced, the potential energy of which is calculated. The initial shape of the yarn, represented by a planar undulated beam supposed to be periodic, is described by a Fourier series. The coefficients of the series are expressed vs. the contact force exerted at the top of the undulations, and vs. the mechanical properties of the solicited yarn. The contact force is then expressed vs. the mechanical properties of the transverse yarn and vs. the vertical displacement of the contact point. The potential energy of the coupling is then built, assuming the continuity of the displacement at the contact points. The equilibrium shape of the yarn submitted to unidirectional traction is obtained numerically as the minimum of the total potential energy. The simulated traction curve reproduces in a satisfactorily manner the observed behaviour. The respective contributions of the flexional and extensional effects of the yarn are analysed. The consideration of the coupling enhances the rigidity of the response of the yarn; one demonstrates the effect of the geometrical and mechanical parameters of the transverse yarn. To cite this article: B. Ben Boubaker et al., C. R. Mecanique 331 (2003).     

Local elastodynamic stresses in the unit cell of a woven fabric composite

Summary A theoretical study of the local elastodynamic stresses of woven fabric composites under dynamic loadings is presented in this article. The analysis focuses on the unit cell of an orthogonal woven fabric composite, which is composed of two sets of mutually orthogonal yarns of either the same fiber (nonhybrid fabric) or different fibers (hybrid fabric) in a matrix material. Using the mosaic model for simplifying woven fabric composites and a shear lag approach to account for the inter-yarn deformation, a one-dimensional analysis has been developed to predict the local elastodynamic and elastostatic behavior. The initial and boundary value problems are formulated and then solved using Laplace transforms. Closed form solutions of the dynamic displacements and stresses in each yarn and the bond shearing stresses at the interfaces between adjacent yarns are obtained in the time domain for any type of in-plane impact loadings. When time tends to infinity, the dynamic solutions approach to their corresponding static solutions, which are also developed in this article. Solutions of certain special cases are identical to those reported in the literature. Lastly, the dynamic stresses and bond shearing stresses of plain weave composites subjected to step uniform impacts are presented and discussed as an example of the general analytical model. Received 3 May 1999; accepted for publication 22 September 1999     

On the finite element analysis of woven fabric impact using multiscale modeling techniques

Finite element modeling of the impact of flexible woven fabrics using a yarn level architecture allows the capturing of complex projectile-fabric and yarn–yarn level interactions, however it requires very large computational resources. This paper presents a multiscale modeling technique to simulate the impact of flexible woven fabrics. This technique involves modeling the fabric using a yarn level architecture around the impact region and a homogenized or membrane type architecture at far field regions. The level of modeling resolution decreases with distance away from the impact zone. This results in a finite element model with much lower computational requirements. The yarns are modeled using both solid and shell finite elements. Impedances are matched across all interfaces created between the various regions of the model to prevent artificial reflections of the longitudinal strain waves. A systematic approach is presented to determine geometric and material parameters of the homogenized zone. The multiscale model is extensively validated against baseline models. The limitations of using shell elements to model the yarn level architecture underneath the projectile are addressed.     

Impact of woven fabric: Experiments and mesostructure-based continuum-level simulations

Woven fabric is an increasingly important component of many defense and commercial systems, including deployable structures, restraint systems, numerous forms of protective armor, and a variety of structural applications where it serves as the reinforcement phase of composite materials. With the prevalence of these systems and the desire to explore new applications, a comprehensive, computationally efficient model for the deformation of woven fabrics is needed. However, modeling woven fabrics is difficult due, in particular, to the need to simulate the response both at the scale of the entire fabric and at the meso-level, the scale of the yarns that compose the weave. Here, we present finite elements for the simulation of the three-dimensional, high-rate deformation of woven fabric. We employ a continuum-level modeling technique that, through the use of an appropriate unit cell, captures the evolution of the mesostructure of the fabric without explicitly modeling every yarn. Displacement degrees of freedom and degrees of freedom representing the change in crimp amplitude of each yarn family fully determine the deformed geometry of the mesostructure of the fabric, which in turn provides, through the constitutive relations, the internal nodal forces. In order to verify the accuracy of the elements, instrumented ballistic impact experiments with projectile velocities of 22-550 m/s were conducted on single layers of Kevlar^® fabric. Simulations of the experiments demonstrate that the finite elements are capable of efficiently simulating large, complex structures.     

Fatigue crack opening stress based on the strip-yield model

The modified strip-yield model based on the Dugdale model and two-dimensional approximate weight function method were utilized to evaluate the effect of in-plane constraint, transverse stress, on the fatigue crack closure. The plastic zone sizes and the crack opening stresses considering transverse stress were calculated for four specimens: single edge-notched tension (SENT) specimen, single edge-notched bend (SENB) specimen, center-cracked tension (CCT) specimen, double edge-notched tension (DENT) specimen under uniaxial loading. And the crack opening behavior of the center-cracked specimen under biaxial loading was also evaluated. Normalized crack opening stresses σ_op/σ_max for four specimens were successfully described by the normalized plastic zone parameter Δω^′_rev/ω^′ considering transverse stress, where Δω^′_rev and ω^′ are the size of the reversed plastic zone at the moment of first crack tip closure and the size of the forward plastic zone for maximum stress, respectively. The normalized plastic zone parameter with transverse stress also was satisfactorily correlated with the behavior of crack closure for CCT specimen under biaxial loading.     

An evaluation of a combined isotropic-kinematic hardening model for representation of complex strain-path changes in dual-phase steel

This paper aims at evaluating an elastoplastic constitutive model which accounts for combined isotropic-kinematic hardening for complex strain-path changes in a dual-phase steel, DP800. The capability of the model to reproduce the transient hardening phenomena under two-stage non-proportional loading has been assessed through numerical simulations of sequential uniaxial tension and notched tension/shear tests. Finite element simulations with shell elements were performed using the explicit non-linear FE code LS-DYNA. Numerical predictions of the stress–strain response were compared to the corresponding experimental data. The results from the experiments demonstrated that prior plastic deformation has certainly influenced the subsequent work-hardening behaviour of the material under biaxial or shear deformation modes. Furthermore, the numerical simulations from the two-stage uniaxial tension–notched tension and uniaxial tension–shear tests predicted the general trends of the experimental results such as transitory hardening and overall work hardening. However, some discrepancies were found in accurately describing the transient hardening behaviour subsequent to strain path changes between the experiments and numerical simulations.     

酚醛层压材料的冲击力学行为及本构模型

为了研究酚醛层压材料的冲击力学行为并获得本构模型,利用万能试验机和整形修正的分离式霍普金森压杆(SHPB)装置,对材料试样进行了应变率范围为10-3~103 s-1的单轴压缩实验,得到了不同加载应变率下的应力应变曲线,对其在准静态、动态载荷下的压缩破坏机理进行了初步探讨。结果表明,酚醛层压材料具有较强的应变率效应,与准静态(1.67×10-3 s-1)时相比,在动态载荷(7×102 s-1)下,峰值应力增加了约10倍;破坏应变减少了约一半;在准静态和动态加载条件下试样力学性能的差异是由于纤维基体界面特性以及不同应变率下破坏模式的不同;采用朱-王-唐本构方程描述了酚醛层压材料力学行为,拟合得到了本构方程的系数,在加载过程中,理论计算值与实验结果吻合较好。     

The influence of biaxial stresses on high-cycle fatigue-crack propagation

Arising from a design study, an examination has been made of the problems associated with evaluating the fatigue behavior of an I-beam section joined to a transverse stiffener, loaded in biaxial bending. A laboratory test rig has been designed to study high-frequency (40–230 Hz) fatigue-crack propagation in the tensile flange of a composite aluminum I-section, for a range of stress biaxiality ratios (λ) from 0 to +1 (equibiaxial tension), λ being varied by adjusting the ratios of loading spans in the two orthogonal axes for the same amount of uniaxial deflection. The results obtained are considered in the context of the significantly contradictory information currently available in the literature. Crack geometry and test procedure (as influenced by load/stress measurement) are found to considerably influence the effect of biaxial stresses on fatigue. Thus, the growth rate of corner-initiated cracks increases with increasing stress biaxiality (monitored in terms of nominal applied stresses), but the reverse is true for center-cracked specimens. When tests are conducted in terms of combinations of local stresses, or data reduced using corrections, there is no significant effect of biaxial stresses on fatigue-crack growth.     

脆性材料在双向应力下的断裂实验与理论分析

研究了脆性材料在双向应力下的断裂特性和失效机理，特别是在平行于裂纹的应力对临界断裂参数的影响方面进行了实验上和理论上的研究．采用玻璃、陶瓷等脆性材料进行了平面双向拉伸和单向拉伸试验，并对实验结果进行比较．观测直通裂纹的启裂和扩展过程，证明了双向应力对裂纹驱动力有明显影响，讨论了裂纹扩展的应变准则．     

Compressive response and failure of braided textile composites: Part 1—experiments

Experimental results obtained by examining the planar biaxial compression/tension response of carbon 2D triaxial braided composites (2DTBC) are reported in this paper. These experiments were motivated by a need to examine the failure of 2DTBC in a state of stress that would be similar to what is experienced by the walls of a tubular member under compressive crush loads. Results obtained from a series of biaxial tests that were conducted with different proportional displacement loading ratio combinations of compression and tension are reported. In all cases, the dominant failure mechanism under such a stress state is the buckling of the bias and axial tows within the composite. Full field surface displacement data is acquired concurrently during all biaxial and some uniaxial tests using the technique of digital speckle photography. Digital images of the specimen surface that is illuminated with a He-Ne laser are acquired at discrete time intervals during the loading history using a high-resolution digital camera. These images are stored and analyzed to obtain the incremental inplane surface displacement field, Δu(x,y) and Δv(x,y). From these, the incremental inplane surface strains Δε_x, Δε_y and Δγ_xy are obtained by numerical differentiation. The present paper, which is the first in a two part series, is devoted to the biaxial experimental results pertaining to 2DTBC failure.     

In-plane shear testing of graphite-woven fabric composites

The losipescu shear test method was used to determine the in-plane shear response of AS4 and Celion carbon fiber/epoxy fabric composite materials. Several weave architectures were studied: AS4 uniweave, AS4 and Celion plain weaves, Celion 5-harness and 8-harness satin weaves. Specimens were tested using traditional strain gage techniques and full-field moiré interferometry. A full-node localized hybrid analysis is introduced to perform efficient reduction of moiré data, producing whole-field strain distributions in the specimen test section. It was found that the fabric yarn size greatly influenced the uniformity of the shear field in the specimen test section. However, consistent shear moduli still can be obtained using the modified losipescu specimen and Wyoming fixture except for fabrics with large fiber yarns.     

Inelastic behavior of an AS4/PEEK composite under combined transverse compression and shear. Part I: experiments

The nonlinear behavior in shear and transverse compression of unidirectional AS4/PEEK and their interaction are investigated experimentally. The composite is rate dependent even at room temperature and its rate exponent is similar to that of neat PEEK. The material is tested under pure shear, pure compression and under biaxial loading histories. The biaxial tests are performed in a custom facility on thin strips of the material. The facility allows freedom to choose the loading path in the biaxial stress and strain spaces of interest. Tests are performed for three biaxial loading paths. In the first, the specimen is sheared then compressed while the shear stress is held constant; in the second, the specimen is compressed then sheared while the compressive stress is held constant; and in the third, the specimen is loaded simultaneously by proportional amounts of compression and shear. It was found that the induced deformation is influenced significantly by the loading history followed. Also, initial loading in shear or compression has only a modest effect on subsequent loading of the other type. An unorthodox yielding behavior for the composite results from this lack of interaction. Finally, the stresses at failure are found to trace an elliptical path in the shear–transverse compression plane, but the failure stress state is not significantly affected by the loading path followed.     

多轴加载下混凝土细观破坏模拟的强度准则探讨

实际工程结构中混凝土材料大多处于双轴或三轴的复杂应力状态,已有的细观力学数值研究工作大多针对单轴加载问题,对于双轴或者三轴加载条件下混凝土破坏模拟的研究相对较少。复杂受力条件下的混凝土材料破坏模拟中,细观组分强度准则选取的合理与否将成为混凝土破坏模式及宏观力学性能数值研究准确和成功与否的关键。本文旨在探讨单轴强度准则,如最大拉应变准则在多轴加载条件下混凝土破坏过程研究中运用的合理性。鉴于此,首先在细观尺度上建立了混凝土试件的二维随机骨料模型,分别采用弹性损伤本构关系模型及塑性损伤本构关系模型来描述细观组分(即砂浆基质)的力学性能,对双轴加载条件下混凝土的细观破坏过程进行数值模拟,对比了单轴强度准则和多轴强度准则下混凝土试件破坏路径及宏观应力-应变关系的差异。数值结果表明,简单的单轴强度准则难以反映双轴加载下混凝土内部应力状态的复杂性,不宜采用单轴强度准则来描述多轴加载下混凝土的破坏行为。     

A continuum constitutive model for the mechanical behavior of woven fabrics

We propose a new approach for developing continuum models for the mechanical behavior of woven fabrics in planar deformation. We generate a physically motivated continuum model that can both simulate existing fabrics and predict the behavior of novel fabrics based on the properties of the yarns and the weave. The approach relies on the selection of a geometric model for the fabric weave, coupled with constitutive models for the yarn behaviors. The fabric structural configuration is related to the macroscopic deformation through an energy minimization method, and is used to calculate the internal forces carried by the yarn families. The macroscopic stresses are determined from the internal forces using equilibrium arguments. Using this approach, we develop a model for plain weave ballistic fabrics, such as Kevlar®, based on a pin-joined beam geometry. We implement this model into the finite element code ABAQUS and simulate fabrics under different modes of deformation. We present comparisons between model predictions and experimental findings for quasi-static modes of in-plane loading.     

Ratchetting under tension–torsion loadings: experiments and modelling

This paper is concerned with the mechanical behaviour of 316 austenitic stainless steel under multiaxial loadings and particular attention is paid to ratchetting under tension–torsion non-proportional loadings. First, a series of uniaxial tests and biaxial tests has been carried out in order to calibrate five different cyclic plasticity models based on an isotropic hardening rule and a non-linear kinematic hardening rule. It is shown that this class of models gives quite good agreement between the experimental and numerical results. Second, another series of ratchetting tests has been carried out under tension–torsion loadings in order to test the prediction capacities of the previous models. It is shown that whereas the models have been calibrated with similar loading paths, four of the five selected models give poor predictions.     

Temperature dependent bulge test for elastomers

The bulge test is a particularly convenient testing method for characterizing elastomers under biaxial loading. In addition, it is convenient to utilize this test for validating material models in simulation due to the heterogeneous strain field induced during inflation. During the bulge test the strain field for elastomers covers uniaxial tension at the border to pure shear and equibiaxial tension at the pole. Elastomeric materials exhibit a hyperelastic material behavior, with a dependency on temperature and loading rate. The temperature effect on the mechanical behavior during biaxial loading is considered in the present study. A bulge test setup combined with a temperature chamber is developed in order to characterize this effect, and an exemplary temperature dependent characterization of a poly(norbornene) elastomer is performed with this setup. The equibiaxial stress–strain curves measured at 60 °C, 20 °C and −20 °C are presented.     

Plane-stress fracture testing of finite sheets under biaxial loads

A procedure which combines the Williams series-type stress- and displacement-field expressions at the crack-tip neighborhood with a suitable numerical scheme away from the crack-tip was employed in the determination of the plane-stress fracture properties of four finite 7076-T6 aluminum sheets containing cracks emanating from a circular hole under four biaxial loads. The compatibility of the analytical and numerical displacements at the nodal points along the boundary of the crack-tip neighborhood was utilized in formulating displacement-continuity expressions containing some undetermined constants which solution depends on the nature of the boundary loading conditions. By linear superposition of the displacement due to remote uniaxial load and the displacements due to remotely applied transverse load in the neighborhood of the crack-tip, biaxial-displacement-continuity expressions containing these important fracture properties—namely, the opening Mode I stress-intensity factorK, the nonsingular stress term associated with the stresses in the direction parallel to the plane of cracksA and the integration termB associated with the displacement in this direction—were evaluated. Because no known biaxial testing of this geometry had been reported prior to this research, the analytical procedure was used to select the optimum geometry required in a biaxial fracture test of a finite-sheet specimen containing cracks emanating from a circular hole. This geometric optimization of the specimen guaranteed uniformity of stress all over the volume of specimen and also made the alteration of the existing MTS test fixtures unnecessary. Four square sheets of 7075-T6 aluminum alloy containing a central hole with two collinear cracks emanating radially at the edge of the hole were then fabricated in accordance with the analytically determined geometric requirements. The biaxial fracture test was then conducted under four biaxial load factors (λ) of 0.0, 0.5, 1.0 and 1.5. The fracture toughness obtained in this research was compared with those reported for uniaxial loading of large panels. It was found that there is a good correlation between the reported fracture toughness and this work.