Deformation induced loss of ellipticity in an anisotropic circular cylindrical tube

When a transversely isotropic circular cylindrical tube is subject to axial extension and inflation, the governing equations of equilibrium can lose ellipticity under certain combinations of deformation and direction of transverse isotropy. In this paper, it is shown how the inclusion of an axial shear deformation moderates the loss of ellipticity condition. In particular, this condition is analysed for a material model consisting of an isotropic neo-Hookean matrix within which are embedded fibres whose properties are characterized by the addition to the strain-energy function of a reinforcing model depending on the local fibre direction.     

Mullins effect on incompressible hyperelastic cylindrical tube in finite torsion

The isotropic softening effect in non-homogeneous deformation resulting from combined effect of torsion, extension and inflation of cylindrical rubber tube is discussed. The effects of deformation induced anisotropy, permanent set and hysteresis are neglected. A general neo-Hookean parent material model is illustrated and subsequently the stress-softening effect on the same hyperelastic material is analyzed. Simple torsion of cylindrical tube with neo-Hookean material model is analyzed and the results obtained are shown in various plots. Analytical results are compared with the experimental results of Rivlin and Saunders. Universal relations are also established for incompressible, isotropic, hyperelastic material for non-homogeneous deformation with isotropic damage function in both virgin and stress-softened cases.     

Single fibre and multifibre unit cell analysis of strength and cracking of unidirectional composites

Numerical simulations of damage evolution in composites reinforced with single and multifibre are presented. Several types of unit cell models are considered: single fibre unit cell, multiple fibre unit cell with one and several damageable sections per fibres, unit cells with homogeneous and inhomogeneous interfaces, etc. Two numerical damage models, cohesive elements, and damageable layers are employed for the simulation of the damage evolution in single fibre and multifibre unit cells. The two modelling approaches were compared and lead to the very close results. Competition among the different damageable parts in composites (matrix cracks, fibre/matrix interface damage and fibre fracture) was observed in the simulations. The strength of interface begins to influence the deformation behaviour of the cell only after the fibre is broken. In this case, the higher interface layer strength leads to the higher stiffness of the damaged material. The damage in the composites begins by fibre breakage, which causes the interface damage, followed by matrix cracking.     

Hyperelastic modelling for mesoscopic analyses of composite reinforcements

A hyperelastic constitutive law is proposed to describe the mechanical behaviour of fibre bundles of woven composite reinforcements. The objective of this model is to compute the 3D geometry of the deformed woven unit cell. This geometry is important for permeability calculations and for the mechanical behaviour of the composite into service. The finite element models of a woven unit cell can also be used as virtual mechanical tests. The highlight of four deformation modes of the fibre bundle leads to definition of a strain energy potential from four specific invariants. The parameters of the hyperelastic constitutive law are identified in the case of a glass plain weave reinforcement thanks to uniaxial and equibiaxial tensile tests on the fibre bundle and on the whole reinforcement. This constitutive law is then validated in comparison to biaxial tension and in-plane shear tests.     

Degradation and healing in a generalized neo-Hookean solid due to infusion of a fluid

The mechanical response and load bearing capacity of high performance polymer composites changes due to degradation or healing associated with diffusion of a fluid, temperature, oxidation or the extent of the deformation. Hence, there is a need to study the response of bodies under such degradation/healing mechanisms. In this paper, we study the effect of degradation and healing due to the diffusion of a fluid on the response of a solid which prior to the diffusion can be described by the generalized neo-Hookean model. We show that a generalized neo-Hookean solid—which behaves like an elastic body (i.e., it does not produce entropy) within a purely mechanical context—creeps and stress relaxes due to degradation/healing when infused with a fluid and behaves like a body whose material properties are time dependent. We specifically investigate the torsion of a degrading/healing generalized neo-Hookean circular cylindrical annulus infused with a fluid. The equations of equilibrium for a generalized neo-Hookean solid are solved together with the convection–diffusion equation for the fluid concentration. Different boundary conditions for the fluid concentration are also considered. We also solve the problem for the case when the diffusivity of the fluid depends on the deformation of the generalized neo-Hookean solid.     

Post microbuckling mechanics of fibre-reinforced shape-memory polymers undergoing flexure deformation

The buckling mechanics of fibre-reinforced shape-memory polymer composites (SMPCs) under finite flexure deformation is investigated. The analytical expressions of the key parameters during the buckling deformation of the materials were determined, and the local post-buckling mechanics of the unidirectional fibre-reinforced SMPC were further discussed. The cross section of SMPC under flexural deformation can be divided into three areas: the non-buckling stretching area, non-buckling compression area and buckling compression area. These areas were described by three variables: the critical buckling position, the neutral plane position and the fibre buckling half-wavelength. A strain energy expression of the SMPC thermodynamic system is developed. According to the principle of minimum energy, the analytical expressions of key parameters in the flexural deformation process is determined, including the critical buckling curvature, critical buckling position, position of the neutral plane, wavelength of the buckling fibre, amplitude of the buckling fibre and macroscopic structural strain of the composite material. The results showed that fibre buckling occurred in the material when the curvature increasing from infinitesimal to the critical value. If the curvature is greater than the critical curvature, the neutral plane of the material will move towards the outboard tensile area, and the critical buckling position will move towards the neutral plane. Consequently, the half-wavelength of the buckling fibre was relatively stabilised, with the amplitude increasing dramatically. Along with the increasing of the shear modulus, the critical curvature and buckling amplitude increase, while the critical half-wavelength of the fibre buckling decrease and the critical strain of the composite material increase. Finally, we conducted experiments to verify the correction of the key parameters describing SMPC materials under flexural deformation. The values determined by the experiments proved that the theoretical prediction is correct. Additionally, the buckling deformation of the carbon fibre generated a large macroscopic structural strain of the composite material and obtained a resulting large flexural curvature of the structure with minimal material strain of the carbon fibre.     

Mechanical modeling of incompressible particle-reinforced neo-Hookean composites based on numerical homogenization

In this paper, the mechanical response of incompressible particle-reinforced neo-Hookean composites (IPRNC) under general finite deformations is investigated numerically. Three-dimensional Representative Volume Element (RVE) models containing 27 non-overlapping identical randomly distributed spheres are created to represent neo-Hookean composites consisting of incompressible neo-Hookean elastomeric spheres embedded within another incompressible neo-Hookean elastomeric matrix. Four types of finite deformation (i.e., uniaxial tension, uniaxial compression, simple shear and general biaxial deformation) are simulated using the finite element method (FEM) and the RVE models with periodic boundary condition (PBC) enforced. The simulation results show that the overall mechanical response of the IPRNC can be well-predicted by another simple incompressible neo-Hookean model up to the deformation the FEM simulation can reach. It is also shown that the effective shear modulus of the IPRNC can be well-predicted as a function of both particle volume fraction and particle/matrix stiffness ratio, using the classical linear elastic estimation within the limit of current FEM software.     

A variational approach to a circular hyperelastic membrane problem

Summary The variational principles of nonlinear elasticity are applied to a problem of axially symmetric deformation of a uniform circular hyperelastic membrane. The supported edge of the membrane is in a horizontal plane and its radius is equal to that of the undeformed plane reference configuration, so that an initially plane unstretched membrane is subjected to a dead load due to its weight.It is shown how the stationary complementary energy principle can be used to obtain an accurate approximate solution for the deformation and stress distribution. It is also shown how the potential energy principle can be applied to the problem and how close bounds for an energy functional can be obtained from the two theorems. Numerical results are presented for realistic properties for a rubberlike material and for two strain energy functions, the semi-linear and the neo-Hookean.     

A Unit‐Cell Approach of Finite Element Analysis for Transverse Impact Damage of 3‐D Biaxial Spacer Weft‐Knitted Composite

Abstract: This paper evaluates the transverse impact damage of a 3‐D biaxial spacer weft‐knitted composite using experimental results and complimentary finite element analysis. The load–displacement curves and damage morphologies during impact loading were obtained to analyse energy absorption and impact damage mechanisms of the knitted composite. A unit‐cell model based on the microstructure of the 3‐D knitted composite was established to calculate the deformation and damage evolution when the composite is impacted by a hemisphere‐ended steel rod. An elastoplastic constitutive equation is incorporated into the unit‐cell model and the critical damage area failure theory developed by Hahn and Tsai has been implemented as a user‐defined material law (VUMAT) for commercial available finite element code ABAQUS/Explicit. The load–displacement curves, impact damages and impact energy absorption obtained from ABAQUS/Explicit are compared with those FROM experiments. The good agreement of the comparisons supports the validity of the unit‐cell model and user‐defined subroutine VUMAT. The unit‐cell model can also be extended to evaluate the impact crashworthiness of engineering structures made out of the 3‐D knitted composites.     

The in-plane shear modulus of fibre reinforced composites as defined by the concept of interphase

In this paper we describe a model to find the approximate equations for determining the in-plane shear modulus of a unidirectional fibre reinforced composite from the constituent material properties. Classical elasticity theory has been applied to the simplified model of a composite unit cell in which the concept of an interphase between fibre and matrix is taken into account. Thus the model considers that the composite material consists of three phases, that is the fibre, the matrix, and the interphase which is the part of the polymer matrix lying close to the fibre surface which possesses different physico-chemical properties from those of the main constituents. Thermal analysis was used for the determination of the thickness and volume fraction of the interphase. The theoretical results are compared with other theoretical expressions and with experimental data. The model introduced in this paper seems to be an improvement for the shear modulus.     

Hyperelastic finite deformation analysis with the unsymmetric finite element method containing homogeneous solutions of linear elasticity

A recent unsymmetric 4-node, 8-DOF plane finite element US-ATFQ4 is generalized to hyperelastic finite deformation analysis. Since the trial functions of US-ATFQ4 contain the homogenous closed analytical solutions of governing equations for linear elasticity, the key of the proposed strategy is how to deal with these linear analytical trial functions (ATFs) during the hyperelastic finite deformation analysis. Assuming that the ATFs can properly work in each increment, an algorithm for updating the deformation gradient interpolated by ATFs is designed. Furthermore, the update of the corresponding ATFs referred to current configuration is discussed with regard to the hyperelastic material model, and a specified model, neo-Hookean model, is employed to verify the present formulation of US-ATFQ4 for hyperelastic finite deformation analysis. Various examples show that the present formulation not only remain the high accuracy and mesh distortion tolerance in the geometrically nonlinear problems, but also possess excellent performance in the compressible or quasi-incompressible hyperelastic finite deformation problems where the strain is large.     

Derivation of separation laws for cohesive models in the course of ductile fracture

The paper addresses the determination of the traction-separation law of the cohesive model on a micromechanical basis. For this task, a specific failure mechanism, i.e. ductile damage consisting of void nucleation, growth and coalescence, is investigated. An approach already described in the literature is to transfer the deformation behaviour of the simplest representative volume element, i.e. a single voided unit cell, to the cohesive interface. After reviewing the existing approach, its main drawback, namely that the unit cell contains both, deformation and damage of a material point whereas the cohesive model should contain the material separation only, is addressed. A new approach is presented, in which the behaviour of a unit cell is partitioned in its elasto-plastic deformation and damage, and only the damage contribution is applied as the traction-separation law for the cohesive model. Instead of modelling the voided unit cell, a single element with Gurson type plastic potential for the damage has been employed as a reference for the behaviour at the microscale. A study with fracture specimens, C(T) and M(T), made of an engineering Aluminium alloy shows that the new approach exhibits a better transferability than the existing one.     

The mechanical behaviour of corrugated-core sandwich panels

A series of experimental investigations and numerical analyses is presented into the compression response, and subsequent failure modes in corrugated-core sandwich panels based on an aluminium alloy, a glass fibre reinforced plastic (GFRP) and a carbon fibre reinforced plastic (CFRP). The corrugated-cores were fabricated using a hot press moulding technique and then bonded to face sheets based on the same material, to produce a range of lightweight sandwich panels. The role of the number of unit cells and the thickness of the cell walls in determining the overall deformation and local collapse behaviour of the panels is investigated. The experiments also provide an insight into the post-failure response of the sandwich panels. The results are compared with the numerical predictions offered by a finite element analysis (FEA) as well as those associated with an analytical model. Buckling of the cell walls has been found to be initial failure mode in these corrugated systems. Continued loading resulted in fracture of the cell walls, localised delamination as well as debonding between the skins and the core. The predictions of the FEA generally show reasonably good agreement with the experimental measurements. Finally, the specific compressive properties of the corrugated structures have been compared to those of other core materials where evidence suggests that these systems compare favourably with their more conventional counterparts.     

In situ monitoring of thermally cycled metal matrix composites by neutron diffraction and laser extensometry

A novel stroboscopic neutron diffraction data collection system has been developed. In combination with scanning laser extensometry this has been used to investigate the thermal cycling behaviour of SiC short fibre reinforced Al matrix composites. Three-dimensional unit cell finite element models have been produced, incorporating matrix deformation both by creep and plasticity. Comparison of the experimental results with model predictions has allowed conclusions to be drawn about the deformation processes which dominate at different parts of the thermal cycle.     

预成型体渗透率预测及其受压缩变形的影响

建立了织物预成型体单胞内纱线间细观流动和纱线内部微观流动的统一的数学模型。基于最小势能原理建立了织物松弛状态下的单胞几何模型,同时对在模具压缩下的单胞变形进行了分析,并建立了不同压缩状态下的单胞几何模型。通过对单胞内树脂流动数学模型的数值求解,获得了流动速度场及压力场,进而预测了预成型体的渗透率。预测1组不同压缩状态下的单胞渗透率,研究了预成型体压缩变形对渗透率的影响。结果显示:随着压缩量的增加,其渗透率逐渐降低。通过实验测量及数据分析,验证了建模和预测方法的正确性。     

Constitutive model and finite element formulation for large strain elasto-plastic analysis of shells

This contribution presents a refined constitutive and finite element formulation for arbitrary shell structures undergoing large elasto-plastic deformations. An elasto-plastic material model is developed by using the multiplicative decomposition of the deformation gradient and by considering isotropic as well as kinematic hardening phenomena in general form. A plastic anisotropy induced by kinematic hardening is taken into account by modifying the flow direction. The elastic part of deformations is considered by the neo-Hookean type of a material model able to deal with large strains. For an accurate prediction of complex through-thickness stress distributions a multi-layer shell kinematics is used built on the basis of a six-parametric shell theory capable to deal with large strains as well as finite rotations. To avoid membrane locking in bending dominated cases as well as volume locking caused by material incompressibility in the full plastic range the displacement based finite element formulation is improved by means of the enhanced assumed strain concept. The capability of the algorithms proposed is demonstrated by various numerical examples involving large elasto-plastic strains, finite rotations and complex through-thickness stress distributions.     

Compressive fracture in undirectional glass-reinforced plastics

The shear mode of compressive failure in unidirectional fibre composites is discussed. A mechanism is described in which the shear deformation is restricted to a band of material inclined to the plane normal to the fibre axes. The relationship between the orientation of the failed band of material and the limiting shear deformation in the band is explained in terms of volumetric strains. Tests are described which demonstrate that, in GRP, this type of failure can propagate from a notch and this notch sensitivity is put forward as an explanation for the apparent inadequacy of the theoretical model. The sequence of events in the propagation of compressive failure is studied by examining serial sections of an arrested failure. It is found that fibre fracture at the boundaries and interlaminar failure within the band follow as a result of increasing shear deformation in the band.     

Large strain rate-dependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations

Two different viscoelastic frameworks adapted to large strain rate-dependent response of elastomers are compared; for each approach, a simple model is derived. Within the Finite Linear Viscoelasticity theory, a time convolution integral model based on an extension to solid of the K-BKZ model is proposed. Considering the multiplicative split of the deformation gradient into elastic and inelastic parts, an internal variable model based on a large strain version of the Standard Linear Solid model is considered. In both cases, the strain energy functions involved are chosen neo-Hookean, and then each model possesses three material parameters: two stiffnesses and a viscosity parameter. These parameters are set to ensure the equivalence of the model responses for uniaxial large strain quasi-static and infinitely fast loading conditions, and for uniaxial rate-dependent small strain loading conditions. Considering their responses for different Eulerian strain rates, their differences are investigated with respect to the strain rate; more specifically, both stiffness and dissipative properties are studied. The comparison reveals that these two models differ significantly for intermediate strain rates, and a closing discussion highlights some issues about their foundations and numerical considerations.