Eli Altus

Buckling of Multiply Delaminated Beams

An analytical model is developed for buckling of multiply delaminated composite beams or long plates under cylindrical bending, and their interactive effects are studied. The prebuckling and buckling equations, formulated for one-dimensional cases with an arbitrary configuration of through-the-width cracks, are solved in a closed-form manner. The effects of crack length and location are studied, showing different trends of behavior for global and local mode shapes, including a decrease in the buckling load when cracked zones overlap. In some cases, where inadmissible interlaminar penetration in the first mode is involved, higher eigenvalues and the corresponding eigenmodes are needed to define the buckling state. In other cases, where all eigen modes exhibit penetration, the present model can be used only as indicated for the behavior, and a nonlinear analysis with contact constraints is called for. Considering the scanty work done to-date in the field of multiply delaminated beams, the proposed model-even though restricted to a contactless situation-can also serve as the basis for more general nonlinear analysis. A parametric study of several beam configurations is presented through examples.

Stresses and failure patterns in the bending of sandwich beams with transversely flexible cores and laminated composite skins

The stresses and failure maps in a sandwich beam that consists of a transversely flexible compressible core between two laminated composite skins, are presented. The stresses and the failure maps are determined using a general, systematic rigorous, and high-order analysis that is based on variational principles, and includes the flexibility effects of the core on the global and local bending behavior of the beam. The analysis uses closed form solutions for any type of skin construction, symmetric or unsymmetric laminated composite layups, any type of core, compressible or incompressible, any type of loading, concentrated or distributed, and any types of boundary and continuity conditions that may differ from one skin to the other, even in the same section. Failure patterns are determined with the aid of the analytical description of the longitudinal stresses in the skins and the principal stresses through the thickness of the core. The stresses in the core and the skins, along with an appropriate failure criteria, for a specified three point bending beam, are demonstrated in the form of principal stresses, failure and failure load maps, that indicate possible failure patterns and locations.

Optimum laser surface treatment of fatigue damaged Ti–6Al–4V alloy

Abstract A laser surface treatment (LST), which enhances fatigue resistance of Titanium 6Al–4V alloy has been studied by 1.8 kW CW–CO 2 Laser. The aim of the study is to find optimal conditions of treatment, which will extend the material resistance to fatigue failure, and explore the mechanisms involved. The LST was applied to ‘three point bend’ specimens loaded cyclically ( R =0.1, 0.7 σ y near surface stresses), at different stages of fatigue life. Temperature surface fields were found by infrared camera. Two basic mechanisms were identified, one is related to healing mechanism (HM), which ‘erases’ prior fatigue damage up to a certain level, and the other is connected to microstructure mechanism (MM). Healing was found to be effective for surface temperatures above 400°C. In most cases, microstructure changes had a negative effect on fatigue resistance except for temperatures lower than 600°C and specific laser conditions. The combination of both mechanisms lead to optimal LST of 2 s and 0.85 kW cm −2 , for which a 50% increase in fatigue life was found to be due to MM, in addition to a full healing. A positive correlation between hardness and fatigue life was also found.

Buckling of stochastically heterogeneous beams, using a functional perturbation method

Abstract The buckling load and its probabilistic nature (average and variance) of Bernoulli beams with stochastic material (bending stiffness) properties is derived analytically by a new functional perturbation method (FPM). A buckling shape function is assumed, based on the homogeneous solution and additional terms to account for the morphology effects. The buckling load in the transcendental equation is treated as a functional of the bending modulus (stiffness or compliance) field. Applying a functional perturbation to the above equation, the buckling load is found analytically to any desired degree of accuracy, as a function of material morphology. The FPM is executed using both stiffness and compliance statistical data. The impact of each of the two data sources on the solution accuracy is examined, showing that compliance based solutions are accurate for small correlation lengths. Statically indeterminate problems can be treated with no additional effort. An example of a simply supported beam is solved in detail. Comparison with previous studies, where stochastic finite element and Monte Carlo simulation were used, showed the relative accuracy and insight capabilities of the method. The clamped-free case is also studied to demonstrate that symmetry conditions, used for homogeneous beams to find the buckling load on the basis of a simply supported case, are not valid for heterogeneous beams.

Statistical modeling of heterogeneous micro-beams

Statistical characteristics (averages and variances) of heterogeneous, small size beams (micro-beams), composed of micro-elements (grains) of randomly distributed stiffness (isotropic) is studied. Only longitudinal variations are considered. Element to beam size (length) ratio is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient. Using standard statistical tools (probability density and correlation functions), it is shown that for indeterminate cases, even average deflections (obtained explicitly) are load dependent, and are not identical with their corresponding homogeneous case. Bounds for the statistical dispersion of deflections and reaction forces are found analytically for some cases of weak heterogeneity. It is shown that dispersion parameters for indeterminate problems can be obtained by superposition (using both averages and dispersion data) of corresponding determinate cases, leading to reciprocal type relations.

Strength reliability of statically indeterminate heterogeneous beams

The strength reliability of linearly elastic (up to failure) beams, made from random heterogeneous microstructures is studied, based on the weakest link approach. Heterogeneity is confined to the longitudinal direction. The problem is statically indeterminate, and the local stress at each point in any cross section is a function of the stiffness morphology of the whole beam. External loading is not random, but reaction forces are, due to their statistical correlation with the beam morphology. The case of one degree of indeterminacy is studied here, for simplicity. The strength and reliability of the beam, being a stochastic function of local stresses, is therefore morphology dependent, in addition to (coupled with) the classical inherent probabilistic nature, associated with surface defects and irregularities. This dependence is found analytically as a function of external loading shape. A simple design formula for the bound of these effects on the beam strength has been found, covering any possible external loading. For example, for a beam of 10 grains (compliance correlation length of 0.1L) and a 10% compliance variance, the bound of the heterogeneity effect on strength is about 8%.

Fatigue, fractals, and a modified miner's rule

A Mechano Chemical Fatigue Model was recently developed. Two basic features of the model are studied here: (a) the fractal property of the damage function (b) how it leads to a «Modified Miners Rule»

Post-buckling analysis of multiply delaminated beams

Abstract An analytical model for the post-buckling behavior of multiply delaminated beams or long plates under cylindrical bending is presented. The model comprises partial non-linear differential equations based on the Von Karman kinematic approach. A special variable separation, based on global and local functions derived from the eigenfunction procedure, is employed to obtain the algebraic non-linear system, which is solved by the Newton-Raphson method and a modified “arc-length” procedure. Two illustrative examples are provided.

A 3-D finite difference solution for orthotropic laminated composites using curvilinear coordinates

Abstract The static equilibrium equations for orthotropic composite laminates are written in body fitted curvilinear coordinate system. A 3-D Finite Difference code has been developed to solve these equations. This combination extends the capability of the Finite Difference method to analyse effectively stress concentrations near arbitrary 2-D defects in composites. The stress field around a circular hole in an angleply composite laminate (−45°, 45°| s ) is analysed as a test case.

Analysis of Bernoulli beams with 3D stochastic heterogeneity

The behavior of stochastically heterogeneous beams, composed of isotropic sub-elements of randomly distributed stiffness is studied. Cross sectional as well as longitudinal heterogeneity are included. Average displacements, reaction forces and their statistical variance are found analytically by a functional perturbation method. Ratio of sub-element to beam characteristic size is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient. The major aim is to study the relation between various microstructure properties (grain size, shape, modulus, statistical correlation lengths etc.) and the overall behavior of linear elastic Bernoulli beams. For the statically determinate case, only cross sectional 2D microstructure statistics is found to affect the elastic response, so that an equal average displacement can be achieved by an equivalent, non-isotropic homogeneous beam. For the indeterminate case, the average values of macro properties are affected by the 3D morphological features. Therefore, the proper equivalent homogeneous beam has to include non-local elastic properties. A simple reciprocal relation, connecting two separate loading systems is found, relating their external forces and displacement statistical variances. Morphological parameters, like two point probability moments, used in the final results are derived analytically, and their physical interpretations are discussed.