c Allerton Press, Inc., 2010. ISSN 0025-6544, Mechanics of Solids, 2010, Vol. 45, No. 3, pp. 445–464.  c O.Ya. Izvekov, V.I. Kondaurov, 2010, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2010, No. 3, pp. 164–187. Original Russian Text 

Scattered Fracture of Porous Materials with Brittle Skeleton O. Ya. Izvekov and V. I. Kondaurov* Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudny, Moscow oblast, 141700 Russia Received January 25, 2010

Abstract—A model of damage accumulation in a porous medium with a brittle skeleton saturated with a compressible fluid is formulated in the isothermal approximation. The model takes account of the skeleton elastic energy transformation into the surface energy of microcracks. In the case of arbitrary deformations of an anisotropic material, constitutive equations are obtained in a general form that is necessary and sufficient for the objectivity and thermodynamic consistency principles to be satisfied. We also formulate the kinetics equation ensuring that the scattered fracture dissipation is nonnegative for any loading history. For small deviations from the initial state, we propose an elastic potential which permits describing the principal characteristics of the behavior of a saturated porous medium with a brittle skeleton. We study the acoustic properties of the material under study and find their relationship with the strength criterion depending on the accumulated damage and the material current deformation. We consider the problem of scattered fracture of a saturated porous material in a neighborhood of a spherical cavity. We show that the cavity failure occurs if the Hadamard condition is violated. DOI: 10.3103/S0025654410030155 Key words: porous medium, damage, surface energy, kinetics, constitutive relations.

INTRODUCTION The microinhomogeneous materials, in particular, porous media, have recently aroused great interest. This is caused by the use of new constructive materials of ceramic and nanomaterial type [1, 2] and new technologies, one of whose examples is the geo-loosening technology [3]. At the microloevel, a porous medium with a brittle skeleton, dry or fluid-saturated, is a strongly inhomogeneous material that can accumulate microcrack-type damage not only under high-rate deformation, as in the case of homogeneous materials, but also under quasistatic loads. But significantly less attention is paid to the damage accumulation in porous media than to the viscous and elastoplastic properties of rocks [4, 5]. The porous materials that strongly differ in their structure and properties have several common specific characteristics of mechanical behavior, which manifest themselves in the linear-elastic behavior under small loads, in cracking and subsequent fragmentation not only under an intensive shear but also under a strong compression. The cracking is accompanied with energy absorption due to microcrack growth and variation in the material structure near the faces of these cracks. In fragmentation of brittle rocks, fracture is superimposed with plastic deformation arising due to friction of the newly formed fragments and is accompanied by intensive transformation of elastic energy into heat. In the present paper, we consider only the process of nucleation and development of microcracks scattered in the bulk of the material. In the authors’ opinion, the continuum description of this process must first reflect the strong degradation of deformation and strength properties under intensive loads of any type such as tension, shear, uniform compression. Under such loads, the model must predict residual deformation, which manifests itself as either dilatancy (an increase in the volume of an unconstraint material element under shear deformations) or compaction (nonelastic compaction of the material under strong compression due to fracture of pore channels). Under constrained deformation, ruling out volume changes, the shear deformation results in the appearance of compressive normal components of the stress tensor and *

e-mail: [email protected],[email protected]

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uniform pressure, which are comparable in magnitude with the acting tangential stress. Finally, the damage accumulation must be accompanied by changes in the medium porosity and permeability. The first two requirements are characteristic of elastoplastic models of rocks. An example of such models is given by constitutive equations [6–8] that use the plastic flow law associated with the plasticity condition depending on both the shear stresses and hydrostatic pressure. This permits describing the decrease in the material stiffness in plastic state, the difference in the loading and unloading processes, the energy dissipation, and the material dilatancy. Most often, these models are formally generalized to a porous medium [4] by introducing an effective stress equal to the sum of the acting stress and the pore pressure. The validity of such a change is not always obvious. Moreover, in the plasticity models, one completely ignores the transformation of thermomechanical energy forms into the latent (surface) energy of the structure transformation, which applies to scattered fracture. Such a transformation can be described by taking account of the surface energy in the thermodynamic potential of the porous medium skeleton. It is assumed that the rate of change in the surface energy is nonzero in any process where the material damage changes. The surface energy density, which depends on the medium degree of fracturing, plays the role of the “initial value” of the thermodynamic potential [9]. In contrast to the classical case of thermoelastic medium, for which the energy in the initial state is an inessential constant, in the model under study, the dependence of the initial energy value on the medium damage serves as the energy measure of structural changes. The proposed description of the damage is a generalization of the continuum fracture model [10, 11] to the case of a porous medium saturated with an elastic fluid. The approximation of interpenetrating and interacting continua is used [4, 12]. In the isothermal approximation for a solid deformable skeleton and saturating fluid, we formulate the mass and momentum conservation laws and the law of compatibility of finite deformations and velocities, which are supplemented with a dissipation inequality. The state and reaction (response) of the medium under study are determined. The state of a point x at time t is determined by the skeleton distortion and the fluid density, its velocity with respect to the skeleton, and the scalar parameter of damage. The medium response, which determines the fluid and skeleton properties, is characterized by elastic potentials, partial stresses, porosity, and dissipative force of fluid– skeleton interaction. These quantities are assumed to be functions of the medium state. In addition, the system of constitutive equations is also supplemented with the equation of kinematic continuum fracture. Applying the general principles of the theory of constitutive equations (invariance under symmetry transformations, independence of the choice of the reference frame, and the thermodynamic consistency [13]) to the material under study, we obtain the general form of the constitutive equations, which is necessary and sufficient for these principles to be satisfied. On the basis of this general theory, we consider the approximation of small deformations of the skeleton and small variations in the pore pressure. For a homogeneous isotropic skeleton, we construct an elastic potential, which in the case without damage coincides with the potential of a liquid-saturated linearly elastic porous body [4, 14]. In the case with damage, the potential includes the latent energy of scattered fracture (the surface energy of microcracks) and the terms characterizing the change in the elastic potential due to the damage accumulation. The decrease in the elastic potential originates from the fact that the material is partially unloaded near the faces of the microcracks, and the increase in the potential is caused by almost complete disappearance of pores under strong compression. The model characteristics are illustrated with an example of the problem of crack origination near a spherical cavity because of the fluid pressure difference and the skeleton stresses at infinity and inside the cavity. The solution of this problem, which has application in oil engineering, illustrates the characteristics of the behavior of the material under study. 1. KINEMATICS AND CONSERVATION LAWS The initial premise is the assumption that the saturated porous medium with a deformable damageable skeleton is the set of two interpenetrating and interacting continua simultaneously occupying the same domain in space. In contrast to the classical continuum mechanics, the Euler description of motion is more preferable here. Let χ be a fixed domain at each of whose points x ∈ χ the particles of the skeleton and of the saturating fluid are superimposed. The boundary ∂χ of this domain does not move. Since, at different time moments, the domain χ is occupied by different skeleton and fluid particles, the boundary ∂χ in the general case is not a material surface for the skeleton and the fluid. Assume that a continuum particle A = s, f at time t is at a point x. In what follows, the index s refers to the skeleton and MECHANICS OF SOLIDS

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the index f refers to the fluid. Let XA (x, t) denote the position of this particle at the initial time moment and let κA (t) denote the spatial domain occupied at the initial time moment by particles of continuum A that are at time t in the domain χ. The time-dependence of the domain κA (t) emphasizes the fact that, at different time moments, there are different sets of particles in the domain χ. The domains κA (t) are called the initial (reference) configurations of the part of continuum A that is in the domain χ at time t. The mappings κA (t) → χ are assumed to be one-to-one and differentiable. These mappings can be written as the motion laws (1.1) XA = XA (x, t), x ∈ χ, XA ∈ κA (t), t > −∞. The motion law (1.1) defines a particle at thex‘ space point under study. Let GA = [∇ ⊗ XA (x, t)]T be the distortion of an element of continuum A. This quantity is a nonsymmetric tensor of rank 2 such that dXA = GA · dx. Because the mappings (1.1) are one-to-one, the tensor GA is nondegenerate, i.e., JA = det GA > 0. For the nondegenerate tensor, we have the polar decomposition GA = RA · UA = VA · RA ,

(1.2)

where UA and VA are symmetric positive definite tension (compression) tensors and RA is an orthogonal rotation tensor. Decompositions (1.2) have the uniqueness property.   Assume that vA = ∂x(XA , t)/∂tX is the mass velocity and MA = ∂XA (x, t)/∂tx is the rate A at which fluid particles are replaced at the point x. Differentiating the motion law (1.1) written in the form XA = XA (x(XA , t), t) with respect to t for XA = const, we obtain the relation MA = −GA · vA . The distortion GA (x, t) and the velocity MA (x, t) are the first derivatives of the mapping (1.1). Therefore, for smooth motions of each continuum, there is a relationship between the distortion and the velocity, i.e., the compatibility equation for total deformations and velocities [15]:  dA GTA dA f ∂f (x, t)  −1T · GA = −∇ ⊗ vA , ≡ + vA · ∇f, (1.3) dt dt ∂t x where dA f/dt is the material derivative along the trajectory of a particle of continuum A. In what follows, we consider the case in which the mass exchange between the fluid and the skeleton can be written as 0 JA , rA = rA

JA = det GA ,

(1.4)

0 are the effective (average) density of the continuum A mass in the current and initial where rA and rA state. Relation (1.4) implies the continuity equation

dA rA + rA ∇ · vA = 0. (1.5) dt If φ(x, t) is the porosity, then the fluid effective density rf is related to its true density ρf by rf = φρf . We assume that the force acting on the volume χ of continuum A is equal to the sum of the volume and contact forces. In turn, the volume force is equal to the sum of the gravity force and the continua interaction forces. Then we have    fA = rA g dV + tA dS − bint A dV, χ

∂χ

χ

where g is the acceleration due to gravity, tA is the stress tensor, and bint A is the force with which continuum B = A acts on medium A per unit volume of the body. The procedure of averaging of the momentum conservation laws considered at the microlevel for the components of the heterogeneous medium shows [12] that the continua interaction forces are balanced: int bint s + bf = 0.

(1.6)

The connection between the forces acting on continuum A and its motion is realized by the momentum and angular momentum conservation laws written in the inertial reference frame. For smooth flows, the equation of motion has the form rA MECHANICS OF SOLIDS

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where TA is the symmetric tensor of partial stresses for continuum A. In what follows, it is convenient to represent the interaction force, the partial stress, and the medium porosity in the form of two terms [15]: 0 dis bint A = bA + bA ,

TA = T0A + Tdis A ,

φ = φ0 + φdis ,

(1.8)

dis dis are the dissipative where b0A , T0A and φ0 are the equilibrium components and Bdis A , TA , and φ components of the interaction force, stress, and porosity, respectively. The dissipative terms are zero in equilibrium, where the mass velocities are zero in a certain neighborhood of the current time instant. The equilibrium fluid-skeleton interaction force has the form

b0f = σf0 · ∇φ0 ,

(1.9)

where σf0 is the fluid equilibrium true stress such that T0f = φ0 σf0 . Relation (1.9) is independent of the rheology of the saturating fluid in contrast to the term bdis A for which the constitutive relation is formulated. Further, we consider the isothermal approximation. In this case, the energy balance equation can be written as         d rA eA dV + rA eA vA · n dS = rA g · vA dV + tA · vA dS − δ(x, t) dV, dt χ

A

χ

∂χ

A

∂χ

A

χ

where eA = uA + 12 vA · vA is the specific total energy of continuum A, uA is the specific intrinsic energy, δ(x, t)  0 is the dissipation equal to the density of heat discharge necessary to maintain a constant temperature. The assumption that δ(x, t)  0 is an analogue of the entropy inequality in the processes that occur with variations in the temperature. Just as the second law of thermodynamics, the dissipation inequality imposes important restrictions on the form of the constitutive relations. For smooth motions, the energy balance equation can be written as the local dissipation inequality  dA uA  + rA TA : ∇ ⊗ vA + bint · w  0, − dt A

A

where w = vf − vs is the velocity of the fluid motion with respect to the skeleton and bint = bint f . In what follows, it is convenient to use the dissipation inequality in the form  dA uA ds Gs − (T · G−1 + σf0 : ∇ ⊗ (φ0 w) + δf  0, rA (1.10) − s ): dt dt A

dis · w is the filtration dissipation. where T = Ts + Tf is the total stress and δf = Tdis f : (∇ ⊗ w) + b Inequality (1.10) can be verified directly taking account of the deformation and velocity compatibility equation (1.3), condition (1.6) that the interaction forces are balanced, and relation (1.9) between the equilibrium force with the fluid true stress and the porosity gradient.

2. CONSTITUTIVE RELATIONS The system of conservation laws (1.3), (1.5), and (1.7) is not closed. To specify the material behavior, it is necessary to have constitutive relations that would complete this system to a closed system and determine the properties of the skeleton, the filtered fluid, the forces of their interaction, and the damage kinetics. The model is based on the following assumptions. The skeleton under small stresses and the fluid have an elastic response; in the skeleton under intensive loads, there arise scattered microcrack-type defects, which decrease the skeleton stiffness and change its filtration properties; and the dissipative interaction force depends on the velocity of the relative motion. These qualitative statements are formalized as follows. The state of a point x at time t is determined by the set of quantities Λ(x, t) = {Gs (x, t), Gf (x, t), ω(x, t), w(x, t)}, MECHANICS OF SOLIDS

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where ω is the scalar parameter of damage. The medium response at the point (x, t) is characterized by elastic potentials, partial stresses, porosity, and dissipative force of continua interaction Υ(x, t) = {uA (x, t), TA (x, t), φ(x, t), bdis (x, t)},

A = s, f.

These quantities are assumed to be functions of state, i.e., Υ(x, t) = Υ+ [Λ(x, t)]. In this formula + + + Υ+ = {u+ A , TA , φ , b } is the set of functions determining the constitutive equations. In detailed form, these relations can be written as uA (x, t) = u+ A [Λ(x, t)], φ(x, t) = φ+ [Λ(x, t)],

TA (x, t) = T+ A [Λ(x, t)],

(2.2)

bdis (x, t) = b+ [Λ(x, t)].

In addition to (2.2), the system of constitutive equations also contains the equations of kinetics of the scattered fracture ds ω = Ω(Λ(x, t)), (2.3) dt where Ω(Λ) is a nonnegative function of the state parameters. The material is assumed to be initially homogeneous; therefore, the set of state parameters does not contain the radius vectors XA of the fluid or skeleton particles. The material definition (2.1)–(2.3) follows the rule of “equal presence,” i.e., it is primarily assumed that the fluid and skeleton properties, the interaction force and the kinetic function depend on the same set of state parameters. Because of this rule, the relative velocity, which is “natural” for the interaction force, is also contained in the set of arguments of all constitutive functions. Further, the use of the thermodynamic consistency principle allows one to show that the potentials, stresses, and porosity are independent of the relative velocity. The solution of Eq. (2.3) is a functional defined on the prehistory of states and generally dependent on the initial data. This means that the material under study, whose properties depend on the damage parameter ω, has the previous deformation memory, and the parameter itself reflects the condensed influence of the state history on the medium current response. Such an approach, called the internal variable method [16], is widely used to study plastic and viscoelastic solids. The system of constitutive relations (2.2) contains not the total interaction force but only its dissipative part. This is related to the fact that the interaction force has the equilibrium component, which is determined by expression (1.9), which is independent of properties of the saturating fluid. To the material (2.1)–(2.3) we apply the following three principles of the theory of constitutive equations: the invariance under symmetry transformations, independence of the choice of the reference frame, and the thermodynamic consistency. In contrast to the one-component material, the saturated porous body is composed by two continua with their own initial states. Therefore, for the group of symmetry of the saturated porous medium we take the set of two groups g = {gs , gf }, which correspond to transformations of the skeleton and fluid initial states under which the constitutive equations remain unchanged. Since the skeleton is a solid deformable anisotropic material and the other continuum is a fluid, the constitutive equations are insensitive to the unimodular (volume preserving) transformations of any initial state of the fluid. For the skeleton, there exists an undistorted initial state such that some of its orthogonal transformations preserve the material response. Therefore, Eqs. (2.1)–(2.3) have the symmetry group g = {g0 , u}. Here g0 ∈ o is a subgroup of the proper orthogonal group, and u is the unimodular group of secondrank tensors with determinant equal to +1. In the cases of isotropic and transversally isotropic skeleton, g0 coincides with the proper orthogonal group and the group of rotations about a selected axis. The general form of transformations of the reference frame is as follows: x∗ = Q(t) · (x − x0 ) + x1 (t),

r ∗ = t − a,

x1 (−∞) = x0 ,

Q(−∞) = I,

(2.4)

where Q(t) is the orthogonal rotation tensor, x1 (t) − x0 is the parallel transition, x0 = const is the point with respect to which the rotation occurs, and a = const is the time shift. For simplicity, we assume that the systems are indistinguishable as t → −∞; i.e., Q(−∞) = I and x1 (−∞) = x0 . The principle of independence of the reference frame means that the relation between the state and the current response is determined by the same functions Υ+ and Ω in all reference frames. MECHANICS OF SOLIDS

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Since dx∗ = Q(t) · dx and dX∗A = dXA when the reference frame is changed, it follows from the definitions of distortion, velocity, and uniqueness of decomposition (1.2) that G∗A = GA · QT ,

∗ VA = VA ,

R∗A = RA · QT ,

w∗ = Q · w.

(2.5)

Under the assumptions that the potentials, porosity, and damage remain the same when the reference frame is changed, the interaction force is transformed as a vector, and the tensors of partial stresses are “objective” tensors, we have u∗A = uA ,

φ∗ = φ,

ω ∗ = ω,

bint∗ = Q · bint ,

T∗A = Q · TA · QT .

(2.6)

The principle of thermodynamic consistency of the constitutive equations requires that the dissipation inequality (1.10) is satisfied for any smooth history of states. For the above principles to be satisfied, it is necessary and sufficient that the constitutive relations of the considered medium with any type of the compressible solid skeleton symmetry have the form ∂uf , (2.7) uf = uf (ρf ), σf = −pI, p = ρ2f ∂ρf ∂F ∂F , (2.8) · Gs , p = rs us = F (V, φ, ω), Ts = −rs ∂GTs ∂φ ∂uf (ρf ) ∂F (V, φ, ω) = 0, (2.9) − rf Π(Vs , φ, ρf , ω) ≡ ρ2f ∂ρf ∂φ ds ω = Ω(V, φ, ω, R · w), (2.10) bdis = RT · b× (V, φ, ω, R · w), dt ∂F Ω, δf = bdis · w, (2.11) δω + δf  0, δω = −rs ∂ω where the porosity φ and the pore pressure p coincide with their equilibrium values φ0 and p0 . From now on, we have V = Vs and R = Rs . The quantity δω is the dissipation of the continuum fracture. Formulas (2.7) mean that the fluid elastic potential is a function of the true mass density and are independent of the skeleton distortion, the relative velocity, and damage. The stress in the fluid is a spherical tensor. The fluid pore pressure is determined by the partial derivative of its elastic potential, and hence is independent of the relative velocity and damage. It follows from formulas (2.8) that the skeleton elastic potential depends on its deformation, porosity and accumulated damage. The skeleton stress tensor is determined by its potential and hence is independent of the relative velocity. The computation of the pore pressure by using the fluid potential and the skeleton potential leads to Eq. (2.9), which determined the porosity dependence on the skeleton deformation, its damage, and the fluid true density. This means that, in particular, even for fixed values of skeleton deformation and fluid density, the porosity can vary due to the damage variations. Formulas (2.10) determine the structure of expressions for the dissipative component of the interaction force and the kinetic equation. In the general case, these functions depend on the relative velocity, i.e., there is a mutual relationship between the medium filtration properties and the skeleton damage. Inequality (2.11) shows that, for any state of the material under study, the total dissipation is equal to the sum of filtration dissipation and continuum fracture dissipation, and their sum must be nonnegative. The obtained relations hold for arbitrary deformations of the medium with skeleton anisotropy of an arbitrary type. These results are not an assumption by a consequence proved in the framework of the general hypotheses formulated above. A sufficiently cumbersome proof of relations (2.7)–(2.11) to a large extent repeats the proof given in [15] in the case of a thermoelastic porous medium. Therefore, we omit it for brevity. If we pass from the skeleton porosity state parameters φ and the tension tensor V to the Lagrange finite strain tensor E = 12 (V−2 − I) and the pore pressure p, then, in these variables, the constitutive relations (2.8)–(2.11) take the form Ψ = Ψ(E, p, ω),

∂Ψ ds ω ∂Ψ · G−1T , = Ω(E, p, ω, R · w), , φ = −rs s ∂E ∂p dt (2.12) ∂Ψ Ω, δω + δf  0, δω = −rs ∂ω

T = −rs G−1 s ·

bdis = RT · b(E, p, ω, R · w),

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where T = Ts + Tf = Ts − φpI is the total stress tensor and Ψ is the Legendre transform of the function F determined by the equation Ψ − p ∂Ψ/∂p = F (Gs , φ, ω). For a porous medium with an incompressible skeleton (ρs = const), it follows from the mass conservation law (1.4) that 1 − φ = (1 − φ0 )Js and Js = det Gs , where φ and φ0 are the current and initial values of the porosity. This means that the porosity is determined by the skeleton distortion. Therefore, the constitutive equations of such a medium do not contain any independent relation for the porosity. In this case, the general form of constitutive relations of a porous medium with incompressible damageable skeleton saturated with an elastic fluid is as follows: ∂uf (ρf ) , ∂ρf

uf = uf (ρf ),

p = ρ2f

Ψ = Ψ(E, ω),

Teff = −rs G−1 s ·

bdis = RT · b(E, ω, R · w),

∂Ψ · G−1T , s ∂E ∂Ψ Ω, δω = −rs ∂ω

ds ω = Ω(E, ω, R · w), dt δω + δf  0,

where Teff = T + pI = (1 − φ)(Ts + pI) is the effective stress tensor. In what follows, we assume that the rate of change in the damage parameter is proportional to the derivative of the elastic potential with respect to this parameter:    rs ∂Ψ(λ) 1 ds ω x, x  0, =− , λ ≡ (Gs , p, ω), x = (x + |x|) = , (2.13) dt τβ ∂ω 2 0, x < 0, where τ > 0 is the characteristic time of microcrack development and β = const > 0 is a constant parameter having the dimension of pressure (specific energy). The fact that the kinetic equation (2.13) contains the function x shows that the skeleton damage does not decrease for any loading history; i.e., it is assumed that the microcracks are not healed at moderate temperatures. For a material with kinetics (2.13), the dissipation of scattered fracture forces is nonnegative for any states, because   rs2 ∂Ψ(λ) ∂Ψ(λ) ∂Ψ Ω=  0. δω (λ) = −rs ∂ω τ β ∂ω ∂ω In addition, it is assumed that the filtration dissipation is positive for nonzero relative velocity: δf = bdis · w > 0,

w = 0.

(2.14)

Assumptions (2.13)–(2.14) are sufficient for conditions (2.12) to be satisfied, but are not necessary. 3. QUASILINEAR APPROXIMATION Now we consider a special case in which the current state of a saturated porous medium with damageable skeleton is close to the initial state. The skeleton is assumed to be homogeneous and isotropic. There is no damage of the material in the initial state. The skeleton particle displacement u = x − Xs , the displacement gradients, the time-derivatives, the fluid velocity, and the variation in the pore pressure are assumed to be small in the sense that vA t0 u = O(δ), ∇ ⊗ u = O(δ), = O(δ), L L ∇pL ∂p t0 p − p0 = O(δ), = O(δ), = O(δ), δ  1. p0 p0 ∂t p0 Then, in the linear approximation, the skeleton distortion is Gs = I − (∇ ⊗ u)T , and the stretch tensor and the orthogonal tensor contained in decomposition (1.2) are equal to V = I − e and R = I − ω, where e = 12 [(∇ ⊗ u)T + ∇ ⊗ u] and ω = 12 [(∇ ⊗ u)T − ∇ ⊗ u] are the symmetric tensor of infinitely small deformations and the antisymmetric tensor of infinitely small rotations of the skeleton element. The Lagrange finite strain tensor satisfies the condition E ≈ e. The compatibility equations (1.3) imply e˙ =

1 2 [∇

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The dot over a symbol denotes the partial derivative with respect to time, which, in the linear approximation, coincides with the derivative along the particle trajectory. The state of an element of a saturated porous medium with an elastic damageable skeleton is determined by the skeleton distortion Gs , the fluid true density ρf (or the pore pressure p), the damageability parameter ω, and the relative velocity w. The material response is characterized by three functions: the skeleton and fluid potentials and the dissipative force of continua interaction. The porosity, total stress, and the kinetic functions are determined by the skeleton potential   ds ω rs ∂Ψ ∂Ψ −1 ∂Ψ −1T · Gs , φ = −rs , =− . (3.1) Ψ = Ψ(E, p, ω), T = −rs Gs · ∂E ∂p dt τ β ∂ω We assume that the fluid saturating the porous medium is weakly compressible. The elastic energy of such a fluid in a neighborhood of the initial state with density ρ0f can be represented as the quadratic decomposition

Δρf Δρf 2 1 0 0 0 , Δρf = ρf − ρ0f , ρf uf (ρf ) = ρf uf + p0 0 + Kf 2 ρf ρ0f where u0f and p0 are the elastic potential and pressure at the initial state and Kf is the modulus of bulk compression. This implies p = p0 + Kf

Δρf ρ0f

or

Δρf Δp = . Kf ρ0f

(3.2)

We assume that, in the initial configuration κ, from which the skeleton deformation is reckoned, the total stress is −p0 I, the average mass density of the skeleton is r0 , and the initial porosity is φ0 . The external action causes small displacements of the skeleton, small variations in the porosity and damage, and the stresses that are small compared with the elastic stress moduli. We choose the skeleton elastic potential in the form βω 2 KI12 1 (Δp)2 + + μJ 2 − − bI1 Δp − α± ωI1 − αJ ωJ − α± p ωΔp, (3.3) 2 2 2 N where K and μ are the bulk compression and shear moduli, b and N are the Biot coefficients, I1 = e : I is the bulk strain, and J = (ε : ε)1/2 is the shear intensity, where ε = e − 13 I1 I is the deviator of the small strain tensor. The quantities γ ± and β > 0, are positive parameters determining the quadratic dependence of the microcrack surface energy on the accumulated damage ω. The constant coefficients α± , αJ , and α± p characterize the variation in the elastic potential in the case of scattered fracture due to the bulk strain, shear, and the pore pressure variation. It is assumed that the coefficients satisfy the conditions: − αJ > 0, α+ > 0, α+ p > 0, α− < 0, and αp < 0. The choice of the potential is based on the following arguments. Suppose that the damage is zero. Then expression (3.3) coincides with the potential of a porous medium with linearly elastic skeleton [4]: r0 Ψ(e, Δp, ω) = γ ± ω +

KI12 1 (Δp)2 + μJ 2 − − bI1 Δp. 2 2 N The damage accumulation is taken into account by introducing the following two terms. The first term, γ ± ω + 12 βω 2 , characterizes the microcrack surface energy. The second term, −α± ωI1 − αJ ωJ −α± p ωΔp, determines the potential variation due to the partial unloading of the material near the microcrack faces. The plus sign in the subscript ± corresponds to the scattered fracture due to dominant shear (I1 > I1∗ ), the minus sign corresponds to the fracture due to intensive bulk compression (I1 < I1∗ ), where I1∗  0 is the bulk strain separating these two types of fracture. Since the elastic energy of the porous skeleton decreases at fracture, we have α+ > 0 in the case of shear fracture and α− < 0 in the case of intensive compression. Here the elastic energy of microcrack formation also decreases. The sign of the coefficient α± p is determined by the fact that the tensile strength decreases as the pore pressure increases, and the material loosens in tension and shear (α+ p > 0). Since the compression r0 Ψ(e, Δp) =

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Fig. 1.

strength increases with increasing pore pressure and the material thickens under intensive compression, we take α− p < 0. The total stress tensor and the porosity increment, which are determined by the skeleton elastic potential (3.3), have the form

αJ ω ∂Ψ(e, Δp, ω) T = r0 = (KI1 − bΔp − α± ω)I + 2μ − ε, (3.4) ∂e J Δp ∂Ψ(e, Δp, ω) = bI1 + + α± (3.5) Δφ = −r0 p ω. ∂p N The damage kinetics for a skeleton with potential (3.3) can be written as 1 ∂ω ± = α± I1 + αJ J + α± p Δp − γ − βω . ∂t τβ

(3.6)

± The value −r0 ∂Ψ(e, Δp, ω)/∂ω equal to α± I1 + αJ J + α± p Δp − γ − βω characterizes the balance between the surface energy r0 Ψω = γ ± ω + 12 βω 2 increasing in the case of microcrack formation and the elastic energy decreasing at fracture:

KI12 1 (Δp)2 + μJ 2 − − bI1 Δp − α± ωI1 − αJ ωJ − α± p ωΔp. 2 2 N If the elastic energy release exceeds the surface energy increment, then the damage increases; otherwise, it remains invariable. This balance can be used for any deformed state. Equation (3.6) determines the threshold values of strains and pore pressure at which the damage accumulation starts. Since damage increase means the combination of the two conditions ω  0 and ω˙ > 0, the fact that the right-hand side of Eq. (3.6) is zero for ω = 0 implies the relation r0 Ψ e =

γ ± − α± p Δp − α± I1 . (3.7) αJ On the half-plane (J  0) of the strain tensor invariants I1 and J, the function (3.7) determines a triangular domain dependent on the pore pressure, in whose interior there is no damage and the skeleton behaves elastically. This domain, shown in Fig. 1, is called the domain of elasticity. The boundary of the domain of elasticity is strongly asymmetric with respect to the axis J, which is related to difference in the skeleton strength properties in tension and compression. We write the equation of the boundary of the elastic domain as

⎧ J1 ⎪ + ⎪ ⎨J0 1 − + , I1∗  I1  I1 , I 1

(3.8) J(I1 , Δp) = I1 ⎪ ⎪ ⎩J1 1 − − , I1−  I1  I1∗ , I1 J=

where J0 is the pure shear threshold value (I1 = 0) above which the fracture starts, I1± are the bulk strain values at which the damage starts to grow, and J1 = J0 (1 − I1∗ /I1+ )/(1 − I1∗ /I1− ). Comparing MECHANICS OF SOLIDS

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expressions (3.7) and (3.8), we obtain the relationship between the quantities contained in Eq. (3.8) and the coefficients of the potential (3.3) and the pore pressure: γ + − α+ γ − − α− γ + − α+ p Δp p Δp p Δp + J0 (Δp) = , J1 (Δp) = , I1 (Δp) = , αJ αJ α+ − γ − − α− γ + − γ − − (α+ p Δp p − αp )Δp − ∗ , I1 (Δp) = . I1 (Δp) = α− α+ − α− In slow processes for which the time τ is small compared to the characteristic time of the problem, it follows from the boundedness of the damage growth rate for the kinetic equation (3.6) that the numerator on the right-hand side of (3.6) is identically zero. In this case, we obtain the model of uniform accumulation of damage, in which the damage is determined by the current strain and the pore pressure ω=

± α± I1 + αJ J + α± p Δp − γ . β

(3.9)

Relation (3.9) is satisfied for ω  0 and ω˙ > 0, which implies ± α± I1 + αJ J + α± p Δp > γ ,

α± I˙1 + αJ J˙ + α± p Δp˙ > 0.

Otherwise, the rate of damage variation is zero.

3.1. Dissipative Force of Interaction between the Fluid and Skeleton Under the assumption that the interaction force is a smooth function of the velocity w in a neighborhood of w = 0, in the linear approximation for the isotropic skeleton, we obtain bdis (e, Δp, ω, w) = Y (e, Δp, ω)w,

(3.10)

where Y is the positive coefficient of equilibrium resistance. In expansion (3.10), we take into account that the force bdis = 0 for w = 0. Substituting (3.10) into the equation of fluid motion (1.8) and taking the interaction forces represendis tation bint A = −p∇φ + bA into account, we obtain ρf

df vf + ∇p − ρf g = −φ−1 Y (e, Δp, ω)w. dt

We denote the permeability coefficient by k(e, Δp, ω) = μf φ2 (e, Δp, ω)Y −1 (e, Δp, ω),

(3.11)

where μf is the fluid dynamic viscosity. One can see that the pore pressure, skeleton deformation, and skeleton damage affect the permeability through both the resistance coefficient and the porosity. As a result, we obtain the generalized Darcy law for a porous medium with deformable brittle skeleton

  df vf −1 . (3.12) W ≡ φw = −μf k(e, Δp, ω) ∇p − ρf g − dt

3.2. Uniaxial Compression As an example, we consider the uniaxial compression deformation e = e(t)e1 ⊗ e1 , where e(t) is a given function monotonically decreasing to zero. To this deformation there corresponds the straight 

line OABM on the half-plane (I1 , J  0), which is shown in Fig. 1. Since I1 = e and J = −e

2 3 for  the uniaxial compression deformation, the equation of the straight line OAM has the form J = −I1 23 .

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Fig. 2.

For a given pore pressure Δp, it follows from (3.9) that the variation in the equilibrium damage with increasing compression deformation is piecewise linear: ⎧ 0, I1A < e < 0, ⎪ ⎪ ⎪ + + ⎪ ˆ + e + αp Δp − γ ⎨α , I1∗  e < I1A , ω(t) = β ⎪ − ⎪ ⎪ α ˆ e + α− (3.13) p Δp − γ ⎪ ⎩ − , e  I1∗ , β   γ + − α+ p Δp , α+ < αJ 23 . α ˆ ± = α± − αJ 23 , I1A = α ˆ+  The restriction on the coefficients α+ < αJ 23 follows from the condition I1A < 0. The case under study is realized at the pore pressure Δp < γ +/α+ p. A If I1 → 0, then the straight line of uniaxial compression OAM almost completely lies in the damage domain. The graphs of damage against the uniaxial strain are shown in Fig. 2 by broken lines 1 and 2 corresponding to small and large pore pressures. In passing from shear fracture to bulk fracture (I1 = I1∗ ), ˆ + (I1∗ − I1A )/β > 0, and the jump value increases the damage becomes discontinuous [ω] = ω − − ω + = α with increasing pore pressure. Relations (3.4) and (3.5) for the total stresses and porosity, with formulas (3.13) for the damage taken into account, imply γ±α γ±α ˆ± ¯± , T22 = T33 = L12 e − b2 Δp + , β β γ ± α± α ˆ2 Δp α ¯± α ˆ± p − , L11 = λ + 2μ − ± , L12 = λ − , Δφ = b1 e + N β β β 2 α ˆ ± α± α ¯ ± α± 2(α± αJ 1 1 p p p) , b2 = b + , + . = α ¯ ± = α± + √ , b1 = b + β β N1 N β 6 T11 = L11 e − b1 Δp +

(3.14)

(3.15)

We note that the Biot modulus relating the stress increment to the pore pressure variation decreases (b1 < b) at the shear fracture (I1 > I1∗ ) and increases (b1 > b) at the bulk fracture (I1 < I1∗ ). The tangential modulus L11 of the material in damaged state is less than the modulus L0 = λ + 2μ in elastic state. This means that the stress increment T11 as the strain e(t) changes is less than that in the undamaged material; i.e., the damage development results in the material softening so that the diagram T11 (e) may even start to “fall.” Just as the damage, the stress T11 in the case of uniaxial compression becomes discontinuous at the strain e = I1∗ . The dependence T11 (e) shown in Fig. 3 by the broken line OABCM is discontinuous at MECHANICS OF SOLIDS

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Fig. 3.

point B and the jump is equal to ˆ + (ˆ α+ − α ˆ − )(eB − eA ) > 0. β(TC − TB ) = α

(3.16)

Indeed, it follows from the first formula in (3.14) that − α2+ − α ˆ 2− )eB + (ˆ α+ α+ ˆ − α− ˆ− − γ +α ˆ+. β(TC − TB ) = (ˆ p −α p )Δp + γ α

Taking account of the expressions for the strains I1∗ = eB =

− γ + − γ − − (α+ p − αp )Δp , α+ − α−

I1A = eA =

γ + − α+ p Δp α ˆ+

and the relation γ − − α− ˆ + eA − (ˆ α+ − α ˆ − )eB , which follows from them, we arrive at relap Δp = α ˆ+ > α ˆ − , and eB < eA taken into account, we obtain tion (3.16). With the inequalities α ˆ + < 0, α TC − TA > 0. 4. ACOUSTIC PROPERTIES OF A BRITTLE DAMAGEABLE MEDIUM The acoustic properties of a saturated porous medium are important both from the practical standpoint — concerned with problems of seismic tomography and sounding of oil reservoirs and other geophysical objects — and in the scientific interest. The study of propagation laws of acoustic waves (characteristic surfaces) allows one to understand [17] what boundary conditions and how many conditions should be posed for a systems of nonstationary partial differential equations. Propagation of waves whose velocities in the medium under study depend on the current strains, accumulated damage, and propagation direction is also connected with the Hadamard condition, which is a necessary condition for a system of partial differential equations to be hyperbolic. For degenerate cases, where the velocity of one of the nonstationary characteristic surfaces in a certain direction tends to zero, the boundary-value problems for the system of equations under study cease to be well posed. This means that the further deformation of the medium cannot be described by the model. Therefore, this restriction on the admissible deformation and accumulated damage plays the role of a strength criterium, which surely cannot be exceeded for the material. Since the Hadamard condition is based on the medium elastic potential, the strength criterium thus obtained is directly related to the properties of the material. In the quasilinear isothermal approximation, the system of dynamical equations of the initially isotropic medium with zero initial stress and zero pore pressure can be written as a system of partial differential equations for the solution vector uα (x, t) = {vs , vf , e, p, ω}. The system contains the compatibility equation for the skeleton strains and velocities, 2e˙ − ∇ ⊗ vs − (∇ ⊗ vs )T = 0,

(4.1)

the continuity equation (1.5) for the fluid, which, with (3.2) taken into account, becomes ˙ Kf−1 p˙ + φ−1 0 φ + ∇ · vf = 0, MECHANICS OF SOLIDS

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the equation of motion of the porous medium element as a whole, rs vs + rf v˙ f − ∇ · T = rg,

r = rs + rf ,

(4.3)

the equation of motion of the fluid in the form of Darcy’s law (3.12), ρf v˙ f + ∇p = ρf g − μf k−1 W,

W = φ(vf − vs ),

(4.4)

and the kinetic equation for the damage, ± τ β ω˙ = α± I1 + αJ J + α± p p − γ − βω .

(4.5)

System (4.1)–(4.5) is closed by two equations of state for the total stress (3.4) and porosity (3.5): αJ ω , T = [KI1 (e) − bp − α± ω]I + (2μ − ξ)ε, ξ = J(e) (4.6) p ω. + α± φ − φ0 = bI1 (e) + p N We differentiate expression (4.6) for porosity with respect to time, substitute the resulting relation into (4.2), and take the compatibility equation (4.1) and the kinetic equation (4.5) into account to obtain p˙ + bM ∇ · vs + φ0 M ∇ · vf = −

α± pM ± α± I1 + αJ J + α± p p − γ − βω , τβ

(4.7)

where the modulus M is determined by the relation M −1 = φ0 Kf−1 + N −1 . From the first formula in (4.6), relating the stress to the strain, pore pressure and damage, we obtain the following expression for the divergence of the stress tensor: ∇ · T = (λ +

1 3 ξ)∇(e

: I) + (2μ − ξ)∇ · e + ξN · (∇ ⊗ e) : N − b∇p − (α± I + αJ N) · ∇ω,

(4.8)

where N(e) = ε/J is the normed strain deviator such that N(e) : N(e) = 1. Equation (4.4) implies the expression rf v˙ f = rj g − μf k−1 φ0 W − φ0 ∇p.

(4.9)

Relations (4.7)–(4.9) permit reducing the system of equations (4.1)–(4.6) of the linear elastic saturated porous medium to the form solved for the time derivative: rs v˙ s − (λ +

1 3 ξ)∇

⊗ e : I − (2μ − ξ)∇ · e − ξN · (∇ ⊗ e) : N

− (b − φ0 )∇p + (α± I + αJ N) · ∇ω = rs g + μf k−1 φ0 W, rf v˙ f + ∇p = rj g − μf k−1 W, p˙ + bM ∇ · vs + φ0 M ∇ · vf = ω˙ =

e˙ − 12 [∇ ⊗ vs + (∇ ⊗ vs )T ] = 0, −α± pM

± α± I1 + αJ J + α± p p − γ − βω

, τβ

(4.10)

± α± I1 + αJ J + α± p p − γ − βω

. τβ

Now we consider the propagation of the weak discontinuity surface σ(x, t) = 0 on which the solution vector uα (x, t) is continuous and its derivatives ∂uα /∂t and ∇uα are discontinuous. The jumps of these derivatives satisfy the compatibility relations       ∂uα ∂uα ∂uα ∂uα = −c , [∇ ⊗ uα ] = n ⊗ , ≡ n · ∇ ⊗ uα , ∂t ∂n ∂n ∂n where n = ∇σ/|∇σ| and c = −|∇σ|−1 ∂σ/∂t are the normal vector and the speed of propagation of the weak discontinuity surface, and ∂uα /∂n is the normal derivative. We denote           ∂vf ∂e ∂p ∂ω ∂vs = V, = U, = E, = P, = W. ∂n ∂n ∂n ∂n ∂n MECHANICS OF SOLIDS

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Then Eqs. (4.10) imply that the following relations hold on the weak discontinuity surface: rs cV+ (λ+ 13 ξ)(E : I)n+(2μ−ξ)E · n+ξ(E : N)N · n−(b−φ0 )P n−(α± n+αJ N · n)W = 0, rf cU−P n = 0,

cE+ 12 (n ⊗ V+V ⊗ n) = 0,

cP −bM V · n−φ0 M U · n = 0,

cW = 0.

(4.11)

We consider system (4.11) as a system of homogeneous equations for the discontinuities of the normal derivatives. For c = 0, it follows from (4.11) that (SI − Ln ⊗ n − ξm ⊗ m) · V = 0, S = rs c2 − μ + W = 0,

1 ξ, 2

L=λ+μ−

φ−1 bM V · n n, U= 0 2 ρf c − M

m = N · n,

(4.12) c2 M

ρf 1 ξ + b(b − φ0 ) , 6 ρf c2 − M

ρf bcM P = V · n, ρf c2 − M

1 E = − (n ⊗ V + V ⊗ n). 2c

(4.13)

Formulas (4.13) imply that the damage normal derivative jump W on the nonstationary weak discontinuity surface is zero. The jumps of the normal derivatives of the fluid velocity U, the pore pressure p, and the strain e are determined by the vector V, and the jump U of the fluid velocity normal derivative on the weak discontinuity wave is collinear with the normal vector n. The homogeneous system (4.12) has a nontrivial solution V = 0 if the determinant of the matrix of coefficients dependent on the strain e, parameter ξ, normal vector n, and speed of propagation c is equal to zero. It follows that S[S 2 − (L + ξm · m)S + ξL(m × n)2 ] = 0.

(4.14)

Indeed, if the vectors n and m = N(e) · n are not collinear, i.e., if the normal n is not an eigenvector of the tensor N(e), then, taking the scalar products of Eq. (4.12) by the vectors n, m, and n × m, where the symbol × denotes the vector cross product, we obtain the system of three equations (S − L)x − ξ(m · n)y = 0, −L(m · n)x + [S − ξ(m · m)]y = 0, x = V · n, y = V · m, z = V · (m × n).

Sz = 0,

(4.15) (4.16)

This implies equality (4.14). If the vectors m and n are collinear, relation (4.14) follows directly from Eq. (4.13). The value S = 0 corresponds to a shear wave whose speed of propagation is independent of the propagation direction and is given by

1 1 2 (4.17) μ− ξ . csh = rs 2 For S = 0, the jump of the normal derivative of the skeleton velocity V is orthogonal to the vectors n and m = N(e) · n, i.e., V · n = V · m = 0. This proves that this wave is a shear wave. Using expression (4.17) for the shear wave speed and introducing the notation 1 M rs M 2 b(b − φ0 )rs , a=μ− ξ− , ρf 2 ρf 1 l0 = λ + μ − ξ + b(b − φ0 )M, l = l0 + ξm · m, 6 q=

Λ = l0 +

q , S+a

we can rewrite the condition S 2 − (L + ξm · m)S + ξL(m × n)2 = 0 in the form S 3 + (a − l)S 2 − [al + q + l0 ξ(m × n)2 ]S + (al0 + q)ξ(n × n)2 = 0.

(4.18)

Relation (4.18) is a cubic equation for S. Hence, for the material under study, there can exist, apart from the shear wave, either one or three waves with a real speed of propagation, depending on the propagation direction n, strain e, and damage ω. In the absence of damage (ξ = 0), relation (4.18) can be reduced to the equation S (0) [S (0) − l − q/(S (0) + a)] = 0, which gives either a shear wave (S (0) = 0) or two quasilongitudinal waves (S (0) = 0) which are determined by the quadratic equation (S (0) )2 − (l − a)S (0) − (al + q) = 0, MECHANICS OF SOLIDS

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which has two real solutions: rs c21,2 ≡ S1,2 = 12 [l − a ± (l + a)2 + 4q ]. For small values of ξ = 0, Eq. (4.18) has three real roots, two of which are close to the solution of the quadratic equation (4.19), and the third root is close to the speed of the shear wave (4.17). Indeed, we seek the roots of Eq. (4.18) for small values of the parameter ξ in the form (0)

S1,2 = S1,2 + ε1,2 ,

(0)

εi  |x1,2 |.

S3 = ε3 ,

In the linear approximation, Eq. (4.18) takes the form    (0) (0) (0) (0) (0) (0) εi S 2 + [S1 S2 + S1 (ε2 + ε3 ) + S2 (ε1 + ε3 )]x − S1 S2 ε3 = 0. S3 − l − a +

(4.20)

Matching the coefficients of Eqs. (4.19) and (4.20), we obtain ε1 + ε2 + ε3 = 0,

(0)

ε1 + ε3 = −ξ(m × n)2 , (0)

(0)

x1 ε1 + x2 ε2 = l0 ξ(m × n)2 .

(0)

Here we take into account that S1 S2 = −(al + q) = −(al0 + q) + Q(ξ), which implies (0)

ε1 =

l0 + S2 (0) S1

−

(0) S2

(0)

ξ(m × n) , 2

ε2 = −

l0 + S1 (0) S1

−

(0) S2

ξ(m × n)2 ,

ε3 = ξ(m × n)2 .

In the general case, Eq. (4.18) depends on the parameter ξ > 0, the normal vector components ni , and the principal value N1 of the normed deviator. Indeed, suppose that the axes of the Cartesian coordinatesystem are the principal axes of the strain tensor. Then the normed deviator takes the form N = Ni ei ⊗ ei , where Ni are the principal values such that N1 + N2 + N3 = 0 and N12 + N22 + N32 = 1. These two equations imply   √  √  1 1 2 2 3 2 3 2 2  N1  , N2 = − N1 ± − N1 , N3 = − N1 ∓ − N12 , − 3 3 2 2 3 2 2 3 and the quantities m · m = N12 n21 + N22 n22 + N32 n23 , m · n = N1 n22 + N2 n22 + N3 n22 , and (m × n)2 = m × n − (m × n)2 are functions of the principal value N1 and the normal vector components ni . 2 Figure 4 presents the graphs of the propagation velocities r√ s ci against the parameter ξ in the direction √ of the normals n = (1, 0, 0) (dotted curves) andn = (1/ 2, 1/ 2, 0) (solid curves) in the case of uniaxial tensile deformation corresponding to N1 = 2/3. We considered a water-saturated sandstone type rock with the following parameters: ρf = 103 kg/m3 , M = 2.25 · 103 MPa, 2λ = μ = 2 · 104 MPa, φ0 = 0.15, ρs = 2.5 · 103 kg/m3 , and b = 0.8. Curves 1 correspond to the fast quasilongitudinal wave whose velocity in the range 0 < ξ < 2 is practically independent of the values of the parameter ξ and the propagation direction. Curves 2 correspond to the shear and quasitransverse waves whose velocities for ξ = 0 are equal to rs c2 = μ and decrease in all directions as ξ increases. Curves 3 correspond to “slow” quasilongitudinal waves. The velocity of these waves in the direction n = (1, 0, 0) is constant and equal √ √ (0) to rs c22 ≡ S2 , while the velocity in the direction n = (1/ 2, 1/ 2, 0) decreases in the range 0  ξ < 1, where it is real, and increases for 1 < ξ < 1.333. At the value ξ = 1.333, the velocities of the slow and quasitransverse waves coincide and then take complex values as the parameter ξ increases. The fact that the velocity of the shear wave 2 is zero for ξ = 2 and takes complex values for ξ > 1.333 means that the Hadamard condition is violated. The earliest moment at which degeneration starts corresponds to the attainment of the critical state in which the material macroscopic fracture occurs. 5. DAMAGE NEAR A SPHERICAL CAVITY We consider the static spherically-symmetric problem of damage accumulation near a spherical cavity. Along with the scientific interest, the problem has practical applications related, first of all, to the influence of damage on the oil-and-gas collector conductivity near production and injection wells and to estimating the rock proximity to a limit state near mine workings. The skeleton is assumed to be initially isotropic and is characterized by the potential (3.3). The gravity force and the pore fluid compressibility are neglected. The damage accumulation is assumed to be an equilibrium process for which relation (3.9) holds. The medium permeability weakly depends on the accumulated damage, MECHANICS OF SOLIDS

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Fig. 4.

We use the spherical coordinates (r, ϕ, θ) whose origin coincides with the center of a cavity of radius a. The displacement vector of skeleton particles u = (u(r), 0, 0) determines the strains er = du/dr and eϕ = eθ = u/r, where er , eϕ , eθ are the nonzero physical components of the small strain tensor. The vector v = (v(r), 0, 0) is the fluid velocity, σr and σϕ = σθ are the components of the total stress tensor, and sr and sϕ = sθ are the components of the stress deviator. On the cavity surface (r = a), the pore pressure is p = p0 , the radial component of the total stress is σr = σ0 , and at infinity (r → ∞), these quantities are equal to p∞ and σ∞ . The system of differential equations contains the total stress equilibrium equation σr − σθ dσr +2 = 0, dr r

(5.1)

the continuity equation, and the equation of motion of the fluid v dv + 2 = 0, dr r

dp = −μf k−1 v. dr

(5.2)

The boundary conditions for system (5.1)–(5.2) have the form p = p0 , σr = σ0 , r = a, p → p∞ , σr → σ∞ , r → ∞.

(5.3) (5.4)

For the weak dependence of permeability on the damage, it follows from Eqs. (5.2) with the boundary conditions (5.3) and (5.4) taken into account that p(r) = p∞ −

aΔp , r

Δp = p∞ − p0 .

(5.5)

The equilibrium damage expressed in terms of the radial displacement of the skeleton particles and the pore pressure has the form

u 1  ± ± a ˆ± u + 2¯ α± + αp p(r) − γ , (5.6) ω= β r MECHANICS OF SOLIDS

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where the function p(r) is given by expression (5.5). Here, just as above, α ˆ ± = α± + καJ and 1 α ¯ ± = α± − 2 καJ . Relations (3.4) for the total stresses become α ˆ±γ ± u α ¯±γ ± u − b1 p(r) + , σϕ = λ1 u + 2(λ2 + μ) − b2 p(r) + , r β r β α ˆ ± α± α ¯ ± α± α ¯2 α ˆ2 α ˆ±α ¯± p p , λ2 = λ − ± , b1 = b + , b2 = b + . L = λ + 2μ − ± , λ1 = λ − β β β β β σr = Lu + 2λ1

Substituting these relations into the equilibrium equation (5.1), we obtain the second-order equation for the displacement of skeleton particles 2  2m f n u − 2 u− 2 + = 0, r r r r 3καJ (γ ± − α± 9κα± αJ p p± ) , n= , m=1+ 2βL βL

u +

  α± aΔp p (α± − 2καJ ) f= b− . L β

The general solution of Eq. (5.7) has the form q2 q1 r f n r + r, +B − u(r) = A a a 2m 2(m − n)

q1,2 =

√ 1 (−1 ± 8m + 1). 2

(5.7)

(5.8)

We assume that at a sufficiently large distance from the cavity, the material is in elastic state. In the medium with undamaged skeleton, with ω = 0, we have m = 1, n = 0, q1 = 1, q2 = −2, f = abΔp/L0 , and L0 = λ + 2μ. The displacement and the radial stress, with the boundary conditions at infinity taken into account, are equal to u0 (r) =

σ∞ + bp∞ B0 abΔp r+ 2 − , 3K r 2L0

σr0 (r) = σ∞ − 4μ

B0 2μ abΔp . − 3 r L0 r

(5.9)

If there is no damage anywhere, then by taking into account the boundary condition (5.3) on the cavity surface, from (5.9) we obtain u0 (r) =

σ∞ + bp∞ D a3 abΔp r+ − , 2 3K 4μ r 2L0

σr0 (r) = σ∞ − D

a3 2μ abΔp , − 3 r L0 r

(5.10)

where D = Δσ − (2μ/L0 )bΔp. Since the bulk and shear strains attain an extremum on the cavity boundary r = a, it follows from the fracture process initiation condition ω = 0, with formulas (5.6) and (5.10) taken into account, that the damage accumulation on the cavity internal boundary r = a begins when the absolute values of the stress difference Δσ and pressure difference Δp attain a threshold value:     (2καJ − α± )b σ∞ + bp∞ 3καJ ± ± Δσ = − I1 (p∞ ) . − αp Δp + α± 4μ L0 K If the pressure and total stress differences exceed the threshold value, then a domain of scattered fracture begins to arise near the cavity. The damaged material occupies the domain a  r  c, where c is the unknown radius of the damage domain, which must be determined by solving the problem. Outside the damage domain (r > c), the material is in elastic state (ω = 0), for which expressions (5.9) hold. The boundary condition on the cavity surface is supplemented with the conjugation conditions: continuity of the displacements and radial stresses and zero damage on the boundary r = c. These conditions lead to the system of equations α ˆ α± ¯ = σ0 + bp0 − g1 + λ1 f − ± p Δp, F (q1 )A¯ + F (q2 )B am β

bΔp f q1 −1 q1 −1 −3 ¯ ¯ ¯ X −1 , + BX − B0 X = g2 − − AX 2L0 2am ¯ (q2 )X q2 −1 + 4μB ¯0 X −3 = σ∞ + bp∞ − g1 − ¯ (q1 )X q1 −1 + BF AF MECHANICS OF SOLIDS

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α ˆ ± α± λb p + L0 β



 λ1 f Δp − X −1 , am

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IZVEKOV, KONDAUROV

Fig. 5.

¯ q1 −1 ψ(q1 ) + BX ¯ q2 −1 ψ(q2 ) = 1 AX β



α ¯±f ± + αp Δp X −1 am

for the dimensionless variables A A¯ = , a

¯= B, B a

¯0 = B0 , B a3

X=

c . a

Here we use the notation xˆ α± + 2¯ ¯± α± 2ˆ α± α , ψ(x) = , β β σ∞ + bp∞ n g2 = − . 3K 2(m − 1)

F (x) = xL + 2λ − g1 =

3Kn , 2(m − 1)

¯ B, ¯ and B ¯0 , which enter the system of equations linearly, we obtain the Eliminating the unknowns A, following equation for the dimensionless radius X of the material damage domain: F (q1 )X 1−q1 P1 + F (q2 )X 1−q2 P2 + F (q1 )X −q1 P3 + F (q2 )X −q2 P4 λ1 f − β −1 α = σ0 + bp0 + ˆ ± α± (5.11) p Δp, am (σ∞ + bp∞ − g1 + 4μg2 )ψ(q2 ) ψ(q1 ) , P2 = −t1 , Φ(x) = F (x) + 4μ, P1 = Φ(q2 )ψ(q1 ) − Φ(q1 )ψ(q2 ) ψ(q2 ) δ1 (Δp)ψ(q2 ) − δ2 (Δp)Φ(q2 ) δ1 (Δp)ψ(q1 ) − δ2 (Δp)Φ(q1 ) , P4 = , P3 = Φ(q2 )ψ(q1 ) − Φ(q1 )ψ(q2 ) Φ(q1 )ψ(q2 ) − Φ(q2 )ψ(q1 )





α ˆ ± α± α ˆ±α f 1 α ¯±f ¯± p ± + b− Δp, δ2 (Δp) = + αp Δp . δ1 (Δp) = L0 − β am β β am In the general case, Eq. (5.11) must be solved numerically. As an example, we studied a material with the parameters K/μ = 3, αJ /μ = 1.42, β/μ = 1, γ +/μ = 0.01, I1+ = 0.01, I1− = −0.025, I1∗ = −0.02, and J0 = 0.001 for some values of the fluid pressure in the cavity for 0  p∞  0.02. The numerically obtained dependences of X = c/a on the total stress difference Δσ = σ∞ − σ0 are shown in Fig. 5. Curves 1 –3 correspond to p0 /μ = 0, 0.01, 0.02. One can see that the solution has an infinite derivative as Δσ → Δα∗ . MECHANICS OF SOLIDS

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SCATTERED FRACTURE OF POROUS MATERIALS WITH BRITTLE SKELETON

463

Fig. 6.

The equation for the critical radius X∗ is obtained by direct differentiation of Eq. (5.11) and has the form −(q1 +1)

F1 P1 (1 − q1 )X∗−q1 + F2 P2 (1 − q2 )X∗−q2 − q1 F1 F3 X∗

−(q2 +1)

− q2 F2 P4 X∗

= 0.

In the absence of the pore pressure, this equation has an explicit solution (q1 −q2 )

X∗

=

F (q1 )(q1 − 1)ψ(q2 ) , F (q2 )(q2 − 1)ψ(q1 )

which shows that, in this case, the critical radius of the damage zone is determined only by the material characteristics, because the functions F (x) and ψ(x) are independent of the stresses σ0 and σ∞ . For a saturated porous medium, the critical radius of the damage zone and the stress difference Δσ∗ at which the cavity fracture occurs depend on the pressure difference. Figure 6 present the graphs of the stresses σr (r) and σϕ (r) and damage ω(r) for c/a = 1.45 and Δp = 0. Curves 1 and 2 correspond to elastic radial and circumferential stresses, curves 3 and 4 correspond to the same stresses in the case of damaged material, and curve 5 presents corresponds to ω(r). The stresses in the damage zone are shown by dotted lines. The computations show that, in a dry porous material, in the presence of the pore pressure, the cavity fracture also occurs as the parameter ξ = αJ ω/J(e) attains the value μ at which the Hadamard condition is violated. Thus, the above model of scattered fracture of a porous medium with a brittle skeleton permits not only determining the stress-strain state and the material damage but also obtaining a strength criterion that restricts the range of loads applied to the body. ACKNOWLEDGMENTS The research was supported by the Programm of the RAS Branch of Power Industry, Machine Building, Mechanics, and Control Processes “Control of Tribological and Strength Properties of Materials and Products by Physical-Mechanical and Chemical Actions,” by the Russian Foundation for Basic Research (project No. 09-05-00542), and the Program of Russian Federation Ministry of Education and Science “Development of Scientific Potential of Higher School (2009–2010).” MECHANICS OF SOLIDS

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REFERENCES 1. R. V. Goldstein and N. M. Osipenko, Brittle Failure of Highly Porous Bodies under Compression, Preprint No. 812 (IPMekh RAN, Moscow, 2006) [in Russian]. 2. R. V. Goldstein, N. M. Osipenko, and A. V. Chentsov, “To Determination of the Strength of Nanodimensional Objects,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 164–181 (2008) [Mech. Solids (Engl. Transl.) 43 (3), 453–469 (2008)]. 3. V. I. Karev and K. B. Ustinov, “Seepage of a Gas Condensate Mixture When Using the Geoloosening Method,” Prikl. Mat. Mekh. 73 (5), 787–798 (2009) [J. Appl. Math. Mech. (Engl. Transl.) 73 (5), 566–573 (2009)]. 4. O. Coussy, Poromechanics (Wiley, New York, 2004). 5. R. de Boer, Trends in Continuum Mechanics of Porous Media (Springer, Berlin, 2005). 6. S. S. Grigorian, “Some Problems of the Mathematical Theory of Deformation and Fracture of Hard Rocks,” Prikl. Mat. Mekh. 31 (4), 643–669 (1967) [J. Appl. Math. Mech. (Engl. Transl.) 31 (4), 665–686 (1967)]. 7. V. N. Nikolaevskii, Mechanics of Porous and Cracked Media (Nedra, Moscow, 1984) [in Russian]. 8. M. Aubertin, L. Li. and R. Simon, “A Multiaxial Stress Criterion for Short- and Long-Term Strength of Isotropic Rock Media,” Int. J. Rock. Mech. Mining Sci. 37, 1169–1193 (2000). 9. L. I. Sedov, Continuum Mechanics, Vol. 2 (Nauka, Moscow, 1970) [in Russian]. 10. V. I. Kondaurov and N. V. Kutlyarova, “Damage and Rheological Instability of Initially Porous Materials,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 99–109 (2000) [Mech. Solids (Engl. Transl.) 35 (4), 83–92 (2000)]. 11. V. I. Kondaurov and V. E. Fortov, Foundations of Thermomechanics of Condensed Media (Izd-vo MFTI, Moscow, 2002) [in Russian]. 12. R. I. Nigmatulin, Foundations of Mechanics of Heterogeneous Media, (Nauka, Moscow, 1978) [in Russian]. 13. C. A. Truesdell, A First Course in Rational Continuum Mechanics (The Johns Hopkins University Press, Baltimore, Maryland, 1972; Mir, Moscow, 1975). 14. M. A. Biot, “Mechanics of Deformation and Acoustic Propagation in Porous Media,” J. Appl. Phys. 33 (4), 1482–1498 (1962). 15. V. I. Kondaurov, “The Thermodynamically Consistent Equations of a Thermoelastic Saturated Porous Medium,” Prikl. Mat. Mekh. 71 (4), 616–635 (2007) [J. Appl. Math. Mech. (Engl. Transl.) 71 (4), 562–579 (2007)]. 16. P. Germain, Course of Continuum Mechanics (Vysshaya Shkola, Moscow, 1983) [in Russian]. 17. B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics, (Nauka, Moscow, 1978) [in Russian].

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