Behavior of laminated shell composite with imperfect contact between the layers
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 , where is the region occupied by the solid, it is bounded by the surface , where are the elements of the composite, separated by the interface . In , the stress and strain are related through the Hooke’s law, , where are the components of the elastic tensor . 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 bywhere is the local function for the first order approach, is -periodic, where and [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 , external radius and thickness . The inner surface 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 and . Then, choosing a test function v, which can be discontinuous across the interface , multiplying the equilibrium Eq. (6) by this test function and integrating among Y, one obtains after integration by parts
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 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)
- et al.
Asymptotic analysis of an adhesive joint with mismatch strain
Eur J Mech – A/Solids
(2012) - et al.
Electro-mechanical moduli of three-phase fiber composites
J Mater Lett
(2008) - et al.
Transport properties in fibrous elastic rhombic composite with imperfect contact condition
Int J Mech Sci
(2011) - et al.
Semi-analytical method for computing effective properties in elastic composite under imperfect contact
Int J Solids Struct
(2013) - et al.
Effective elastic properties of a periodic fiber reinforced composite with parallelogram-like arrangement of fibers and imperfect contact between matrix and fibers
Int J Solids Struct
(2013) - et al.
Homogenization of structures with generalized periodicity
Compos: Part B
(2012) - et al.
Imperfect interfaces as asymptotic models of thin curved elastic adhesive interphases
Mech Res Commun
(2013) - et al.
Effective properties of regular elastic laminated shell composite
Compos Part B
(2016) Thin interphase/imperfect interface in elasticity with application to coated fiber composites
J Mech Phys Solids
(2002)- et al.
Effective elastic shear stiffness of a periodic fibrous composite with non-uniform imperfect contact between the matrix and the fibers
Int J Solids Struct
(2014)
Fem modeling of multilayered textile composites based on shell elements
Compos Part B
Higher order model for soft and hard elastic interfaces
Int J Solids Struct
Derivation of the adhesively bonded joints by the asymptotic expansion method
Int J Eng Sci
Numerical implementation of imperfect interfaces
Comp Mat Sci
An asymptotic approach to the adhesion of thin stiff films
Mech Res Commun
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