On debonding at interface in the Eshelby’s elliptical inclusion under remote loading
Introduction
The study of inclusion problem plays an important role in the field of applied mathematics and mechanics [1]. It was pointed out that in the three-dimensional case, if uniform eigenstrians are applied in an ellipsoidal inclusion, the stress field in the inclusion is also uniform. Similar situation happens in the inclusion problem of plane elasticity. Physically, the so called eingenstrain problem is initiated by an inaccurate manufacture of the machine part or the thermal expansion of inclusion.
Depending on the assumed elastic constants in the matrix and inclusion, the inclusion problems can be categorized into three types, (a) inhomogeneous inclusion, (b) homogeneous inclusion and (c) inhomogeneity [2]. The equivalent method (MID) is formulated on the analysis for two cases, or the inhomogeneous inclusion case and the homogeneity case. A review of recent works on the inclusion problems was carried out [3]. The solution for an elliptical inhomogeneity in plane elasticity by using the equivalent method was provided [4].
A general solution for the elastic curvilinear inclusion problem was given [5]. Based on the usage of complex potential and conformal mapping, the elastic field of an elliptical inhomogeneity in plane elasticity was derived explicitly [6]. For the partially debonded circular inclusion in piezoelectric materials, a closed form solution was obtainable [7]. The problem for a reinforce layer bonded to an elliptic hole under remote loading was investigated [8].
Uniform stress state inside an inclusion of arbitrary shape in a three phase composite was investigated [9]. Within the framework of 2D or 3D linear elasticity, a general approach based on the superposition principle was proposed to study the problem of a finite elastic body with an arbitrarily shaped and located inclusion [10]. An innovative solution for inhomogeneous elliptical inclusion in plane elasticity was achieved [11]. Two boundary value problems, or the problem for elliptic inhomogeneity and the problem for Eshelby’ elliptical inclusion, were formulated [12]. The equivalence condition for two problems in a closed form was achieved. The stress and strain fields inside and outside the inclusion resulted from the polynomial eigenstrains were obtained [13].
This technical note studies the debonding problem in the Eshelby’s elliptical inclusion with remote stresses. The first step in the derivation is to insert the eigenstain in the inclusion. In this case, a compression state for the normal stress exists along the interface of matrix and inclusion. The next step is to increase the remote loading gradually. Once the normal stress at one point on the interface becomes vanishing, the debonding at that point will be starting. The stress components along the interface depend on the two factors, or the applied eigenstrins , and remote loading , . Therefore, the debonding problem in general is very complicated. Two particular cases for the debonding problem are solved in this note.
The present note relies on the existing result in Ref. [12] significantly. Particularly, Eqs. (46) and (47) in [12] plays an important role in the formulation. Below, the suggested solution technique is first obtained in the note. Particularly, the complex variable technique provides an efficient tool in the note.
Section snippets
Fundamental in plane elasticity using complex potential
The following analysis depends on the complex variable function method in plane elasticity [14]. In the method, the stresses (), the resultant forces (X, Y) and the displacements (u, v) are expressed in terms of two complex potentials and such thatwhere z = x + iy denotes complex variable, G is the shear modulus of elasticity, is for the plane stress problems,
Conclusions
The debonding problem is rather important in the machine design. Once the remote stress reaches some limitation, the tight connection between the matrix and the inclusion will not be working.
Comparing the present note with relevant reference [7], significance differences can be found between two sources. In this note, (1) the eigenstarin is assumed in the inclusion, (2) the debonding is caused by gradually increased remote loading, (3) the computed result is the limit remote debonding loading,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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STRESSES IN COMPOSITE PLATES WITH RIVETED BARS USED FOR AIRCRAFT CONSTRUCTION
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