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.     

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.     

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.     

On the transverse compression response of Kevlar KM2 using fiber-level finite element model

Flexible textile composites like woven Kevlar fabrics are widely used in high velocity impact (HVI) applications. Upon HVI they are subjected to both longitudinal tensile and transverse compressive loads. To understand the role of transverse properties, the single fiber and tow transverse compression response (SFTCR and TTCR) of Kevlar KM2 fibers are numerically analyzed using plane strain finite element (FE) models. A finite strain formulation with a minimum number of 84 finite elements is determined to be required for the fiber cross section to capture the finite strain SFTCR through a mesh convergence study. Comparison of converged numerical solution to the experimental results indicates the dominant role of geometric stiffening at finite strains due to growth in contact width. The TTCR is studied using a fiber length scale FE model of a single tow comprised of 400 fibers transversely loaded between rigid platens. This study along with micrographs of yarn after mechanical compaction illustrates fiber spreading and fiber–fiber contact friction interactions are important deformation mechanisms at finite strains. The TTCR is also studied using homogenized yarn level models with properties from the literature. Comparison of TTCR between fiber length scale and homogenized yarn length scale models indicate the need for a nonlinear material model for homogenized approaches to accurately predict the transverse compression response of the fabrics.     

Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements

This paper presents experimental studies on the mechanical behavior of fiber fabrics using a biaxial tensile device based on two deformable parallelograms. The cross-shaped specimens are well adapted to fabrics because of their lack of shear stiffness. Tension versus deformation curves, for different strain ratios, are obtained in the case of composite woven reinforcements used in aeronautic applications. It is shown that the tensile behavior of the fabric is strongly nonlinear due to the weaving undulations and the yarn contraction, and that the phenomenon is clearly biaxial. A constitutive model is described and identified from the experimental data. The essential role played by the yarn crushing will be pointed out.     

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.     

Equivalent mechanical properties of textile monolayers from discrete asymptotic homogenization

The determination of the effective mechanical moduli of textiles from mechanical measurements is usually difficult due to their discrete architecture, which makes micromechanical analyses a relevant alternative to access those properties. Micropolar continuum models describing the effective mechanical behavior of woven fabric monolayers are constructed from the homogenization of an identified repetitive pattern of the textile within a representative unit cell. The interwoven yarns within the textile are represented as a network of trusses connected by nodes at their crossover points. These trusses have extensional and bending rigidities to allow for yarn stretching and flexion, and a transverse shear deformation is additionally considered. Interactions between yarns at the crossover points are captured by beam segments connecting the nodes. The woven fabric is modeled after homogenization as an anisotropic planar continuum with two preferred material directions in the mean plane of the textile. Based on the developed methodology, the effective mechanical properties of plain weave and twill are evaluated, including their bending moduli and characteristic flexural lengths. A satisfactory agreement is obtained between the effective moduli obtained by homogenization and numerical values obtained by finite element simulations performed over periodic unit cells.     

Numerical investigation of the slope discontinuities in large deformation finite element formulations

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.     

Reduced order model of offshore wind turbine wake by proper orthogonal decomposition

A wide range of previously designed methods for faster parametrization of partial differential equations requires them to be solved using existing finite volume, finite element, and finite difference solvers. Due to the requirement of high degrees of freedom to accurately model the physical system, computational costs often becomes a bottle-neck. It poses challenges to conducting efficient repeated parametric sampling of the input parameter that disrupts the whole design process. Model reduction techniques adopted to high fidelity systems provide a basis to accurately represent a physical system with a lower degree of freedom. The present work focuses on one such method for high-fidelity simulations that combines finite volume strategy with proper orthogonal decomposition and Galerkin projection to test reduced-order models for high Reynolds number flow applications. The model is first benchmarked against flow around a cylinder for which extensive numerical and experimental data is available in the literature. The models are then tested to full-scale NREL 5MW offshore wind turbines to evaluate wake evolution in the downstream direction. The simulations results show relative errors of wind turbines for the first seventy modes approach 4.7% in L²-norm for velocities.     

预张力纤维织物超高速碰撞热-力学特性分析

高性能纤维织物承力层承担充气舱的内压载荷,并为充气舱提供空间碎片防护。充气舱内压载荷将导致纤维织物承力层产生预张力,并对纤维织物的空间碎片超高速碰撞特性产生显著影响,从而影响其空间碎片防护性能。为分析预张力对纤维织物超高速碰撞过程中热-力学特性的影响,采用Johnson-Cook强度模型和Mie-Grüneisen状态方程建立了纤维材料热-力耦合材料模型,利用有限元法-光滑粒子流体动力学耦合算法对纤维织物的纱线编织结构进行离散建模,并通过施加张力载荷实现纤维织物靶板的预拉伸,进而建立了预张力纤维织物超高速碰撞数值模型,分析并得到了预张力作用下纤维织物超高速碰撞热-力学特性及空间碎片防护性能。结果表明:在弹丸超高速碰撞下,随着预张力的提高,纤维织物穿孔面积增大,碎片云扩散角减小,弹丸动能吸收率降低,碰撞区域温度降低。预张力的存在显著降低了纤维织物的空间碎片防护性能。     

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).     

Multi-scale modelling of strongly heterogeneous 3D composite structures using spatial Voronoi tessellation

3D composite materials are characterized by complex internal yarn architectures, leading to complex deformation and failure development mechanisms. Net-shaped preforms, which are originally periodic in nature, lose their periodicity when the fabric is draped, deformed on a tool, and consolidated to create geometrically complex composite components. As a result, the internal yarn architecture, which dominates the mechanical behaviour, becomes dependent on the structural geometry. Hence, predicting the mechanical behaviour of 3D composites requires an accurate representation of the yarn architecture within structural scale models. When applied to 3D composites, conventional finite element modelling techniques are limited to either homogenised properties at the structural scale, or the unit cell scale for a more detailed material property definition. Consequently, these models fail to capture the complex phenomena occurring across multiple length scales and their effects on a 3D composite’s mechanical response. Here a multi-scale modelling approach based on a 3D spatial Voronoi tessellation is proposed. The model creates an intermediate length scale suitable for homogenisation to deal with the non-periodic nature of the final material. Information is passed between the different length scales to allow for the effect of the structural geometry to be taken into account on the smaller scales. The stiffness and surface strain predictions from the proposed model have been found to be in good agreement with experimental results.The proposed modelling framework has been used to gain important insight into the behaviour of this category of materials. It has been observed that the strain and stress distributions are strongly dependent on the internal yarn architecture and consequently on the final component geometry. Even for simple coupon tests, the internal architecture and geometric effects dominate the mechanical response. Consequently, the behaviour of 3D woven composites should be considered to be a structure specific response rather than generic homogenised material properties.     

Nonlinear transient dynamic response of damped plates using a higher order shear deformation theory

Damped transient dynamic elasto-plastic analysis of plate is investigated. A finite element model based on a C ⁰ higher order shear deformation theory has been developed. Nine noded Lagrangian elements with five degrees of freedom per node are used. Selective Gauss integration is used to evaluate energy terms so as to avoid shear locking and spurious mechanisms. Von Mises and Tresca yield criteria are incorporated along with associated flow rules. Explicit central difference time stepping scheme is employed to integrate temporal equations. The mass matrix is diagonalized by using the efficient proportional mass lumping scheme. A program is developed for damped transient dynamic finite element analysis of elasto-plastic plate. Several numerical examples are studied to unfold different facets of damping of elasto-plastic plates.     

Buckling analysis of plain knitted fabric sheets under simple shear in an arbitrary direction

Knitting structures make plain knitted fabric different from woven fabric. With the aid of a micro-constitutive model the buckling of a knitted fabric sheet subjected to simple shear in an arbitrary direction is investigated. The large deformation of the fabric sheet in its critical configuration is considered. The theory of stability for finite deformations is applied to the analysis. All the stress boundary conditions of the knitted fabric sheet are satisfied. An equation for determining the buckling direction angle is derived. It is shown that there are two possible buckling modes: a flexural mode and a barreling mode. The buckling conditions for the two modes are also obtained, respectively. A numerical calculation reveals that only the flexural mode can occur, which agrees with experimental observations.     

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     

具有不同功能梯度分布函数的板件的三维分析

发展了一种新颖的功能梯度结构分析的细观元法。细观元法对结构的常规有限单元内部设置密集细观单元以反映材料特性梯度变化,又通过协调条件将各细观元结点自由度转换为同一常规有限元自由度,再上机计算。这种细观元法既能充分反映材料功能梯度变化特性;而其计算单元又和常规有限元一样,是一种针对功能梯度结构分析的有效数值方法。文中通过细观元技术进行计算、分析,给出了具有不同功能梯度分布函数板件的力学量三维分布形态。     

Use of B-spline surface to model large-deformation continuum plates: procedure and applications

The absolute nodal coordinate formulation (ANCF) has been used in the analysis of large deformation of flexible multibody systems that encompass belt drive, rotor blade, and cable applications. As demonstrated in the literature, the ANCF finite elements are ideal for isogeometric analysis. The purpose of this investigation is to establish a relationship between the B-splines, which are widely used in the geometric modeling, and the ANCF finite elements in order to construct continuum models of large-deformation geometries. This paper proposes a simplified approach to map the B-spline surfaces into ANCF thin plate elements. Matrix representation of the mapping process is established and examined through numerical examples successfully. The matrix representation of the mapping process is used because of its suitability of computer coding and to minimize the calculation time. The error estimation is carried out by analyzing the gap between the points of each ANCF element and the corresponding points of the portion of the B-spline surface. The Hausdorff distance is used to study the effect of the number of control points, the degree of interpolation, and the knot multiplicity on the mapped geometry. It is found that cubic interpolation is recommended for optimizing the accuracy of mapping the B-spline surface to ANCF thin plate elements. It is found that thin plate element in ANCF missing a number of basis functions which considered a source of error between the two surfaces, as well as it does not allow to converting the ANCF thin plate elements model to B-spline surface. In this investigation, an application example of modeling large-size wind turbine blade with uniform structure is illustrated. The use of the continuum plate elements in modeling flexible blades is more efficient because of the relative scale between the plate thickness and its length and width and the high flexibility of its structure. The numerical results are compared with the results of ANSYS code with a good agreement. The dynamic simulation for mapped surface model shows a numerical convergence, which ensures the ability of using the proposed approach for applications of dynamics for design and computer-aided design.     

Transient elastodynamic analysis of two-dimensional structures using Coons-patch macroelements

Recently, Coons’ interpolation was used for the construction of large finite elements with degrees of freedom appearing mostly along the boundaries of a structure. So far, these so-called Coons-patch macroelements were successfully applied to the analysis of two-dimensional and axisymmetric elastic structures as well as potential problems including Poisson equation and acoustics. Now, this paper continues the research by investigating their applicability and performance in calculating the propagation of elastic waves within continua due to sudden loads. Explicit (central difference) and implicit (θ-Wilson) time-integration schemes have been successfully applied to four typical model problems in conjunction with the proposed Coons-patch macroelements—without and with substructuring—and the results are successfully compared with conventional finite elements having the same number of nodes along the boundary. Finally, theoretical issues between the proposed global technique and well-established computational methods are discussed.     

Dynamics of finite element structural models with multiple unilateral constraints

Fast and accurate simulation of mechanical structures with complex geometry requires application of the finite element method. This leads frequently to models with a relatively large number of degrees of freedom, which may also possess non-linear properties. Things become more complicated for systems involving unilateral contact and friction. In classical structural dynamics approaches, such constraints are usually modeled by special contact elements. The characteristics of these elements must be selected in a delicate way, but even so the success of these methods cannot be guaranteed. This study presents a numerical methodology, which is suitable for determining dynamic response of large scale finite element models of mechanical systems with multiple unilateral constraints. The method developed is based on a proper combination of results from two classes of direct integration methodologies. The first one includes standard methods employed in determining dynamic response of structural models possessing smooth non-linearities. The second class of methods includes specialized methodologies that simulate the response of dynamical systems with unilateral constraints. The validity and effectiveness of the methodology developed is illustrated by numerical results.     

A finite deformation theory of strain gradient plasticity

Plastic deformation exhibits strong size dependence at the micron scale, as observed in micro-torsion, bending, and indentation experiments. Classical plasticity theories, which possess no internal material lengths, cannot explain this size dependence. Based on dislocation mechanics, strain gradient plasticity theories have been developed for micron-scale applications. These theories, however, have been limited to infinitesimal deformation, even though the micro-scale experiments involve rather large strains and rotations. In this paper, we propose a finite deformation theory of strain gradient plasticity. The kinematics relations (including strain gradients), equilibrium equations, and constitutive laws are expressed in the reference configuration. The finite deformation strain gradient theory is used to model micro-indentation with results agreeing very well with the experimental data. We show that the finite deformation effect is not very significant for modeling micro-indentation experiments.