Deformation, Fracture, and Fragmentation in Brittle Geologic Solids

Deformation, Fracture, and Fragmentation in Brittle Geologic Solids by J. D. Clayton

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A reprint from the International Journal of Fracture, Vol. 163, pp. 151–172, 2010.

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ARL-RP-299

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Deformation, Fracture, and Fragmentation in Brittle Geologic Solids J. D. Clayton Weapons and Materials Research Directorate, ARL

A reprint from the International Journal of Fracture, Vol. 163, pp. 151–172, 2010.

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A reprint from the International Journal of Fracture, Vol. 163, pp. 151–172, 2010. 14. ABSTRACT

A model is developed for mechanical behavior and failure of brittle solids of geologic origin. Mechanisms considered include elastic stretch and rotation, thermal expansion, and deformation associated with micro-cracks. Decohesion on preferred cleavage planes in the solid, and subsequent effects of crack opening and sliding, are modeled. Explicit volume averaging over an element of material containing displacement discontinuities, in conjunction with the generalized divergence theorem, leads to an additive decomposition of the deformation gradient into contributions from thermoelasticity in the intact material and displacement jumps across micro-cracks. This additive decomposition is converted into a multiplicative decomposition, and the inelastic velocity gradient is then derived in terms of rates of crack extension, opening, and sliding on discrete planes in the microstructure. Elastic nonlinearity at high pressures, elastic moduli degradation from micro-cracking, dilatancy, pressure and strain rate-sensitive yield, and energy dissipation from crack growth and sliding are formally addressed. Densities of micro-cracks are treated as internal state variables affecting the free energy of the solid. The mean fragment size of particles of failedmaterial arises from geometric arguments in terms of the evolving average crack radius and crack density, with smaller fragments favored at higher loading rates. The model is applied to study granite, a hard polycrystalline rock, under various loading regimes. Dynamic stress–strain behavior and mean fragment sizes of failed material are realistically modeled. Possible inelastic anisotropy can be described naturally via prescription of cleavage planes of varying strengths. 15. SUBJECT TERMS

micromechanics, crystal plasticity, fracture, fragmentation, rock, granite 17. LIMITATION OF ABSTRACT

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Int J Fract (2010) 163:151–172 DOI 10.1007/s10704-009-9409-5

ORIGINAL PAPER

Deformation, fracture, and fragmentation in brittle geologic solids J. D. Clayton

Received: 26 February 2009 / Accepted: 16 September 2009 / Published online: 9 October 2009 © Springer Science+Business Media B.V. 2009

Abstract A model is developed for mechanical behavior and failure of brittle solids of geologic origin. Mechanisms considered include elastic stretch and rotation, thermal expansion, and deformation associated with micro-cracks. Decohesion on preferred cleavage planes in the solid, and subsequent effects of crack opening and sliding, are modeled. Explicit volume averaging over an element of material containing displacement discontinuities, in conjunction with the generalized divergence theorem, leads to an additive decomposition of the deformation gradient into contributions from thermoelasticity in the intact material and displacement jumps across micro-cracks. This additive decomposition is converted into a multiplicative decomposition, and the inelastic velocity gradient is then derived in terms of rates of crack extension, opening, and sliding on discrete planes in the microstructure. Elastic nonlinearity at high pressures, elastic moduli degradation from micro-cracking, dilatancy, pressureand strain rate-sensitive yield, and energy dissipation from crack growth and sliding are formally addressed. Densities of micro-cracks are treated as internal state variables affecting the free energy of the solid. The mean fragment size of particles of failed material arises from geometric arguments in terms of the evolving average crack radius and crack density, with smaller fragments favored at higher loading rates. The model J. D. Clayton (B) Impact Physics, U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005-5066, USA e-mail: [email protected]

is applied to study granite, a hard polycrystalline rock, under various loading regimes. Dynamic stress–strain behavior and mean fragment sizes of failed material are realistically modeled. Possible inelastic anisotropy can be described naturally via prescription of cleavage planes of varying strengths. Keywords Micromechanics · Crystal plasticity · Fracture · Fragmentation · Rock · Granite 1 Introduction An understanding of dynamic mechanical behavior of rock materials is of importance to construction and mining industries, among others, and hence has received much attention from geophysics and solid mechanics communities (Goodman 1989). Mechanical behavior of rock shares a number of features with other brittle solids such as polycrystalline ceramics and concrete, with a significant fraction of coarse aggregate in concrete often consisting of granite or limestone. Common features include degradation of elastic coefficient as a result of micro-cracking, inelastic deformation arising from contributions of micro-cracking and subsequent granular fl w in failed material, shear-induced expansion (i.e., dilatancy or dilatation), as well as criteria for inelastic deformation (e.g., yield or localization) dependent upon normal stress or hydrostatic pressure (Brace et al. 1966; Rudnicki and Rice 1975; Amitrano 2006; Foster et al. 2007; Paliwal and Ramesh 2008).

123

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Various classes of models have been developed to address dynamic deformation and failure of brittle solids—e.g., rocks, concrete, and ceramics—at high rates. Models based on continuum damage mechanics concepts and employing scalar variables accounting for evolving crack densities include, for example, those of Holcomb (1978), Moss and Gupta (1982), Margolin (1984), Rajendran (1994), Ai and Ahrens (2006), Clayton (2008) and Paliwal and Ramesh (2008). Models accounting for potentially large deformations resulting from micro-crack opening on discrete failure planes include representative theories of Zienkiewicz and Pande (1977), Espinosa et al. (1998) and Bazant et al. (2000a,b). Direct numerical simulations of dynamic failure and fragmentation of microstructures have also become prevalent in recent years (Warner and Molinari 2006; Kraft et al. 2008; Vogler and Clayton 2008). Micromorphic continuum theories for heterogeneous particulate materials featuring pressure sensitive yield have been developed (Regueiro 2009). Efforts at modeling fragment size distributions of brittle materials have been undertaken using statistical thermodynamic arguments (Englman et al. 1987; Grady and Winfree 2001; Clayton 2008) as well as direct numerical simulations (Miller et al. 1999). In this work a new model is developed for brittle solids of geologic origin. Materials of interest are assumed to exhibit preferred cleavage planes whose separation may be activated by shear stress or tensile pressure. For example, quarried granite often exhibits three mutually perpendicular cleavage planes, in order of increasing fracture strength: the rift plane, the grain plane, and the hardway plane (Simmons et al. 1975; Sano et al. 1992; Almeida et al. 2006). These planes traverse the heterogeneous polycrystalline microstructure, and do not necessarily correspond to crystallographic planes in single crystals embedded within the material. For example, the mineral composition of granite includes quartz, feldspar, and muscovite or biotite micas (Ichikawa et al. 2001; Wenk et al. 2008). Preferred cleavage planes may include basal planes in sheets of mica or interfaces between constituents (Horii and Nemat-Nasser 1985; Ichikawa et al. 2001). Previous constitutive models for granite have incorporated concepts from continuum viscoplasticity (Maranini and Yamaguchi 2001) or micromechanical elasticity solutions for sliding cracks, wing cracks, and elliptic or circular cracks (Holcomb 1978; Moss and Gupta 1982; Horii and Nemat-Nasser 1985;

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Li et al. 2000). Rubin and Lomov (2003) developed an elastic-viscoplastic model accounting for anisotropic tensile failure and applied this model to granite subjected to large deformations. Ai and Ahrens (2006) adapted the ceramic material model of Johnson and Holmquist (1999) to model impact loading of granite. In the present paper, deformation mechanisms considered are reversible elastic stretch and rotation, thermal expansion or contraction, and irreversible deformation associated with micro-cracks. Decohesion on preferred cleavage planes in the solid, and subsequent effects of crack opening and sliding are modeled. Rigorous volume averaging in the reference configura tion (Hill 1972; Nemat-Nasser 1999) over an element of material containing displacement discontinuities, in conjunction with the generalized divergence theorem of vector calculus, leads to an additive decomposition of the deformation gradient into contributions from thermoelasticity in the intact material and displacement jumps across micro-cracks (Clayton 2005a, 2006). The description is converted into a multiplicative decomposition reminiscent of that of metal plasticity (Lee 1969). The inelastic velocity gradient is derived in terms crack growth, opening, and sliding on discrete planes in the microstructure. The form of the inelastic velocity gradient is analogous to that of crystal plasticity theory (Asaro 1983), but with shearing rates resulting in the present context from mode II or III crack extension and sliding rather than relative slip of planes due to dislocation glide in the crystal lattice. The shear discontinuity arising from crack opening and sliding is analogous to the Burgers vector of a Somigliana dislocation in the context of geophysics (Eshelby 1973). Inelastic volume increases result from mode I crack opening due to normal stress and dilatation induced by shear. A typical source of dilatation is thought to be misfi between asperities on opposite faces of a sliding crack (Goodman 1989). The general model incorporates elastic nonlinearity important at high pressures, i.e., third-order elastic constants (Winkler and Liu 1996), elastic moduli degradation from cracking (Budianski and O’Connell 1976; Margolin 1983; Sano et al. 1992; Rajendran 1994), pressure- and strain rate-sensitive yield (Amitrano 2006; Kraft et al. 2008), and dissipation from crack growth and sliding leading to temperature rise at high strain rates. Internal state variables include the density of micro-cracks and the average crack size on each cleavage plane. The mean

Deformation, fracture, and fragmentation in brittle geologic solids

fragment size of particles in failed material arises from the evolving average crack size and crack density. Geometric arguments (Seaman et al. 1985; Grady 1988) demonstrate that the mean fragment size is proportional to the ratio of average crack size to dimensionless crack density. Hence a material whose inelastic deformation is accommodated by a few larger cracks will exhibit a larger mean fragment size than one whose deformation is accommodated by many smaller ones. The model is able to predict smaller fragment sizes with increasing applied strain rates, as has been observed in dynamic fragmentation experiments on geologic materials (Grady 1982; Li et al. 2005). The present model differs from other micromechanics-inspired models in the literature (e.g., Espinosa et al. 1998; Bazant et al. 2000a,b) in a number of ways. First, the number and orientation of cleavage planes and crack propagation directions in the present work are associated directly with features of the microstructure of the solid, rather than assigned generically. This enables consideration anisotropic inelastic and failure behavior resulting from the texture, i.e., orientation of cleavage and slip planes, of the geologic polycrystal. Second, kinematics of the model are derived via volume averaging of the microscopic deformation gradient in the reference configuratio (Hill 1972), rather than assuming a mathematical form of the strain rate contribution from micro-cracks a priori. Thus, rotation resulting from micro-cracks is considered naturally, in addition to stretch. Third, thermodynamic consistency is achieved via consideration of the energy balance and entropy inequality in the formal context of internal state variable theory (Coleman and Gurtin 1967). In particular, the dissipative nature of damage progression is ensured by proper prescription of thermodynamically admissible evolution equations for crack density and average crack size on each cleavage plane. Finally, the present model incorporates aspects of nonlinear thermoelastic deformation (Imam and Johnson 1998) to account for the response of the solid to temperature rise resulting from dissipation at high rates, as well as high pressures and large compressions attained in granite in shock physics experiments (Millett et al. 2000). Possible importance of dissipation and thermal expansion in shocked concrete has been noted elsewhere (Grady 1995). The rest of this paper is structured as follows. Kinematics and averaging concepts are addressed in Sect. 2. Thermodynamic and kinetic relations for

153

thermoelasticity and damage evolution, including micro-cracking and fragmentation, are addressed in Sect. 3. Applications of the model to granite deformed in uniaxial stress compression, shear, tension, and uniaxial strain compression are addressed sequentially in Sect. 4. Conclusions follow in Sect. 5. Remarks regarding novelty of the present work are in order. Some averaging concepts used to motivate large deformation kinematic and stress definitions were developed in previous works (Hill 1972; Nemat-Nasser 1999; Clayton and McDowell 2003, 2004; Clayton 2006), and some algorithms used in polycrystal plasticity (Clayton 2005a,b, 2006) are modifie to address inelastic rock behavior in the present paper. Kinetic equations for micro-cracking and fragmentation posited in this paper are new. Furthermore, application of the model to granite rock is completely new; no model results discussed in Sect. 4 have been published prior to the present study. Standard notation of nonlinear continuum mechanics applies (Eringen 1962). Scalars and components of vectors and tensors are written in italic font, while vectors and tensors of higher rank are written in bold. Einstein’s summation convention is used for repeated indices. The · symbol denotes the scalar product of vectors (a · b = a a ba = a 1 b1 + a 2 b2 + a 3 b3 ), while ⊗ indicates the outer product ((a ⊗ b)ab = a a bb ). Juxtaposition implies summation over one set of adjacent indices ((AB)a.c = Aab Bbc ). Summation over two sets of indices is denoted by : (A : B = Aab Bab ). Superposed ·, T , and −1 denote a material time derivative, transpose, and inverse, respectively, and subscripted commas denote partial differentiation. 2 Kinematics 2.1 Average deformation gradient Consider a volume element of fi ed mass of material that may include internal surfaces across which local deformation is discontinuous, as illustrated by the 2-D cross section in Fig. 1. In the present treatment, coordinate systems in reference and deformed configuration are assumed to consist of a single set of basis vectors such that covariant derivatives and partial coordinate derivatives are equivalent. Let x and X denote, respectively, local fin scale spatial and reference coordinates within the material element or along its boundaries,

123

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J. D. Clayton

and let this element occupy volume V in the reference configuration The local deformation gradient f within continuously deforming (i.e., undamaged) regions of the solid is then ∂x f= . (1) ∂X bulk

The total deformation gradient F for the volume element is define in terms of the motion of its external boundary S (Hill 1972; Nemat-Nasser 1999; Clayton and McDowell 2003, 2004), and includes contributions from the bulk as well as contributions from damage entities contained within the element: F = V −1 x ⊗ Nd S. (2) S

In (2), reference surface S with unit outward normal N encloses the element of reference volume V , and x denotes spatial coordinates of the deformed external surface of this element, as shown in Fig. 1. Surface S does not include internal surfaces associated with damage entities contained within V . The total deformation is decomposed additively (Clayton and McDowell 2003, 2004; Clayton 2005a, 2006) according to F = Fˆ + FD , (3) where the damage-induced deformation FD =v FD +s FD ,

(4)

stems from j volumetric-type defects (e.g., open voids of arbitrary shape) as ( j) v D −1 F =V x( j) ⊗ Nv d Sv( j) , (5) j S v

and k surface-type defects (e.g., closed or open cracks or shear discontinuities): s D F = V −1 (6) x(k) ⊗ Ns(k) d Ss(k) . k S s

Deformation associated with continuously deforming regions of the bulk solid is the volume average (7) Fˆ = V −1 fd V . V ( j) Nv

is the outward unit normal to internal surIn (5), ( j) face j of referential area Sv . The sign convention for the unit normal to the surface of a spherical void, ( j) for example, is that Nv is directed radially outward from the center of the void towards the surrounding material. In (6), Ns(k) is the outward unit normal to (k) internal surface k of referential area Ss . The displace(k) ment discontinuity across Ss is denoted by the jump (k) x(k) = x(k)+ − x(k)− , and Ns points from the negative to the positive side of surface k. Formally, (5) ( j)+ ( j)− ( j) reduces to (6) when Nv = −Nv = Ns . Differential surface area elements NdS in (2) are mapped to their current configuratio representation nds via the Piola transform j f .a−1A N A d S = n a ds, where j = ( j) ( j) det f. Similar transformations apply for Nv d Sv in (k) (k) (5) and Ns d Ss in (6), enabling surface integrals in (2), (5), and (6) to be evaluated over the deformed body when f is known. When the body is simply connected and f is continuous within V , then application of Gauss’s theorem to the right of (2) results in the D right side of (7): when damage ais absent, F = 0 and a ˆ F = F ⇔ S x N A d S = V x.,A d V (Hill 1972). No additive decomposition of spatial position x is implied by (3). From a computational mechanics standpoint, additive decomposition (3) is challenging (Clayton 2006). Hence, (3) is converted to a multiplicative decomposition that can be used in standard computational frameworks for geometrically nonlinear polycrystal plasticity (Clayton 2005a,b). Let ˆ + Fˆ −1 FD ) = FF ˆ D, F = F(1

(8)

−1 Fˆ FD

D

with F = 1 + and 1 the unit tensor. Henceforth, damage is assumed to consist only of surface-type defects, such that FD =s FD in (3) and (4). Specifically considered are fla micro-cracks, each with constant reference normal N(k) and uniform time−1 (k) dependent displacement jump δx(k) = Fˆ x on each surface of area S (k) . Thus, D F = 1+ V −1 S (k) δx (k) ⊗ N(k) k

Fig. 1 Deforming volume element of a solid body containing displacement discontinuities

123

= 1+

k

γ (k) s(k) ⊗ N(k) ,

(9)


with s(k) = δx(k) /δx (k) the direction of crack opening or sliding and γ (k) = δx (k) S (k) V −1 a dimensionless measure of deformation on plane k. The magnitude of the jump is δx (k) = (δx(k) · δx(k) )1/2 . 2.2 Rate kinematics The inelastic velocity gradient obtained from (9) is D D−1 D L =

F˙ F (k) = γ˙ (k) s(k) ⊗ N(k) + γ (k) s˙ ⊗ N(k) k

k

× 1+ ≈

−1

(10)

γ (k) s(k) ⊗ N(k)

k

γ˙ (k) s(k) ⊗ N(k) .

k

The fina expression in (10) agrees exactly with the prek (crack opening/sliding ceding expression when s˙ = 0 directions fi ed a priori) and 1+ γ (k) s(k) ⊗ N(k) ≈ 1 (small inelastic deformation); otherwise it is treated as a fundamental kinematic assumption, analogous to the plastic velocity gradient of crystal plasticity theory (Asaro 1983). In other words, the fina equality in (10) can be derived directly from (9) only when reference crack directions are fi ed and inelastic deformations are small. The model is not, however, restricted to small deformation applications, and can be used to address large deformation problems if (10) and (11) below are assumed as fundamental definitions. Such assumptions are analogous to those used in metal plasticity, wherein rate equations for the plastic velocity gradient are prescribed directly in terms of shearing rates on preferred crystallographic planes. It is convenient to restrict the description to cracks on a finit set of planes in the solid. A crack system is define as a set of cracks sharing the same direction of opening/extension on cleavage planes of the same orientation. Index k hereafter refers to a crack system rather than an individual crack, analogous to a slip system of crystal plasticity, and an integer value of k is assigned to each pair of (sk , Nk ). Parentheses are removed from index k to denote crack systems as opposed to individual cracks. In terms of crack systems, velocity gradient (10) and scalar deformation measure become

155 D

L =

γ˙ k sk ⊗ Nk ,

k

γ k = n k δx k S k V −1 = δx k ωk S k .

(11)

The number of cracks per unit reference volume on system k is ωk = n k V −1 , with n k the absolute number of equivalent cracks belonging to system k. Presumably the number of crack systems is far smaller than the total number of cracks, enabling a drastic reduction in degrees of freedom k entering the model in (11) as compared to (10). Different crack systems may share either the same crack plane or the same opening/sliding direction, but micro-cracks that share both the same direction and plane belong to the same crack system. Unlike in crystal plasticity theory, here sk · Nk = 0, in genD D D D eral, meaning that J˙ = d(detF )/dt = J trL = 0

and the inelastic deformation is not generally restricted to be isochoric. For example, for mode I opening or dilatation on a single crack system, s · N = 1 and D D−1 J˙ J = γ˙ .

Following other models of brittle solids (Margolin 1983, 1984; Rajendran 1994; Espinosa et al. 1998), considered are penny-shaped micro-cracks of radius a k = (S k /π)1/2 , though generalization of the theory to other crack shapes does not present any conceptual difficultie . Crack deformation rates become k γ˙ k = δx k d(ωk S k )/dt + δ x˙ ωk S k k k k k = π δx k a k a k ω˙ k + 2a˙ k ωk + π δ x˙ ω a a ,

sliding or opening generation and growth

(12) where the firs term on the right side of (12) accounts for generation and growth of free surfaces, and the second term accounts for the rate of crack sliding or opening displacement. It is convenient to introduce a dimensionless crack density parameter ωk that leads to an equivalent version of (12): k ωk = ωk (a k )3 , γ˙ k = π δx k a k−1 (ω˙ − a˙ k a k−1 ω k ) k (13) +π δ x˙ ωk a k−1 .

2.3 Thermoelastic deformation The bulk deformation of the solid is treated as thermoelastic: Fˆ = FE Fθ ,

(14)

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J. D. Clayton

where FE accounts for rigid body rotation of the solid and elastic deformation associated with applied mechanical stresses, and Fθ accounts for thermal expansion or contraction due to temperature changes (Imam and Johnson 1998; Clayton 2005a,b). A measure of mechanical elastic strain to be used later is 2EE = FET FE − 1,

(15)

and the thermal deformation is spherical for materials of cubic or isotropic symmetry: θ F˙ Fθ −1 = αT θ˙ 1,

(16)

where αT is the coefficien of thermal expansion that is assumed constant for simplicity. Following from (8), (11), (14), and (16), the total spatial velocity gradient is ˙ −1 = F˙ E FE−1 L = FF

elasticity

+ αT θ˙ 1 +

thermal expansion/ contraction

k

γ˙ k sk ⊗ nk ,

(17)

micro−cracking

ˆ k and nk = Nk Fˆ −1 the crack system with s k = Fs direction and crack plane normal mapped to the spatial configuration neither generally of unit length.

3 Thermomechanics and kinetics 3.1 Average stress and governing equations The net firs Piola-Kirchhoff stress P, work conjugate to F of (2), is define by (Hill 1972; Nemat-Nasser 1999; Clayton and McDowell 2003, 2004) P = V −1 t0 ⊗ Xd S, (18) where the traction measured per unit reference area acting on external boundary S is t0 , and X are reference coordinates along S. At the macroscopic scale of observation, local balances of mass, linear momentum, and angular momentum are aA P..,A + B a = ρ0 Aa ,

a bA b aA F.A P = F.A P ,

(19)

with ρ0 the reference mass density of a material element, ρ the mass density in the deformed state with

123

3.2 General constitutive relations The Helmholtz free energy is assumed to exhibit the functional dependencies (23) ψ = ψ EE , θ, {ωk } , where {ωk } = {ω1 , ω2 , . . . ωn } is the set of dimensionless crack densities on all n crack systems, included, for example, to account for degradation of elastic moduli with damage accumulation. Each ωk is treated as an internal state variable. It is convenient to consider the energy balance and entropy inequality in the unloaded intermediate configuratio of the body, i.e., the evolving reference configuratio for the reversible elastic response (Eckart 1948). The mass density of an elastically unloaded −1 element of material is ρ = ρ J E = ρ0 J , with D D J E = detFE and J = J θ J = detFθ detF . Multi−1 plying (20) by J gives

ρ e˙ = J E σ : L − ∇ · q + ρr, (24) with the symmetric Cauchy stress σ = J −1 FP T , intermediate covariant derivative operators ∇ α = −1A

S

ρ0 = ρ J,

J = detF, B the body force, and A the acceleration. The local balance of energy is written as follows: ρ0 e˙ = P : F˙ − ∇0 · Q + ρ0 r, (20) where e is the internal energy per unit mass, Q is the heat flu per unit reference area, r is scalar heat source per unit mass, and ∇0 is the gradient operator in reference coordinates. The entropy inequality is Q ρ0 r ρ0 η˙ ≥ −∇0 · + , (21) θ θ with η the entropy per unit mass and θ the absolute temperature. Introducing the free energy per unit mass ψ = e − θ η, (21) becomes ˙ . P : F˙ − θ −1 Q · ∇0 θ ≥ ρ0 ψ˙ + θη (22)

∇0 A F .α

and ∇ α = ∇ α + J D Fθ F ,

−1

−1A

(J F .α ),A , residual

deformation F = and intermediate heat flu −1 q = J FQ. Similarly, (22) becomes J E σ : L − θ −1 q · ∇θ ≥ ρ ψ˙ + θ˙ η . (25) Expanding (23) using the chain rule (Coleman and Gurtin 1967) and substituting (17) into (25) gives ∂ψ ˙ E + ρ χ − η − ∂ψ θ˙ −ρ E :E ∂θ ∂E ∂ψ k τ k γ˙ k − ρ k ω˙ − θ −1 q · ∇θ ≥ 0, (26) + ∂ω k


where = J E FE−1 σ FE−T is an elastic second Piola-Kirchhoff stress, χ = −3J E αT p/ρ with the Cauchy pressure p = −trσ /3, and scalar resolved Kirchhoff stresses conjugate to micro-crack deformations are τ k = J E σ : sk ⊗ nk . Stress–strain and entropy–temperature relations deduced from (26) are =ρ

∂ψ ∂ψ , η=− + χ. E ∂θ ∂E

(27)

Introducing the conduction law q = −k∇θ , where k is symmetric and positive definite, the entropy inequality becomes ∂ψ ˙ k k k τ γ˙ − ρ k ω + θ −1 ∇θ · k ∇θ ≥ 0, (28) ∂ω k and entropy production from conduction is always nonnegative since θ −1 ∇θ · k∇θ ≥ 0. After introducing specifi heat capacity per unit mass c = ∂e/∂θ, energy balance (24) can be written ρcθ˙ =

k k

τ γ˙ − ρ

k

∂ψ ∂χ ∂ 2ψ + θ − θ ∂ω k ∂ω k ∂θ ∂ω k

−θ β : E˙ E + ∇ · k ∇θ + ρr, where thermal stress coefficient ∂χ ∂ 2ψ β=ρ − . ∂EE ∂θ ∂EE

k ω˙

(29)

(30)

The following more specifi form of the free energy is now considered for isotropic solids:

ρψ = μEE : EE + K (trEE )2 /2 +C

αβχδεφ

C

αβχδεφ

= ν 1 δ αβ δ χ δ δ εφ + ν 2 δ αβ δ χ ε δ δφ + δ χ φ δ δε + δ χ δ δ αε δ βφ + δ αφ δ βε + δ εφ δ αχ δ βδ + δ αδ δ βχ + ν 3 δ αχ δ βε δ δφ + δ βφ δ δε + δ βδ δ αε δ χ φ + δ αφ δ χ ε + δ αδ δ βε δ χ φ + δ βφ δ χ ε (32) + δ βχ δ αε δ δφ + δ αφ δ δε .

Elastic coefficient are assumed independent of temperature but do depend upon micro-crack densities: μ = μ {ωk } , K = K {ωk } , ν 1 = ν 1 {ωk } , ν 2 = ν 2 {ωk } , ν 3 = ν 3 {ωk } . (33) Functional forms for the firs two of (33) are available for penny shaped micro-cracks (Budianski and O’Connell 1976; Margolin 1983; Rajendran 1994). Elastic coefficient of the undamaged material are μ = μ(0), K = K (0), ν1 = ν 1 (0), ν2 = ν 2 (0), and ν3 = ν 3 (0). Elastic modulus and Poisson’s ratio are related to bulk and shear moduli by usual relations E = 9μK /(μ + 3K ) and ν = (3K − 2μ)/(6K + 2μ). Stress–strain relationships following from (27) and (31) are

= 2μEE + (K trEE )1 + (C : EE ) : EE /2.

(34)

3.4 Kinetics of micro-cracking and dilatancy

3.3 Representative free energy

157

E E E αβ E χEδ E εφ /6 − ρcθ ln(θ/θ0 ). (31)

In (31), EE = EE − (tr EE /3)1 is the deviatoric elastic strain, μ is the effective shear modulus, K is the effective bulk modulus, c is the reference specifi heat at constant volume, θ0 is a reference temperature at which the thermal energy (fina term in (31)) vanishes, and third-order elastic coefficient satisfy (Teodosiu 1982)

Micro-crack systems are partitioned into those of pure mode I opening or closing, with sk · Nk = 1, and mode II or III separation, with sk · Nk = 0. First consider the former case (i.e., mode I). The rate of deformation associated with opening (γ˙ k > 0) or closing (γ˙ k < 0) on system k is dictated by ⎧ ⎫ 1/m ⎪ k k ⎪ ⎪ τ k / τ k +β γ˙ i , γ k > −γ0k⎪ ⎨γ˙0 τ /g ⎬ i γ˙ k = 1/m ⎪ ⎪ ⎪ ⎪ +β γ˙ i , γ k ≤ −γ0k ⎩γ˙0 τ k /g k ⎭ i

∀k with sk · Nk = 1,

(35) " k# k k where the bracket notation 2 τ = τ + |τ | ensures that opening occurs only with the applied normal Kirchhoff stress is tensile when the cumulative normal deformation γ k = γ˙ k dt ≤ −γ0k , where γ0k is a positive constant denoting the degree of crack opening on system k in the reference configuration e.g., possible opening associated with initial porosity. This constraint prohibits inter-penetration of matter that would

123

158

J. D. Clayton

otherwise occur across the crack face in compressive loading. Constant γ˙0 is a positive reference strain rate required on dimensional grounds, m is the rate sensitivity, and g k is the evolving resistance to deformation. The crack opening resistance is assumed to consist of the cohesive strength gck = g|t=0 for crack generation and the subsequent resistance of the surrounding body to irreversible crack separation. Dilatancy is addressed via term β i |γ˙ i |, where β is non-negative and summation proceeds over all i shear (i.e., mode II or III) crack systems sharing the same normal Nk as the mode I crack system of interest k. Mode I cracking clearly involves volume changes and is affected by hydrostatic pressure. Crack growth is irreversible in the present model since healing of cracks (e.g., resealing under high pressure and temperature) is not considered. Crack separation may be irreversible or reversible. Reversibly separated cracks close within an element of a solid when all mechanical stresses are removed from the external boundaries of that solid. Likewise, reversibly closed cracks open when external stresses are removed. Reversible crack separation or closure is captured by the change in elastic moduli with crack density enabled by specifi forms of (33). Irreversible crack separation remains when external loads are removed from a volume element of material containing cracks. Irreversible cracks result from local plasticity in the vicinity of crack tips, contact friction between asperities on opposite faces of cracks, and also may include cracks that traverse the entire volume element. Irreversibly closed damage entities may also include permanently collapsed pores resulting from large compression (Brace 1978). It is assumed that resistance to crack separation and crack density on a given system obey the simple proportionality relationship k

k

g ω = constant,

(36)

meaning that strength decreases linearly with increasing crack density. Differentiation of (36) leads to k g˙ k = −g k (ω˙ /ωk ).

(37)

Now consider mode II/III cracking and sliding. The analog of (35) for such shear micro-cracking is 1/m γ˙ k = γ˙0 τ k − μˆ −τ Nk /g k τ k / τ k , τ Nk = J E σ : nk ⊗ nk , ∀k with sk · Nk = 0,

123

(38)

with μˆ the friction coefficien that takes effect when the normal stress acting on the plane is compressive. A nonzero friction coefficien μˆ contributes to pressure sensitivity of the deviatoric inelastic response. The larger the value of m, the greater the strain rate sensitivity. In contrast, rate independence is achieved in the limit that m approaches zero. Resistance g k again consists of the cohesive strength gck = g|t=0 for crack generation/extension and the subsequent resistance of the body to crack separation and interfacial sliding. Relationships (36) and (37) are assumed to hold in modes II and III as well as mode I. Thus evolution of resistance to crack deformation is dictated by evolution of dimensionless crack densities on each system. From (35) and (38) non-negative dissipation from inelastic deformation of cracking, τ k γ˙ k ≥ 0, is ensured since γ˙ k and its driving force τ k share the same algebraic sign, so long as the rate of dilatation does not exceed the rate of crack closure when normal stress is compressive in the firs of (35). Hence (28) is always satisfie when free energy k decreases with damage so that ω˙ ρ(∂ψ/∂ωk ) ≤ 0. Normalized crack densities evolve in conjunction with inelastic deformation. Though more general relationships are conceivable, here it is assumed that k ω˙ /ωk =

n

kj h γ˙ j ,

(39)

j=1 kj

where h is an interaction matrix with dimensionless entries, reflectin the rate of change of crack density of system k due to deformation on system j. For a steady deformation rate on a single system, (39) implies exponential evolution, ω = ω|γ =0 exp(h |γ | ). Crack radii evolve in a more general manner: ⎡ a˙ k /a k = γ˙1 ⎣

n

⎤ pˆ hˆ k j γ˙ j /γ˙1 ⎦ ,

(40)

j=1

where γ˙1 is a normalization constant and pˆ accounts for rate sensitivity of micro-crack growth that may differ from the rate sensitivity implicit in (39). Interaction matrix hˆ k j accounts for crack extension on system k due to deformation on system j. The absolute number of cracks per unit volume may increase or decrease, the latter occurring if multiple cracks coalesce. The average crack radius may also increase or decrease, the latter occurring if numerous new small cracks are initiated in a material containing pre-existing larger cracks.


3.5 Fragmentation The volume element of material is assumed to fragment when the micro-cracks coalesce or percolate through the element. Such a fragmented state is attained when a total dimensionless crack density = ωk (41) k

exceeds a critical measure C (Budianski and O’Connell 1976; Rajendran 1994). A mean fragment size can be estimated from geometric arguments. At percolation, let the total surface area of r cubical fragments of mean size b be proportional to the summed area of all micro-cracks (Grady 1988). Thus, ( ) 3 6r b 6a k k 2 b= = 6V / α πn (a ) = 2 απ N a 3 /V 6r b k 6 a = , (42) απ where α is a proportionality constant and N is the total number of micro-cracks of effective radius a: ( )1/3 k 3 k k a= (a ) ω / ω . (43) k

k

When all cracks are of equal size, a = a k . If closepacked spherical fragments of mean diameter b, rather than cubes, are assumed, then (42) still holds. Values of α less than unity can be interpreted as accounting for cracks whose surface areas do not contribute to the fragment dimension or percolation, for example small micro-cracks within larger individual fragments (Rosenberg 1971). 4 Application to granite rock 4.1 Material The foregoing model is applied to enable an improved understanding of dynamic deformation and failure of granite. In addition to its prominence in geology (e.g., mining and earthquake engineering), granite is of importance to the construction industry because of its frequent use as a structural material, both alone and as coarse aggregate in concretes. Granite is a heterogeneous, polycrystalline igneous rock with greater than 65 mol. % silica (SiO2 ) and often significant alumina content, i.e., >10 mol. % Al2 O3 (Maqsood et al. 2003).

159

The mineral content of a typical granite (Wenk et al. 2008) includes grains of quartz (trigonal crystal system, grain size of 0.1–3.0 mm), biotite (monoclinic symmetry, grain size of 0.05–1.5 mm), and feldspar (most generally triclinic, grain size of 0.25–2.5 mm). Average grain sizes of 2–3 mm for individual constituents of granite have also been reported elsewhere (Millett et al. 2000; Golshani et al. 2006). Feldspars consist of plagioclase and orthoclase, and have a complex chemical composition and atomic structure (Seo et al. 1999; Ichikawa et al. 2001). For example, albite is one member of the plagioclase series, and exhibits triclinic symmetry (Seo et al. 1999). Biotite, often called black mica, is a sheet silicate. Biotite is prone to basal cleavage via separation of lamellar sheets. Muscovite mica has also been reported in granite (Ichikawa et al. 2001). Natural porosities in granite are usually small, i.e., 0–1.0% (Goodman 1989; Sano et al. 1992; Maqsood et al. 2003). In the absence of fracture, granite is often classifie as a hard, polycrystalline, isotropic rock (Goodman 1989), though depending on sample origin, mechanical properties can be anisotropic as a result of pre-existing micro-cracks, as discussed below. Quarried granite often exhibits three mutually perpendicular primary cleavage planes: the rift plane, the grain plane, and the hardway plane (Simmons et al. 1975; Sano et al. 1992; Ichikawa et al. 2001). In quarried specimens, most (initial) cracks lie within the rift plane. Secondary cracks lie within the grain plane, while the hardway plane usually contains few, if any, micro-cracks (Golshani et al. 2006). The particular distribution of cracks among the three planes varies with composition and microstructure of the granite (Sano et al. 1992). Relative strengths of cleavage planes result from both the distribution of microstructural constituents (e.g., grains and phases) as well as initial crack densities that tend to be larger on the rift plane (Sano et al. 1992). Ichikawa et al. (2001) noted prevalent intergranular crack growth between phases, for example between quartz and feldspar and between quartz and biotite. Distributions of secondary micro-cracks on planes oriented at 30◦ –60◦ to the orthogonal cleavage planes have also been reported (Sano et al. 1992), attributed to boundary cracks of weakly oriented feldspar grains and intragranular cracks within quartz crystals. In recovered shock-loaded samples, Rosenberg (1971) found transgranular micro-fractures in quartz and feldspar grains, micro-fractures and kink bands in mica, and few grain boundary cracks. Fracture toughness can

123

160 Table 1 Primary (H, G, R) and secondary (S) micro-crack systems in granite

J. D. Clayton k

Direction

Plane

k

Direction

Plane

1

[100]

(001)H

10

[111]

2

[010]

(001)H

11

[111]

Direction

Plane

(011)S

19

[001]

(001)H

(101)S

20

[010]

(010)G (100)R

3

[100]

(010)G

12

[111]

(101)S

21

[100]

4

[001]

(010)G

13

[111]

(101)S

22

[011]

(011)S

5

[010]

(100)R

14

[111]

(101)S

23

[011]

(011)S

6

[001]

(100)R

15

[111]

(110)S

24

[101]

(101)S

7

[111]

(011)S

16

[111]

(110)S

25

[101]

(101)S

8

[111]

(011)S

17

[111]

(110)S

26

[110]

(110)S

9

[111]

(011)S

18

[111]

(110)S

27

[110]

(110)S

Fig. 2 Volume element of material with orthogonal cleavage planes

vary significantly among different varieties of quarried granite, e.g., in excess of 200% (Amaral et al. 1999; Golshani et al. 2006).

4.2 Micro-crack systems Micro-crack systems for granite are listed in Table 1. These include pure shearing modes accommodated by systems 1–18 and pure opening modes accommodated by systems 19–27. Primary planes are designated in Miller indices by the notation (100) for rift (R), (010) for grain (G), and (001) for hardway (H), respectively, as shown in Fig. 2. Fractures on secondary (S) planes of type {110}, inclined by 45◦ to the primary planes, are considered, corresponding to inter- and intra-granular fractures mentioned above (Sano et al. 1992). The mean spacing between fracture planes depends upon the microstructure, for example the spacing between grain boundaries or cleavage planes within layers of biotite. In the model, the spacing b of (42) between all

123

k

primary crack planes is assumed to remain the same in all three directions for simplicity, leading to a kind of simple cubic structure, and to cube-shaped fragments when micro-cracks coalesce on all planes at percolation. Note however that the magnitude of b evolves with deformation according to (42) and (43). Crack systems 1–18 consist of mode II/III opening or sliding in directions (systems 1–6) and directions (systems 7–18). Systems 1–6 correspond to the shortest slip distance in a simple cubic structure, while the slip distance across a unit cell of the√ material for systems 7–18 exceeds this by a factor of 3. The present theory differs from others that enable crack opening in any direction on a predefine number of potential crack planes (Espinosa et al. 1998). Here relative sliding of crack faces is restricted to certain preferred directions, resulting in a possibly anisotropic inelastic response. As will be shown later, a macroscopically isotropic response can be achieved by averaging the responses of a large number of randomly oriented volume elements or “grains” of the kind shown in Fig. 2, as is typically done in polycrystal metal plasticity (Clayton 2006). Systems 19–21 consist of opening on each of the three primary cleavage planes of type {100}, while systems 22–27 account for opening modes on each of the six geometrically distinct {110} planes. Assumptions of preferred planes and directions for micro-cracking in granite rock are warranted by the following arguments. The assignment of fracture planes follows from discussion and supporting references in Sect. 4.1: characteristic failure planes are the rift plane, grain plane, hardway plane, and secondary planes. Directions for mode I micro-cracking are then define uniquely as normal vectors to these fracture planes. Directions for mode II/III micro-cracking

Deformation, fracture, and fragmentation in brittle geologic solids Table 2 Mechanical and thermodynamic properties of granite

161

Parameter

Value

Definition

Reference

μ [GPa]

23.6

Shear modulus

Winkler and Liu (1996)

Bulk modulus


30.0 K [GPa]

Ai and Ahrens (2006)

29.9 55.6

Ai and Ahrens (2006)

K 1 [GPa]

45.9

Third-order elastic constant

ρ0 [kg/m3 ]

2,650

Mass density


c [J/kgK]

755

Specifi heat

Maqsood et al. (2003)

k [W/mK]

2.0

Thermal conductivity

Maqsood et al. (2003)

αT [1/K]

6 × 10−6

Thermal expansion

Touloukian et al. (1977)

(i.e., shearing modes) are assigned a priori as a modeling assumption. In a real material, shear fracture could conceivably take place in any direction on a given fracture plane. In the present model, shear fractures on a given plane result from the combined effect (vector sum) of fracture in two (orthogonal) directions on each plane. Thus, the net direction of shear fracture on a given plane is not geometrically restricted, though the stress–strain response of a single grain will generally be anisotropic and will depend on the choice of crack directions. The assignment fracture planes and directions used here is very similar to that described in a version of the micro-plane model for concrete and brittle solids (Bazant et al. 2000a) as well as other theories of anisotropic rock plasticity (Zienkiewicz and Pande 1977). Because sufficien directionally independent crack systems are provided, any inelastic opening or shearing mode can be accommodated. For example, in a Cartesian coordinate system with axes aligned normal to each of the cleavage planes, each of the opening modes 19–21 provides for a diagonal component of the inelastic velocity gradient, while each of the shearing modes 1–6 provides for an off-diagonal component. Systems 22–27 are geometrically redundant, but will affect the yield surface depending on the choice of parameters dictating their fracture kinetics. Any number of grains of various orientations can be assigned to a material element, with their response homogenized in order to model an aggregate of polycrystalline rock exhibiting a lesser degree of anisotropy than that of a single grain. Shear along preferred directions suggested here, when taking place in quantized amounts (i.e., integer multiples of the interplanar spacing) preserves the relative separation distances and orientations of the cleavage planes. In the limit of very small inter-planar


spacing, a discontinuity of arbitrary size and shape (i.e., a Somigliana dislocation) can be built up from the summation of small micro-cracks that are of constant magnitude and direction (Eshelby 1973). The inelastic velocity gradient in (11) then results from the flu of such Somigliana dislocations. Crack planes and opening directions are fi ed in the reference configuration however, their orientation vectors are updated in the spatial frame with the thermoelastic deformation of the body, as indicated following (17). Thus, crack planes and directions are transformed appropriately when the solid undergoes large rigid body rotations. 4.3 Thermoelasticity with damage Thermoelastic properties are listed in Table 2. Specifi heat c in (31) corresponds to specifi heat at constant volume; differences between isochoric and isobaric specifi heat are negligible in temperature and pressure regimes considered in this work. Recall that an isotropic elastic response is assumed, even if damage is present (Budianski and O’Connell 1976; Margolin 1983; Rajendran 1994). Isotropic thermal conductivity ab is also assumed: k = kδ ab . A more detailed model would account for orthorhombic symmetry (e.g., nine independent second-order elastic constants) resulting from orthogonal planes containing oriented circular cracks (Sano et al. 1992). The effect of the nonlinear elasticity is most important in volumetric compression and is neglected in shear, since in the present context the material will deform plastically by shear fracture before large elastic deviatoric strains are attained. Hence, (32) reduces to (Clayton 2005b) C

αβχδεφ

= −2K 1 δ αβ δ χ δ δ εφ , K 1 = −ν1 − 2ν2 − 8ν3 /9.

(44)

123

162

J. D. Clayton

Elastic moduli degrade with damage according to the model of Rajendran (1994), in which Margolin’s formulae (Margolin 1983) for solids with non-interacting penny shaped cracks are used when one or more principal stresses is compressive, and Budianski and O’Connell’s self-consistent model (Budianski and O’Connell 1976) is used when all principal stresses are tensile such that contact across crack faces does not occur. A crack density parameter with dimensions of stress−1 is introduced as

and effective bulk modulus may be necessary, as is the case for concrete with substantial initial porosity (Clayton 2008). In the present application, any initial mode I cracks are approximated either as stationary or fully closed; that is, all effects of initial porosity are included implicitly in the choice of K and K 1 . Hence it is assumed that γ0k = 0 in (35).

= 16/(45E),

Table 3 lists parameters for kinetics of micro-cracking on systems 1–18 that become active during compressive loading. Because a single reference containing all required material characterization data is not presently available, parameters necessarily are acquired from consideration of several sources. The value of the rate sensitivity of the yield strength m, appropriate for granite deformed in compression at strain rates in the regime of 10–100/s, follows from Li et al. (2005) and references therein. The value for γ˙0 corresponds to a typical strain rate in experiments (Li et al. 2005) and complementary calculations discussed later. Values of frictions coefficien have been suggested in the range 0.2 ≤ μˆ ≤ 1.1 for brittle polycrystalline ceramic and geologic materials (Christensen et al. 1974; Rajendran 1994; Espinosa et al. 1998; Astrom 2007; Foster et al. 2007; Paliwal and Ramesh 2008). Christensen et al. (1974) observed a friction coefficien in Westerly granite nearly independent of shearing displacement rate. Here, the midpoint of the range suggested by Astrom (2007) for rocks is used: μˆ = 0.6. For crack kinetics, it is assumed for simplicity that activity of all systems affects the softening behavior of a particular family in the same way, so that (39) and (40) reduce to ⎡ ⎤ pˆ n n k k j k k j ω˙ /ω = h γ˙ , a˙ /a = γ˙1 ⎣ γ˙ /γ˙1 ⎦ ,

(45)

where is a dimensionless crack density summed over all systems as define in (41). Stresses and effective moduli entering (34) define with respect to the elastically unloaded intermediate configuratio are of present interest. When no principal values of are positive, the shear modulus degrades with cumulative cracking but the bulk modulus does not: −1 μ = 3(1 + ν) + 1/μ , K = K.

(no principal values of positive)

(46)

For other cases wherein one or more principal stresses is positive, formulae for effective shear and bulk moduli in terms of and principal stresses are listed by Rajendran (1994) and are not repeated here. In all cases, elastic coefficient μ and K remain the same or decrease with increasing , leading to positive dis˙ ≥ 0. The latter holds when all sipation in (28) for kj entries of h in (39) are non-negative: ( ) ∂ψ k ∂μ E E 1 ∂ K E 2 ˙ ω =− −ρ E :E + (trE ) 2 ∂ ∂ωk ∂ k ⎡ ⎛ ⎞⎤ n n kj ⎝ω k ×⎣ h γ˙ j ⎠⎦ ≥ 0. k=1

j=1

(47) Possible effects of cracking on third-order elastic coefficient of granite are not quantitatively known and are not included in the present application of the model. Notice from (46) and the preceding discussion that bulk modulus K and the higher-order elastic coeffi cient K 1 remain constant under compressive hydrostatic loading. If the bulk modulus exhibits an increase as the initial porosity is compressed out of the material (i.e., as open tensile cracks become fully closed above some threshold pressure (Brace et al. 1966)), a more detailed equation of state coupling porosity

123

4.4 Micro-crack kinetics

j=1

j=1

(48) where the constant hˆ = hˆ k j is absorbed into the parameter γ˙1 . Initial crack radii and crack densities on each family of systems H, G, R, and S follow from representative experimental measurements (Sano et al. 1992; Golshani et al. 2006). Normalized initial crack densities are then computed from (12). Ratios of cohesive strengths gck are assumed inversely proportional to their associated crack densities via the same proportionality constant in (36) such that

Deformation, fracture, and fragmentation in brittle geologic solids Table 3 Kinetic parameters for micro-crack systems in granite

163

Parameter Value Plane Definition

Reference

m

1/3

all

Rate sensitivity

Li et al. (2005) (dynamic)

all

Reference strain rate

Arbitrary (dynamic)

0.016 γ˙0 [1/s]

50

Li et al. (2000) (quasi-static)

10−4

(Quasi-static)

μˆ

0.6

all

Friction coefficien

Astrom (2007)

gck [GPa]

0.12

H

Cohesive strength

Model fi

0.06

G

0.02

R

0.12

S Initial crack density

Golshani et al. (2006)

Initial crack radius

Golshani et al. (2006)

Crack density evolution coefficien

Model fi Model fi

ωk0

0.008 H 0.017 G 0.044 R 0.008 S

a0k [mm]

0.48

H

0.56

G

0.54

R

0.61

S

h

67

all

γ˙1 [1/s]

103

all

Rate coefficien for crack radius

pˆ

−0.1

all

Rate exponent for crack radius

Model fi

α

0.07

all

Crack fraction at fragment surface

Model fi

β

3

all

Dilatation coefficien

Model fi

C

9/16

all

Crack density at failure

Budianski and O’Connell (1976)

gck ωk0 = gck ωk0 = gck ωk0 = gck ωk0 . (∀k) H

G

R

S

(49)

Thus, in addition to h in the f rst of (48), only a single value of cohesive strength is needed from consideration of experimental data, and the remainder follow from (49). Cohesive strength controls initial yield, while h controls strain softening with cumulative damage, as is clear from (37) and (48). 4.5 Dynamic compression: uniaxial stress The predicted stress–strain response under unconfine uniaxial compressive loading is shown in Fig. 3a along with experimental data (Li et al. 2005). The model is implemented at a single 3-D integration point using mixed implicit-explicit integration algorithms for nonlinear crystal elasto-plasticity with thermal expansion, adiabatic heating, and potential strain softening (Clayton 2005b). Prominent new aspects incorporated in the computational algorithms include sliding

friction in (38), reduction in shear modulus with damage in (46), and numerical integration of evolution Eq. (48). Loading conditions correspond to compression along the Cartesian 1-axis, i.e., F11 = 1 − ε˙ t, with a constant applied strain rate of ε˙ , and the cumulative strain is ε = ε˙ dt. The magnitude of the non-zero Cauchy stress component is σ = |σ11 |. Initial texture (i.e., orientations of fracture planes relative to loading direction) and anisotropy were not addressed in the experimental study. Hence an isotropic elastic-inelastic response is assumed. Isotropy is achieved by averaging the Cauchy stress over a large number of grains of random orientations subjected to the same macroscopic deformation F11 . Since the response did not change appreciably upon increasing the sample domain from 100 to 300 orientations, as is clear from Fig. 3a, 100 orientations are used to generate results in subsequent figure unless noted otherwise. Elastic constants of Winkler and Liu (1996) given in Table 2 are used in the present and subsequent calculations unless noted otherwise.

123

164

J. D. Clayton

(a) 0.25

Granite Uniaxial compression

σ [GPa]

0.20 0.15 0.10 model 100 orientations, 50 /s model 300 orientations, 50 /s Li et al. 49.1 /s

0.05 0.00

0.000 0.002 0.004 0.006 0.008 0.010

ε

(b) 0.25

Granite Uniaxial compression

model 50 /s model 45 /s model 40 /s model 35 /s model 30 /s model 25 /s model 20 /s model 15 /s model 10 /s Li et al. 51.2 /s Li et al. 47.6 /s Li et al. 42.8 /s Li et al. 38.9 /s

σ [GPa]

0.20 0.15 0.10 0.05 0.00 0.000 0.002 0.004 0.006 0.008 0.010

ε

Fig. 3 Uniaxial stress versus compressive strain for a ε˙ ≈ 50/s and b range of strain rates

The f t in Fig. 3a is reasonably accurate for both hardening and softening portions of the curve. However, it is noted that in the experiments, fluctuation about the applied rate of ε˙ = 49.1/s took place, especially during latter stages of the test; such fluctuation in loading rate are not addressed in the boundary conditions used here. The model slightly under-predicts the elastic stiffness, suggesting that elastic modulus obtained from acoustic velocities by Winkler and Liu (1996) in Westerly granite may be lower than that exhibited by the Bukit Timah granite in the dynamic compression tests of Li et al. (2005). Simulations are performed adiabatically. Predicted temperature rises are small, i.e., θ < 10K at ε = 0.01, and thermal expansion does not appreciably affect results shown in Fig. 3.

(b)

1.8 1.6 1.4 1.2 1.0 0.8

50 /s 45 /s 40 /s 35 /s 30 /s 25 /s 20 /s 15 /s 10 /s

2.5 decreasing strain rate

1.5

50 /s 45 /s 40 /s 35 /s 30 /s 25 /s 20 /s 15 /s 10 /s

decreasing strain rate

1.0

0.6 0.4

0.5

0.2 0.000 0.002 0.004 0.006 0.008 0.010

ε

123

2.0

a [mm]

(a)

Ω

Fig. 4 Model predictions for a dimensionless total crack density and b average crack radius

Predictions of the model for uniaxial compressive loading over a range of strain rates 10/s ≤ ε˙ ≤ 50/s are shown in Fig. 3b. Experimental data (Li et al. 2005) are also shown (the data set from Fig. 3a is not duplicated in Fig. 3b). Both model and experiment demonstrate a general trend of increasing compressive strength with increasing loading rate, though the experimental data is notably stochastic. Predicted peak stresses σ p follow experimental observations of Li et al. (2005): σ p = A˙ε1/3 with the strain at which peak stress occurs increasing with increasing rate in model and experiment. A regression to the present results gives A = 0.063 GPa, very close to the experimental value of A = 0.065 GPa (Li et al. 2005). Recall from (38) that shear strength depends both on frictional resistance to sliding (μ) ˆ and cohesive strength or resistance to crack growth/extension (g k ). In geologic materials, friction may be highly variable. For example, depending on the material and environment, a positive jump in shear stress may occur when the slip rate is suddenly increased, and steady state friction stress at constant normal stress may increase or perhaps more often decrease with an increase in relative velocity of sliding crack faces (Ruina 1983; Foster et al. 2007). In the present model, the friction coeffi cient is approximated as independent of loading rate, as suggested by experiments of Christensen et al. (1974) on Westerly granite. However, the total strength of the material in dynamic compression, as a result of ratedependent cohesive strength g k , varies with strain rate via m = 1/3 in (38), resulting in an increase in compressive stress with increasing loading rate as observed in the compression experiments (Li et al. 2005). For geologic materials exhibiting different behavior, a variable friction coefficien could be assigned in the model as justifie by experimental data. Crack growth is shown in Fig. 4 for boundary conditions discussed above, corresponding to stress–strain

0.000 0.002 0.004 0.006 0.008 0.010

ε

Deformation, fracture, and fragmentation in brittle geologic solids 70 model [present work] model [Grady] experiment [Li et al.]

b [mm]

60 50 40 30 20 10

20

.

30

40

50

ε [1/s]

Fig. 5 Mean fragment size versus strain rate: models (Eq. (42) and Grady 1982) and experiment (Li et al. 2005)

curves of Fig. 3b. Shown in Fig. 4a is total dimensionless micro-crack density computed from (41) for each grain and averaged over 100 randomly oriented grains. Crack density increases with increasing strain in a manner only slightly dependent on rate. At small strains ε < 0.003, the material behaves primarily elastically and the crack density increases slowly with strain. Shown in Fig. 4b is average crack radius over all systems computed from (43) for each grain and averaged over the same 100 orientations. Average crack size a increases more steeply with increasing strain as the strain rate is decreased. Since normalized crack density depends only mildly on strain rate as indicated in Fig. 4a, the number of cracks increases drastically with increasing rate, since by (12), the absolute number of cracks on each system is proportional to the inverse of the cube of the crack radius on that system. For example at ε = 0.01, the number of cracks per unit volume /a 3 = 0.11 mm−3 at a rate of 10/s while /a 3 = 3.32 mm−3 at a rate of 50/s. While quantitative experimental data are not available for crack radii versus applied strain, predictions appear reasonable since crack sizes lie within dimensions of the experimental sample: cylinders of length and diameter on the order of 100 mm. Mean fragment sizes are plotted in Fig. 5 versus applied compressive strain rate. Experimental data are from the aforementioned compression tests, wherein numbers of fragments after the test were counted (Li et al. 2005). In most experiments (Li et al. 2005), relatively few large fragments were generated (e.g.,