Raman spectroscopy investigation of stiffness change and residual strains due to matrix cracking

The matrix cracking models developed for cross-ply composite laminates have been poorly extended in the past to more complex geometries used in practice, and they are still under development. In this paper, a new detailed analysis of the effect of matrix cracking on the behaviour of cross-ply and [0/45]_s laminates under uniaxial tension is attempted. The model used in this work is applicable both to cross-ply laminates and unbalanced systems. It gives exact closed-form expressions for all thermomechanical properties of a general symmetric laminate with cracks in arbitrary layers. The theoretical approach is backed by experimental data obtained by microscopic strain-state variation measurements within a specimen, with using the technique of laser Raman spectroscopy. Glass-fibre-reinforced epoxy systems were investigated. Embedded aramid fibres-sensors within the 0° ply and near the 0°/θ ° interface were necessary due to the poor Raman signal of glass. Using experimental Raman data, the residual strain and the stiffness reduction are determined as functions of increase in crack density. The stiffness reduction is predicted with a high accuracy, whereas the measured residual strains are larger than predicted. The good results for the reduction in the elastic modulus show that the basic assumption of the model is accurate. The difference is explained by the viscoelastic-viscoplastic behaviour of the off-axis layer in shear, which in creases the “apparent” residual strain. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 771–786, November–December, 2006.     

Transverse cracks in cross-ply laminates. 1. Stress analysis

During service loading of cross-ply laminates, transverse cracks occur in plies. The cracks parallel to the fiber direction are extended over the full thickness of transverse plies and often cross the entire test specimen width. It is widely recognized that the changes of laminate thermomechanical constants, caused by the transverse cracking of composite laminates, can be significant. Theoretical stress analysis in the cross-ply laminates in the vicinity of cracks is performed using numerical (FE) and analytical methods. The effect of transverse cracks on the degradation of elastic properties will be discussed in Part 2 [1]. Approximate analytical micromechanical models based on shear lag predictions, variational analysis, and numerical 2D finite element calculations were verified in their predictive abilities. The three variational models used are based on the principle of minimum complementary energy and have different degrees of accuracy with respect to the stress assumptions used (Hashin's, 2D 0° and 2D 0°/90° models). Using FEM, the plane stress and strain state were analyzed. The effect of material properties and layer thickness on the stress distribution in a 90° layer was evaluated by varying the crack spacing. The crack opening displacement (COD), normalized with respect to the far field strain, is proposed as a measure of reduction of the mechanical properties. Since the CODs are rather insensitive to the crack spacing (crack density) in a wide region, they will be used in modeling the stiffness reduction in these laminates [1].Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 796–820, November–December, 1997.     

Estimation of laminate stiffness reduction due to cracking of a transverse ply by employing crack initiation-and propagation-based master curves

The applicability range of toughness-and strength-based criteria for progressive cracking of a transverse layer in a cross-ply composite laminate subjected to tensile loading is considered. Using a deterministic cracking model, approximate relations for the crack density as a function of stress are derived for initiation-and propagation-controlled types of cracking. The master-curve approach is applied to progressive cracking in glass/epoxy laminates. The accuracy of estimation of laminate stiffness reduction by using crack density master curves is evaluated. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 5, pp. 633–646, September–October, 2008.     

Detection of matrix cracks in composite laminates by using the modal strain energy method

As the foremost mechanism of damage development, matrix cracking is the critical damage found in the early stage of structural failure of composites. This study aims to nondestructively detect matrix cracks in composite laminates by using an experimental modal analysis (EMA). An AS4/PEEK composite was used to fabricate cross-ply [0₂/90₁₂/0₂] and quasi-isotropic [(±45/0/90)₄] _s laminates. The damage in the form of a matrix crack in the laminates was created by using a tensile load. The EMA was conducted on the laminates to obtain the modal displacements before and after damage. The displacements were then employed to compute the modal strain energy and to define the damage index, which is used for detecting matrix cracks. Limited by the mesh points of measurements, we used the differential quadrature method to calculate the partial differentials in the strain energy formula. The results obtained were validated by using the X-ray radiography method and three-point bending tests. The experimental results showed that the damage index well identified the location of breadthwise matrix cracks inside the laminates. However, the resolution of the damage index became poor if the spans of matrix cracks were short or the matrix cracks were scattered over the laminates.     

Analysis of the free-edge effect in composite laminates by the boundary finite element method

Because of the risk of delamination due to high interlaminar stresses in the vicinity of free edges of composite laminates, there is a strong interest in efficient methods for the analysis of this free-edge effect. By the example of a symmetric [0°/90°]_s cross-ply laminate, the Boundary Finite Element Method is presented as a very efficient numerical method, which combines the advantages of the finite element method and the boundary element method. Analogously to the boundary element method, only the boundary is discretized, while the element formulation is finite element based. The resultant stress field is shown to be in very good agreement qualitatively and quantitatively with the comparative finite element analysis. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 355–366, March–April, 2000.     

Transverse cracks in cross-ply laminates 2. Stiffness degradation

From the results of stress analysis between two transverse cracks in cross-ply laminate [1], a model for the stiffness reduction based on generalized plane strain assumptions has been developed. Simple analytical expressions are obtained for the longitudinal modulus and the Poisson's ratio as a function of the transverse crack density. Apart from the crack density, these expressions depend only on the elastic and geometrical properties of constituent laminae and the average crack opening displacement (ACOD) normalized in the proper way. Calculations of the ACOD are performed and analyzed with the FEM and analytical models used for the stress analysis in [1]. The predicting capabilities of approximate models are discussed in comparison with experimental data and FEM results. In order to predict the stiffness degradation for a wide variety of laminates, a simple procedure requiring only one FEM calculation for some average laminate with average crack spacing is proposed and has been proved effective.Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 211–233, March–April, 1998.     

Circular Delaminations - the Effect of Thermal Strains on the Strain Energy Release Rate

The analysis presented focuses on the extent to which the thermal loading resulting from a difference between the curing and service temperatures and from variations in the thermal-expansion coefficients from layer to layer can affect the growth of delaminations. Square laminate plates containing circular delaminations are examined. The reinforcement combinations for the delaminated layer and the core plate considered are [0°]/[90°], [90°]/[0°], [0°]/[±45°] _s , and [±45°] _s /[0°]. The analysis is carried out using the FEM. The strain energy release rate components are calculated using the modified crack-closure integral method. The results obtained show that, for all the reinforcement combinations studied, with the exception of [±45°] _s /[0°], a temperature drop considerably reduces all the strain energy release rate components, but only minor changes in their proportion occur. In the case of the [±45°] _s /[0°] plates, all these components increase, and noticeable changes in their proportion are also observed.     

Bending Stiffness of Damaged Cross-Ply Laminates

Mechanics of Composite Materials - The bending stiffness of carbon/epoxy and glass/epoxy cross-ply laminates with intralaminar cracks in the surface 90° plies and local delaminations were...     

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the random variableZ=N/D whereN= ₁ ⁿ m _iX_i+ _n ^+1s m _iX_i,D= _n ^+1s l _iX_i+ _s ^+1r n _iX_i, and theX _i's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quardratic forms is evaluated at various points and then compared with simulated values.     

A new graphic characterization of non singular quadrics of PG(r,q)

LetK be ak-set of class [0, 1,m,n]₁ of anr-dimensional projective Galois space PG(r, q) of orderq. We prove that: Ifr = 2s (s 2),k = _2s–1 and if through each point ofK there are exactlyq ^2(s–1) tangent lines and at most _2s–3 n-secant lines, thenK is a non singular quadric of PG(2s,q). Ifr = 2s–1 (s2),k=_2(s–1) +q ^s–1 and if at each point ofK there are exactlyq ^2s–3 –q ^s–2 tangents and at most _2(s–2)+q ^s–2 n-secant lines, thenK is a hyperbolic quadric of PG(2s–1,q).     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

The Sensitivity of Residual Stresses of Cross-Ply Laminates to Manufacturing and Material Parameters

By using a finite-element model elaborated, the sensitivity of residual stresses of polyester/glass cross-ply laminates to manufacturing and material parameters is investigated. The development of residual stresses in the laminates and the significance of the parameters for the problem are discussed. It is found that the main attention in calculating residual stresses should be focused on the properties of resin, which must be measured with care. The most important parameters related to the resin are, of course, its stiffness, thermal expansion, and chemical shrinkage, while the properties of fibers can be obtained from material handbooks with a sufficient accuracy. In curing a thin laminate in an autoclave, the simulation of chemical reactions and the parameters needed in thermal analysis are quite insignificant, because, in practice, the autoclave temperature and the properties of the mold determine the laminate temperature history.     

Dynamic and static nonlinear analysis of generally laminated imperfect plates having nonuniform boundary conditions and resting on elastic foundation

A multimode solution to the dynamic von Kármán-type nonlinear equations is presented for the titled plates. The plate edges subjected to inplane forces are elastically restrained against rotation. The variation of rotational stiffness is assumed identical along parallel edges. Generalized double Fourier series with the time-dependent coefficients and the method of harmonic balance are used in the formulation of the solution. Nonlinear bending and postbuckling of the laminate are treated as special cases. Numerical results are presented for nonlinear free vibration and postbuckling of antisymmetric angle-ply and cross-ply laminates.     

Interpolation of operator ideals with an application to eigenvalue distribution problems

We derive results on the interpolation of complete quasinormed operator ideals, mainly for the absolutelyp-summing and thes-number idealsS _p ^s defined by Pietsch. By estimating theK-Functional of Peetre, we get that the interpolation ideal (S _p1 ^s ,S _p2 ^s ),_,p is contained inS _p ^s and is even equal to it in the case of the approximation numbers. A similar fact is proved for absolutely (p, q)-summing operators, interpolating the first index. We show further that the absolutelyp-summing operators onc ₀ are contained in the complex interpolation space ( _p1 (c _o), _p2 (c _o))_[].The previous results are then applied to prove summability properties for the eigenvalues of operators in Banach spaces, which are products ofS _p1 ^s -type and absolutelyp _j -summing operators. Roughly speaking, the summability order is the harmonic sum of thep _i - andp _j -indices, wherep _j 2. In the case of Hilbert spaces, this reduces to the well-known Weyl-inequality. The method uses an abstract interpolation estimate for ideal quasinorms which may be useful also for other operator ideals.     

Estimates for the -Neumann problem and nonexistence of <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript> Levi-flat hypersurfaces in

The Markov moment problem is to characterize sequences admitting the representation s_n=₀¹x^nf(x)dx, where f(x) is a probability density on [0,1] and 0f(x)c for almost all x. There are well-known characterizations through complex systems of non-linear inequalities on {s_n}. Necessary and sufficient linear conditions are the following: s₀=1, and for all and . Here, is the forward difference operator. This result is due to Hausdorff. We give a new proof with some ancillary results, for example, characterizing monotone densities. Then we make the connection to de Finettis theorem, with characterizations of the mixing measure.in final form: 18 June 2003     

The second and the third smallest distances on the sphere

Let s₁ (n) denote the largest possible minimal distance amongn distinct points on the unit sphere . In general, let s_k(n) denote the supremum of thek-th minimal distance. In this paper we prove and disprove the following conjecture of A. Bezdek and K. Bezdek: s₂(n) = s₁([n/3]). This equality holds forn > n₀ however s₂(12) > s₁(4).We set up a conjecture for s_k(n), that one can always reduce the problem of thek-th minimum distance to the function s₁. We prove this conjecture in the casek=3 as well, obtaining that s₃(n) = s₁([n/5]) for sufficiently largen.The optimal construction for the largest second distance is obtained from a point set of size [n/3] with the largest possible minimal distance by replacing each point by three vertices of an equilateral triangle of the same size . If 0, then s₂ tends to s₁([n/3]). In the case of the third minimal distance, we start with a point set of size [n/5] and replace each point by a regular pentagon.     

Finite Order Rank-One Convex Envelopes and Computation of Microstructures with Laminates in Laminates

Finite order rank-one convex envelopes are introduced and it is shown that the i-th order laminated microstructures, or laminates in laminates, can be solved by any of the k-th order rank-one convex envelopes with k i. It is also shown that in finite element approximations of microstructures, replacing the non-quasiconvex potential energy density by its k-th order rank-one convex envelope, one can generally obtain sharper numerical results. Especially, for crystalline microstructures with laminates in laminates of order no greater than k + 1, numerical results with up to the computer precision can be obtained. Numerical examples on the first and second order rank-one convex envelopes for the Ericksen-James two-dimensional model for elastic crystals are given. A numerical example on finite element approximations of a crystalline microstructure by using the first order rank-one convex envelope and the periodic relaxation method is also presented. The methods turn out to be very successful for microstructures with laminates in laminates.     

Transformation de Fourier et temps d'occupation browniens

Summary In this paper, we study oscillatory stochastic integrals of the form where is a non zero parameter andg a square integrable function. We study integrability properties of () and its behavior as a function of , using stochastic calculus techniques: martingale theory, representation of Itô for a random variable of the Wiener space, lemma of Garsia-Rodemich-Rumsey .... We also obtain limit theorems in law related to the variables () based upon an asymptotic version of a theorem of Knight on orthogonal continuous martingales.We consider the random measure, image by the Brownian motion of the unbounded measure 1_[0,] (s)g(s) ds; we prove the existence and the continuity of an occupation time density.Finally, under a stronger integrability condition ong, we show the existence of a density for the law of (), using Malliavin's calculus.     

Series expansions in Fréchet spaces and their duals, construction of Fréchet frames

Frames for Fréchet spaces X_F with respect to Fréchet sequence spaces Θ_F are studied, and conditions implying series expansions in X_F and are determined. If is a Θ₀-frame for X₀ and Θ_F (resp. X_F) is given, we construct a sequence {X_s}_s∈N0, X_s⊂X_s−1, s∈N, (resp. {Θ_s}_s∈N0, Θ_s⊂Θ_s−1, s∈N), so that is a pre-F-frame or F-frame for X_F with respect to Θ_F under different assumptions given on X₀, Θ₀ and Θ_F (resp. X_F).