Finite element modelling of thermally bonded bicomponent fibre nonwovens: Tensile behaviour

Nonwovens are polymer-based engineered textiles with a random microstructure and hence require a numerical model to predict their mechanical performance. This paper focuses on finite element (FE) modelling the elastic–plastic mechanical response of polymer-based core/sheath type thermally bonded bicomponent fibre nonwoven materials. The nonwoven fabric is treated as an assembly of two regions having distinct mechanical properties: fibre matrix and bond points. The fibre matrix is composed of randomly oriented core/sheath type fibres acting as load-transfer link between bond points. Random orientation of individual fibres is introduced into the model in terms of the orientation distribution function (ODF) in order to determine the material’s anisotropy. The ODF is obtained by analysing the data acquired with scanning electron microscopy (SEM) and X-ray micro computed tomography (CT). On the other hand, bond points are treated as a deformable bicomponent composite material composed of the sheath material as matrix and the core material as fibres having random orientations. An algorithm is developed to calculate the anisotropic material properties of these regions based on properties of fibres and manufacturing parameters such as the planar density, core/sheath ratio and fibre diameter. Having distinct anisotropic mechanical properties for two regions, the fabric is modelled with shell elements with thicknesses identical to those of the bond points and fibre matrix. Finally, nonwoven specimens are subjected to tensile tests along different loading directions with respect to the machine direction of the fabric. The force–displacement curves obtained in these tests are compared with the results of FE simulations.     

A generalized orthotropic hyperelastic material model with application to incompressible shells

In the present paper a new orthotropic hyperelastic constitutive model is proposed which can be applied to the numerical simulation of a wide range of anisotropic materials and particularly biological soft tissues. The model represents a non‐linear extension of the orthotropic St. Venant–Kirchhoff material and is described in each principal material direction by an arbitrary isotropic tensor function coupled with the corresponding structural tensor. In the special case of isotropy this constitutive formulation reduces to the Valanis–Landel hypothesis and may therefore be considered as its generalization to the case of orthotropy. Constitutive relations and tangent moduli of the model are expressed in terms of eigenvalue bases of the right Cauchy–Green tensor C and obtained for the case of distinct and coinciding eigenvalues as well. For the analysis of shells the model is then coupled with a six (five in incompressible case) parametric shell kinematics able to deal with large strains as well as finite rotations. The application of the developed finite shell element is finally illustrated by a number of numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.     

The relationship between the elasticity tensor and the fabric tensor

An algebraic relationship between the fourth rank elasticity tensor of a porous, anisotropic, linear elastic material and the fabric tensor of the material is considered. The fabric tensor is a symmetric second rank tensor which characterizes the geometric arrangement of the porous material microstructure. In developing this result it is assumed that the matrix material of the porous elastic solid is isotropic and, thus, that the anisotropy of the porous elastic solid is determined by the fabric tensor. It is then shown that the material symmetries of orthotropy, transverse isotropy and isotropy correspond to the cases of three, two and one distinct eigenvalues of the fabric tensor, respectively.     

Macroscopic model with anisotropy based on micro–macro information

Physical experiments can characterize the elastic response of granular materials in terms of macroscopic state variables, namely volume (packing) fraction and stress, while the microstructure is not accessible and thus neglected. Here, by means of numerical simulations, we analyze dense, frictionless granular assemblies with the final goal to relate the elastic moduli to the fabric state, i.e., to microstructural averaged contact network features as contact number density and anisotropy. The particle samples are first isotropically compressed and then quasi-statically sheared under constant volume (undrained conditions). From various static, relaxed configurations at different shear strains, infinitesimal strain steps are applied to “measure” the effective elastic response; we quantify the strain needed so that no contact and structure rearrangements, i.e. plasticity, happen. Because of the anisotropy induced by shear, volumetric and deviatoric stresses and strains are cross-coupled via a single anisotropy modulus, which is proportional to the product of deviatoric fabric and bulk modulus (i.e., the isotropic fabric). Interestingly, the shear modulus of the material depends also on the actual deviatoric stress state, along with the contact configuration anisotropy. Finally, a constitutive model based on incremental evolution equations for stress and fabric is introduced. By using the previously measured dependence of the stiffness tensor (elastic moduli) on the microstructure, the theory is able to predict with good agreement the evolution of pressure, shear stress and deviatoric fabric (anisotropy) for an independent undrained cyclic shear test, including the response to reversal of strain.     

Experimental and theoretical studies of the elastic behavior of knitted-fabric composites

This paper presents experimental and theoretical studies of the elastic behavior of knitted-fabric composites. In the experimental studies, two types of weft-knit preforms based upon plain-stitch and rib-stitch fabrics were first fabricated and fabric composites were consolidated by using a hand lay-up process. Tensile and rail shear tests were performed, and Young's moduli along the warp and weft directions and shear modulus determined. In order to correlate the preform microstructure with composite elastic properties, geometric models for plain-stitch and rib-stitch fabric composites were developed. Modeling of the elastic behavior was conducted by using an averaging method. The predicted elastic constants are in reasonably good agreement with experimental values. Finally, the limitation and potential of knitted-fabric composites are discussed.     

Apparent stiffness tensors for porous solids using symmetric Galerkin boundary elements

Representative volume elements (RVEs) from porous or cellular solids can often be too large for numerical or experimental determination of effective elastic constants. Volume elements which are smaller than the RVE can be useful in extracting apparent elastic stiffness tensors which provide bounds on the homogenized elastic stiffness tensor. Here, we make efficient use of boundary element analysis to compute the volume averages of stress and strain needed for such an analysis. For boundary conditions which satisfy the Hill criterion, we demonstrate the extraction of apparent elastic stiffness tensors using a symmetric Galerkin boundary element method. We apply the analysis method to two examples of a porous ceramic. Finally, we extract the eigenvalues of the fabric tensor for the example problem and provide predictions on the apparent elastic stiffnesses as a function of solid volume fraction.     

Fibre orientation and mechanical behaviour in reinforced thermoplastic injection mouldings

Process variables, fibre orientation distribution and mechanical properties were inter-related for injection-moulded short-glass fibre-reinforced polypropylene and polyamide and long-glass fibre-reinforced polyamide. The properties of the reinforced grades were also contrasted with those of the base polymer. A rectangular mould with triple pin-edge gates on the same side to facilitate a single melt flow-front or double flow-front advancing adjacently, was employed. Mouldings were evaluated for fibre orientation distribution, and tensile, dynamic mechanical and fracture properties. The relative magnitudes of the shell and the core fibre-orientation persuasions depended on the melt and the mould temperatures, and the injection ram speed. An increase in G _c, K _c, ultimate tensile strength (UTS) and elastic moduli values and a decrease in tan values were observed with fibre reinforcement. The properties showed marked sensitivity to the position of the specimens in the moulding because of the associated variations in the fibre orientation. The prediction of UTS based on a rule of mixture relationship for strength and the Halpin-Tsai equations for elastic moduli had limited success. The predictions were improved by employing measured data representative of regions of high fibre orientation, e.g. knit-line. The estimation was weakened in the material systems containing significant production-induced voids. An inference of the fracture toughness values was that the composites contained flows of the order of 0.2 mm in size, which is conceivable, either as voids or as defects introduced in the machining of the specimens.     

Modelling fabric reinforced composites under impact loads

The paper describes recent progress on the materials modelling and numerical simulation of the in-plane response of fibre reinforced composite structures. A continuum damage mechanics model for fabric reinforced composites under in-plane loads is presented. It is based on methods developed for UD ply materials (Compos. Sci. Technol., 43 (1992) 257), which are generalised here to fabric reinforcements. The model contains elastic damage in the fibre directions, with an elastic–plastic model for inelastic shear effects. Test data on a glass fabric/epoxy laminate show the importance of inelastic effects in shear. A strategy is described for determining model parameters from the test data. The fabric model is being implemented in an explicit FE code for use in crash and impact studies and preliminary results are presented on a plate impact simulation.     

Inversions related to the stress-strain-fabric relationship

The stress-strain-fabric relationship is an extension of the anisotropic form of Hooke's law to include a dependence of the elastic coefficients upon a second-rank tensor called the fabric tensor. The fabric tensor represents features of the material microstructure associated with the type and the degree of the anisotropy. The inversion considered first in this work is that in which the stress-strain-fabric relation is constructed from the strain-stress-fabric relation and vice versa. Next, a semi-inversion of the relationship between the fourth-rank tensor of elastic coefficients and the fabric tensor is developed. This latter inversion permits the determination of the fabric tensor from a fourth-rank tensor of elastic constants. Explicit, approximate forms of these results, including a numerical example, are given for the case when the fabric tensor is normalized and terms of order three and higher in the fabric tensor are neglected.     

Aspects of the formulation and finite element implementation of large strain isotropic elasticity

The paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation. On the theoretical side, an Eulerian setting of isotropic elasticity is discussed exclusively in terms of the Finger tensor as a strain measure. Noval aspects are a direct representation of the Eulerian elastic moduli in terms of the Finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the Finger tensor into spherical and unimodular parts. The isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity. On the computational side, the implementation of the constitutive model in three possible finite element formulations is discussed. After pointing out algorithmic techniques for the treatment of incompressible elasticity, several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity.     

The "music" of core-shell spheres and hollow capsules: influence of the architecture on the mechanical properties at the nanoscale

We report on the first measurement of elastic vibrational modes in core-shell spheres (silica-poly(methyl methacrylate), SiO2-PMMA) and corresponding spherical hollow capsules (PMMA) with different particle size and shell thickness using Brillouin light scattering, supported by numerical calculations. These localized modes allow access to the mechanical moduli down to a few tens of nanometers. We observe reduced mechanical strength of the porous silica core, and for the core-shell spheres a striking increase of the moduli in both the SiO2 core and the PMMA shell. The peculiar behavior of the vibrational modes in the hollow capsules is attributed to antagonistic dependence on overall size and layer thickness in agreement with theoretical predictions.     

Numerical and experimental stiffness characterisations applied to soft textile composites for tensile structures

This paper presents numerical and experimental stiffness characterisation methods for soft composite textile membranes used in fabric roof structures. The studied material is a polyester plain-woven fabric coated with PVC. We present three numerical textile composite micro-structure models. They are integrated in stiffness calculation software programs which are used to identify linear elastic characteristics for a coated fabric sample. The first two models are based on the laminated thin plate theory; the fabric is represented by a stacking of unidirectionally-reinforced layers, or by the ‘Crimp Model’. The third one considers a geometrical approach to the basic cell of the fabric; the elastic characteristics are calculated by assembly of the meshing elements. In addition, an inverse and experimental stiffness identification method, based on biaxial tensile tests conducted (in orthotropic directions), is proposed. Load-controlled tests are conducted on cross-shaped samples with different loading ratios in warp and weft directions: 1/1,1/2,2/1.     

Wedge disclinations in the shell of a core–shell nanowire within the surface/interface elasticity

The elastic behaviors of a two-axes dipole of wedge disclinations and an individual wedge disclination located inside the shell of a free standing core–shell nanowire is studied within the surface/interface elasticity theory. The corresponding boundary value problem is solved using complex potential functions, defined through modeling the disclination dipole by two finite walls of infinitesimal edge dislocations. The stress field, disclination strain energies and image forces acting on the disclinations, are calculated and studied in detail. It is shown that the stresses are rather inhomogeneous across the nanowire cross section, change their signs and reach local maxima and minima far from the disclination lines in the bulk or on the surface of the nanowire. For negative values of the surface/interface modulus and relatively small values of the ratio of the shell and core shear moduli, the surface/interface effect manifests itself through non-classical stress oscillations along the shell free surface in the case of a disclination dipole and core–shell interface in both the cases of a disclination dipole and an individual disclination. The non-classical solution for the strain energy deviates from the classical solution with different effects caused by the surface/interface moduli on the wedge disclination dipole and an individual disclination. When the core is softer than the shell, the dipole with radial orientation of its arm has an unstable equilibrium position in the shell. In general, if the surface/interface modulus is positive, the surface/interface effects are rather weak; however, if it is negative, the effect can be very strong, especially near the shell surface.     

Evaluating the development of elastic anisotropy with plastic flow

It has been observed that many initially isotropic materials show the development of anisotropic elastic response after plastic flow. It is desirable to be able to model this change in the elastic properties as a function of the extent of plastic flow. This is particularly important when considering the traveling of waves in some glassy polymers that exhibit large differences in the wave moduli along the different directions resulting from unequal plastic flow in these directions. A thermodynamically based model of plasticity is developed and used to evaluate the elastic moduli associated with infinitesimal elastic deformations around the unloaded configuration. It is shown that for this model there are at least four independent material functions describing the elastic moduli of an initially isotropic material. These moduli are functions of the isotropic invariants of the right plastic Cauchy stretch tensor.     

Fabric tensor based boundary element analysis of porous solids

In this paper, the boundary element analysis of porous solids (sintered materials, foams, etc.) is studied utilizing a fabric tensor. The fabric tensor provides a measure of anisotropy in the solid, as well as information concerning the geometry and distribution of the pores. The homogenized, orthotropic elastic properties of a porous solid can then be predicted with the fabric tensor. To illustrate the analysis, the effect of porosity on a trabecular bone-titanium bimaterial is studied. The boundary element analysis uses an anisotropic, bimaterial Green's function so the interface does not require discretization. It is shown that the anisotropic Stroh variables are independent of the structural density and dependent on the eigenvalues of the fabric tensor. An example calculation is presented where the effect of porosity on the in-plane maximum shear stress in a trabecular bone-titanium bimaterial is substantial.     

Simulation of the hydration kinetics and elastic moduli of cement mortars by microstructural modelling

The ability of the VCCTL microstructural model to predict the hydration kinetics and elastic moduli of cement materials was tested by coupling a series of computer simulations and laboratory experiments, using different cements. The novel aspects of this study included the fact that the simulated hydration kinetics were benchmarked using real-time measurements of the early-age phase composition during hydration by in situ X-ray diffraction. Elastic moduli are measured both by strain gauges (static approach) and by P-wave propagation (dynamic approach). Compressive strengths were measured by loading mortar prisms until rupture. Virtual samples were generated by VCCTL, using particle size distribution and phase composition as input. The hydration kinetics and elastic moduli were simulated and the numerical results were compared with the experimental observations. The compressive strength of the virtual mortars were obtained from the elastic moduli, using a power-law relation. Experimentally measured and simulated time-dependence of the major cement clinker phases and hydration product phases typically agreed to within 5%. Also, refinement of the input values of the intrinsic elastic moduli of the various phases enabled predictions of effective moduli, at different ages and different water-to-cement mass ratios, that are within the 10% uncertainty in the measured values. These results suggest that the VCCTL model can be successfully used as a predictive tool, which can reproduce the early age hydration kinetics, elastic moduli and mechanical strength of cement-based materials, using different mix designs.     

Parametric hypoplasticity as continuum model for granular media: from Stokesium to Mohr-Coulombium and beyond

Fluid-particle systems, in which internal forces arise only from viscosity or intergranular friction, represent an important special case of strictly dissipative materials defined by a history-dependent 4th-rank viscosity tensor. In a recently proposed simplification, this history dependence is represented by a symmetric 2nd-rank fabric tensor with evolution determined by a given homogeneous deformation. That work suggests an essential physical link between idealized suspensions (“Stokesium”) and granular media (“Mohr-Coulombium”) along with possible models for the visco-plasticity of fluid-saturated and dry granular media. The present paper deals with the elastoplasticity of dilatant non-cohesive granular media composed of nearly rigid, frictional particles. Based on the underlying physics and past modeling by others, a continuum model based on parametric hypoplasticity is proposed, which involves a set of rate-independent ODEs in the state-space of stress, void ratio and fabric. As with the standard theory of hypoplasticity, the present model does not rely on plastic potentials but, in contrast to that theory, it is based explicitly on positive-definite elastic and plastic moduli. The present model allows for elastic loading or unloading within a dissipative yield surface and also provides a systematic treatment of Reynolds dilatancy as a kinematic constraint. Some explicit forms are proposed and comparisons are made to previous hypoplastic models of granular media.     

Analysis of the effect of the winding angle of glass fiber on the frequencies of free vibrations of a shell made of glass-reinforced plastic under nonsymmetric boundary conditions

Calculated results for the frequencies of free vibrations of a cylindrical anisotropic shell made of oriented glass-reinforced fiber are presented, which are based on the Rayleigh–Ritz method. Elastic properties of a shell depend on the orientation of glass fiber and are determined by six elastic moduli. Boundary conditions correspond to fixation of one end and hinge support of the other end of a shell. Numerical results in the form of dependences of the frequencies of free vibrations on the winding angle of glass fiber for various parameters of wave formation and ratios of geometric dimensions of a shell are obtained for fabric glass-reinforced plastic.