Elsevier

Composite Structures

Volume 176, 15 September 2017, Pages 539-546
Composite Structures

Behavior of laminated shell composite with imperfect contact between the layers

https://doi.org/10.1016/j.compstruct.2017.05.058Get rights and content

Abstract

The paper focuses on the calculation of the effective elastic properties of a laminated composite shell with imperfect contact between the layers. To achieve this goal, first the two-scale asymptotic homogenization method (AHM) is applied to derive the solutions for the local problems and to obtain the effective elastic properties of a two-layer spherical shell with imperfect contact between the layers. The results are compared with the numerical solution obtained by finite elements method (FEM). The limit case of a laminate shell composite with perfect contact at the interface is recovered. Second, the elastic properties of a spherical heterogeneous structure with isotropic periodic microstructure and imperfect contact is analyzed with the spherical assemblage model (SAM). The homogenized equilibrium equation for a spherical composite is solved using AHM and the results are compared with the exact analytical solution obtained with SAM.

Introduction

Composite materials have emerged as the materials of choice in various branches of industry – aerospace, automotive, sport, etc. – for increasing the performance and reducing the weight and cost. However, defects induced during the manufacturing process or accumulated due to environmental and operational loads lead to the reduction in the mechanical performance and material strength and are recognized as a general problem in this type of composites, [1]. Most typically, such defects can be found at the interfaces between the layers creating an imperfect contact condition, [2], [3].

The effect of the contact imperfectness on elastic properties of composites attracted the attention of researchers from 1970’s, [4], [5]. In [6], [7], [8], [9], the authors obtained analytical expressions for the effective elastic properties of rectangular fibrous composites with imperfect contact between the matrix and the reinforcement. On the other hand, the multilayered curvilinear shell structures have received special attention in the last years. In [10], [11], [12], [13] several mathematical methods have been used to derive analytical expression for the elastic properties of laminated shell composites. As a particular case, in [14], the expression of the effective coefficients for a curvilinear shell composite with perfect contact at the interface is obtained.

Several mathematical models and techniques have been developed to evaluate the elastic properties of curvilinear laminated shell composites with imperfect contact at the interfaces. In papers as [10], [12], [15], [16], [17], [18], [19], [20], the assemblage model, finite elements method and the two-scale asymptotic homogenization method are used to derive in one way or another the effective behavior of the elastic properties of particular composites with imperfect contact at the interface.

In this paper a spherical shell structure is studied. In [21] the authors considered the effect in the elastic properties of a spherical laminated shell composite under the influence of stress and strain distributions for two composites with perfect and imperfect contact at the interface using AHM and SAM. In [21] the imperfect contact condition is modeled considering a thin interphase between the layers of the composite, i.e., a three phase composite is used in the analysis. Here the same effect in the spherical shell structure is studied except that the imperfect contact condition is modeled as a linear spring type and the FEM is used to validate the results obtained via AHM and SAM. The purpose of studying this kind of spherical structures obeys to the development and application of mathematical methods for the study of the cornea and other similar soft tissues.

In the present paper, first the AHM technique is used to evaluate the elastic properties of a two-layer laminated shell with imperfect contact of the spring type at the interface. The general analytical expressions of the effective coefficients are derived from the solution of the local problem. We focus on a two-layer spherical shell subjected to internal pressure assuming that the layers are isotropic. To validate the model, the effective coefficients of the spherical structure are compared with FEM calculations. The elastic fields (stresses, strains and displacements) are also compared with ones calculated by the method of Bufler [22] for the analysis of a spherical assemblage model (SAM). The approach is based on the transfer matrix method and yields closed form calculation of the equivalent elastic properties of a periodically laminated hollow sphere made of alternating layers of isotropic elastic materials with imperfect contact. The effective displacement, radial and hoop stresses computed via AHM are compared with the elastic fields calculated by FEM and SAM.

Section snippets

The linear elastic problem

A curvilinear elastic periodic composite is studied. The geometry of the structure is described by the curvilinear coordinates system x=(x1,x2,x3)ΩR3, where Ω=Ω1Ω2 is the region occupied by the solid, it is bounded by the surface Ω=Σ1Σ2, where Σ1Σ2=,Ωα α=1,2 are the elements of the composite, separated by the interface Γε. In Ω, the stress σ and strain are related through the Hooke’s law, σij=Cijklkl, where Cijkl are the components of the elastic tensor C. For a linear periodic solid

Homogenization of two-layer laminated shell composites with imperfect contact

In order to obtain an equivalent problem to (1), (2), (3) with not fast oscillating coefficients, the two–scales Asymptotic Homogenization Method (AHM) is used. The general expression of the truncated expansion is given byum(ε)=vm+εN̂mpvp+Nmlkvl,k+o(ε),where vmvm(x),NmlkNmlk(x,y) is the local function for the first order approach, N(1)mlk(x,y) is Y-periodic, where Y=[0,1] and N̂mp=-ΓlkpNmlk [14]. Substituting the expansion (4) into the Eqs. (1), (2), (3) a recurrent family of problem is

The spherical assemblage model with imperfect contact (SAM)

In this section, a spherical assemblage model consisting of N different thin elastic layers is studied using the transfer-matrix method.

The transfer-matrix method is a classic approach [23]. Here, we first review its application to a periodic laminated hollow sphere proposed in [22] and next extend the obtained results to the case of imperfect contact between the layers.

The spherical assemblage has internal radius Ri, external radius Re and thickness h=2t. The inner surface r=Ri is loaded by a

The finite element method

In this section, a numerical method based on the finite element is proposed to solve problem (6), (7), (8). Since this technique is quite standard, it is rapidly outlined here.

For the sake of simplicity, we denote Y-=[0,γ) and Y+=(γ,1]. Then, choosing a test function v, which can be discontinuous across the interface Γε, multiplying the equilibrium Eq. (6) by this test function and integrating among Y, one obtains after integration by parts-Y-Ci3lk+Ci3m3Nmlkyvydy+Ci3lk+Ci3m3Nmlky(γ-)v(γ-)

Rectangular shell composite

Here a flat shell structure is considered in order to validate formula (12). The unit cell is composed of two layers of isotropic materials like aluminum and reinforced carbon fiber. The properties are Young’s modulus equal to 150 Gpa and Poisson’s ratio 0.3. The volume fraction of both materials are set to 0.5. The matrix K that characterizes the imperfect contact takes the same values as given in Section 3.1. In Table 1 the effective coefficients are obtained using formula (12). The same

Conclusions

In this paper three different approaches areused to study the elastic properties of a spherical shell composite. The two-scale Asymptotic Homogenization Method is used to obtain the general expression of the local problems and the effective coefficients of elastic composites with imperfect contact at the interface. The expression of such effective coefficients is given in (12). The results are compared for different cases of imperfections and the limit case reported in [14] for perfect contact

Acknowledgements

The authors gratefully acknowledge to the project SHICHAN, supported by FSP (Cooperation Scientifique Franco-Cubaine) PROJET No 29935XH and to the project Composite Materials from University of Havana. SHIHMAS Interfaces rigides et souples dans les matriaux et structures htrognes (Soft and Hard Interfaces for Heterogeneous Materials and Structures), 2016 sponsored by French Embassy in La Habana is also grateful. Thanks to Departamento de Matemáticas y Mecánica, IIMAS-UNAM, for its support and

References (30)

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