Fracture of disordered solids in compression as a critical phenomenon: II. Model Hamiltonian for a population of interacting cracks

HAL Id: hal-00110575

https://hal.archives-ouvertes.fr/hal-00110575

Submitted on 20 Nov 2018

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Fracture of disordered solids in compression as a critical

phenomenon: II. Model Hamiltonian for a population of

interacting cracks

Renaud Toussaint, Steven R Pride

To cite this version:

Renaud Toussaint, Steven R Pride. Fracture of disordered solids in compression as a critical

phe-nomenon: II. Model Hamiltonian for a population of interacting cracks. Physical Review E :

Sta-tistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2002, 66 (3), pp.036136.

�10.1103/physreve.66.036136�. �hal-00110575�

Fracture of disordered solids in compression as a critical phenomenon.

II. Model Hamiltonian for a population of interacting cracks

Renaud Toussaint*and Steven R. Pride†

Ge´osciences Rennes, Universite´ de Rennes 1, 35042 Rennes Cedex, France

共Received 14 November 2001; revised manuscript received 13 June 2002; published 27 September 2002兲 To obtain the probability distribution of two-dimensional crack patterns in mesoscopic regions of a disor-dered solid, the formalism of Paper I requires that a functional form associating the crack patterns共or states兲 to their formation energy be developed. The crack states are here defined by an order parameter field repre-senting both the presence and orientation of cracks at each site on a discrete square network. The associated Hamiltonian represents the total work required to lead an uncracked mesovolume into that state as averaged over the initial quenched disorder. The effect of cracks is to create mesovolumes having internal heterogeneity in their elastic moduli. To model the Hamiltonian, the effective elastic moduli corresponding to a given crack distribution are determined that includes crack-to-crack interactions. The interaction terms are entirely respon-sible for the localization transition analyzed in Paper III. The crack-opening energies are related to these effective moduli via Griffith’s criterion as established in Paper I.

DOI: 10.1103/PhysRevE.66.036136 PACS number共s兲: 62.20.Mk, 46.50.⫹a, 46.65.⫹g, 64.60.Fr

I. INTRODUCTION

In triaxial-stress experiments on rocks in the brittle re-gime, the onset of a macroscopic localization of deformation is usually observed around peak stress关c.f. Be´suelle 关1兴 for a review兴. Such departure from a macroscopically uniform de-formation regime is intrinsically beyond the capacities of a mean-field theory, and so a specific model is developed here that takes the orientational nature of crack-to-crack interac-tions into account.

This is the second paper in a series of three dedicated to exploring how the physical properties of disordered solids evolve as they are led to failure in a state of compression. The goal of this paper is to obtain a reasonable form for the Hamiltonian Ej(␧,␧m) which is defined as the average work required to lead an intact region at zero deformation to the crack state denoted by j when the maximum applied strain is

␧mand where the final strain␧ is possibly different than␧m due to a final unloading. This Hamiltonian must be expressed in terms of the spatial distribution of the local order param-eter that is the variable used to characterize the population of cracks in each mesovolume of a huge disordered-solid sys-tem.

Most existing lattice models explore the dynamics of sca-lar order parameters either representing the breakdown of elastic spring or beam networks under tensile stress关2兴, or of fuse networks关3兴. The analogies between such scalar models and fracture of disordered media has been widely discussed

关4兴. One advantage of our approach is the ability to explore

interactions based on a fully tensorial description of the stress perturbations produced by each crack. Another is its ability to yield analytical rather than only numerical results.

Using Ejin the partition function established in Paper I, it is possible to explore the crack patterns that emerge in com-pressive settings for which isolated cracks appear in an in-trinsically stable manner no matter their size 关5兴, and for which macroscopic localization is a collective phenomenon due to the energetic organization of small cracks as opposed to an instability associated with the largest defects. In the present paper, we retain the leading-order effects of oriented-crack populations interacting in two-dimension 共2D兲. The overriding importance of the long-range elastic interactions leaves hope that 3D generalizations would not yield qualita-tively different critical behavior.

II. PRINCIPLES OF THE MODEL A. Order-parameter definition

We now elaborate on the crack model introduced in Paper I. Each mesovolume of a huge rock system is discretized into a square network of diamond-shaped cells of size ⌳ 共grain sizes兲 and only a single crack is potentially present in each cell. A crack is located at the center of the cell and has a length d somewhere within the support 关0,dm兴, where dm is the maximum crack length共a fixed parameter of the system兲 required to satisfy dm⬍⌳. In the perturbative treatment of the crack interactions developed herein, ⑀⫽(dm/⌳)D is taken to be a small number, where D is the number of space dimensions 共in the present model, D⫽2). The local order parameter␸(x) associated with each cell x is taken to have an amplitude␺⫽兩␸兩⫽(d/dm)Dand has a sign that indicates whether the crack is oriented at ⫹45 ° or ⫺45 ° relative to the principal stress direction 共the so-called ‘‘axial direc-tion’’兲. The model is summarized in Fig. 1

The restriction that cracks are either at ⫾45 ° and have lengths less than the grain dimensions is of course a great simplification compared to what is found inside of real rocks. However, we only need to characterize the essential features of a crack population that contribute to localization phenom-ena and, to this extent, it appears overly complicated to *Present address: Department of Physics, University of Oslo, P.O.

Box 1048 Blindern, 0316 Oslo 3, Norway. Email address: Renaud.Toussaint@fys.uio.no

model the amazing variety of crack geometries encountered in real rocks关6,7兴. The localization transition involves spon-taneous breaking of symmetries both under translation and parity共inversion of the minor stress axis兲 as is seen from the structure of the shear bands emerging in the post-peak-stress regime关8兴. The essential feature of any proposed order pa-rameter is that it must reflect and quantify the amount of local symmetry breaking and our simple model with cracks at either⫾45 ° does just this.

Furthermore, there is evidence both from acoustic-emissions monitoring 关9兴 and from direct observation after unloading关10兴, that cracks developing prior to peak stress do not exceed an extent of a few grains diameters. This is prin-cipally because the grain contacts that break are much weaker than the grains and have a finite length so that cracks arriving in compression do so stably 关5兴. Crack coalescence is not explicitly allowed for. However, since several neigh-boring cells in a line may all contain cracks of the same sign, the long-range elastic effect of long 共coalesced兲 cracks is effectively allowed for. Our picture of the final shear bands experimentally observed in the post-peak-stress regime is that they were created by unstable sliding along a band weakened in the pre-peak-stress regime by a concentration of coherently oriented cracks 关8兴. Our model allows small cracks to stably concentrate en e´chelon along conjugate bands relative to the principal-stress direction; however, it does not model the final unstable rupture along a given band.

B. Formation energy of a crack pattern

It has been established in Paper I that to a reasonable approximation, the work required to form a crack state, as

averaged over the initial disorder, separates into one part representing the work required to break the grain contacts, and a second part representing the elastic energy stored in the cracked solid. This was expressed in Eqs.共26兲–共28兲 of Paper I as Ej⫽ q 2␧m:共C0⫺Cj兲:␧m⫹ 1 2␧:Cj:␧. 共1兲

The first term of Eq.共1兲 is the energy spent in the irreversible formation of the crack state j averaged over quenched disor-der and was obtained through an application of Griffith’s principle. The parameter q derives from the quenched-disorder distribution and lies in the range 关1/2,1兴 共see Sec. III B 2 of Paper I兲. The second term is the reversibly-stored elastic energy with Cj being the elastic-stiffness tensor of state j.

Our principal task is therefore to model the way that cracks and collective crack-states affect the overall elastic moduli of a mesovolume. This requires detailed knowledge of the stress 共or strain兲 field throughout the mesovolume in the presence of arbitrary crack populations, and we treat this need using the following approximations. First, since the cracks in the model are isolated one to each cell, their main effect regarding the far-field stress is to change the elastic moduli of their embedding cell. Such a change is modeled assuming the cracks to be penny-shaped ellipsoidal cavities. We ignore how such ellipses change shape when the applied stress is unloaded/reloaded since linear elasticity alone cap-tures the principle effect of how the rock becomes weaker due to strategic placement of cracks in cells. Since a crack occupies a limited extent of a cell, the modification of the moduli is small compared to the moduli of the intact cell so that the resulting far-field stress field can be developed as a Born series. It is in the third term of this development that crack-to-crack interactions are first allowed for. Higher-order interactions 共three cracks simultaneously interacting and so on兲 are negligible to the extent that⑀⫽(dm/⌳)Dcan be con-sidered small.

III. ELASTIC ENERGY

The goal of this section is to determine the elastic energy

E_jelstored in a mesovolume occuping the region⍀ and con-taining the crack state j 共which denotes the spatial distribu-tion of␸(x) at all points x of⍀) when a displacement cor-responding to a uniform strain tensor ␧(0) is applied on the external surface ⳵⍀ of the mesovolume.

A. Elastic energy of a weakly heterogeneous solid

The effect of the crack field ␸(x) is to perturb the stiff-ness tensor of each cell as C(x)⫽C0⫹␦c关␸(x)兴, where C0 denotes the moduli of an uncracked cell 共assumed uniform for all cells兲, and where␦c(x) is a small perturbation due to

the possible presence of a crack as characterized by␸(x). It is established in the Appendix that the nonzero components of ␦c are typically smaller than those of C0 by a factor ⑀

⫽(dm/⌳)DⰆ1. Our problem is to resolve an elasticity

FIG. 1. Part of the diamond network of cells that comprise a mesovolume. Each cell has the linear dimension ⌳ and is only allowed to contain one crack. The maximum length of any crack is dm and this length is assumed to be sufficiently small that

(dm/⌳)DⰆ1. The amplitude of the order parameter is by definition

␺⫽兩␸兩⫽(d/dm)D, where d is the length of the crack found in the

cell, while the sign of␸ indicates the orientation of the crack as shown.

RENAUD TOUSSAINT AND STEVEN R. PRIDE PHYSICAL REVIEW E 66, 036136 共2002兲

boundary-value problem in a region ⍀ containing a weakly heterogeneous stiffness tensor C(x).

The displacement boundary conditions are given as

᭙x苸⳵⍀, u共x兲⫽␧(0)_{•x, 共2兲}

where x denotes distance from the center of the mesovolume. Elastostatic equilibrium requires that

⳵j␶i j⫽Ci jkl

0 _⳵

j⳵kul⫹⳵j关␦ci jkl⳵kul兴⫽0 共3兲 throughout ⍀, where summation over repeated indices is assumed both here and throughout. Due to the linearity of the problem, we use the elastostatic Green tensor G⫽Gi jxˆixˆjfor a uniform material which is a solution of

C_{i jkl}0 ⳵j⳵kGlm共x,x

⬘

兲⫹␦im␦D共x⫺x

⬘

兲⫽0, 共4兲

᭙x苸⳵⍀, Gi j共x,x

⬘

兲⫽0. 共5兲 The components Gi j(x,x

⬘

) define the ith component of the displacement at x induced by a unit point force acting along the j axis at x

⬘

. Here,␦i jis the Kronecker symbol, and␦Dis the D-dimensional Dirac distribution.

The solution for the displacements when no cracks are present is simply u(0)_(x)⫽␧(0)_{•x throughout all of ⍀. Thus,}

it is a straightforward excercise to demonstrate that the total displacement u in the presence of the cracks satisfies the following integral equation:

ui共x兲⫽ui

(0)_共x兲⫹

冕

⍀Gi j共x,x

⬘

兲⳵k⬘关␦cjklm⳵l⬘um兴共x

⬘

兲d

D_x

_⬘

_,

共6兲

where⳵i⬘denotes the partial derivative relative to the coor-dinate x_i

⬘

. Using⑀as the argument of a series expansion, we write the displacements as u⫽u(0)⫹u(1)⫹•••u(n) ⫹O(⑀(n⫹1)_{), where each u}(m) _{is O(⑀}m_{). Collecting terms at} each order of ⑀ in Eq. 共6兲 gives the following recursion re-lation:

u_i(n⫹1)共x兲⫽

冕

⍀Gi j共x,x

⬘

兲⳵k⬘关␦cjklm⳵l⬘um

(n)_兴共x

_⬘

_兲dD_x

_⬘

_.

共7兲

The boundary conditions used to define G guarantee that for all n⬎0, the displacements u(n) are zero on the boundary

⳵⍀.

The quantity we are specifically seeking to establish is the elastic energy density Eel_⫽ᐉ⫺D_兰

⍀12␶(x):␧(x)d

D_{x, where} we recall that ᐉ is the linear dimension of a mesovolume. The definitions of the strain ␧i j⫽

1

2(⳵iuj⫹⳵jui) and stress

␶i j⫽(Ci jkl

0 _⫹␦

ci jkl)␧kl give immediately the following rela-tions: ␧i j (n)_⫽1 2共⳵iuj (n)_⫹⳵ jui (n)_{兲, 共8兲} ␶i j (n)_⫽C i jkl 0 _␧ kl (n)_⫹␦c i jkl␧kl (n⫺1)_{, 共9兲} E(n)⫽ 1 2ᐉDa

兺

⫽0 n

冕

⍀␶ (n⫺a)_:␧(a)_dD_{x, 共10兲} with the convention␧(⫺1)⫽0. In the last expression, the fact that u(a)⫽0 on the boundary for all a⬎0 guarantees that after integrating by parts,

冕

⍀␶

(n⫺a)_:␧(a)_dD_x⫽

冕

⍀␶

(n⫺a)_:“u(a)_dD_x

⫽

冕

⳵⍀n•␶

(n_⫺a)•u(a)_dD_⫺1x⫽0, where we used the facts that the stress tensor is symmetric and solenoidal. The nth term of the total elastic energy is then

E(n)⫽1

2␶

(n)_:␧(0)_{, 共11兲}

where the upper bar denotes a volume average over ⍀. The first term of the elastic energy is independent of the␸ field, and corresponds to the physically unimportant amount of energy

E(0)_⫽1

2␧

(0)_:C(0)_:␧(0) _共12兲

stored in the intact state.

For the higher orders n⭓1, ␶(n)_{is expressed by Eq.共9兲,}

and the same argument as above using the fact that u(n)⫽0 on⳵⍀ eliminates a term: E(n)⫽␧(n):C(0):␧(0)⫹1 2␧ (n_{⫺1):␦c:␧}(0)_⫽1 2␧ (n_{⫺1):␦c:␧}(0)_. 共13兲

The second term of the developement,

E(1)⫽1

2␧

(0)_:¯:␦c␧(0)_{, 共14兲}

represents only a local dependance on ␦c共and therefore on

the crack field兲 since it does not involve nested integrals over two different positions. It will be shown to represent only the contribution of the average crack porosity to the stiffness of the rock.

The third term of the development is where the desired crack-to-crack interactions arrive. Using the symmetry of␦c

under the inversion of its two first or last indices, and Eq.共7兲 to have an integral form of u(1), Eq.共13兲 transforms to

2ᐉDE(2)⫽

冕

⳵bua (1) 共x兲␦cabcd共x兲␧cd (0) dDx ⫽

冕 冕

␧cd (0)_␦c abcd共x兲⳵bGa j共x,x

⬘

兲 ⫻⳵k⬘␦cjklm共x

⬘

兲␧lm (0)_dD_xdD_x

_⬘

_{. 共15兲}

This term accounts for the way that a crack present at x

⬘

energetically interacts with a different crack at x. This is the nonlocal interaction term that is ultimately responsible for the localization transition. The higher terms of the Born de-velopment can be neglected for our purposes.

B. Elastic energy as explicit function of the crack field

To establish the terms of the Born-approximated elastic energy as explicit functions of both the crack state␸and the imposed strain␧(0), a few definitions are first introduced.

The principal axes of ␧(0) are along (eˆradial,eˆaxial) as

de-noted in Fig. 1. Our square network of cells is rotated⫹45 ° from this orthonormal basis. We work here in the coordinates (eˆ1,eˆ2) of the square network so that the applied-strain takes

the form

␧(0)_⫽1

2

冉

⌬ ␥

␥ ⌬

冊

, 共16兲

where⌬⫽␧_radial⫹␧_axialand␥⫽⫺(␧_radial⫺␧_axial) are the im-posed dilatation and shear strain.

For convenience, we assume the intact material to be iso-tropic. Taking␭⫹2␮as the stress unit, where (␭,␮) are the Lame´ parameters of the material, and using the usual

tensor-to-matrix mapping of the indices (11)→1; (22)

→2; (12)→3, the fourth-order stiffness tensor of the intact material takes the form

C0⫽

冉

1 2␣⫺1 0 2␣⫺1 1 0 0 0 1⫺␣

冊

, 共17兲 where ␣⫽_␭⫹2␭⫹␮_␮ 共18兲

is a material-dependent constant in the range关0.5,1兴. The deviation ␦c of this tensor due to the possible

pres-ence of a crack in a cell separates into an isotropic contribu-tion independent of the crack’s orientacontribu-tion, and into an an-isotropic orientation-dependent contribution. In the Appendix, we demonstrate that

␦c共x兲⫽⑀关A␸共x兲⫹B兩␸共x兲兩兴, 共19兲 A⫽

冉

␩2⫺␩1 0 0 0 ␩₁⫺␩₂ 0 0 0 0

冊

, 共20兲 B⫽

冉

⫺␩1 ⫺共2␣⫺1兲␩2 0 ⫺共2␣⫺1兲␩2 ⫺␩1 0 0 0 ⫺共1⫺␣兲␩₃

冊

, 共21兲

where (␩1,␩2,␩3) are positive constants expressed in the

Appendix in terms of the Lame´ parameters.

Making the necessary contractions over the indices, we easily obtain the trivial 共crack independent兲 energy E(0) us-ing Eqs. 共12兲, 共16兲, and 共17兲. For later convenience, this re-sult is best written in matrix form as

E(0)⫽1 2共⌬,␥兲•M0•共⌬,␥兲 T_{, 共22兲} M0⫽

冉

␣ 0 0 1⫺␣

冊

. 共23兲

Using the auxiliary field

␺共x兲⫽兩␸共x兲兩 共24兲

denoting the amplitude of each crack, one similarly obtains

关using Eqs. 共14兲 and 共19兲–共21兲兴

E(1)⫽⑀ 2关␧ (0)_:A:␧(0)_␸¯⫹␧(0)_:B:␧(0)_␺¯兴 ⫽1₂共⌬,␥兲•M1•共⌬,␥兲T, 共25兲 with M1⫽⫺⑀␺¯

冉

␬2 0 0 ␬3

冊

, 共26兲 ␬2⫽ ␩1 2 ⫹ 2␣⫺1 2 ␩2, 共27兲 ␬3⫽共1⫺␣兲␩3. 共28兲

The term proportional to␸¯ has algebraically canceled due to the symmetry of the problem under parity; inversion of the minor axis eˆradialflips the orientation of cracks, and therefore

changes the sign of ␸¯ , while the energy remains necessarily unchanged. The surviving term is negative and proportional to␺¯ , and accounts for the softening of the mesovolume due to the presence of cracks. This dependence on the total num-ber of cracks is the only order-parameter dependent effect to first order.

Last, the crack-interaction term of principal interest can be readily expressed from Eqs.共15兲 and 共19兲 as

⫺2ᐉD_E(2)_⫽⑀2_␧ cd (0)_␧ kl (0)_A abcdAi jklfaib j ⫹2⑀2_␧ cd (0)_␧ kl (0)_A abcdBi jklgaib j ⫹⑀2_␧ cd (0)_␧ kl (0)_A abcdBi jklhaib j, 共29兲 where the fourth-order tensors f,g,h are functionals of␸ and defined as

faib j⫽

冕

dDx

冕

dDx

⬘

Gai共x,x

⬘

兲⳵b␸⳵j⬘␸, 共30兲

gaib j⫽

冕

dDx

冕

dDx

⬘

G兵ai其共x,x

⬘

兲⳵b␸⳵j⬘␺, 共31兲

RENAUD TOUSSAINT AND STEVEN R. PRIDE PHYSICAL REVIEW E 66, 036136 共2002兲

h_{aib j}⫽

冕

dD_x

冕

_dD_x

_⬘

_G

ai共x,x

⬘

兲⳵b␺⳵j⬘␺. 共32兲 In the second term, the reciprocity of the Green function

Gai(x,x

⬘

)⫽Gai(x

⬘

,x) is used as well as the notation G兵ai其

⫽(Gai⫹Gia)/2.

The Green tensor needed here satisfies the Dirichlet con-ditions of Eq.共5兲 and can be obtained, in principle, from the infinite-space Green tensor via the image method. However, this transforms E(2)_{into an infinite series共one term for each}

image兲, and makes the functional integrations of Paper III analytically hopeless. To remedy this problem, the Green function with a periodic instead of zero boundary condition is used as ersatz. Since u(1) is only affected close to the boundaries by this replacement, this approximation will be considered valid for the evaluation of the volume integral

E(2)_.

The double integrals of Eqs. 共30兲–共32兲 are most easily expressed using the 2D finite-Fourier transform

F ˜_共k兲⫽

冕

⍀d D_xF共x兲e⫺ik•x_{, 共33兲} F共x兲⫽ 1 ᐉD

兺

_k F˜共k兲e ik•x_{, 共34兲}

where the sum over the wave vectors k is over 兵k

⫽2␲ni/ᐉeˆi; ᭙i,ni苸Z其 with an upper cutoff given by maxni⬎ᐉ/⌳ that reflects the fact that the order parameter cannot vary on scales smaller than cell sizes ⌳. Since the Green function used is defined with periodic boundary con-ditions, it satisfies G_ai(x,x

⬘

)⫽G_ai(x⫺x

⬘

). Its Fourier trans-form is easily established, and upon recalling that (␭⫹2␮) is adopted as the stress unit, reads

G˜共k兲⫽ 1

共1⫺␣兲k2共I⫺␣kˆkˆ兲, 共35兲

kˆ⫽ k

储k储, 共36兲

where I is the identity tensor. This is real and symmetric, as is G(x) itself. Since ␸ and ␺ are real fields, one has ␸˜ (⫺k)⫽␸˜*(k) and ␺˜ (⫺k)⫽␺˜*(k). Using these relations, together with the identity 兰_⍀dDxeik•x⫽ᐉD␦k, the integrals

of Eqs.共30兲–共32兲 become the following sums

f_{aib j}⫽1 ᐉD_k

兺

_⫽0 兩␸˜共k兲兩 2_kˆ jkˆb共␦ia⫺␣kˆikˆa兲 共37兲 gaib j⫽ 1 ᐉD

兺

k⫽”0 Re关␸˜共k兲␺˜*_{共k兲兴kˆ}jkˆb共␦ia⫺␣kˆikˆa兲, 共38兲 haib j⫽ 1 ᐉD

兺

k⫽”0 兩␺˜_共k兲兩2_kˆ jkˆb共␦ia⫺␣kˆikˆa兲, 共39兲

where Re denotes the real part of a complex quantity and where kˆi denotes the ith component of kˆ⫽k/储k储. With the following definitions associated with the orientation of k:

␪k⫽共eˆ1,k兲, 共40兲

uk⫽cos共2␪k兲⫽cos2␪k⫺sin2␪k⫽kˆ1 2_⫺kˆ

2 2

, 共41兲

vk⫽sin共2␪k兲⫽2 cos␪ksin␪k⫽2kˆ1kˆ2, 共42兲

the remaining contraction in Eq.共29兲 over the eight indices (abcdi j kl) is performed. The calculation is a bit long but without surprise and finally produces

E(2)⫽1 2共⌬,␥兲•M2•共⌬,␥兲 T_{, 共43兲} M2⫽ ⫺⑀2 共1⫺␣兲ᐉ2D

冉

a b b c

冊

, 共44兲

where the components a, b, and c are defined,

a⫽

兺

k⫽”0 a_k; b⫽

兺

k⫽”0 b_k; c⫽

兺

k⫽”0 c_k, 共45兲 ak⫽共1⫺␣uk 2_兲␬ 1 2_兩␸˜ k兩2⫹2共1⫺␣兲uk␬1␬2Re共␸˜k␺˜k*兲 ⫹共1⫺␣兲␬2 2_兩␺˜ k兩2, b_k⫽⫺␣u_kvk␬1␬3Re共␸˜k˜␺k*兲⫹共1⫺␣兲vk␬2␬3兩␺˜k兩2, ck⫽共1⫺␣vk 2_兲␬ 3 2_兩␺˜ k兩2,

with␬₂,␬₃defined in Eqs.共27兲, 共28兲 and␬₁a new material-dependent constant,

␬1⫽ˆ

␩1⫺␩2

2 . 共46兲

IV. SURFACE FORMATION ENERGY

Next, we must account for the energy EI_j that irreversibly went into creating the cracks of a given crack state j at a maximum deformation␧_m. In Paper I, this contribution was obtained using Griffith’s criterion as

EI_j⫽q

2ᐉ D_␧

m:共C0⫺Cj兲:␧m, 共47兲 where q derives from the quenched disorder and is bounded as 0.5⭐q⬍1. The derivation of this statement implicitly as-sumed that all cracks were the same length. In the present treatment, cracks are allowed to have any length in the range 0⭐d⭐dm. It is a straightforward exercise to demonstrate that if the breaking energies for each possible length d are all sampled from the same quenched-disorder distribution, then Eq.共47兲 again holds. We forego such a demonstration. In the notation of the present paper we may thus state that

E关␸兴I⫽q共E(0)⫺Eel关␸兴兲⫽⫺q共E(1)⫹E(2)兲关␸兴, 共48兲

where E(1) and E(2) are the terms of the Born-development given by Eqs.共25兲 and 共43兲 upon replacing the current strain parameters⌬ and␥by the maximum-achieved strain⌬mand

␥m.

V. TEMPERATURE

Although not required as part of the Hamiltonian model, we now give an explicit in ⌬m,␥m approximate expression for the temperature by using Eq. 共59兲 of Paper I. This tem-perature was derived in Paper I assuming only a single crack size. Unfortunately, the result does not easily generalize to multiple crack sizes and so we simply take d⫽dmto obtain the estimate ᐉD_T共⌬ m,␥m兲⫽⫺ 共1⫺q兲dm D_e 1共⌬m,␥m兲 ln兵关␨/e1共⌬m,␥m兲兴q/(1⫺q)⫺1其 , 共49兲

where e₁d_mD is how much the first-Born elastic energy in a mesovolume is reduced when a crack of length d_m is

intro-duced 关c.f. Eqs. 共25兲 and 共26兲兴. The energy density

e1(⌬m,␥m) is defined as e1共⌬m,␥m兲⫽ 1 2共␬2⌬m 2_⫹␬ 3␥m 2_{兲, 共50兲}

while ␨ is a dimensionless ‘‘fracture toughness’’ parameter defined as

␨⬅_共␭⫹2⌫_␮兲d

m

. 共51兲

There is a phase transition when (␨/e₁)q/(1⫺q)_{⫽2 and T} di-verges so that all crack states become equally probable.

We now consider whether such a phase transition is ex-pected in laboratory experiments on rocks. The order of mag-nitude values ⌫⬃102 _J/m2_{, d}

m⬃10⫺5m, ␬2⬃1, and (␭

⫹2␮)⬃1010Pa are appropriate for typical grains in rocks so that ␨⬃10⫺3. When a rock fails in shear, the accumulated strain is on the order of a percent or two, so that the maxi-mum value of e1of interest is also on the order of 10⫺4. We

thus find that at shear failure, ␨/e1⬃10 and so we a priori

expect the localization transition to occur prior to the temperature-divergence transition. This is more quantita-tively demonstrated in Paper III.

VI. SUMMARY

Collecting together both the elastic energy and the surface formation energy, we obtain at last the Hamiltonian to be used in performing ensemble averages over crack states in the next paper. We write this Hamiltonian in the final form

Ej共␧,␧m兲⫽ER共␧兲关␸兴⫹EI共␧m兲关␸兴,

ER共␧兲关␸兴⫽E0共⌬,␥兲⫹Eav共⌬,␥兲关␸兴⫹Eint共⌬,␥兲关␸兴, EI共␧m兲关␸兴⫽⫺q兵Eav共⌬m,␥m兲关␸兴⫹Eint共⌬m,␥m兲关␸兴其, where (⌬,␥) are the isotropic and shear strain components of the current strain tensor␧, and (⌬m,␥m) are similar quan-tities referring to the maximum achieved strain␧m. The en-ergy E0 is the trivial elastic energy of the uncracked state

E0共⌬,␥兲⫽1

2兵␣⌬

2_{⫹共1⫺␣兲␥}2_其,

where ␣ is a dimensionless elastic constant in the range

关0.5,1兴 defined by Eq. 共18兲.

The next term in the Born development is Eav⫽E(1), which depends only on the volume average␺¯ , which is the fraction of cracked cells in the crack state ␸ and is thus entirely independent of the spatial fluctuations of ␸. Its de-pendence on the strain (⌬,␥) is

Eav共⌬,␥兲关␸兴⫽⫺1

2关␬2⌬

2_⫹␬

3␥2兴⑀␺¯ .

We defined ⑀⫽(dm/⌳)Dto be a small parameter, where D

⫽2 is the number of space dimensions in the model, and

dm, ⌳, and ᐉ as respectively the linear sizes of the largest crack, a unit cell, and a mesovolume. The three coefficients

␬i are positive dimensionless material-dependent constants defined by Eqs. 共27兲, 共28兲, and 共46兲.

The interaction energy Eint⫽E(2)involve a quadratic ma-trix operator Pkthat, for each nonzero wave vector k, mixes

together the Fourier modes of both ␸ and␺:

Eint共⌬,␥兲关␸兴⫽ ⫺⑀ 2 2共1⫺␣兲ᐉ2Dk

兺

_⫽0 共Rk T_•P k•Rk⫹Ik T_•P k•Ik兲, Rk⫽关Re共␸˜k兲;Re共␺˜k兲兴T, Ik⫽关Im共␸˜k兲;Im共␺˜k兲兴T, P_k⫽

冋

Lk Mk Mk Nk

册

,

where Re and Im represent the real and imaginary part of a complex number. The components L_k, M_k, N_k depend both on the applied-strain parameters 共maximum or actual ones兲, and the wave vector k. In anticipation of Paper III, it is convenient to introduce

␻⫽␬_␬3

1

␥

⌬

as the shear-strain variable and to define the parameter

c⫽␬2/␬1⫽1⫹

2␮共␭⫹␮兲共␭_⫹2␮兲

␭3 ,

RENAUD TOUSSAINT AND STEVEN R. PRIDE PHYSICAL REVIEW E 66, 036136 共2002兲

where c⬎1. The components of the matrix Pkare then Lk共⌬,␻兲⫽⌬2␬1 2_共1⫺␣u k 2_兲, Mk共⌬,␻兲⫽⌬2␬1 2_u k关共1⫺␣兲c⫺␣vk␻兴, Nk共⌬,␻兲⫽⌬2␬1 2 关共1⫺␣兲c2_{⫹2共1⫺␣兲cv} k␻ ⫹共1⫺␣vk 2_兲 ␻2_兴,

where uk⫽cos(2␪k) andvk⫽sin(2␪k) are functions that

char-acterize the orientation of k through its polar angle ␪_k. Note that all terms contributing to the Hamiltonian have been written in a dimensionless form in which energy den-sity Ej, like stress, is measured in units of (␭⫹2␮).

APPENDIX: EFFECTIVE MODULI OF A CRACKED CELL

A crack is modeled here as an elongated ellipse having a major axis of length d and a minor axes of length w in the limit that w/dⰆ1 which corresponds to a so-called ‘‘penny-shaped’’ crack. Its long axis is by convention oriented along

eˆ₁ if locally ␸⬎0, and along eˆ₂ if ␸⬍0. The unit cell is a square whose sides are colinear with (eˆ1;eˆ2), and has a size

⌳Ⰷd since the crack is taken to be small. The interior of the

crack is supposed to be much more compliant than the em-bedding matrix and all plastic deformation will be ignored; i.e., there is no residual stress or strain allowed for in the cracked system when it is unloaded to zero applied stress.

Denoting as usual the volume average of a quantity with an overbar, we seek to determine the elastic-stiffness tensor

C of a cell as defined through the relation

␶

¯

i j⫽Ci jkl␧¯kl. 共A1兲 The region inside the crack is occupied by a uniform mate-rial of stiffness C1 while the intact matrix surrounding the crack is occupied by a material of stiffness C1. Upon denot-ingv the volume fraction of the crack in the cell, we obtain directly ␶ ¯_{i j⫽共1⫺v兲C} i jkl 0 _␧¯ kl 0_⫹vC i jkl 1 _␧¯ kl 1 . 共A2兲

Eshelby关11兴 demonstrates that the strain␧1inside an elliptic inclusion is uniform while Wu 关12兴 relates this strain to the strain at infinity by a tensor T,

␧¯i j

1_⫽T

i jkl␧kl⬁. 共A3兲 With cracks considered as small inclusions in their embed-ding cell (vⰆ1), the approximation␧⬁⯝␧¯ is valid to lead-ing order in the above, so that

␧¯i j

0_⫽W

i jkl␧¯kl, 共A4兲

共1⫺v兲Wi jkl⫽共␦ik␦jl⫺vTi jkl兲. 共A5兲

Using Eqs.共A3兲 and 共A4兲 for the average deformation in and out of the inclusion, Eq.共A2兲 has the desired linear form of Eq. 共A1兲 with an effective stiffness tensor given by

C_{i jkl}⫽C_{i jkl}0 ⫺v共C_{i jmn}0 ⫺C_{i jmn}1 兲T_mnkl⯝C_{i jkl}0 ⫺vC_{i jmn}0 T_mnkl.

共A6兲

This approximation is justified under the hypothesis that the material inside the inclusion共air兲 is far more compliant than the host material共solid silicate兲. These relations are valid in any space dimension D . The two-dimensional case of inter-est to us here can be obtained from the three-dimensional Wu-Eshelby results by working with a three-dimensional el-lipsoidal inclusion having semiaxes of linear dimension a

⫽d/2; b⫽w/2; and c⫽h/2 embedded within a cell of

di-mension ⌳⫻⌳⫻h in the limit that hⰇ⌳. In this limit, the three-dimensional problem becomes one in two dimensions. Wu expresses his tensor T in terms of a tensor S defined by Eshelby, Ti ji j⫽ 1 2共1⫺2S_{i ji j}兲 when i⫽” j, 共A7兲

冉

T1111 T1122 T1133 T2211 T2222 T2233 T3311 T3322 T3333

冊

⫽

冉

1⫺S1111 ⫺S1122 ⫺S1133 ⫺S2211 1⫺S2222 ⫺S2233 ⫺S3311 ⫺S3322 1⫺S3333

冊

⫺1 . 共A8兲

The Eshelby 关11兴 tensor components are defined

S1111⫽Qa2Iaa⫹RIa, 共A9兲

S1122⫽Qb2Iab⫺RIa, 共A10兲 S1212⫽ Q 2共a 2_⫹b2_兲I ab⫹ R

2共Ia⫹Ib兲, 共A11兲 with similar expressions for the remaining components ob-tained through the permutation of a,b,c and 1,2,3. In the notation of the present paper, the various parameters of Eqs.

共A9兲–共A11兲 are defined

Q⫽ 3

8␲共1⫺␴p兲

and R⫽ 1⫺2␴p

8␲共1⫺␴p兲

, 共A12兲 where␴p⫽␭/2(␭⫹␮) is the Poisson’s ratio of the solid ma-terial共assumed isotropic兲, and

I_a⫽2␲ab

冕

0

⬁ du

共a2_⫹u兲D, 共A13兲

I_aa⫽2␲ab

冕

0

⬁ du

Iab⫽ 2 3␲ab

冕

0

⬁ du

共a2_⫹u兲共b2_⫹u兲D, 共A15兲

with D⫽

冑

(a2⫹u)(b2⫹u) and Ic⫽Iac⫽Ibc⫽Icc⫽0. Simi-lar expressions are obtained for Iband Ibbby replacing a and

b in the above. These elliptic integrals are evaluated to the

leading order in the small aspect ratio␦⫽b/a which gives

Ia⫽4␲␦, Ib⫽4␲共1⫺␦兲, 共A16兲 Iaa⫽ 4␲ 3a22␦, Ibb⫽ 4␲ 3b2, Iab⫽ 4␲ 3a2共1⫺2␦兲. 共A17兲

Defining parameters q and r by

q⫽4␲Q⫽ 3 2共1⫺␴p兲 and r⫽4␲R⫽ 1⫺2␴p 2共1⫺␴p兲 , 共A18兲

we obtain that to the leading order in 1/␦,

T1212⫽ 3 4q 1 ␦, 共A19兲 T2211⫽ q 3⫺r r

冉

1⫹q 3⫺r

冊

1 ␦, 共A20兲 T₂₂₂₂⫽ 1 r

冉

1⫹q 3⫺r

冊

1 ␦. 共A21兲

All remaining components of T are either O(1) and there-fore negligible, or are unimportant for the components of C related to directions 1 and 2.

To get finally the deviation ␦c of the effective elastic

moduli of the cracked cell through Eq. 共A6兲, we note first that v⫽4␲ 3 abc ⌳2_h⫽ 2␲ 3 a2 ⌳2 b a⫽ ␲ 6 d2 ⌳2␦⫽ ␲ 6⑀␺␦, 共A22兲 where we recall that d⫽2a is the crack’s length, w⫽2b its width, and␦its aspect ratio. It is through this expression that the small parameter ⑀⫽(dm/⌳)2Ⰶ1 enters the Born series. Note that ␺⫽兩␸兩⫽(d/d_m)2 _{characterizes the extent of the}

crack. The third dimension of h⫽2c goes to infinity in order to obtain the two-dimensional limit of this three-dimensional system.

Replacing q and r by their expressions in terms of the Lame´ parameters␭,␮, and using by convention (␭⫹2␮) as the stress unit, the crack-induced perturbations of the cell moduli are ␦c2222⫽⫺vC2222 0 _T 2222⫽⫺ ␲ 6⑀␺ ␭⫹2␮ ␮ , 共A23兲 ␦c1111⫽⫺vC1122 0 T2211⫽⫺ ␲ 6⑀␺ ␭2 ␮共␭⫹2␮兲, 共A24兲 ␦c1122⫽⫺vC1122 0 T2222⫽⫺ ␲ 6⑀␺ ␭ ␮, 共A25兲 ␦c2211⫽⫺vC2222 0 _T 2211⫽⫺ ␲ 6⑀␺ ␭ ␮, 共A26兲 ␦c₁₂₁₂⫽⫺v共C₁₂₁₂0 T₁₂₁₂⫹C₁₂₂₁0 T₂₁₁₂兲⫽⫺␲ 6⑀␺ 1 2 ␮ ␭⫹␮, 共A27兲

with all other terms being zero except those obtained by the necessary symmetries under exchange of the two first or two last indexes.

Using the dimensionless constants ␣ defined in Eq. 共18兲 and introducing the positive dimensionless coefficients␩i,

␩1⫽ ␲ 6 ␭3_{⫹␮共␭⫹2␮兲}2 ␭␮共␭⫹2␮兲 , ␩2⫽ ␲ 6 ␭⫹2␮ ␭ , ␩3⫽ ␲ 12 ␭⫹2␮ ␭⫹␮ , 共A28兲

we obtain at last the deviation of the elastic moduli of a cell containing a crack with long axis oriented along eˆ1

共corre-sponding to a positive␸), ␦c⫽

冉

␩2⫺2␩1 ⫺共2␣⫺1兲␩2 0 ⫺共2␣⫺1兲␩2 ⫺␩2 0 0 0 ⫺共1⫺␣兲␩3

冊

⑀␺. 共A29兲

The expression for both possible orientations of the cracks is straightforward. Orienting the crack along eˆ2 instead eˆ1 is

equivalent to exchanging the one and two indices in the com-ponents of ␦c, which results in an exchange of the

compo-nents ␦c1111 and ␦c2222, all remaining components of ␦c

being unaffected by this change. Separating both expressions of ␦c into symmetric and antisymmetric parts, and noting

that ␦c⫽0 trivially when ␸⫽0 共no crack兲, we obtain the

general expression used in Eqs. 共19兲–共21兲.

RENAUD TOUSSAINT AND STEVEN R. PRIDE PHYSICAL REVIEW E 66, 036136 共2002兲

关1兴 P. Be´suelle, Ph.D. thesis, Universite´ Joseph Fourier–Grenoble I, 1999, and references therein for a review.

关2兴 S. Arbabi and M. Sahimi, Phys. Rev. B 47, 695 共1993兲; M. Sahimi and S. Arbabi, ibid. 47, 703共1993兲; 47, 713 共1993兲. 关3兴 G. G. Batrouni and A. Hansen, Phys. Rev. Lett. 80, 325 共1998兲. 关4兴 Statistical Models for the Fracture of Disordered Media, ed-ited by H. J. Herrmann and S. Roux共North-Holland, Amster-dam, 1990兲.

关5兴 D. A. Lockner, in Rock Physics and Phase Relations, A Hand-book of Physical Constants, edited by T. J. Ahrens共American Geophysical Union, Washington, DC, 1995兲, and references therein.

关6兴 R. L. Kranz, Tectonophysics 100, 449 共1983兲.

关7兴 M. L. Batzle, G. Simmons, and R. W. Siegfried, J. Geophys. Res. B 85, 7072共1980兲.

关8兴 M. S. Paterson, Experimental Rock Deformation—the Brittle Field共Springer-Verlag, Berlin, 1978兲.

关9兴 D. A. Lockner, J. D. Byerlee, V. Kuksenko, A. Ponomarev, and A. Sidorin, in Fault Mechanics and Transport Properties of Rocks, edited by B. Evans and T. F. Wong 共Academic Press, San Diego, 1992兲.

关10兴 K. Mair, I. Main, and S. Elphick, J. Struct. Geol. 22, 25 共2000兲. 关11兴 J. D. Eshelby, Proc. R. Soc. London, Ser. A 241, 376 共1957兲. 关12兴 T. T. Wu, Int. J. Solids Struct. 2, 1 共1966兲.