Micromechanics of Fracture and Damage

By Luc Dormieux and Djimedo Kondo

This book deals with the mechanics and physics of fractures at various scales.  Based on advanced continuum mechanics of heterogeneous media, it develops a rigorous mathematical framework for single macrocrack problems as well as for the effective properties of microcracked materials.  In both cases, two geometrical models of cracks are examined and discussed: the idealized representation of the crack as two parallel faces (the Griffith crack model), and the representation of a crack as a flat elliptic or ellipsoidal cavity (the Eshelby inhomogeneity problem).

The book is composed of two parts:

- The first part deals with solutions to 2D and 3D problems involving a single crack in linear elasticity. Elementary solutions of cracks problems in the different modes are fully worked. Various mathematical techniques are presented, including Neuber-Papkovitch displacement potentials, complex analysis with conformal mapping and Eshelby-based solutions.

- The second part is devoted to continuum micromechanics approaches of microcracked materials in relation to methods and results presented in the first part. Various estimates and bounds of the effective elastic properties are presented. They are considered for the formulation and application of continuum micromechanics-based damage models.

• Language: English
• Publisher: Wiley
• Release date: Mar 31, 2016
• ISBN: 9781119292180

Book preview

Preface

And it shall come to pass, while my glory passeth by, that I will put thee in a clift of the rock, and will cover thee with my hand while I pass by.

Exodus 33:22

An examination of the literature devoted to cracked media reveals that there are two main options for the geometrical modeling of cracks:

– the first option [GRI 21] consists of the idealized representation of the crack as two parallel faces (segments in plane strain/stress conditions or plane surfaces in three dimensions [3D]). The usual approach consists of adopting stress free boundary conditions on the crack faces. The two faces asymptotically coincide in this mathematical idealization and the displacement undergoes a discontinuity across the crack line (respectively, surface). Indeed, the displacement vectors of two material points located on each face at the same geometrical point in the initial configuration can differ from one another. Clearly, the discontinuity of the displacement field is a consequence of the idealization of the crack as a geometrical entity having a measure equal to zero in the integral sense. For the same reason, the stresses at a crack tip are singular, which has led to the introduction of the well-known concept of stress intensity factors. This first model is referred to throughout the book as the Griffith crack model. It will be presented in two-dimensional conditions (plane strain/stress), as well as in 3D conditions;

– as a second option, the crack is represented as a flat cavity. For instance, it will be a flat ellipse in plane strain/stress conditions, or a flat spheroid (or ellipsoid) in 3D, characterized by an infinitesimal aspect ratio. Consequently, the mathematical measure (in the integral sense) of the crack is infinitesimal but non-zero. This point of view represents the cracked medium as a heterogeneous material and the crack itself as an inhomogeneity in the sense of the homogenization theory. This geometrical description will therefore be referred to as the inhomogeneity model. As long as the aspect ratio has a small but non-zero value, the latter model warrants the ability to define a continuous extension of the displacement field in the crack cavity, as done classically in micromechanics of porous media. It also avoids the occurrence of stress singularities.

The very existence of two geometrical models for the same physical entity raises the question of their consistency. As pointed out above, one model induces mathematical singularities while the other model preserves the continuity of the displacement field and the absence of stress singularity, provided that the aspect ratio remains infinitesimal but non-zero. This of course might erroneously suggest that the two models are not compatible. In fact, the consistency must be examined in an asymptotic sense, when the crack aspect ratio tends to 0. It will be shown that the two models yield perfectly consistent estimates in terms of effective elastic properties. A thorough comparison of the local stress, strain and displacement fields is also proposed.

The book is organized as follows:

– Chapter 1 presents some mathematical tools of the theory of linear elasticity, which will be useful in forthcoming developments. Beginning with plane elasticity, the method of the Airy function is recalled. Biharmonic stress functions can be generated in a systematic way by means of the complex potential approach, which is also briefly presented. The method of the Airy function will be implemented in the framework of each of the two geometrical models;

– in view of application to the inhomogeneity model, Chapter 2 first introduces the Green’s function. This paves the way for a presentation of the so-called inclusion and inhomogeneity Eshelby problems. Indeed, the solutions of the latter requires the determination of the Hill tensor, which is defined from the derivatives of the Green’s function. Eshelby’s inclusion problem is a first step toward the concept of polarization. This motivates the introduction of the Green operator. These tools will be essential for the derivation of variational bounds on the effective elastic properties of microcracked media;

– Chapter 3 deals with the Griffith crack model in two-dimensional conditions. To begin with, the stress singularity at the crack tip and the stress intensity factors are introduced. Then, the complete solutions to mode I and mode II loadings are derived, based on the use of a displacement potential technique (Papkovitch–Neuber potential), which is directly presented in the context of its implementation to crack problems. This yields the corresponding stress intensity factors. Similarly, Chapter 5 deals with the Griffith crack model in 3D conditions. Again the complete 3D solutions in mode I and in shear mode are derived;

– Chapter 4 is devoted to the inhomogeneity model of crack in two-dimensional conditions. The cross-section of the crack is assumed to be a flat ellipse. Two different mathematical techniques are implemented, namely the complex potential approach of the Airy stress function and the solution to the Eshelby inhomogeneity problem. The same Eshelby-based technique is used in Chapter 6 in order to deal with 3D flat spheroidal cracks;

– Chapter 7 introduces the concept of energy release rate and presents the classical thermodynamic reasoning leading to the related criterion for crack propagation;

– the second part of the book is devoted to the effective properties of microcracked media and to damage modeling. It opens with Chapter 8, which proposes a brief introduction to the homogenization of heterogeneous elastic media. The two geometrical models for microcracks (Griffith crack and inhomogeneity model) are successively considered. These two routes are explored in Chapter 9 (Griffith crack) and in Chapters 10 and 11 (inhomogeneity model);

– Chapter 12 is devoted to the variational approach to effective properties. It first presents the energy-based definition of the effective stiffness. Then, the Hashin– Shtrikman–Willis variational approach is detailed. The discussion emphasizes the respective roles of the inhomogeneity shape (flat spheroid in the present case) and of the crack spatial distribution;

– eventually, a micromechanics-based damage constitutive law can be formulated and this is the aim of Chapter 13, which serves as a conclusion to this book. Uniqueness and stability issues concerning the damage model will be discussed.

Rock of Ages, cleft for me,

Let me hide myself in Thee.

Luc DORMIEUX

Djimédo KONDO

February 2016

PART 1

Elastic Solutions to Single Crack Problems

1

Fundamentals of Plane Elasticity

The purpose of this chapter is to present the solution to plane elasticity problems, based on the use of complex-valued potentials. An isotropic linear elastic behavior is considered (except in section 1.8).

1.1. Complex representation of Airy’s biharmonic stress function

Let U be an Airy stress function, from which the stress components in plane elasticity conditions are derived according to:

[1.1]

1_Inline_10_9.jpg

Let 1_Inline_10_11.gif denote the projection on the plane 1_Inline_10_12.gif of a stress tensor σ defined by [1.1]. It is readily proven that Πσ is given by:

[1.2]

1_Inline_10_10.jpg

This expression is useful for the derivation of the components of σ in polar coordinates as a function of the partial derivatives of U. To do so, we recall that:

[1.3] 1_Inline_11_12.jpg

and

[1.4]

1_Inline_11_13.jpg

Introducing [1.3] and [1.4] into [1.2], we obtain:

[1.5]

1_Inline_11_14.jpg

Equations [1.5] are the counterpart in polar coordinates of equations [1.1]. The compatibility condition of the strains, which reads:

[1.6] 1_Inline_11_15.jpg

is ensured, in the case of an isotropic linear elastic behavior, by the condition

[1.7] 1_Inline_11_16.jpg

As a matter of fact, under plane stress or strain conditions, the assumption of linear isotropy allows to write the state equations in the form:

[1.8] 1_Inline_12_13.jpg

Under plane stresses, the elastic compliances Aij are:

[1.9]

1_Inline_12_14.jpg

Under plane strains, these relations become:

[1.10]

1_Inline_12_15.jpg

In both plane strains and plane stresses, the Aij satisfy:

[1.11] 1_Inline_12_16.jpg

Combining [1.1] with [1.10] and using [1.11], we see that condition [1.6] reduces to [1.7]. Such a biharmonic function U is now considered. Let P = ΔU. By definition, P is a harmonic function. Let Q denote the conjugate function, defined up to a constant by:

[1.12] 1_Inline_12_17.jpg

This implies that the complex-valued function f(x + iy) = P (x, y) + iQ(x, y) is holomorphic, which means that the limit (with z = x + iy)

1_Inline_12_18.gif

exists. Indeed, at the first order in dx and dy:

[1.13]

1_Inline_13_15.jpg

Using [1.12] with [1.13] yields

[1.14]

1_Inline_13_16.jpg

so:

[1.15]

1_Inline_13_17.jpg

Following [MUS 53], consider now a primitive ϕ(z) =p + iq of f(z)/4:

1_Inline_13_18.gif

where p and q are two conjugate harmonic functions. Therefore, we have:

[1.16]

1_Inline_13_19.jpg

We can see that

1_Inline_13_20.gif

is harmonic, and that

1_Inline_13_21.gif

Finally, let χ(z) denote the holomorphic function whose real part is p1:

1_Inline_14_15.jpg

Following these definitions, we have:

[1.17]

1_Inline_14_16.jpg

For future purposes, let us determine the partial derivatives of U. Observing that 1_Inline_14_21.gif we first obtain:

[1.18]

1_Inline_14_17.jpg

In turn, 1_Inline_14_22.gif yields:

[1.19]

1_Inline_14_18.jpg

It is convenient to summarize these results in the form:

[1.20]

1_Inline_14_19.jpg

with the notation ψ(z) = χ′(z).

1.2. Force acting on a curve or an element of arc

Let us consider a curve oriented by the tangent unit vector 1_Inline_14_23.gif

1_Inline_14_20.gif

where s denotes the curvilinear abscissa. The positive direction of the normal unit 1_Inline_15_20.gif is defined such that 1_Inline_15_19.gif is oriented like 1_Inline_15_21.gif This being the case, we have:

1_Inline_15_11.gif

Using [1.1], the components of the stress vector 1_Inline_15_18.gif read:

[1.21]

1_Inline_15_12.jpg

The elementary force 1_Inline_15_17.gif acting on ds is represented by a complex dF with real and imaginary parts Txds and Tyds. Using [1.21], this yields:

[1.22]

1_Inline_15_13.jpg

By integration, we obtain the resultant force 1_Inline_15_16.gif acting on a given arc oriented from A to B. Introducing [1.20] into [1.22], the components Fx and Fy are given by:

[1.23]

1_Inline_15_14.jpg

The boundary conditions on a loaded arc are an important application of this result. In the following, let f(z) be defined as:

[1.24]

1_Inline_15_15.jpg

where the point A is fixed and z denotes the affix of point Bz. f(z) is a complex representation of the resultant force acting between A and Bz on the considered arc. f(z) is defined up to constant.

For instance, consider a uniform pressure acting on the loaded arc:

1_Inline_16_13.gif

or

1_Inline_16_14.gif

Introducing this result into [1.24], we obtain:

[1.25]

1_Inline_16_15.jpg

1.3. Derivation of stresses

Consider the choice ds = dy in [1.22], for which 1_Inline_16_21.gif is equal to 1_Inline_16_18.gif so that 1_Inline_16_19.gif is equal to 1_Inline_16_20.gif This implies that Tx = σxx and Ty = σxy:

[1.26]

1_Inline_16_16.jpg

In turn, if ds = −dx, 1_Inline_16_21.gif is along 1_Inline_16_22.gif so that 1_Inline_16_23.gif Hence, we have Tx = σxy and Ty = σyy:

[1.27]

1_Inline_16_17.jpg

Combinations of these relations successively yield:

[1.28]

1_Inline_17_13.jpg

where [1.16] has been used, and

[1.29]

1_Inline_17_14.jpg

The stress components in cartesian and polar coordinates being related by:

[1.30]

1_Inline_17_15.jpg

it is readily seen from [1.28] and [1.29] that:

[1.31]

1_Inline_17_16.jpg

The stresses are not modified if ϕ(z) is replaced by ϕ(z) + iCz + γ and if ψ(z) is replaced by ψ(z) +γ′, where γ and γ′ are complex-valued constants and C is a real-valued constant. Let us assume that the origin z = 0 is part of the domain of study. If the boundary conditions prescribe stresses only, the arbitrariness of the definition of ϕ(z) and ψ(z) allows us to choose them in such a way that:

[1.32]

1_Inline_17_17.jpg

When the domain of study is infinite, another possibility is to define ϕ(z) and ψ(z) by conditions at infinity of the form:

[1.33]

1_Inline_17_18.jpg

1.4. Derivation of displacements

In plane strains, the isotropic linear elastic constitutive equation reads:

[1.34]

1_Inline_18_13.jpg

Observing that σxx = ∂²U/∂y² = P − ∂²U/∂x², and using [1.16] together with [1.28], we obtain:

[1.35] 1_Inline_18_14.jpg

which can be integrated in the form (see [1.16]):

[1.36] 1_Inline_18_15.jpg

We recall that the partial derivatives of U have been determined previously (see equations [1.18] and [1.19]).

Similarly, note that σyy = ∂²U/∂x² = P − ∂²U/∂y². Again, we use [1.16] and [1.28], which yields:

[1.37] 1_Inline_18_16.jpg

A primitive of [1.37] reads:

[1.38] 1_Inline_18_17.jpg

Equations [1.36] and [1.38] define the displacement up to a rigid body motion. Finally, a combination of these equations together with [1.20] gives:

[1.39]

1_Inline_19_12.jpg

where κ = 3 − 4ν.

1.5. General form of the potentials ϕ and ψ

Considering applications, the domain of study S is the complex plane, except a bounded region with closed contour L. Therefore, the studied domain is non-simply connected. We aim to determine the general form of the complex-valued functions ϕ and ψ. Without loss of generality, it can be assumed that the point z = 0 is located within the region bounded by L, that is z = 0 S.

Owing to [1.28], we first note that the real part of ϕ′(z) is single-valued, but this is possibly not the case for the imaginary part. Therefore, the integral of ϕ′(z) on a closed contour surrounding L is a priori not 0 and denoted by 2iπA (A ∈ imgR1.jpg ). There exists a single-valued holomorphic function F(z) defined on S such that:

1_Inline_19_13.gif

By integration, we obtain:

1_Inline_19_14.gif

where zo is some fixed point in S. Again, if (z) is not single-valued, there exists a complex-valued constant B such that (z) − B log(z) is single-valued:

[1.40]

1_Inline_19_15.jpg

where ϕ∗(z) is a single-valued holomorphic function defined on S. A similar reasoning starting from [1.29] shows that there exists a complex-valued constant C such that:

[1.41] 1_Inline_20_12.jpg

where ψ∗(z) is a single-valued holomorphic function defined on S.

We now recall [1.39], and take advantage of the fact that the displacement is single-valued. An anticlockwise integration around L yields:

1_Inline_20_13.gif

from which the following identities are derived:

[1.42] 1_Inline_20_14.jpg

We now apply [1.23] to the whole contour L:

[1.43]

1_Inline_20_15.jpg

where Fx and Fy denote the components of the resultant force acting on the contour. In order for the unit normal 1_Inline_20_17.gif to point outward with respect to S, note that the contour must be oriented clockwise. Using [1.40], [1.41] and [1.42], we find that:

1_Inline_20_16.gif

Eventually, combining this result with [1.42], ϕ(z) and ψ(z) take the form:

[1.44]

1_Inline_21_9.jpg

Let us finally add the assumption that the stresses are bounded at infinity. This being the case, consider the Laurent series expansions of ϕ∗(z) and ψ∗(z) in S:

1_Inline_21_10.gif

We can easily see that [1.28] requires an = 0 for n ≥ 2. In the same line of reasoning, [1.29] requires bn = 0 for n ≥