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## Matching asymptotic method in presence of singularities in antiplane elasticity

**Introduction**

We use matching asymptotic expansions to treat the anti-plane elastic problem associated with a small defect located at the tip of a notch. In a first part, we develop the asymptotic method for any type of defect and present the sequential procedure which allows us to calculate the diﬀerent terms of the inner and outer expansions at any order. That requires in particular to separate in each term its singular part from its regular part.

Specifically, the chapter is organized as follows. Section 1.2 is devoted to the description of the MAM on a generic anti-plane linear elastic problem where the body contains a defect near the tip of a notch. We first decompose the solution into two expansions: one, the outer expansion, valid far enough from the tip of the notch, the other, the inner expansion, valid in a neighborhood of the tip of the notch. These expansions contain a sequence of inner and outer terms which are solutions of inner and outer problems and which are interdependent by the matching conditions. Moreover each term contains a regular and a singular part. We explain how all the terms and the coeﬃcients entering in their singular and regular parts are sequentially determined. The section finishes by an example where the exact solution is obtained in a closed form and hence where we can verify the relevance of the MAM.

In Section 1.3, we introduce another decomposition of the expansions of inner and outer problems. That leads to solve two sequences of inner or outer problems which are independent of each other. That allows us to solve these problems once and for all, the inner ones being characteristic of the defect whereas the outer ones are characteristic of the whole structure without its defect. This new method is illustrated by solving the inner problems in the case of a cavity or of a crack. In both cases the solution is obtained in a closed form with the help, in the case of crack, of the theory of complex potentials.

### The Matched Asymptotic Method

**The real problem**

Here, we are interested in a case where a small geometrical defect of size ‘ (like a crack or a void) is located near the corner of a notch, see Figure 1.1. The geometry of the notch is characterized by its angle !, see Figure 1.2. The tip of the notch is taken as the origin of the space and we will consider two scales of coordinates: the “macroscopic » coordinates x = (x1; x2) which are used in the outer domain and the “microscopic » coordinates y = x=‘ = (y1; y2) which are used in the neighborhood of the tip of the notch where the defect is located, see Figure 1.2. In the case of a crack, the axis x1 is chosen in such a way that the crack corresponds to the line segment (0; ‘) f0g. The unit vector orthogonal to the (x1; x2) plane is denoted e3.

When the length ‘ of the defect is small by comparison with the characteristic length of the body (in this section, this characteristic length is not precised), then it is necessary to make an asymptotic analysis of the problem rather than to try to obtain directly an approximation by classical finite element methods. In the case of a crack for instance, because of the overlap of two singularities (one at the tip of the notch and the other at the tip of the crack), it is diﬃcult and even impossible to obtain accurate results without using a relevant asymptotic method. Here we will use the matched asymptotic expansion technique which consists in making two asymptotic expansions of the field u‘ in terms of the small parameter ‘. The first one, called the inner expansion, is valid in the neighborhood of the tip of the notch, while the other, called the outer expansion, is valid far from this tip. These two expansions are matched in an intermediate zone.

**The outer expansion**

Far from the tip of the notch, i.e. for r ‘, we assume that the real displacement field u‘ can be expanded as follows u‘(x) = ‘i ui(x): (1.8)

In (1.8), even if this expansion is valid far enough from r = 0 only, the fields ui must be defined in the whole outer domain 0 which corresponds to the sound body, see Figure 1.2-left. Inserting this expansion into the set of equations constituting the real problem, one obtains the following equations that the ui’s must satisfy:

**Verification in the case of a small cavity**

This subsection is devoted to the verification of the construction of the MAM presented in the previous subsections on an example where the exact solution is obtained in a closed form and hence can be directly expanded. Specifically, we consider a Laplace’s problem posed in a domain which consists in an angular sector delimited by two arc of circles. The radius of the outer circle is equal to 1 while the radius of the inner circle is ‘, see Figure 1.3. Thus,= fx = r cos e1 + r sin e2 : r 2 (‘; 1); 2 (0; !)g:

#### Another method for determining the inner and outer expansions

**Another decomposition which allows to treat independently the inner and outer problems**

Throughout this section, we assume that we are in a situation such that the singularity in ln(r) vanishes, see Remark 1.3. Accordingly, the first outer term u0 contains no singular part (u0S = 0; u0 = u0) and u0 is the unique solution in H1( 0) of (1.9). That solution depends on the loading characterized by h and f. Note that (1.9) is posed on the domain without the defect and hence u0 does not depend on the defect. There is, in general, no particular method to find it and hence we will assume that u0 has been determined. Consequently, by virtue of the previous analysis, the coeﬃcients bnn for n 0 are assumed to be known.

Le us consider now the other outer terms, i.e. ui for i 1. From the previous part, we know that the outer term ui is decomposed into its singular part uiS and its regular part ui. Moreover ui is determined in terms of uiS by virtue of (1.31), see also Proposition 1.2. Recalling that there is no logarithmic singularity, by virtue of (1.26) uiS is given by uSi(x) = anir n cos(n ):

**Table of contents :**

**1 Matching asymptotic method in presence of singularities in antiplane elasticity **

1.1 Introduction

1.2 The Matched Asymptotic Method

1.2.1 The real problem

1.2.2 The basic ingredients of the MAM

The outer expansion

The inner expansion

Matching conditions

1.2.3 Determination of the different terms of the inner and outer expansions

The singular behavior of the ui’s and the vi’s

The problems giving the regular parts ui and vi

The construction of the outer and inner expansions

The practical method for determining the coefficients ai

1.2.4 Verification in the case of a small cavity

1.3 Another method for determining the inner and outer expansions

1.3.1 Another decomposition which allows to treat independently the inner and outer problems

1.3.2 The construction of the coefficients ai+n

1.3.3 Example of calculation of the sequence of coefficients

1.3.4 The analytic form of the U

n’s in the case where

0 is an angular sector

1.3.5 The analytic form of the V n’s in the case of a circular cavity

1.3.6 The analytic solution for the V n’s in the case of a crack

Conclusion

1.4 Conclusion and Perspectives

1.5 Appendix: the Hilbert problem

1.5.1 Stress intensity factor K

1.5.2 The jump of the displacement V at position (a; 0)

**2 Application to the nucleation of a crack in mode III**

2.1 Introduction

2.2 The case of a non cohesive crack

2.2.1 Setting of the problem

2.2.2 The issue of the computation of the energy release rate

2.2.3 Numerical results obtained for G` by the FEM

2.2.4 Application of the MAM to the non cohesive case

2.2.5 Evaluation of the energy release rate by the MAM

2.3 Application to the nucleation of a non cohesive crack

2.3.1 The two evolution laws

2.3.2 The main properties of the G-law and the FM-law

2.3.3 Computation of the crack nucleation by the MAM

2.4 The case of a cohesive crack

2.4.1 Dugdale cohesive model

2.4.2 Study of the crack nucleation

The variational formulation of the evolution of the two crack tips

Calculation of the energy release rate G

Calculation of the energy release rate G`

2.4.3 Approximation by the MAM

Outer problem

Inner problem

2.4.4 Resolution of the inner problem in a closed form

The condition on the tip of the cohesive zone: vanishing of the stress intensity factor ~K1

Calculation of [[~v1]]()

The first stage of the nucleation of the crack, when = 0

Beyond the first stage of the crack nucleation

2.4.5 Comparison of the crack nucleation criteria

**3 Generalization to plane elasticity **

3.1 Introduction

3.2 Case of a non cohesive crack

3.2.1 Numeric results obtained for P(`) and G(`) by FEM

3.3 Case of a cohesive crack

3.3.1 Analysis of characteristic equations

3.3.2 Derivation of the singular solutions

3.3.3 Problem of elasticity in 2D

3.3.4 The stress intensity factor

3.3.5 The jump of the displacement

3.3.6 Conclusion

3.A Appendix

3.A.1 Proof of Proposition 3.1

3.A.2 Proof of Proposition 3.3

3.A.3 Proof of Proposition 3.4

3.A.4 Proof of Proposition 3.5

3.A.5 Lemma 3.2 and its proof