A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables. Part I—Theory

Journal of the Mechanics and Physics of Solids 70 (2014) 281–296

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A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables. Part I—Theory Alessandro Tengattini a,b, Arghya Das a, Giang D. Nguyen c, Gioacchino Viggiani b, Stephen A. Hall d, Itai Einav a,n a

School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia Univ. Grenoble Alpes, 3SR, F-38000 Grenoble, France c School of Civil, Environmental and Mining Engineering, The University of Adelaide, Adelaide, SA 5005, Australia d Division of Solid Mechanics, Lund University, Lund, Sweden and European Spallation Source AB, Lund, Sweden b

a r t i c l e i n f o

abstract

Article history: Received 6 August 2013 Received in revised form 18 April 2014 Accepted 31 May 2014 Available online 16 June 2014

This is the first of two papers introducing a novel thermomechanical continuum constitutive model for cemented granular materials. Here, we establish the theoretical foundations of the model, and highlight its novelties. At the limit of no cement, the model is fully consistent with the original Breakage Mechanics model. An essential ingredient of the model is the use of measurable and micro-mechanics based internal variables, describing the evolution of the dominant inelastic processes. This imposes a link between the macroscopic mechanical behavior and the statistically averaged evolution of the microstructure. As a consequence this model requires only a few physically identifiable parameters, including those of the original breakage model and new ones describing the cement: its volume fraction, its critical damage energy and bulk stiffness, and the cohesion. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Cemented granular materials Constitutive behavior Microstructures Energy methods Fracture mechanisms

1. Introduction A number of naturally occurring and artificial materials are classified as cemented granular materials (CGMs) (Fig. 1). The well-known naturally occurring examples include calcite, quartz and clay cemented sands (Dvorkin and Yale, 1997) and sedimentary rocks like sandstones, conglomerates and breccias (Topin et al., 2007). Artificially cemented materials widely used in engineering include asphalts, mortars, concrete, bio-, mortar-, and polymer-grouted soils (Anagnostopoulos, 2005). Other CGMs are solid propellants, high explosives and wheat endosperm (Topin et al., 2007). All the CGMs described above share a common texture of grains being bridged by cement matrix that partially or completely fills the voids; for this reason, CGMs often share microscopic processes that control their macroscopic behavior. This paper focuses on lightly to medium cemented granular materials. The mechanical behavior of such materials is controlled by the grain properties and their (re-)organization, as for uncemented granular materials, plus the effect of the cement, which, even in small amounts, can significantly alter the stress distribution within the grains and therefore the mechanical response of the aggregate (e.g. Wong and Wu, 1995; Alvarado et al., 2012). The inelastic behavior of this class of materials is thus governed by three main processes (Menendez et al., 1996): grain crushing, cement damage and fragment reorganization. n

Corresponding author. Tel.: þ61 (0)2 9351 2113; fax: þ 61 (0)2 9351 3343. E-mail address: [email protected] (I. Einav).

http://dx.doi.org/10.1016/j.jmps.2014.05.021 0022-5096/& 2014 Elsevier Ltd. All rights reserved.

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Fig. 1. Examples of cemented granular materials: (a)–(c) heavily cemented granular materials (Commons, 2006; Winter, 2012; Wong et al., 2007), and (d)–(f) lightly cemented granular materials (Alvarado et al., 2012; Ismail et al., 2002; Rong et al., 2012). (a) Conglomerate, (b) Concrete, (c) Asphaplt, (d) Saltwash Sandstone, (e) Calcite cemented sand and (f) Biocemented Sand.

In this paper, we propose a continuum constitutive model for CGMs, starting from the micro-mechanical observation of grain and cement failures. Each of these phenomena will be captured by a separate internal variable with meaningful statistical interpretations. The injection of physical understanding of the micro-mechanics underlying the macroscopic response minimizes the number of required parameters, which will here have a clear physical meaning and are all measurable. The formulation follows the thermo-mechanical framework, and thus the proposed model obeys the laws of thermodynamics. The choice of the internal variables is justified in Section 3, followed by the study of the elastic stored energy in Section 4. The thermo-mechanical framework is revised in Section 5 and a possible model within the proposed framework is introduced in Section 6. Finally, the range of phenomena that the proposed model is able to reproduce is explored through a sensitivity analysis in Section 7.

2. Previous work on CGM characterization and modelling In 1993, Gens and Nova noticed that there was a general lack of conceptual frameworks and mathematical models able to integrate the behavior of CGMs in a consistent and unified manner (Gens and Nova, 1993). Although several experiments and computational methods have been designed to study CGMs, the observation of Gens and Nova is likely still relevant today.

2.1. Experimental observations Some experimental works have focused on the influence of microscopic features on the macroscopic response, in particular the effect of cement content and type and the composition and fabric of the granular phase (Schnaid et al., 2001; Coop and Atkinson, 1993; Airey, 1993; Ismail et al., 2002). Other authors focused instead on the study of the micromechanisms involved (Menendez et al., 1996; David et al., 2001; Wong and Baud, 2012). The addition of cement to granular materials is known to increase the size of the yield strengths, and enhance the shear and bulk elastic moduli (e.g. Coop and Atkinson, 1993; Airey, 1993; Clough and Nader, 1982; Huang and Airey, 1998); on the other hand, the critical state appears to be independent of the level of cementation (Coop and Atkinson, 1993; Airey, 1993; Huang and Airey, 1998). Shear at low confining pressures is led by progressive fracture of cement bridges among the grains, which releases local degrees of freedom, allowing fragment reorganization. This progressive loss of the contribution of the cement to the stability of the

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system leads to an overall softening, along with well defined shear plane failures and volumetric dilation (Menendez et al., 1996; David et al., 2001). The addition of cement tends to increase brittleness at failure, softening and dilatant behavior. As the confining pressure increases, the material shows a transition to a more ductile, compressive behavior, accompanied by a more diffuse cataclastic flow (Menendez et al., 1996; Schnaid et al., 2001; David et al., 2001; Clough and Nader, 1982; Huang and Airey, 1998; Lade and Overton, 1989; Abdulla and Kiousisr, 1997; Asghari et al., 2003). 2.2. CGM models Several constitutive models based on the elastoplastic framework have been proposed previously to represent the complex behavior of CGMs described above. In the last two decades several authors (Gens and Nova, 1993; Nova et al., 2003; Vatsala et al., 2001; Lagioia and Nova, 1995; Cecconi et al., 2002) have developed a number of constitutive models for CGMs based on the elastoplastic framework, successfully predicting the behavior of bonded geomaterials like calcarenite. In those models, the lack of correlations between the underlying microscopic degradation mechanisms and the only inelastic internal variable (plastic strain) results in the use of a wide set of parameters that are hard to identify physically, let alone to calibrate. The choice of physically meaningful internal variables has a major relevance when building a constitutive model, as discussed in Krajcinovic (1998): an internal variable inferred from the phenomenological evidence and selected to fit a particular stress strain-curve may provide a result that pleases the eye but seldom contributes to the understanding of the processes represented by the fitted curve. In recent times an increasing number of numerical models have been proposed that establish the overall material behavior by describing as much interaction as possible at the microstructural level. Outstanding results have been achieved through the use of Discrete Element Method (DEM, Estrada et al., 2010a, 2010b) and Lattice Element Method (LEM, Topin et al., 2007; Schlangen and Mier, 1992; Delenne et al., 2009), but, as acknowledged by the authors themselves, despite the remarkable insight those models provide, they are still unable to cover the full complexity of CGMs due to a lack of proper description of one or more of the cited microscale failure mechanisms. Inadequate description of micromechanisms can be somewhat exaggerated when integrated to predict macroscopic constitutive response. Perhaps the biggest limitation of such numerical methods is that the description of all the micromechanical effects is computationally unreasonable for real scale engineering problems, unless perhaps cast in a multi-scale framework (but even then will remain highly computationally expensive). 3. Internal variables The choice of appropriate internal variables has always been a determining part in the construction of constitutive models. To be useful, the internal variables should be identifiable, measurable and related to the dominant modes of irreversible rearrangements of the material microstucture (Krajcinovic, 1998). As shown in Menendez et al. (1996) the main inelastic processes involved at the grain scale in CGMs are grain crushing, cement disintegration and fragment reorganization. Given the different mechanical and geometrical properties, grains and cement will play a different role in the evolution of the microstructure. The use of a single variable is therefore an oversimplification. Several effective approaches have been proposed to describe the evolution of the capability of a low porosity material to transport momentum. In its simplest classical description, damage on a section given a unit normal can be seen as the ratio between the fractured (lost) area and the initial area of the undamaged section (Kachanov, 1986). Taking into account an uncemented granular system it is clear how this approach is less appropriate than the Breakage Mechanics one (Einav, 2007a, 2007b). As more extensively detailed in Einav (2007b), in fact, describing the grain crushing through the progressive loss of material integrity is questionable, as the cracks in the grains should rather be regarded as the evolution of a system to a less porous and stronger state through reorganization of the fragments. The addition of cement does not alter this feature of the granular phase, since even in the extreme case of undefinitely resilient cement as grain crushing proceeds the clusters constituted by bridged fragments will behave like pseudo-grains. As shown in Einav (2007b), tracking the evolution of the grain size distribution is an effective way to describe grain crushing. The focus in Einav (2007b) is on how the energy is being distributed among the grains, nominally with respect to their size. The process of grain crushing is then examined statistically. For example, individual grains with a similar size may crush at different times and release a different amount of energy. However, the overall energy release from a statistical volume element is affected by the integrated effects of all of these, while according to Einav (2007b) the evolution of the grain size distribution is controlled by how the energy scales with respect to the grain size. The role of the cement is inherently different from the one of the grains, as it endows the system with tensile resistance, reduces grain contact forces and enhances shear strength of the granular phase. Also, when cracks develop in the cement, its role in the system is redimensioned, in particular in lightly cemented granular material where its modest volume fraction renders the mechanical contribution of its fragments to the force network negligible to a first approximation. The ideal description for the evolution of the role of the cement seems to be therefore a progressive removal of the phase from the system. While continuum damage mechanics is in principle applicable, it seems unfit to capture the influence of particlescale disorder, Fig. 2a, which plays an important role. While in fact on a compact material the evolution of the cracking

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Fig. 2. Idealized cemented granular material: (a) schematic section of CGMs, highlighting the porous nature of CGMs, and therefore the lack of relevance of the classical continuum definitions of damage, (b) X-ray image of calcite cemented glass ballottini, model material developed in this project and well fitting the conceptual scheme.

system can attempt to describe the full complexity of the interacting and coalescing fractures, in CGMs the importance of pores and bare contacts would be in this framework disregarded. The proposed approach is therefore to take into account the evolution of the individual cement bridges connecting grains. Every time a bridge is partially cracked, we can in principle substitute it with another, with a smaller sectional area, but made of undamaged material, as sketched in Fig. 3. At this point it is possible to define an effective cement size distribution through which we can describe the evolution of damage. The experimental measurability and definition of the effective area are further detailed in Section 3.1.2. The rate of fragment reorganization can be captured effectively by the rate of plastic strain, as it is responsible for the unrecoverable macroscopic strain that lends itself as an ideal measure of this process.

3.1. Scalar variables The aim of the present work is the introduction of the simplest possible model capable of capturing the main features of CGMs, therefore scalar internal variables will be used to describe grain crushing and cement damage, disregarding, in first approximation, inherent and induced anisotropy of the failure processes.

3.1.1. Grain crushing A suitable scalar measure of the breakage can be related to the evolution of the grain size distribution (gsd), following the approach proposed in the Continuum Breakage Mechanics theory (Einav, 2007b, 2007a). While the initial gsd is routinely measured in geotechnical engineering materials (ASTM, 2007), the ultimate gsd is generally unknown, even if according to Turcotte (1986), Sammis et al. (1987), McDowell et al. (1996), and Ben-Nun and Einav (2010) it can be taken as fractal, with a fractal dimension α of around 2.5–3.0. The internal variable Breakage, B, can be introduced as the area ratio (Fig. 4a): B¼

Bt ; Bp

ð1Þ

where Bp is the breakage potential, that is the total area included between the initial and the final distribution and Bt is the area within the initial and current distribution, meaning amount of crushing that the grains have undergone (Einav, 2007b). g By integrating the probability density function pdf f ðxg ; BÞ along the grain size, xg, from the smallest dimension Xgm to the g biggest XM we get the cumulative probability density function cdpf F g ðxg ; BÞ. As shown in Einav (2007b), given the knowledge of the initial, current and final distributions F g0 ðxg ; BÞ, F g ðxg ; BÞ and F gu ðxg ; BÞ, Eq. (1) can be rewritten as R X gM B¼R

X gm

ðF g ðxg Þ  F g0 ðxg ÞÞ d logðxg Þ

X gM X gm

ðF gu ðxg Þ F g0 ðxg ÞÞ d logðxg Þ

;

ð2Þ

Assuming the commonly accepted simplifying hypothesis that this internal variable does not change along the grain size range (fractional independence of the Breakage variable) it is possible to deduce the current cdpf and pdf of the grain sizes F g ðxg ; BÞ ¼ F g0 ðxg Þð1  BÞ þ F gu ðxg ÞB:

ð3Þ

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Fig. 3. Sketch of the cement equivalence assumption.

Fig. 4. Scalar internal variables: (a) Breakage, B (Einav, 2007b), and (b) damage, D , where the meaning of effective cement size is detailed in Section 3.1.2.

and g

g

g

f ðxg ; BÞ ¼ f 0 ðxg Þð1 BÞ þ f u ðxg ÞB:

ð4Þ

The assumption of fractional independence of B will hold in first approximation also when cement is added, if the cementation is homogeneously distributed along the whole grain size range.

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In the current formulation no distinction is made between breakage induced by local compressive forces or by tensile fracture of interlocked asperities. While conceptually important, capturing this distinction would require additional state variables tracking the evolution of grain shape factors such as roughness and roundness.

3.1.2. Cement disintegration The cement size distribution may be evaluated through the analysis of X-ray tomography images or analogous techniques. The practical measurability is currently under study (Fig. 2b), but one possible approach is to measure the evolution of the minimum uncracked cross sectional area of the cement bridge, when the level of detail is enough, or, otherwise, to correlate damage with the evolution of the extension of the cement volume and its mean grey scale level (which indicates material density and thus porosity). The cement size distribution can also be deduced by the amount of cement added in artificially cemented materials or guessed starting from the tensile resistance of the material. The choice of the most physically meaningful equivalent cement size is a relevant experimental issue, similar to the one arising when defining the grain size, in particular for non-spherical particles. The grain size can in fact be associated with the maximum, the minimum, or with any other form of length dimension such as the diameter of a mass equivalent sphere. Similarly, the cement size can refer to the diameter of the equivalent circle of the minimum, the maximum or the averaged cross section. Each definition will correspond to a different measurable distribution and therefore to different values of the corresponding model parameters. A logical choice of the scalar Damage variable, D, in parallel with the scalar Breakage variable B, falls on the area ratio D ¼ Dt =Dp in Fig. 4b, that can be similarly evaluated from the cdpfs of cement F c0 ðxc ; DÞ, F c ðxc ; DÞ and F cu ðxc ; DÞ as R X cM Xc

D ¼ R Xmc

M

X cm

ðF c ðxc Þ  F c0 ðxc ÞÞ d logðxc Þ ðF cu ðxc Þ F g0 ðxc ÞÞ d logðxc Þ

;

ð5Þ

where the equivalent cement diameter xc ranges between Xcm and XcM. Again, by assuming the fractional independence of D, the current pdf and cpdf can be evaluated as F c ðxc ; DÞ ¼ F c0 ðxc Þð1 DÞ þ F cu ðxc ÞD; c

c

ð6Þ

c

f ðxc ; DÞ ¼ f 0 ðxc Þð1  DÞ þ f u ðxc ÞD:

ð7Þ

3.2. Statistical homogenization The well known influence of the evolution of the grain size distribution on the mechanical properties of the materials is commonly introduced within a continuum mechanics formulation through statistical homogenization. The effect of the evolution of the integrity of the cement bridges on the microscopic variables can be integrated analogously. If the specimen size is larger than the Representative Elementary Volume REV, it is possible to make use of the pdf as weighting average function of the microscopic variable, H, on the volume. Taking the pdf of grains and cement f ðI; xÞ, where I can be either B or D and x respectively xg or xc, as shown in (Eqs. (4) and 7), the statistically homogenized variable can be evaluated through Z XM HðxÞf s ðx; IÞ dx; ð8Þ H s  〈H〉s ¼ Xm

where s can stand for initial (0), ultimate (u), or current (no symbol).

4. Helmholtz free energy in CGMs Cemented granular materials can occur in nature by deposition/growth of cement on already settled grains or by solidification of a grain/cement paste. In both cases the cement phase can be seen as a component bridging the grains and modifying the mechanical properties of the granular phase, capable of storing additional elastic energy. The simplest form of interaction in terms of elastic potentials is the additivity of the internal stored energies of the cement and the granular phase

Ψ ¼ Ψ g φg þ Ψ c φc ;

ð9Þ

where φg and φc are the grain and cement volume fractions, respectively, and Ψ and Ψ the respective specific energies, where we note that φc þ φg ¼ 1  ϕ, while ϕ denotes the porosity of the material. g

c

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4.1. Helmholtz free energy stored in the granular phase The description of the Helmholtz Free Energy stored within a granular material has already been introduced in Continuum Breakage Mechanics (Einav, 2007b). Despite the existence of the cement bridges connecting the grains, as shown in Topin et al. (2007), the stress distribution, on which Breakage Mechanics relies, is maintained if φc is small, as expected in CGMs. It follows that the formulation developed in Einav (2007b) can be adopted in our problem. It is therefore possible to express the Helmholtz Free Energy through statistical homogenization via (Eqs. (4) and 8) as g

g

g

b 〉 ð1 BÞ þ 〈Ψ b 〉 B ¼ Ψ g ð1 BÞ þ Ψ g B; Ψ g  〈Ψb 〉 ¼ 〈Ψ u 0 u 0

ð10Þ

b g denotes the density of Helmholtz Free Energy of a grain of size xg. Again we are using the notation 0 for initial, u where Ψ for ultimate, and no symbol for current. Such an energy is expected to scale with the grain size xg so that it can be decomposed as g

b ðεe ; xg ; ϕ; …Þ ¼ ςg ðxg ÞΨ g ðεe ; ϕ; …Þ; Ψ ij ij r Ψ

ð11Þ

where ςΨ is the energy split function which distributes the stored energy among the grains according to their size x and Ψr is the energy density stored in a reference grain size xgr, which is a function of the elastic strain εeij , the porosity ϕ and other variables, as shown in Rubin and Einav (2011). The elastic strain of the grain phase can be considered at the given scale as equivalent to the cement strain and therefore to the total strain, as in a system in which cement and grain phases work in parallel. A system in series might also be proposed, by assuming that the stress in the two phases is equal. Although the latter may well describe the single grain to cement contact, it is important to remember that the stress is distributed in a rather more complex way due to the statistically irregular connectivity between the elements. The hypothesis of strain compatibility is therefore adopted here; this is also consistent with the previous developments of Breakage Mechanics. The aim of this paper is to integrate the concept of breakage and damage to describe CGMs in the simplest way. Therefore from now on the material dependence on porosity or other possible internal variables will be neglected. The energy split function has been proven (Einav, 2007b) to be well represented in crushable granular materials by a non-dimensional power function in the form  g n x ςgΨ ¼ g ; ð12Þ xr g

g

g

where n ¼2 for a system of spheres, which implies a linear energy scaling with the grain surface. This equation is based on the understanding of how stresses distributes in granular media: small particles have a smaller coordination number and therefore statistically they carry less load than the bigger ones. As aforementioned, such a structure is expected to be maintained when a small amount of cement connects the particles. As shown in Einav (2007b), it is possible to rewrite Eq. (11) as

Ψ g ¼ Ψ gr ðεeij Þð1  ϑg BÞ; where

ð13Þ

ϑ is the grading index: g

R X gM

g

g

f u ðxg Þðxg Þ2 dxg

M

f 0 ðxg Þðxg Þ2 dxg

ϑ ¼ 1  RXXmg g

X gm

g

;

ð14Þ

when the reference grain size is conveniently chosen as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z Xg M g g 2 g g g f 0 ðx Þðx Þ dx : xr ¼ X gm

ð15Þ

4.2. Helmholtz free energy stored in the cement phase As for the grains, it is possible to express the Helmholtz Free Energy stored in the cement through statistical homogenization as c

c

c

b 〉 ð1  DÞ þ 〈Ψ b 〉 D ¼ Ψ c ð1  DÞ þ Ψ c D: Ψ c  〈Ψb 〉 ¼ 〈Ψ u 0 0 u

ð16Þ

Again we can divide the energy into a part describing the influence of total strain plus damage and one that scales the energy within the different cement sizes: c

b ðεe ; xc Þ ¼ ςc ðxc ÞΨ c ðεe Þ: Ψ ij r ij Ψ

ð17Þ

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We can suppose that the energy split function can be described by a non-dimensional power law  c n x ςcΨ ¼ c ; xr

ð18Þ

where, again, for simplicity and in accordance with (15) we take the reference cement area xcr as ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z c xcr ¼

n

XM

X cm

c

f 0 ðxc Þðxc Þn dxc

ð19Þ

where a discussion on the value of the coefficient n is left to be determined. Analogous to the breakage variable, a grading index is introduced for the damage:   R X cM ðxc Þn c f u ðxc Þ c n dxc 0 ðxr Þ  ϑc ¼ 1  R c  : ð20Þ c n XM c c ðx Þ f ð x Þ dxc n 0 0 c ðxr Þ The cement grading index ϑ ¼ 1 if the ultimate distribution csd (for D¼ 1) is a uniform distribution about zero cement size, regardless of n, which implies a complete disintegration of the cement phase in the ultimate state. This complete disintegration is a reasonable ultimate distribution in several natural and artificial materials, as shown in Coop and Atkinson (1993), and allows the introduction of the simplest model. Further analyses will be developed once a reliable measure of the ultimate cement distribution is deduced from microscale studies. We can thus evaluate the Helmholtz Free Energy for the cement as c

Ψ c ¼ Ψ cr ðεij Þð1 DÞ:

ð21Þ

4.3. Total Helmholtz free energy Eq. (9) can now be expressed using (Eqs. (13) and 21) to give the total Helmholtz Free Energy for a unit volume of CGMs as

Ψ ¼ Ψ gr ðεeij Þð1  ϑg BÞφg þ Ψ cr ðεeij Þð1  DÞφc :

ð22Þ

Possible choices for Ψ r and Ψ r are explored in Section 6. g

c

5. Thermo mechanical constitutive modeling of CGMs A convenient form to represent the two laws of thermodynamics of rate independent materials in isothermal conditions is ~; ~ ¼ Ψ_ þ Φ W

~ Z0; Φ

ð23Þ

~ the increment of energy dissipation and W ~ the increment where Ψ_ is the rate of change of the Helmholtz Free Energy Ψ , Φ of mechanical work done on the RVE boundaries. The use of the tilde symbol ‘ ’ over W and Φ is deliberately different from the proper notation of the increment,‘’. This is to highlight that, while Ψ is a state function, that has a proper differential, W and Φ are not, and only their increments can be defined. The rate of work can be evaluated through ~ ¼ σ ij : ε_ ij ; W

ð24Þ

where σij and ε_ ij are the stress and strain rate tensors applied to the boundaries of the RVE. A convenient way to express the Helmholtz Free Energy of CGMs has been introduced in Section 6.1, from which it follows that Ψ_ can be evaluated as

Ψ_ ¼

∂Ψ _ ∂Ψ _ ∂Ψ e Bþ D þ e ε_ ij : ∂B ∂D ∂εij

ð25Þ

The inelastic processes in CGMs are, as previously described, grain crushing, cement damage and fragment reorganization. The rate of dissipation can therefore be seen as the sum of these three components: ~ ¼Φ ~ þΦ ~ þΦ ~ ; Φ B D P

~ Z0; Φ ~ Z 0; Φ ~ Z0: Φ B D P

ð26Þ

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As CGMs are rate independent materials the dissipations are considered to be first order homogeneous functions of the internal variables, such that it is possible to evaluate the rate of dissipation as ~ ~ ~ ~ ~ ¼ ∂Φ B_ þ ∂Φ D _ þ ∂Φ ε_ P ¼ EB B_ þ ED D _ þ ∂Φ ε_ P ; Φ ij P ij _ _ ∂B ∂D ∂ε_ ∂ε_ P ij

ð27Þ

ij

where we designate the breakage energy EB and the damage energy ED as EB ¼

~ ∂Φ ; ∂B_

ED ¼

~ ∂Φ : ∂D_

ð28Þ

The physical meaning of those variables will be presented in the following. We note that, according to Eq. (26), EB and ED will be positive, for B_ Z 0 and D_ Z 0, respectively. Combining (Eqs. (23)–25), and (27) we obtain ! !     ~ ∂Ψ ∂Φ ∂Ψ _ ∂Ψ _ e _ B  ED þ D ¼ 0: ð29Þ σ ij  e ε ij þ σ ij  P ε_ Pij  EB þ ∂εij ∂B ∂D ∂ε_ ij Employing a standard step, stress is identified as conjugated to the elastic strain (Ziegler, 1977)

σ ij ¼

∂Ψ ∂Ψ g ∂Ψ c ¼ φ þ e φ ¼ σ gij φg þ σ cij φc ∂εeij ∂εeij ∂εij g

c

ð30Þ

where we introduced σ gij and σ cij for grain and cement stresses, which represent the fractions of the specific stress respectively in the grains and cement. Similarly, from Eq. (29), we can also deduce

σ ij ¼

~ ∂Φ ; ∂ε_ Pij

ð31aÞ

EB ¼ 

∂Ψ g g ¼ Ψ 0 ðεeij ; BÞ  Ψ u ðεeij ; BÞ; ∂B

ð31bÞ

ED ¼ 

∂Ψ c c ¼ Ψ 0 ðεeij ; DÞ  Ψ u ðεeij ; DÞ: ∂D

ð31cÞ

(Eqs. (31b) and 31c) provide the physical meaning of EB and ED as the energies necessary to evolve, respectively, the grain and cement size distributions from the initial to their ultimate distributions. 5.1. Coupled model As shown in Einav (2007b), models where the evolution of the inelastic processes is coupled are not only mathematically easier to handle, but also more physically meaningful, as they can describe how the processes influence and enhance each other. The proposed mathematical form for coupling of the rearrangement, damage and breakage is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ ðΦ ~ n Þ2 þðΦ ~ n Þ2 þ ðΦ ~ n Þ2 þ ðΦ ~ n Þ2 ; ð32Þ Φ Pv Ps B D ~ ;Φ ~ ; and Φ ~ is added to highlight the conceptual difference of those terms from the where the superimposed “n” in Φ P B D ones previously introduced. In Eq. (32) we have also divided the square of the dissipation due to irreversible rearrangement ~ n Þ2 ¼ ðΦ ~ n Þ2 þðΦ ~ n Þ2 . We can therefore rewrite Eq. (31) as into its volumetric and shear components: ðΦ P Pv Ps n

n

p¼

~ ∂Φ ~ ~ ∂Φ Φ Pv ¼ Pv ; ~ ∂ε_ P ∂ε_ Pv Φ v

ð33aÞ

q¼

~ n ∂Φ ~ ~n ∂Φ Φ Ps Ps ¼ ; ~ ∂ε_ P ∂ε_ Ps Φ s

ð33bÞ

EB ¼

~ n ∂Φ ~ ~n ∂Φ Φ B ¼ B ; ~ ∂B_ ∂B_ Φ

ð33cÞ

ED ¼

~ n ∂Φ ~ ~n ∂Φ Φ D ¼ D ; _ ~ ∂D_ ∂D Φ

ð33dÞ

where p and q are the respectively the mean and deviatoric stress components. By combining (Eqs. (32) and 33) we can obtain the following yield surface in the generalized stress space: !2 !2 !2 !2 ED EB p q n y ¼ þ þ þ  1 r 0: ð34Þ ~ n =∂ε_ P ~ n =∂ε_ P ~ n =∂D ~ n =∂B_ _ ∂Φ ∂Φ ∂Φ ∂Φ D

B

P

v

P

s

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By adopting flow rules associated with the yield surface in the generalized stress space we obtain flow rules uncoupled to the yield surface in pure stress space.

6. Thermo mechanical constitutive model for CGMs 6.1. Total Helmholtz free energy To complete the description of the elastic potentials of our model, it is now necessary to define the stored energy for the reference size. The elastic energy is here divided into volumetric and shear components Ψv and Ψs as in Einav (2007c). Experimental evidence shows that the bulk modulus in CGMs is pressure dependent (Airey, 1993; Huang and Airey, 1998). Therefore, as in the classical Breakage Mechanics (Einav, 2007c), the following potential is adopted: ðΨ r Þv ¼ pr

2 A3 ; 3K

ð35Þ

where A¼

! K e ε þ1 ; 2 v

ð36Þ

where K is the non-dimensional material constant representing the bulk stiffness, and pr the reference pressure, conveniently taken to be equal to 1 kPa. In Schnaid et al. (2001) the reduction of the pressure dependency of the shear modulus after the addition of cement is reported based on experiments. In Airey (1993), Alvaraldo et al. (2012), and Coop and Willson (2003) the pressure independence of the apparent shear stiffness inside the elastic field is shown through mechanical tests on several CGMs. Such a difference in the pressure dependence of the elastic properties may be attributed to different mechanisms of stress transmissions. The isotropic stress will still be carried through force chain-like structures, as in non-cemented granular material. On the other hand the shear stress is no longer transmitted only thanks to particle jamming and friction but also through cement bridges, as in a compact material. Such an interpretation implicitly assumes cemented contacts, so a transition towards a pressure dependent elastic shear potential with the evolution of damage might be auspicable future refinements. As a first approximation, the same potential is adopted for the two phases such that 3 ðΨ r Þs ¼ Gε2s ; ð37Þ 2 where G is the shear stiffness. The stiffness ratio sr between the two shear moduli is assumed to be equal to the bulk moduli c g ratio: sr ¼ K =K ¼ Gc =Gg . From the above the total Helmholtz Free Energy is given as

Ψ ¼ Ψ v þ Ψ s;

ð38Þ

where

Ψ v ¼ φg pr 3 2

2 ðAg Þ3  2 ðAc Þ3 g  1  ϑ B þ φc pr ð1 DÞ; g 3 K 3 Kc 



3 2

Ψ s ¼ φg Gg ε2s 1  ϑg B þ φc Gc ε2s ð1 DÞ:

ð39aÞ

ð39bÞ

From Eq. (31), it follows that p¼

 ∂Ψ g  ¼ φg pr ðAg Þ2 1  ϑ B þ φc pr ðAc Þ2 ð1 DÞ ¼ φc pc þ φg pg ; ∂εev

ð40aÞ

q¼

 ∂Ψ g  ¼ φg 3Gg εes 1  ϑ B þ φc 3Gc εes ð1  DÞ ¼ φc qc þ φg qg ; ∂εes

ð40bÞ

EB ¼ 

  ∂Ψ pr g 3 2 g g e 23 þ G ; ¼ ϑ φg ðA Þ ð ε Þ g s 3 2 ∂B K

ð40cÞ

ED ¼ 

  ∂Ψ pr c 3 2 3 þ Gc ðεes Þ2 : ¼ φc c ðA Þ 3 2 ∂D K

ð40dÞ

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291

Fig. 5. Schematic representations of the physical meanings of the postulated dissipation criteria for (a) Breakage (Einav, 2007b) and (b) Damage. We follow Einav (2007b) and use EnB as the residual breakage energy, and introduce an equivalent EnD as a residual damage energy.

6.2. Dissipation As motivated in Einav (2007b) and shown in Fig. 5a, the rate of change of the residual breakage energy can be compared to rate of breakage dissipation, and thus it is possible to derive the following dissipation for breakage: pffiffiffiffiffiffiffiffiffiffiffiffi EB EBC B_ ; ð41aÞ ΦnB ¼ ð1  BÞ cos ðωB Þ where EBC is the critical breakage energy, and ωB is the coupling angle between the grain crushing and the volumetric plastic dissipation, as in Einav (2007a). This parameter, as shown in Nguyen and Einav (2009) and Nguyen et al. (2012), separates the energy dissipated in a breakage event through friction after the plastic reorganization of the fragments, from the energy dissipated by the elastic energy release and atomic debonding. As in Einav (2007a) the volumetric plastic dissipation is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p EBC =EB ΦnPv ¼ ε_ p : ð41bÞ ð1  BÞ sin ðωB Þ v The plastic volumetric dissipation is here associated only with the evolution of grain crushing and not with the cement damage. While a release of degrees of freedom in the system is associated with an increment of breakage δB, after an increment of damage δD only a few cement bridges are actually removed and several of the cemented contacts only see their equivalent section reduced. Even when a cement bridge is completely eliminated, the volumetric plastic dissipation associated with such event is mostly negligible when compared to the breakage one. From the assumption that the energy dissipated through damage coincides with the reduction of the residual elastic stored energy in the cement phase, as shown in Fig. 5b it follows, in analogy with breakage, that pffiffiffiffiffiffiffiffiffiffiffiffiffi ED EDC _ ΦnD ¼ D; ð41cÞ ð1  DÞ where EDC is the critical damage energy. We assume a Coulomb-type increment for the rate of shear plastic dissipation for cohesive frictional materials, where the cohesion c is assumed to be reduced as the damage proceeds:

ΦnPs ¼ ðMp þ cð1  DÞÞε_ Ps :

ð41dÞ

From Eq. (34) the flow rules can therefore be expressed as ∂yn ð1 BÞ2 EB cos 2 ðωB Þ B_ ¼ λ_ ¼ 2λ_ ; EB EBC ∂EB

ð42aÞ

2 n _ ¼ λ_ ∂y ¼ 2λ_ ð1  DÞ ED ; D ∂ED ED EDC

ð42bÞ

ε_ Pv ¼ λ_

∂yn EB ð1  BÞ2 sin 2 ðωB Þ ¼ 2λ_ ; pEBC ∂p

ð42cÞ

ε_ Ps ¼ λ_

∂yn q ¼ 2λ_ : ∂q ðMp þ cð1  DÞÞ2

ð42dÞ

From Eq. (41) the yield surface can thus be rewritten as

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y¼

EB ð1  BÞ2 ED ð1 DÞ2 q2 þ þ  1 r 0: EBC EDC ðMp þcð1 DÞÞ2

ð43Þ

The derivation of the yield surface in pure stress space is provided in Appendix A.

7. Parametric study In this section we examine the constitutive model using a synthetic sensitivity analysis to clarify the physical meaning of the newly introduced parameters and to make comparisons with Breakage Mechanics theory and its variables. From (Eqs. (42a) and 42b) it is possible to deduce that D ¼ 1

ð1  BÞΛ ; 1  ð1  BÞð1  ΛÞ

ð44Þ

EDC : EBC

ð45Þ

where

Λ ¼ cos 2 ðωB Þ

Eq. (44) shows how the two internal variables B and D are related. The relative magnitude of B and D is governed by the coefficient Λ, as shown in Fig. 6. Λ is a measure of the tendency of the granular phase to crush. A high value of Λ corresponds to a low critical breakage energy, and a low coupling angle with volumetric plastic strain rate, i.e. a higher rate of breakage for a given imposed strain rate (Einav, 2007b). Imposing Λ ¼ 1=16, we can partially isolate the effect of the damage variable D on the yield surface. As shown in Fig. 7 the evolution of damage induces softening at low pressure regimes, without significantly affecting the yield surface at high confinements. The increase of the critical pressure, yield limit in isotropic compression tests, is solely due to the small increment of B. As expected the system will eventually collapse onto the Mohr–Coulomb failure line once breakage and damage are fully developed (B ¼1, and D ¼1).

Fig. 6. Relative evolution of the internal variables for varying Λ.

Fig. 7. Influence of Damage on the yield surface, with Λ ¼ 1/16.

A. Tengattini et al. / J. Mech. Phys. Solids 70 (2014) 281–296

293

Fig. 8. Influence of Breakage on the yield surface, with Λ ¼ 16.

Fig. 9. Expansion of the yield surface due to cementation: (a) volume fraction φc and (b) cohesion c.

Fig. 10. Effect of cohesion c on k0 consolidation responses: (a) volumetric stress strain curve and (b) stress paths.

For Λ ¼ 16, we can observe how a process dominated by B induces hardening and a nonlinear increment of the critical pressure, and how it does not affect significantly the low pressure regime (Fig. 8). As is well known, the addition of cement results in an expansion of the yield surface. In the present model such an effect can be described through an increase in the cement fraction φc and the cohesion c independently, as shown in Fig. 9. The addition of cement is also known to increase the brittleness of the system. As shown in the k0 consolidation tests reported in Fig. 10 the increased cohesion leads to enhanced material resistance but also increased brittleness. As shown in Fig. 10a the ultimate behavior tends to converge, implying a critical state not affected by the degree of cementation, as suggested by experimental evidence (Coop and Atkinson, 1993; Huang and Airey, 1998; Airey, 1993). An important experimental observation in CGMs is the brittle–ductile transition, with the increase of confining pressure. As shown in Eq. (44), the relation Breakage/Damage is independent of the loading path. Given the physical meaning of the internal variables, the probability of a fracture event in the two phases may be dominated by their fracture properties rather than by the confining pressure. With the level of confinement, though, the effect that an increment of Breakage or Damage

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Fig. 11. Sketch of constant p tests: stage (a) follows isotropic compression to various confinements, while stage (b) follows the shear of the material at constant confinement. Note that the tests with p ¼3500 kPa lies beyond the critical pressure limit.

Fig. 12. Effects of the confinement p in constant pressure tests: (a) deviatoric stress–strain curve and (b) volumetric vs axial strain curve.

Fig. 13. Effects of k ¼ ECD =ECB in isotropic compression tests: (a) volumetric stress–strain curve and (b) evolution of the internal variables B, D with pressure.

c

g

Fig. 14. Effects of sr ¼ K =K ¼ Gc =Gg in isotropic compression tests: (a) volumetric stress–strain curve and (b) evolution of the internal variables B, D with pressure.

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295

has on the system is different. In the model the brittle–ductile transition is explained through the different influences that B and D have on the rate of plastic strain (the fragment reorganization), which is pressure dependent. Constant effective pressure tests at different confining pressures, as sketched in Fig. 11, demonstrate the capability of the model to reproduce this transition, as shown in Fig. 12. Associated with the brittle behavior, CGMs show a dilatant behavior that can be captured through the evolution of porosity as shown in Rubin and Einav (2011). To maintain the simplicity of the introduced model such an extension of the formulation will be analyzed in a forthcoming paper. The effects of the relative magnitude of the critical energies of the two phases on the yield point and the inelastic volumetric response in isotropic compression is shown in Fig. 13. As expected the ratio k influences the critical pressure and the inelastic domain, but does not affect the elastic response. The choice of the coupled dissipation is evident in Fig. 13b, where it is possible to observe how the internal variables B and D start evolving at the same pressures but with a relative magnitude determined by the critical energy ratio. So, even if both the processes start contemporaneously, for contrasted critical energies one process will dominate over the other. The stiffness ratio sr influences not only the overall stiffness, but also the critical pressure, as shown in the isotropic compression test reported in Fig. 14. Unlike the critical energy ratio, though, sr does not affect the relative evolution of the internal variables. The effect of the grading index ϑg has already been discussed in Einav (2007a) (it measures distance of the current grain size distribution in the system from the ultimate distribution). The coupling angle ωB , whose effect on the system is thoroughly explained in Einav (2007a), Nguyen and Einav (2009), and Das et al. (2011), can be regarded, to a first approximation, as a measure of the collapsibility of the material. 8. Conclusions A novel constitutive model for cemented granular material has been proposed in this paper. The unique choice of the internal variables allows us to track the evolution of the material structure along the loading path by connecting the microscopic processes to the macroscopic constitutive response through statistical homogenization. This perspective implies a clear physical meaning to the parameters, and thus reduces their number, which will facilitate their evaluation. The model introduced is based on typical physical observations for CGMs where the amount and stiffness of the cement maintains the granular nature of the material. Unlike materials with very high amounts of cement, those characterized by light cementation can be appropriately described within the current model as the formulation collapses into the classical Breakage Mechanics theory as φc approaches zero. The theoretical limits of the presented model are particle reinforced materials, where the physics is governed by different phenomena and the model loses its physical meaning. To include their behavior, the model would have to involve the adjustment of the energy scaling features and possibly include different failure mechanisms (Wong and Baud, 2012) that are beyond the scope of the current paper. The introduced model requires only 8 mechanical parameters and 3 geometrical indexes, all of which have a precisely defined physical meaning. This should be compared with the 13 mechanical parameters of well established models such as Lagioia and Nova (1995). It is important to distinguish the geometrical indexes from mechanical parameters because their determination does not require mechanical tests. Each of the mechanical parameters can be determined through classical geotechnical tests and does not require the wide range of tests necessary to determine a yield surface.

Acknowledgments Arghya Das and Alessandro Tengattini wish to thank the University of Sydney International Scholarship scheme. Giang Nguyen and Itai Einav would like to acknowledge the Australian Research Council for the Discovery Projects funding scheme (Projects DP110102645 and DP120104926). Appendix A. Yield surface in pure stress space To express the yield surface in pure stress space it is possible to deduce the following set of relations from Eq. (40): ! 2 ðAg Þ3 3 g q2 g G þ E B ¼ ϑ φg ; ð46Þ 3 Kg 2 L E D ¼ φc

! 2 ðAc Þ3 3 c q2 G ; þ 3 Kc 2 L

ð47Þ

where L can be evaluated as L ¼ φg ð1  ϑ BÞ3Gg þ φc ð1  DÞ3Gc ; g

and the relation between Ag and Ac and

ð48Þ

ε in Eq. (36) can be expressed in terms of p–q as

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εev ¼

 bþ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b  4ac ; 2a

ð49Þ

where g

c

 ðK Þ2 ðK Þ2 g  þ φc ð1  DÞpr ; a ¼ φg 1  ϑ B pr 4 4 g

c

ð50aÞ

b ¼ φg ð1  ϑ BÞpr K þ φc ð1  DÞpr K ;

ð50bÞ

c ¼ φg ð1  ϑ BÞpr þ φc ð1  DÞpr  p:

ð50cÞ

g

g

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