Orthotropic Phases Research Articles

The Contribution to the Development of a Two-Dimensional Asymptotic Theory of the Three-Point Bending Behaviour of Multi-Layered Beams: Applications to Orthotropic Phase Sandwich Beams

The objective of this work is to present a methodology for analyzing the behavior in bending of the structure of sandwich beams base on the second order of asymptotic method. This work is in continuation with the work of Talla [1]. This work includes the knowledge of all the physical elastic constant of the sandwich beams. This result confirms the fact that the second order of asymptotic method doesn’t bring a significative change in the behavior of the solution until a certain point. The curves have been obtained by the software named python. This result was predictable because the asymptotic methods deal with small variation due to the presence of the epsilon parameter, which is very small.

Contribution to the Development of a Two-Dimensional Asymptotic Theory of the Three-Point Bending Behaviour of Multi-Layered Beams: Applications to Orthotropic Phase Sandwich Beams

The objective of this work is to present a methodology for analyzing the behavior in bending of the structure of sandwich beams base on the second order of asymptotic method. This work is in continuation with the work of Talla [1]. This work includes the knowledge of all the physical elastic constant of the sandwich beams. This result confirms the fact that the second order of asymptotic method doesn’t bring a significative change in the behavior of the solution until a certain point. The curves have been obtained by the software named python. This result was predictable because the asymptotic methods deal with small variation due to the presence of the epsilon parameter, which is very small.

Characterization of wave propagation in periodic viscoelastic materials via asymptotic-variational homogenization

A non-local dynamic homogenization technique for the analysis of wave propagation in viscoelastic heterogeneous materials with a periodic microstructure is herein proposed. The asymptotic expansion of the micro-displacement field in the transformed Laplace domain allows obtaining, from the expression of the micro-scale field equations, a set of recursive differential problems defined over the periodic unit cell. Consequently, the cell problems are derived in terms of perturbation functions depending on the geometrical and physical-mechanical properties of the material and its microstructural heterogeneities. A down-scaling relation is formulated in a consistent form, which correlates the microscopic to the macroscopic transformed displacement field and its gradients through the perturbation functions. Average field equations of infinite order are determined by substituting the down-scaling relation into the micro-field equation. Based on a variational approach, the macroscopic field equation of a non-local continuum is delivered and the local and non-local overall constitutive and inertial tensors of the homogenized continuum are determined. The problem of wave propagation is investigated in case of a bi-phase layered material with orthotropic phases and axis of orthotropy parallel to the direction of layers as a representative example. In such a case, the local and non-local overall constitutive and inertial tensors are determined analytically. Finally, in order to test the reliability of the proposed approach, the dispersion curves obtained from the non-local homogenized model are compared with the curves provided by the Floquet-Bloch theory.

Second-gradient homogenized model for wave propagation in heterogeneous periodic media

The paper is focused on a homogenization procedure for the analysis of wave propagation in materials with periodic microstructure. By a reformulation of the variational-asymptotic homogenization technique recently proposed by Bacigalupo and Gambarotta (2012a), a second-gradient continuum model is derived, which provides a sufficiently accurate approximation of the lowest (acoustic) branch of the dispersion curves obtained by the Floquet–Bloch theory and may be a useful tool for the wave propagation analysis in bounded domains. The multi-scale kinematics is described through micro-fluctuation functions of the displacement field, which are derived by the solution of a recurrent sequence of cell BVPs and obtained as the superposition of a static and dynamic contribution. The latters are proportional to the even powers of the phase velocity and consequently the micro-fluctuation functions also depend on the direction of propagation. Therefore, both the higher order elastic moduli and the inertial terms result to depend by the dynamic correctors. This approach is applied to the study of wave propagation in layered bi-materials with orthotropic phases, having an axis of orthotropy parallel to the direction of layering, in which case, the overall elastic and inertial constants can be determined analytically. The reliability of the proposed procedure is analysed by comparing the obtained dispersion functions with those derived by the Floquet–Bloch theory.

Three-dimensional elasticity solution for sandwich beams/wide plates with orthotropic phases: The negative discriminant case

In an earlier paper, Pagano (1969) [Pagano NJ. Exact solutions for composite laminates in cylindrical bending. J Compos. Mater. 1969; 3: 398–411] presented the three-dimensional elasticity solution for orthotropic beams (applicable also to sandwich beams) for the cases of: (1) a phase with positive discriminant of the qudratic characteristic equation, which is formed from the orthotropic material constants and further restricted to positive real roots and (2) an isotropic phase, which results in a zero discriminant. The roots in this case are all real, unequal, and positive (positive discriminant) or all real and equal (isotropic case). This purpose of this article is to present the corresponding solution for the cases of (1) negative discrimnant, in which case the two roots are complex conjugates and (2) positive discriminant but real negative roots. The case of negative discriminant is frequently encountered in sandwich construction, where the orthotropic core is stiffer in the transverse than the in-plane directions. Example problems with realistic materials are solved and compared with the classical and the first-order shear sandwich beam theories.

An Elasticity Solution for the Global Buckling of Sandwich Beams/Wide Panels With Orthotropic Phases

There exist several formulas for the global buckling of sandwich plates, each based on a specific set of assumptions and a specific plate or beam model. It is not easy to determine the accuracy and range of validity of these rather simple formulas unless an elasticity solution exists. In this paper, we present an elasticity solution to the problem of global buckling of wide sandwich panels (equivalent to sandwich columns) subjected to axially compressive loading (along the short side). The emphasis on this study is on the global (single-wave) rather than the wrinkling (multiwave) mode. The sandwich section is symmetric, and all constituent phases, i.e., the facings and the core, are assumed to be orthotropic. The buckling problem is formulated as an eigenboundary-value problem for differential equations, with the axial load being the eigenvalue. The complication in the sandwich construction arises due to the existence of additional “internal” conditions at the face-sheet/core interfaces. Results are produced for a range of geometric configurations, and these are compared with the different global buckling formulas in the literature.

Micromechanical modelling of wood fibre composites

A concentric cylinder model for an N-phase composite with orthotropic properties of constituents was previously presented by the authors. With only minor modifications the model allows for including also free hygroexpansion terms in the elastic stress–strain relationship in order to deal with orthotropic phase swelling. Thus the effect of wood fibre ultrastructure and cell wall hygroelastic properties on wood fibre composite hygroexpansion may be analysed.Multiscale modelling was performed to calculate the hygroexpansion coefficients of both the fibre cell wall and the aligned wood fibre composite.Furthermore, the fibre's helical structure leads to an extension–twist coupling and thus a free fibre will deform axially and also rotate upon loading in longitudinal fibre direction. Within the composite, however, the fibre rotation will be restricted. Therefore, the decision was to compare the composite performance in the two extreme cases(i) free rotation(ii) no rotation of the fibre in the composite.

Three-Dimensional Elasticity Solution for Sandwich Plates With Orthotropic Phases: The Positive Discriminant Case

A three-dimensional elasticity solution for rectangular sandwich plates exists only under restrictive assumptions on the orthotropic material constants of the constitutive phases (i.e., face sheets and core). In particular, only for negative or zero discriminant of the cubic characteristic equation, which is formed from these constants (case of three real roots). The purpose of the present paper is to present the corresponding solution for the more challenging case of positive discriminant, in which two of the roots are complex conjugates.

Wrinkling of Wide Sandwich Panels∕Beams With Orthotropic Phases by an Elasticity Approach

There exist many formulas for the critical compression of sandwich plates, each based on a specific set of assumptions and a specific plate or beam model. It is not easy to determine the accuracy and range of validity of these rather simple formulas unless an elasticity solution exists. In this paper, we present an elasticity solution to the problem of buckling of sandwich beams or wide sandwich panels subjected to axially compressive loading (along the short side). The emphasis on this study is on the wrinkling (multi-wave) mode. The sandwich section is symmetric and all constituent phases, i.e., the facings and the core, are assumed to be orthotropic. First, the pre-buckling elasticity solution for the compressed sandwich structure is derived. Subsequently, the buckling problem is formulated as an eigen-boundary-value problem for differential equations, with the axial load being the eigenvalue. For a given configuration, two cases, namely symmetric and anti-symmetric buckling, are considered separately, and the one that dominates is accordingly determined. The complication in the sandwich construction arises due to the existence of additional “internal” conditions at the face sheet∕core interfaces. Results are produced first for isotropic phases (for which the simple formulas in the literature hold) and for different ratios of face-sheet vs core modulus and face-sheet vs core thickness. The results are compared with the different wrinkling formulas in the literature, as well as with the Euler buckling load and the Euler buckling load with transverse shear correction. Subsequently, results are produced for one or both phases being orthotropic, namely a typical sandwich made of glass∕polyester or graphite∕epoxy faces and polymeric foam or glass∕phenolic honeycomb core. The solution presented herein provides a means of accurately assessing the limitations of simplifying analyses in predicting wrinkling and global buckling in wide sandwich panels∕beams.

Composite materials whose phases have partially equal elastic moduli

Hill [J. Mech. Phys. Solids 11 (1963) 357, 12 (1964) 199] discovered that, regardless of its microstructure, a linearly elastic composite of two isotropic phases with identical shear moduli is isotropic and has the effective shear modulus equal to the phase ones. The present work generalizes this result to anisotropic phase composites by showing and exploiting the fact that uniform strain and stress fields exist in every composite whose phases have certain common elastic moduli. Precisely, a coordinate-free condition is given to characterize this specific class of elastic composites; an efficient algebraic method is elaborated to find the uniform strain and stress fields of such a composite and to obtain the structure of the effective elastic moduli in terms of the phase ones; sufficient microstructure-independent conditions are deduced for the orthogonal group symmetry of the effective elastic moduli. These results are applied to elastic composites consisting of isotropic, transversely isotropic and orthotropic phases.

Duality relations, correspondences and numerical results for planar elastic composites

This paper addresses three topics relating to planar elastic composites with anisotropic phases. First, the duality relations of Berdichevski are generalized to a wider class of planar elastic media. These yield phase interchange relations for the effective compliance tensors of certain two phase media. Second, a simple derivation is given of the correspondence between a specific class of planar elastic problems, and the associated pairs of antiplane elastic problems. This correspondence allows one to determine the effective compliance tensor from the effective shear matrices of the associated antiplane problems. Third, a numerical algorithm is developed and implemented for computing the effective moduli of two phase composites with orthotropic phases. The numerical algorithm is based on an integral equation and can, more generally, be applied to solving for the fields in elastic bodies comprised of several phases differing in their anisotropies. A series of examples demonstrates the accuracy of the numerical method and the agreement with the theoretical findings.

Computation of flexural and torsional homogeneous properties and stresses in composite beams with orthotropic phases

A simplified analytical calculation of homogeneous properties and stresses of a beam made up of several orthotropic materials is given. Tests have been realized with existing solutions such as: classic theory, three-dimensional finite elements and experimental data. It is shown that the analytical method proposed gives a good result for beams with a simple cross-section: thin-wall, closed partitioned section filled up or not by foam, etc.

On effective thermal expansion coefficients of composites with anisotropic phases

Bounds on thermal expansion coefficients of composite materials with anisotropic phases are derived using extremum principles of thermoelasticity. Numerical results are presented for two-phase composites with isotropic and orthotropic phases.