Solvability of boundary value problems in a theory of plane-strain elasticity with boundary reinforcement

doi:10.1016/j.ijengsci.2008.09.003

International Journal of Engineering Science

Volume 47, Issues 11–12, November–December 2009, Pages 1331-1338

https://doi.org/10.1016/j.ijengsci.2008.09.003 Get rights and content

Abstract

We investigate the solvability of a recent mathematical model describing plane deformations of an elastic solid whose boundary is partially reinforced by a thin elastic coating.

Introduction

Problems dealing with plane deformations of elastic solids with reinforced or coated boundaries have been dealt with extensively in the literature (see, for example [1], [2], [3], [4], [5], [6], [7], [8] and the references contained therein). The main objectives of these analyses is to understand the mechanics of coated surfaces as well as to attempt to simulate the mechanical response of materials subjected to various industrial surface processing techniques such as shot-peening. In addition, since these ‘reinforced surfaces’ essentially incorporate the effects of surface stresses, this class of problems is also of great interest to researchers working in the emerging area of nanomechanics in which the effects of surface stresses have been included in continuum models in an attempt to understand the size-dependency of material properties at the nano-scale (see, for example [9]). These examples provide compelling physical motivation for the formulation of a well-posed mathematical model predicting the mechanical response of materials which incorporate (in some form) the effects of surface reinforcement.

The study of the solvability of a new mathematical model describing a linear theory of plane-strain elasticity with boundary reinforcement was initiated by the authors in [7]. In that paper, the corresponding fundamental boundary value problems are formulated (including a detailed derivation of the reinforcement (boundary) conditions) and solvability results are proved using the boundary integral equation method. This is a crucial step for, without it, there is no guarantee that a solution of the mathematical model actually exists despite the fact that the physics clearly demonstrates such a solution. A priori knowledge of solvability is the basis of numerical solution and is crucial to the correct formulation of any mathematical model. The results in [7], however, are limited to the case when the reinforced part of the boundary consists of a finite number of sufficiently smooth closed curves. The more general case in which the reinforced boundary can be represented by the union of a finite number of open curves is of considerable practical interest since it allows for the modelling of a much wider class of physical problems involving elastic coatings or the effects of surface stress [3]. For example, the case where a surface is partially reinforced or partially coated with a thin film occurs in many different industrial applications (see, for example [5], [6]). Unfortunately, this more general case is associated with an (already) nonstandard boundary condition (characterizing the effect of the reinforcement), this time, posed over open arcs as opposed to closed curves. The additional resulting end-point conditions to be satisfied at the ends of each arc preclude the extension of the methods used in [7] as well as the subsequent establishment of critical results on the solvability of the corresponding mathematical model.

In the present work, we draw on results established by the authors in [10], and formulate the corresponding mixed boundary value problems with an alternative (lower order) form of the reinforcement boundary condition. This form is particularly attractive in that it automatically satisfies all end-point conditions and leads itself well to analysis by the boundary integral equation method. In fact, using the formulation from [10], the boundary value problems are shown to reduce to systems of singular integral equations (as opposed to systems of singular integro-differential equations [7]) for which Noether’s theorems reduce to Fredholm’s theorems. This important fact alone allows us to finally establish the required solvability results for this more general model.

Section snippets

Preliminaries

In what follows, Greek and Latin indices take the values 1, 2 and 1, 2, 3, respectively, we sum over repeated indices, $M_{m \times n}$ is the space of $(m \times n)$ -matrices, $E_{n}$ is the identity element in $M_{n \times n}$ , a superscript T indicates matrix transposition and $(\dots),_{α} \equiv \partial (\dots) / \partial x_{α}$ . Also, if X is a space of scalar functions and $ν$ a matrix, $ν \in X$ means that every component of $ν$ belongs to X. Let S be a multiply-connected domain in $R^{2}$ whose boundary $\partial S$ is described by the union of a finite number of sufficiently smooth

Boundary condition on the reinforcement

The conditions on the (reinforced) subset $Γ$ of the boundary $\partial S$ couple the response of the solid in S to that of the coating on $Γ$ . These conditions are obtained through equilibrium considerations and represent the linearized versions of those obtained by a variational method in [8]. Let $Γ$ be parametrized by arclength s and let $θ (s)$ be the angle defining the direction of $\underset{̲}{τ} (s) = (τ_{1}, τ_{2})$ , the unit tangent at $s \in Γ$ in the sense that $\underset{̲}{τ} (s) = \cos θ (s) \underset{̲}{i} + \sin θ (s) \underset{̲}{j} .$ Denote by $P (s)$ and $B (s)$ , the normal and

Lower order form of boundary condition (7)

Considerable difficulties are encountered when the methods used in [7] are applied to the corresponding mixed boundary value problems incorporating the boundary condition (7) on the (open) reinforcement. These difficulties are due mainly to the fact that the differential operators in (7) are of fourth order and hence twice the order of those occurring in the governing equation (2) of plane-strain. Since the boundary integral equation method [7] reduces the mixed problems to systems of singular

Interior problem

Let S be the bounded domain enclosed by $\partial S$ . Write $\partial S = \partial S_{1} \cup Γ$ where $\partial S_{1}$ represents the non-reinforced part of $\partial S$ . Further, divide $\partial S_{1}$ into open curves $\partial S_{u}$ and $\partial S_{T}$ and let $Γ$ represent, as before, the union of a finite number of sufficiently smooth open curves $L_{i}$ with end-points $a_{i}$ and $b_{i}$ such that $L_{j}$ and $L_{k}$ $(i, j, k = 1, \dots, m)$ have no points in common for $j \neq k$ . The interior mixed boundary value problem of plane-strain elasticity with boundary reinforcement is posed as follows. Find $u \in C^{2} (S) \cap C^{1} (\bar{S} ⧹ γ)$ such

Reduction to singular integral equations

Using the results for the classical interior and exterior mixed problems of plane-strain [11], we can always reduce (13), (14) to simpler problems with homogeneous conditions on $\partial S_{u}$ and $\partial S_{T}$ . Consequently, without loss of generality, we consider the boundary value problems (13), (14) with $u^{(0)} (x) = t^{(0)} (x) = 0$ , henceforth referred to as boundary value problems $(13)_{0}$ and $(14)_{0}$ , respectively.

Solvability of the boundary value problems

We can now prove the following result concerning the solvability of the interior mixed reinforcement problem $(13)_{0}$ .

Theorem 3

The interior mixed reinforcement problem $(13)_{0}$ has a unique solution for any prescribed matrix t which, with its first derivative, belongs to the space $H^{*} (Γ)$ . This solution is given by (15) with $ϕ$ the unique solution of (17).

Proof

By Theorem 2, the homogeneous system $(18)^{0}$ has only the trivial solution. Since the index [14] of (18) is zero, Fredholm’s theorems now imply that (18) is

Conclusions

A priori knowledge of the solvability of any mathematical model is the basis of numerical solution and is crucial to the correct formulation of any mathematical model. Otherwise, we simply seek the proverbial ‘needle in a haystack’ without knowing the needle is actually in the haystack. In this paper, we consider a recent mathematical model describing the plane deformations of an elastic solid whose boundary is partially reinforced by a thin elastic coating represented by the union of a finite

Acknowledgements

The authors would like to dedicate this paper to Tony Spencer F.R.S. This work was supported by the Natural Sciences and Engineering Research council of Canada.

References (14)

B. Budiansky et al.
On neutral holes in tailored layered sheets
J. Appl. Mech.
(1993)
E.H. Mansfield
Neutral holes in a plane sheet – reinforced holes which are elastically equivalent to the uncut sheet
Quart. J. Mech. Appl. Math.
(1953)
J.R.M. Radok
Problems of plane elasticity for reinforced boundaries
J. Appl. Mech.
(1955)
G.N. Savin
Stress Concentration Around Holes
(1961)
G.N. Savin, N.P. Fleishman, Rib-reinforced Plates and Shells, Israel Program for Scientific Translations, Jerusalem,...
A.A. Wells
On the plane-stress distribution in an infinite plate with a rim-stiffened elliptic opening
Quart. J. Mech. Appl. Math.
(1950)
P. Schiavone et al.
Integral equation methods in plane-strain elasticity with boundary reinforcement
Proc. Roy. Soc. London A
(1998)

There are more references available in the full text version of this article.

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Solvability of boundary value problems in a theory of plane-strain elasticity with boundary reinforcement

Abstract

Introduction

Section snippets

Preliminaries

Boundary condition on the reinforcement

Lower order form of boundary condition (7)

Interior problem

Reduction to singular integral equations

Solvability of the boundary value problems

Conclusions

Acknowledgements

On neutral holes in tailored layered sheets

J. Appl. Mech.

Neutral holes in a plane sheet – reinforced holes which are elastically equivalent to the uncut sheet

Quart. J. Mech. Appl. Math.

Problems of plane elasticity for reinforced boundaries

J. Appl. Mech.

Stress Concentration Around Holes

On the plane-stress distribution in an infinite plate with a rim-stiffened elliptic opening

Quart. J. Mech. Appl. Math.

Integral equation methods in plane-strain elasticity with boundary reinforcement

Proc. Roy. Soc. London A