Stress state characterization of delamination cracks in [0/90] symmetric laminates by BEM

The presence in a laminate of plies oriented at 90° with respect to the preferred direction of load generates almost immediately the appearance in these plies of cracks transverse to the load (parallel to the fibres in the lamina). These cracks reach the interface with the neighbouring ply, which in this paper will be considered oriented 0° with respect to the direction of the load. This may cause the bifurcation of the crack, which now appears propagating as a delamination crack between the two plies. The objective of this study is to characterize the stress state at the tip of both, the transverse crack in the 90° ply reaching the interface with the 0° ply, and the delamination crack for different lengths of the debonding. The analysis is performed by means of the Boundary Element Method allowing contact, without or with friction, to take place between the faces of the crack. The plies are considered as equivalent homogeneous bodies under a generalized plane strain state. The results are compared with those predicted by the open and contact models of Interfacial Fracture Mechanics. Accurate knowledge of the stress state at the neighbourhood of the tips of the cracks studied is necessary to generate failure criteria based on Fracture Mechanics parameters to predict the appearance and growth of the type of damage described.     

Stress analysis of notched anisotropic finite plates under mechanical and hygrothermal loads

Summary Notch-induced stress concentrations in anisotropic composite materials depend on their directional material properties, especially for uniaxially reinforced composites with high-modulus fibres. The design of notched high-performance composites requires therefore a special proof of their notched strength, which includes the structural parameters of the fibre/matrix combination, fibre orientation and layer arrangement. The assessment of the effects of the finite outer boundary is of practical importance when dimensioning critical notched regions. An anisotropic plate with finite dimensions and a hole in its center will be used here to model stress concentrations. The calculation is based on conformal mappings combined with complex-valued stress functions. The outer boundary is described using point-matching and the least-squares method. The solutions are conducive to the assessment of the essential influencing factors of material properties, geometry and loads. Notched finite plates made of fibre/matrix composites, mainly carbon-fibre reinforced polymers, will be presented as illustrations. Received 29 June 1998; accepted for publication 22 October 1998     

An experimental study of spallation criterion in phenolic-resin based woven roving glass fibre reinforced composite material

An experimental study of spallation in phenolic-resin based woven roving glass fibre reinforced composite material under uniaxial strain condition is reported in this paper. Experiments were carried out with a 101 mm bore single-stage light gas gun. The stress waves induced by plate impact propagate along the normal to glass fibre mats. The experiments show that the threshold of incipient spallation in this material depends strongly on the duration of stress wave and that this incipient spallation appears essentially in one layer. The criterion of spallation in this material given in this paper is in good agreement with experimental results. Besides, the spallation in phenolic-resin based obliquely laminated woven roving glass fibre reinforced composite material is also discussed in this paper.     

Toughness in obliquely-stressed fibrous composites

An earlier treatment of toughness for continuous, uniform, fibre reinforced materials given by M.R. Piggott (1970) is extended to the case where the stress is not parallel to the fibres. Experiments on pairs of fibres crossing cracks obliquely are used to reveal the effect on fibre strength of fibre flexure at the crack. The theory indicates that, so long as splitting parallel to the fibres does not occur, the fracture surface energy γ_φ for a material stressed at an angle φ to the fibres is given with sufficient accuracy for brittle fibres by the approximate formula γ_φ = γ_o(1?2.4A tan φ), where γ_o is the surface energy for fracture normal to the fibre direction, and A is a non-dimensional parameter depending on the force exerted by the matrix on the fibres, and involving, in particular, the ratio of matrix flow stress to the fibre ultimate tensile strength. For ductile fibres, the work of fracture increases with the angle φ at a rate depending on fibre breaking stress. The form of fracture surface and the onset of splitting are also discussed.     

Spherical indentation of composite laminates with controlled gradients in elastic anisotropy

Three-dimensional elastic analyses and experiments of indentation of thick laminated plates of carbon fiber reinforced epoxy are presented. Pointwise, the material is characterized as a linear elastic orthotropic material. The in-plane orientation of the carbon fibers is systematically varied as a function of depth. The influence of fiber orientation as a function of depth on the indentation response is considered along with the relationship between the indenter force vs depth. The fiber orientation profiles considered are those of a continuous linear variation between 90° at the outer surfaces and 0° at the center plane of the laminate, and a cross ply laminate involving alternating 90° and 0° layers through thickness. Experimentally, it is found that for the case of a cross-ply laminate, the indentation produces delaminations localized at the interfaces that separate planes of dissimilar orientation. For this case, stress concentrations at interfaces between plies of dissimilar orientation coincide with the observed sites of delamination. For the graded case, evidence of enhanced nonlinear deformation is found, without the nucleation of cracks. Computations show that for the graded material, tensile stresses perpendicular to fibers are suppressed significantly, possibly explaining the absence of matrix cracks in this material. Measured and computed indenter force-depth variations were found to be in good agreement. Experiments and computations also reveal that the orientation-graded material is more compliant when subjected to indentation than the conventional cross-ply laminate.     

An experimental study of dynamic delamination of thick fiber reinforced polymeric matrix composites

Dynamic delamination of thick fiber reinforced polymeric matrix composite laminates is investigated using optical techniques and high-speed photography. The laminates used in this work are graphite/epoxy fiber reinforced, 65 percent fiber volume fraction, composite plates consisting of 48 plies (6 mm plate thickness). Two different laminate layups are tested: a quasi-isotropic arrangement and a unidirectional arrangement. The experimental setup consists of 152 mm×152 mm square plates impact loaded in an outof-plane configuration using a high-speed gas gun. Impact speeds range from 1 m/s to 30 m/s. Real-time imaging of the laminate out-of-pane displacement is performed using the lateral shearing interferometer of coherent gradient sensing (CGS) in conjunction with high-speed photography. Onset of dynamic delamination can be observed, and quantities such as delamination speeds (in some cases up to 1800 m/s) are measured and reported. A brief comparison is made with dynamic fracture experiments of the same material conducted in a separate study.     

On the propagation of elastic waves in a multiperiodically reinforced medium

By a multiperiodically reinforced medium (multiperiodic composite) we mean a composite in which the matrix material is reinforced by two or more families of periodically spaced fibres. Moreover, at least along one direction the periods corresponding to different families are different. An example of this composite is shown in Fig. 1, where along the x ¹-axis we deal with two different periods . The aim of the contribution is twofold. First, we propose a macroscopic (averaged) model of a multiperiodic composite, describing the effect of period lengths on the overall dynamic behaviour of the medium, in contrast to the known homogenized models. Second, we apply this model to the analysis of elastic waves propagating across a composite reinforced by two pairs of families of parallel periodically spaced fibres with different periods along certain direction.     

A continuum theory for fibre-reinforced composites

The effective stiffness theory for fibre reinforced materials with a hexagonal layout of fibres is presented. The theory is illustrated by the dispersion curves of plane steadystate time-harmonic waves. The limiting phase velocities at vanishing wave numbers serve in the determination of the elastic moduli of the equivalent homogeneous transversely isotropic medium.     

Plane strain and generalized plane stress problems for fibre-reinforced materials

In this paper the basic equations governing the plane strain or generalized plane stress deformations of a linear elastic material reinforced by a single family of parallel inextensible fibres are deduced. It is found that a single system of equations will cover all cases. The solutions for plane, half-plane and strip problems are evaluated and compared with those for an ideal fibre-reinforced material.     

A double inclusion model for microcrack arrest in fibre reinforced brittle materials

A model for the prediction of microcrack growth in a fibre reinforced brittle matrix composite material is suggested. The model is based on composite material theory and linear elastic fracture mechanics. The microcracks in question are so-called large microcracks, i.e. microcracks which are bridged by the reinforcing fibres. The crack bridging fibres are “smeared” out to form a homogeneous medium. This homogeneous medium constitutes together with the matrix crack an ellipsoidal so-called “double inclusion.” Matrix cracking as well as interfacial debonding can be analysed and this analysis can be synthesized and interpreted as a determination of the strength of the reinforced matrix. The model is compared with some experimental results, and good agreement is found. The model can serve as a tool for the design of brittle matrix composite materials because it identifies the significance of fibre geometry, volume fraction of fibres, and adhesion between fibres and matrix.     

Diffraction of stress waves by a free or reinforced hole in an orthotropic plate

Photoelastic plates made of an orthotropic material are used to model the dynamic stress state near free and reinforced circular holes under blast loading. The diffraction of stress waves by holes in a thin-walled plate is studied. Experimental data are used to analyze the dynamic stress concentration in a plate with a hole in which quasilongitudinal and quasitransverse waves propagate __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 73–78, July 2007.     

On dynamical fracture mechanics in the case of polyharmonic loading by SH-waves

We solve a fracture-mechanics problem for a cracked material under polyharmonic loading (waves of various lengths propagating through an elastic material). The frictional contact interaction of the finite crack edges in a plane is analyzed for the case of normal incidence of two harmonic shear waves with different frequencies. The forces of contact interaction and displacement discontinuity are determined. The effect of the wave frequency on the stress intensity factor for different normalized wave numbers is examined     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.     

Effects of SH waves in a functionally graded plate

A computational method is presented to investigate SH waves in functionally graded material (FGM) plates. The FGM plate is first divided into quadratic layer elements (QLEs), in which the material properties are assumed as a quadratic function in the thickness direction. A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the wave number domain for an arbitrary FGM plate. The displacements and stresses in the frequency domain and time domain are obtained using inverse Fourier integration. Furthermore, a simple integral technique is also proposed for evaluating modified Bessel functions with complex valued order. Numerical examples are presented to demonstrate this numerical technique for SH waves propagating in FGM plates.     

Optimal Design of Prestressed Laminate/Ceramic Plate Assemblies

It is well known that the dynamic compressive strength of ceramics is enhanced by confining compressive stress. The paper shows how a biaxial in-plane compressive stress can be induced in a ceramic layer by encapsulation in a fiber-prestressed composite laminate. Optimal prestress distribution in individual plies of the laminate is found, together with the resulting residual stresses. Large compressive stress, 600–1000MPa can be supported in an alumina layer by aramid/epoxy or carbon/epoxy laminates of the same total thickness as the ceramic. Stress relaxation due to either temperature changes or matrix creep is found to be of minor significance. Examples present results for specific material systems and laminate layups. Sommario. E'ben noto che la resistenza dinamica dei materiali ceramici viene migliorata dal confinamento degli sforzi di compressione. Questo lavoro mostra come uno stato piano di compressione biassiale possa essere indotto in uno strato di materiale ceramico imprigionandolo in un laminato composito con fibre presollecitate. Viene determinata la distribuzione ottimale delle presollecitazioni nei singoli strati del laminato, nonché l'entità degli sforzi residui risultanti. Compressioni elevate, dell'ordine di 600–1000Mpa, possono essere indotte in uno strato di allumina da laminati epossidici rinforzati con fibre aramidiche o carboniche dello stesso spessore. Il rilassamento degli sforzi dovuto a cambiamenti di temperatura o al comportamento viscoso della matrice risulta avere importanza limitata. Gli esempi forniscono risultati relativi a specifici materiali e orientamenti dei laminati.     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.     

Slip, stick, and reverse slip characteristics during dynamic fibre pullout

Inertial effects in the mechanism of fibre pullout (or push-in) are examined, with emphasis on how the rate of propagation of stress waves along the fibre, and thence the pullout dynamics, are governed by friction and the propagation of companion waves excited in the matrix. With a simple shear lag model (assuming zero debond energy at the fibre/matrix interface), the effect of uniform frictional coupling between the fibre and the matrix is accounted for in a straightforward way. Analytical solutions are derived when the pullout load increases linearly in time. The process zone of activated material is generally divided into two or three domains along the axis of the fibre. Within these domains, slip in the sense implied by the load, slip in the opposite sense (reverse slip), and stick may be observed. The attainable combinations define three regimes of behavior, which are realized for different material parameter values. The elastodynamic problem is also solved more accurately using a plane stress finite element method, with friction represented by an interfacial cohesive zone. The predictions of the shear lag theory are broadly confirmed.     

Magneto elastic stress subjected to uniform magnetic field over a thin infinite plate containing an elliptical hole with an edge crack

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin infinite plate containing an elliptical hole with an edge crack subjected to uniform magnetic field. Using a rational mapping function, each solution is obtained as a closed form. The linear constitutive equation is used for these analyses. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. In the present paper, it raises a plane stress state for a thin plate, the deformation of the plate thickness and the shear deflection. Therefore the magneto elastic stress is analyzed using Maxwell stress. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition expressed by Maxwell stress components is completely satisfied without any linear assumptions on the boundary. First, magnetic field and stress analyses for soft ferromagnetic material are carried out and then those analyses for paramagnetic and diamagnetic materials are carried out. It is stated that those plane stress components are expressed by the same expressions for those materials and the difference is only the magnitude of the permeability, though the magnetic fields H_x, H_y are different each other in the plates. If the analysis of magnetic field of paramagnetic material is easier than that of soft ferromagnetic material, the stress analysis may be carried out using the magnetic field for paramagnetic material to analyze the stress field, and the results may be applied for a soft ferromagnetic material. It is stated that the stress state for the magnetic field H_x, H_y is the same as the pure shear stress state. Solutions of the magneto elastic stress are nonlinear for the direction of uniform magnetic field. Stresses in the direction of the plate thickness and shear deflection are caused and the solutions are also obtained. Figures of the magnetic field and stress distribution are shown. Stress intensity factors are also derived and investigated for the crack length.     

Dynamic stress analysis of a functionally graded material plate with a circular hole

This paper is to study the two-dimensional dynamic stress of a functionally graded material (FGM) plate with a circular hole under plane compressional waves at infinity. With using the method of piece-wise homogeneous layers, the dynamic stress distribution of the FGM plate having radial arbitrary material parameters is derived based on the complex variable method. As examples, numerical results are presented for the FGM plate having given radial shear modulus, density and Poisson’s ratio. It is found that the dynamic stress around the circular hole in the FGM plate can be effectively reduced by choosing the proper change ways of the radial material parameters for different frequency incident wave.