- Email: [email protected]
Micromechanics of granular materials -numerical simulation of the effects of heterogeneities
Int. J. Rock Mech. Min. ~ci. & Geomech. Abstr. Vol.30, No.7, pp. 1281-1284, 1993
0148-9062/93 $6.00 + 0.00 Pergamon Press Ltd
Printed in Great Britain
Micromechanics of Granular Materials -Numerical Simulation of the Effects of Heterogeneities ZHONG LIU*# LARRY R. MYER# NEVILLE G.W. COOK*#
INTRODUCTION The principal objective of this research is to gain an understanding of the relation between grain stresses and microfmcturing and macroscopic deformation and fracture in heterogeneous granular aggregates. The generation and propagation of microcracks in relation to the loading conditions, grain sizes and shapes is essential for understanding the stress-strain relationships and quasibrittle fracture of granular materials under differential compression. Granular materials, typically comprising a great n u m b e r of particles or grains, have a correspondingly large number of degrees of freedom. The complex interaction of individual micro-features of these materials, at some stage, results in macro-failure. At the macro-scale, the entire system exhibits well-defined deformation under uniform external loading. This type of behavior is idealized in the notion of stress-strain relations in continuum theories. From a physical point of view the stress-strain relations under uniform external stresses are a statistical average resulting from the almost independent behavior of different microscopic parts of the large system. In other words, the macro-behavior represents the statistics of the microscopic components of the granular material. Experiments have shown that local stresses in individual grains are important in the overall response of granular materials[Ill2][3]. At a microscopic level, local stresses are related to: 1) tensile strength of grains; 2) grain geometry; 3) pores and; 4) grain contacts. How can we quantify the micro-structure and the distribution of stresses? Once quantified, how are these parameters related to the macro-mechanical behavior? Which parameter is most important or, in other words, dominant in defining the macro-mechanical properties such as the stre,~s-strain relationship and failure of materials? Micro-mechanical models have been used to study macroscopic properties in terms of microscopic processes. Some of these models are based on a continuum containing micro-cracks[4] [5] [6] while others are based on a granular or discrete element approach[7][8]. In this paper we will use a two-dimensional numerical model based on the elastic interaction and tensile fracture of grains and grain bonds to study the effects of heterogeneity in grain strengths on macroscopic strainhardening and strain-softening deformation, and fracture *Department of Materials Science and Mineral Engineering, #Earth Sciences Division, Lawrence Berkeley Laboratory, University of California at Berkeley, Berkeley, CA94720. 1281
under differential distributions within combined Boundary Minimization Scheme section.
compressive stresses. Stress grains are calculated using a Element Method(BEM)-Energy which is outlined in the following
NUMERICAL M E T H O D O L O G Y Local stresses in individual grains are important in the overall response of granular materials and fracture propagation. We need to know in some detail information about the stress distributions in grains. Solutions for idealized bodies, like discs or spheres loaded diametrically, are presented in most texts on the theory of elasticity[9]. The exact solution for a single-line load on a cylinder was obtained by Poritsky[10]. In soil mechanics, individual grains are assumed to be unbreakable. The deformations most frequently treated are in the form of either strain resulting from elastic deformations of grains with spherical[l 1], cubical, and other shapes, or from sliding of grains[12]. Some of these situations have been investigated with the aid of photoelasticity[13]. They are, however, not adequate for a quantitative study of the micromechanical interactions that produce complex macroscopic deformation. The system of granular materials is too complicated to use idealized solutions. Numerical methods can be used to calculate the stresses in grains of the granular materials. In our research, the BEM (direct boundary integral method) is used to calculate the stress distribution within each grain. In the BEM, boundary stresses and displacements are chosen as variables. For every grain in the system, the boundary is divided into n segments, so there are 2n unknowns: ~ , c~, u~, uJn, with j = 1, 2 . . . . . n. c~, u~ are the normal stress and the normal displacement of segment j. c~, u~ are the tangential stress and displacement of segment j. By using the integral representation: u = ~c(GO'fi-fiEu)ds
(1)
and choosing 2n test solutions[14], we can get an equation B{t~B} = A{uB} , (2) which can be used to calculate the unknown stresses and displacements along the boundary. B and A in equation (2) are matrices of influence coefficients. Matrices B and A are dependent only on the geometry of the grain boundary. If two grains have the same shape with different boundary stresses and displacements, there is no change
1282
R O C K M E C H A N I C S IN T H E 1990s
"£
11 21
21
22
B~ B~ fi
B=
2n
22
B~
BsII • • •
n2 Bsn
"~f
B~
I~
Bu
11
11
12
(L
A =
B,.
12
A~
A~
21
21
A~
22
11
11
12
ntln ......
nl
"n2
A~
A~
A~
/ 2 2 V z 2n ,-
B~
in
In
22
2n
n'2" . . . . . .
2n
L In
where
In
(4)
A~ A~
... n2
A~, A~ for either the B or the A matrix. {os} is a stress vector along the boundary and {us} is a displacement vector along the boundary: o 2
Alln ......
.-ot
eo
..,
... ..,
u:
(6)
For granular materials, the detailed stress distributions within grains need to be computed to investigate the mechanisms of nonelastic behavior and microfracture. Granular materials typically comprise a great number of grains. Every grain is divided into hundreds of segments. There are hundreds of thousands of variables in a system. We can not directly use the BEM to deal with this problem. The number of variables must be reduced. In the granular material system, external energy is •input into grains through contacts. Stress distribution within grains can be described in terms of contact stresses and displacements along the boundary. Based on this, for every grain, only contact stresses and contact displacements are chosen as variables. Assuming there are m contact points in a grain, because all the stresses of free elements along the boundary are zero, {oB } in equation (2) can be rearranged as: {oB}' = (oI ~ o~° .... o,me o ~ 0 ... 0) r , (7) in which, all the non-zero stresses are put together. Then {%}' can be divided into two sub-vectors: contact stresses{oBc} and {0} (representing free boundaries), where {OBC}lx2m
----(0
lc
O l nc
0 2c
...
O sme
Onme) r and
{0}lx~2~_2m) = (0 ,.- 0)T" Corresponding to the new order of the stress vector (on}', the displacement vector{uu} is rearranged as two sub-vectors: contact displacement vector {uBc} and displacement vector {ueF} of free stress elements. Corresponding to the new order of the stress vector and the displacement vector, matrices A and B in equation (2) can also be rearranged as: A' =
A:~2m
2mx2(n-m)
(8)
21
• A2(n-m)x2m A2(n-m)x2(n-m)J B2mx2m B" =
mx2Cn-m)
21
~B2(n-m)x2m B2(n-m)x~n-m)J
•
/ 2 2 V z 21
Bc{°Bc}=Ac{ uBc} ,
A_~ A~
Ash ...... 12
-,
(12)
Substitutingcq.12intocq. I0, yields
A~ A~
A~ . . . . . .
(11)
From eq. (11),
{us,}=tA ~ BIOBc~-tA ) A{UBc}
B~
Anti ......
nl
(3)
..... , nn ml
Ass A~ . . . . . .
"~f "fi" "d2" Ass A~ A~
t{ {O~c}= ~ {unc}+~ {uBv} .
Im
•.. B~ B~ In In B~ B~
12
12
2n
B~ B=
... n2
n2
11
A~
...
nll oo
B~ l ~ "~: "if B~ B ~
From equation (2), (8) and (9), we get
111
12
12
(9)
(13)
() (
Bc: B -~ ~ B (14) llx\ 12 122\ -l and Ac = ~A/- A~AJ (15) In our simulation, both stresses and displacements of contact elements can be chosen as the system variables. Using eq. (13), the strain energy of the entire system of grains is calculated. Following the Finite Element Method, we use the principle of minimum energy and eq. (13) to obtain a global stiffness matrix for the particle system, in which only displacements or stresses of those contact elements are variables. After the global system of equations is solved for the stresses and displacements of the contact elements, the stress distribution within the grains of granular materials can be calculated from eq. (1). S I M U L A T I O N S OF G R A I N F A I L U R E
Using the numerical formulism described above, we have used rectangular and hexagonal packing of cylindrical grains of the same size to study the effects of stochastic distributions of grain strength, uniform debonding strengths and confining stresses on the ultimate strength, fracture and macroscopic deformation. The system consists of about 300 grains with the boundary of each grain divided into 100 elements. Within each grain, stresses at 50 points are calculated to find the maximum stress within the grain. With load increasing, the maximum stress point is searched through the whole system. When the maximum stress reaches the tensile strength of the grain, the grain is broken, and its Yotmg's modulus is reduced 25% for the next step of the calculation. If contact stresses at interfaces are tensile stresses and reach debonding strengths, these interfaces are broken and contact elements of these interfaces are traction free. If two grains contact each other during the loading, matrices A and B of these two grains must be recalculated due to the changed contact situation. Loads applied to the boundaries of aggregates are transmitted across grain contacts and through grains. Loads applied at one boundary are transmitted from grain to grain to the other boundaries where, for static equilibritan, equal and opposite forces must applied. From our calculation, it is found that stress concenlrations arise at grain contacts in granular systems because external forces applied at the system boundary are trat~itted f3xan grain to grain over small contact areas. The inteasity of a stress concentration at a contact depends prima~ y on the size of the corresponding contact area, m i c t ~ s l r u ~ e s local to the area near that grain, and the orientation of the normal to that area relative to boundary loads. The stress distribution in the whole particle system is calculated. At the beginning, when no grain fracture has
ROCK MECHANICS IN THE 1990s occurred in the system, there are no big changes in the maximum principal tensile stresses in grains. When grain fracture occurs, a macro-fracture begins to propagate in the system and a stress concentraton develops near the tip of this macro-fracture. This stress concentration along with the tensile strength distribution in grains strongly determines the occurrence of the next microcrack. From our calculation, it is also found that, as each grain or bond fractures, the effective macroscopic elastic modulus of the material decreases and the stress at which the next fracture occurs changes. The effective Young's moduli and limiting stresses are shown below in Figures 1 and 2 for homogeneous and heterogeneous grain strength distributions. Stresses are normalized with respect to the mean tensile strength of the grains and a hexagonal grain packing is assumed. The envelopes of these lines constitute the stress-strain curves.
Fig. 4. tteterogeneous grain strength distributions result in both macroscopic strain-hardening and strain-softening as shown in Fig. 2, as well as a more diffuse distribution of grain failures as shown in Fig. 5. The ultimate compressive strength of granular materials is dependent on the tensile strength distribution in the grains of the material. The compressive peak strength of granular materials, in which all grains have the same tensile strength is the largest, as shown in Fig. 3. It is clear that, with the range of tensile grain strength becoming larger, the compressive peak strength decreases.
o
.,--5
4 "5 ~ 3
5
[
i
]
--
i
i
F
Uniform Strength.
4
Intrinsic /
1283
~2 "~ ~.~
J
0
o
. . . . . . . . . . . .
1.0
0.'/5-1
0.5-1
I ....
0.25-1
t
.
.
.
.
0.0-1
Range of Grain Tensile Strength. (minimum to maximum tensile strength) Figure 3. Effects of the range of tensile strengths in grains on the ultimate compressive strengths of granular materials. i
0
i
[
i
0.002 0.004 0.006 0.008 0,01 0.012 Strain,
Figure 1. Stress-strain curves show decreasing effective Young's moduli with increasing grain and bond damage and strain softening for homogeneous strength distribution.
Stochastic strength distribution. d
z
1.5 Intrinsic M°du'~///~
o.s
0
0.001 0.002 0.003 0.004 0.005 0.006 Strain.
Figure 2. Stress-strain curves show both strain-hardening and strain-softening with increasing grain and bond damage for stochastic strength distribution. From Fig. 1, it is clear that a homogeneous strength distribution results in only macroscopic strain-softening and intense localization of grain fractures as shown in
For uniform grain and bond strengths, the deformation is extremely brittle and failure occurs by localization and propagation of grain fractures along a diagonal spanning the whole network as shown in Fig. 4, in which the stress concentration near the fracture tip controls the propagation of grain fractures. For a heterogeneous distribution of grain strengths in the system, at low-confining pressures grain failures tend to be distributed throughout the sample and propagation of grain fracture tends to be along the loading direction as shown in Fig. 5. As the microcracks propagate and interact, a macrofracture or failure surface is created in the granular material. The effects of packing arrangement on the directions of fracture surfaces are obvious. For a hexagonal structure with uniform tensile strength, the angle between the fracture surface and vertical external load axis is about 30 degrees as shown in Fig. 4. The angle is about 45 degrees for a cubic structure with uniform tensile strength. The peak compressive strength of a hexagonal structure is larger than that of a cubic structure. CONCLUSIONS From the above discussion, the following conclusions can be drawn: When microcracks occur in granular materials they generate tensile stress concentrations in adjacent grains near the crack tip, as would occur in a continuum. These stress concentrations and the stochastic tensile strength distribution drive the propagation of macrofractures in hexagonal and cubic lattices.
1284
ROCK MECHANICS IN THE 1990s whose direction tends to follow the lattice, that is, the plane of the fracture lies at 45 0 or 300 to the direction of the ~ ~ load for cubic o r hexagonal ~ . With large stochastic strength distributions, under lowconfining pressures, the angles between the ~ t boundary stress and the ~ f r a c t m ~ surface are near zero degree.
Granular materials with homogeneous or small
stochastic tensile grain strength distributions have only macroscopic strain-softening. Granular materials with large heterogeneous tensile grain strength distribution have both macroscopic strain-hardening and strainsoftening. As the range of tensile strengths in grains becomes larger, the peak ¢t~wessive strength ~ . Acknowledgmenls--Tifis work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Engineering and Geosciences Division, of the US Department of Energy under Contract No DE-ACO3-76SF(g~8.
Figure 4. Interaction between failed grains results m localization of a macroscopic shear fracture and strain-softening deformation for hexagonal lattice. tlLt0t~l
Figure 5. Initial failures of grains with a stochastic distribution of strengths produce a macroscopic extensile fracture and both strain-hardening and strain-softening deformation.
Macrofracmres are affected by boundary loads, the packing arrangement, and the range of tensile strengths in grains. I f the range of tensile strength distributions is small, that is the grains have uniform tensile strength, grain fracture occurs locally as shown in F]gure 4. If the range of tensile strengths in grains is large, grain fracture occurs randomly over the system, Stress concentrations due to these fractured grains do not hav~ a big effect in determining which grain is broken next, and the ten-Ale strength distribution is dominant ia determining microfracture propagation at the early age. For uniform grain strengths, stress concentration due to fractured grains results in macroscopic shear ftacaa~s
REFERENCES 1. Hallbauer D.K, Wagner H, and Cook N.G.W., Some observations concerning the microscopic and mechanical behavior of quartzite specimens in stiff, ttiaxial compression tests, lnt g Rock Mech. Min. Sci., 10, 713-726 (1973). 2. Zhang J, Wong TF, and Davis DM, Micromechanics of pressure-induced grain crushing in porous rocks, J. Geophys. Res., 95, 341-352 (1990). 3. Myer LR, Kemeny JM, Zheng Z, Suarez R, Ewy RT, and Cook NOW, Extensile ¢rackin~ in porous rock under differential compressive stress, AppL MecK Rev. 45, ng, 263-280(1992). 4. Costin LS, A microcrack model for the deformation and failure of brittle rock, J. Geophys. Res., 88, 9485-9492 (1983). 5. Sammis CG and Ashby MF, The failure of brittle porous solids ander compressive stress states, Acta Metall., 34, 511- 526 (1986). 6. Kemeny JM and Cook NGW, MicTomechanics of deformation in rocks, in Toughening Mechanisms in QuasiBrittle Materials, SP Shaw(ed), Klewer Academic, The Netherlands, 155-188(1991). 7. Cundall, P.A., Distinct element models of rock and soil structure, in Analytical and Computational Methods in Engineering Rock Mechanics, E.T. Brown(ed), Allen & Unwin, London. 129-163(1986). 8. Cundall, P.A and Strack, O.D.L, A discrete numerical model for granular assemblies. Geotechnique, 29, 47-65(1979). 9. Timoshenko, S. P. and Gooclier, J. N., Theory of Elasticity, McGraw-Hill. New York. (1970). 10. Poritsky, H., Stresses and deflections of cylindrical bodies m contsct with application to contact of gears and locomotive wheels. J. Appl. Mech.. 17, 191-201(1950). 1I. Mindtin, R.D. Mechanics of granular media. Proc. U.S. Natl. Congr. Applied MecK by Naghdi, P.M., 2rid, Ann Arbor. Mich.. 13-20(1954). 12. Rowe, P.W., The st~ess-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. R. Soc. London, Set. A, 269, 500-527(I962). 13. ~llagber, J.J., Sowers, G..I~. ~ F'riQdl21all,M., Photomechanical Model Studies Relating to F,~cture in Granulax Rock Aggregates. GeoL Soc. Am., Abstr., 2, 285(1970). 14. S. L. Crouch and A. M. Starfield, Boundary F,/emem Methods in Solid Mechanics. George Allen & Unwin, London(1983).
0148-9062/93 $6.00 + 0.00 Pergamon Press Ltd
Printed in Great Britain
Micromechanics of Granular Materials -Numerical Simulation of the Effects of Heterogeneities ZHONG LIU*# LARRY R. MYER# NEVILLE G.W. COOK*#
INTRODUCTION The principal objective of this research is to gain an understanding of the relation between grain stresses and microfmcturing and macroscopic deformation and fracture in heterogeneous granular aggregates. The generation and propagation of microcracks in relation to the loading conditions, grain sizes and shapes is essential for understanding the stress-strain relationships and quasibrittle fracture of granular materials under differential compression. Granular materials, typically comprising a great n u m b e r of particles or grains, have a correspondingly large number of degrees of freedom. The complex interaction of individual micro-features of these materials, at some stage, results in macro-failure. At the macro-scale, the entire system exhibits well-defined deformation under uniform external loading. This type of behavior is idealized in the notion of stress-strain relations in continuum theories. From a physical point of view the stress-strain relations under uniform external stresses are a statistical average resulting from the almost independent behavior of different microscopic parts of the large system. In other words, the macro-behavior represents the statistics of the microscopic components of the granular material. Experiments have shown that local stresses in individual grains are important in the overall response of granular materials[Ill2][3]. At a microscopic level, local stresses are related to: 1) tensile strength of grains; 2) grain geometry; 3) pores and; 4) grain contacts. How can we quantify the micro-structure and the distribution of stresses? Once quantified, how are these parameters related to the macro-mechanical behavior? Which parameter is most important or, in other words, dominant in defining the macro-mechanical properties such as the stre,~s-strain relationship and failure of materials? Micro-mechanical models have been used to study macroscopic properties in terms of microscopic processes. Some of these models are based on a continuum containing micro-cracks[4] [5] [6] while others are based on a granular or discrete element approach[7][8]. In this paper we will use a two-dimensional numerical model based on the elastic interaction and tensile fracture of grains and grain bonds to study the effects of heterogeneity in grain strengths on macroscopic strainhardening and strain-softening deformation, and fracture *Department of Materials Science and Mineral Engineering, #Earth Sciences Division, Lawrence Berkeley Laboratory, University of California at Berkeley, Berkeley, CA94720. 1281
under differential distributions within combined Boundary Minimization Scheme section.
compressive stresses. Stress grains are calculated using a Element Method(BEM)-Energy which is outlined in the following
NUMERICAL M E T H O D O L O G Y Local stresses in individual grains are important in the overall response of granular materials and fracture propagation. We need to know in some detail information about the stress distributions in grains. Solutions for idealized bodies, like discs or spheres loaded diametrically, are presented in most texts on the theory of elasticity[9]. The exact solution for a single-line load on a cylinder was obtained by Poritsky[10]. In soil mechanics, individual grains are assumed to be unbreakable. The deformations most frequently treated are in the form of either strain resulting from elastic deformations of grains with spherical[l 1], cubical, and other shapes, or from sliding of grains[12]. Some of these situations have been investigated with the aid of photoelasticity[13]. They are, however, not adequate for a quantitative study of the micromechanical interactions that produce complex macroscopic deformation. The system of granular materials is too complicated to use idealized solutions. Numerical methods can be used to calculate the stresses in grains of the granular materials. In our research, the BEM (direct boundary integral method) is used to calculate the stress distribution within each grain. In the BEM, boundary stresses and displacements are chosen as variables. For every grain in the system, the boundary is divided into n segments, so there are 2n unknowns: ~ , c~, u~, uJn, with j = 1, 2 . . . . . n. c~, u~ are the normal stress and the normal displacement of segment j. c~, u~ are the tangential stress and displacement of segment j. By using the integral representation: u = ~c(GO'fi-fiEu)ds
(1)
and choosing 2n test solutions[14], we can get an equation B{t~B} = A{uB} , (2) which can be used to calculate the unknown stresses and displacements along the boundary. B and A in equation (2) are matrices of influence coefficients. Matrices B and A are dependent only on the geometry of the grain boundary. If two grains have the same shape with different boundary stresses and displacements, there is no change
1282
R O C K M E C H A N I C S IN T H E 1990s
"£
11 21
21
22
B~ B~ fi
B=
2n
22
B~
BsII • • •
n2 Bsn
"~f
B~
I~
Bu
11
11
12
(L
A =
B,.
12
A~
A~
21
21
A~
22
11
11
12
ntln ......
nl
"n2
A~
A~
A~
/ 2 2 V z 2n ,-
B~
in
In
22
2n
n'2" . . . . . .
2n
L In
where
In
(4)
A~ A~
... n2
A~, A~ for either the B or the A matrix. {os} is a stress vector along the boundary and {us} is a displacement vector along the boundary: o 2
Alln ......
.-ot
eo
..,
... ..,
u:
(6)
For granular materials, the detailed stress distributions within grains need to be computed to investigate the mechanisms of nonelastic behavior and microfracture. Granular materials typically comprise a great number of grains. Every grain is divided into hundreds of segments. There are hundreds of thousands of variables in a system. We can not directly use the BEM to deal with this problem. The number of variables must be reduced. In the granular material system, external energy is •input into grains through contacts. Stress distribution within grains can be described in terms of contact stresses and displacements along the boundary. Based on this, for every grain, only contact stresses and contact displacements are chosen as variables. Assuming there are m contact points in a grain, because all the stresses of free elements along the boundary are zero, {oB } in equation (2) can be rearranged as: {oB}' = (oI ~ o~° .... o,me o ~ 0 ... 0) r , (7) in which, all the non-zero stresses are put together. Then {%}' can be divided into two sub-vectors: contact stresses{oBc} and {0} (representing free boundaries), where {OBC}lx2m
----(0
lc
O l nc
0 2c
...
O sme
Onme) r and
{0}lx~2~_2m) = (0 ,.- 0)T" Corresponding to the new order of the stress vector (on}', the displacement vector{uu} is rearranged as two sub-vectors: contact displacement vector {uBc} and displacement vector {ueF} of free stress elements. Corresponding to the new order of the stress vector and the displacement vector, matrices A and B in equation (2) can also be rearranged as: A' =
A:~2m
2mx2(n-m)
(8)
21
• A2(n-m)x2m A2(n-m)x2(n-m)J B2mx2m B" =
mx2Cn-m)
21
~B2(n-m)x2m B2(n-m)x~n-m)J
•
/ 2 2 V z 21
Bc{°Bc}=Ac{ uBc} ,
A_~ A~
Ash ...... 12
-,
(12)
Substitutingcq.12intocq. I0, yields
A~ A~
A~ . . . . . .
(11)
From eq. (11),
{us,}=tA ~ BIOBc~-tA ) A{UBc}
B~
Anti ......
nl
(3)
..... , nn ml
Ass A~ . . . . . .
"~f "fi" "d2" Ass A~ A~
t{ {O~c}= ~ {unc}+~ {uBv} .
Im
•.. B~ B~ In In B~ B~
12
12
2n
B~ B=
... n2
n2
11
A~
...
nll oo
B~ l ~ "~: "if B~ B ~
From equation (2), (8) and (9), we get
111
12
12
(9)
(13)
() (
Bc: B -~ ~ B (14) llx\ 12 122\ -l and Ac = ~A/- A~AJ (15) In our simulation, both stresses and displacements of contact elements can be chosen as the system variables. Using eq. (13), the strain energy of the entire system of grains is calculated. Following the Finite Element Method, we use the principle of minimum energy and eq. (13) to obtain a global stiffness matrix for the particle system, in which only displacements or stresses of those contact elements are variables. After the global system of equations is solved for the stresses and displacements of the contact elements, the stress distribution within the grains of granular materials can be calculated from eq. (1). S I M U L A T I O N S OF G R A I N F A I L U R E
Using the numerical formulism described above, we have used rectangular and hexagonal packing of cylindrical grains of the same size to study the effects of stochastic distributions of grain strength, uniform debonding strengths and confining stresses on the ultimate strength, fracture and macroscopic deformation. The system consists of about 300 grains with the boundary of each grain divided into 100 elements. Within each grain, stresses at 50 points are calculated to find the maximum stress within the grain. With load increasing, the maximum stress point is searched through the whole system. When the maximum stress reaches the tensile strength of the grain, the grain is broken, and its Yotmg's modulus is reduced 25% for the next step of the calculation. If contact stresses at interfaces are tensile stresses and reach debonding strengths, these interfaces are broken and contact elements of these interfaces are traction free. If two grains contact each other during the loading, matrices A and B of these two grains must be recalculated due to the changed contact situation. Loads applied to the boundaries of aggregates are transmitted across grain contacts and through grains. Loads applied at one boundary are transmitted from grain to grain to the other boundaries where, for static equilibritan, equal and opposite forces must applied. From our calculation, it is found that stress concenlrations arise at grain contacts in granular systems because external forces applied at the system boundary are trat~itted f3xan grain to grain over small contact areas. The inteasity of a stress concentration at a contact depends prima~ y on the size of the corresponding contact area, m i c t ~ s l r u ~ e s local to the area near that grain, and the orientation of the normal to that area relative to boundary loads. The stress distribution in the whole particle system is calculated. At the beginning, when no grain fracture has
ROCK MECHANICS IN THE 1990s occurred in the system, there are no big changes in the maximum principal tensile stresses in grains. When grain fracture occurs, a macro-fracture begins to propagate in the system and a stress concentraton develops near the tip of this macro-fracture. This stress concentration along with the tensile strength distribution in grains strongly determines the occurrence of the next microcrack. From our calculation, it is also found that, as each grain or bond fractures, the effective macroscopic elastic modulus of the material decreases and the stress at which the next fracture occurs changes. The effective Young's moduli and limiting stresses are shown below in Figures 1 and 2 for homogeneous and heterogeneous grain strength distributions. Stresses are normalized with respect to the mean tensile strength of the grains and a hexagonal grain packing is assumed. The envelopes of these lines constitute the stress-strain curves.
Fig. 4. tteterogeneous grain strength distributions result in both macroscopic strain-hardening and strain-softening as shown in Fig. 2, as well as a more diffuse distribution of grain failures as shown in Fig. 5. The ultimate compressive strength of granular materials is dependent on the tensile strength distribution in the grains of the material. The compressive peak strength of granular materials, in which all grains have the same tensile strength is the largest, as shown in Fig. 3. It is clear that, with the range of tensile grain strength becoming larger, the compressive peak strength decreases.
o
.,--5
4 "5 ~ 3
5
[
i
]
--
i
i
F
Uniform Strength.
4
Intrinsic /
1283
~2 "~ ~.~
J
0
o
. . . . . . . . . . . .
1.0
0.'/5-1
0.5-1
I ....
0.25-1
t
.
.
.
.
0.0-1
Range of Grain Tensile Strength. (minimum to maximum tensile strength) Figure 3. Effects of the range of tensile strengths in grains on the ultimate compressive strengths of granular materials. i
0
i
[
i
0.002 0.004 0.006 0.008 0,01 0.012 Strain,
Figure 1. Stress-strain curves show decreasing effective Young's moduli with increasing grain and bond damage and strain softening for homogeneous strength distribution.
Stochastic strength distribution. d
z
1.5 Intrinsic M°du'~///~
o.s
0
0.001 0.002 0.003 0.004 0.005 0.006 Strain.
Figure 2. Stress-strain curves show both strain-hardening and strain-softening with increasing grain and bond damage for stochastic strength distribution. From Fig. 1, it is clear that a homogeneous strength distribution results in only macroscopic strain-softening and intense localization of grain fractures as shown in
For uniform grain and bond strengths, the deformation is extremely brittle and failure occurs by localization and propagation of grain fractures along a diagonal spanning the whole network as shown in Fig. 4, in which the stress concentration near the fracture tip controls the propagation of grain fractures. For a heterogeneous distribution of grain strengths in the system, at low-confining pressures grain failures tend to be distributed throughout the sample and propagation of grain fracture tends to be along the loading direction as shown in Fig. 5. As the microcracks propagate and interact, a macrofracture or failure surface is created in the granular material. The effects of packing arrangement on the directions of fracture surfaces are obvious. For a hexagonal structure with uniform tensile strength, the angle between the fracture surface and vertical external load axis is about 30 degrees as shown in Fig. 4. The angle is about 45 degrees for a cubic structure with uniform tensile strength. The peak compressive strength of a hexagonal structure is larger than that of a cubic structure. CONCLUSIONS From the above discussion, the following conclusions can be drawn: When microcracks occur in granular materials they generate tensile stress concentrations in adjacent grains near the crack tip, as would occur in a continuum. These stress concentrations and the stochastic tensile strength distribution drive the propagation of macrofractures in hexagonal and cubic lattices.
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ROCK MECHANICS IN THE 1990s whose direction tends to follow the lattice, that is, the plane of the fracture lies at 45 0 or 300 to the direction of the ~ ~ load for cubic o r hexagonal ~ . With large stochastic strength distributions, under lowconfining pressures, the angles between the ~ t boundary stress and the ~ f r a c t m ~ surface are near zero degree.
Granular materials with homogeneous or small
stochastic tensile grain strength distributions have only macroscopic strain-softening. Granular materials with large heterogeneous tensile grain strength distribution have both macroscopic strain-hardening and strainsoftening. As the range of tensile strengths in grains becomes larger, the peak ¢t~wessive strength ~ . Acknowledgmenls--Tifis work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Engineering and Geosciences Division, of the US Department of Energy under Contract No DE-ACO3-76SF(g~8.
Figure 4. Interaction between failed grains results m localization of a macroscopic shear fracture and strain-softening deformation for hexagonal lattice. tlLt0t~l
Figure 5. Initial failures of grains with a stochastic distribution of strengths produce a macroscopic extensile fracture and both strain-hardening and strain-softening deformation.
Macrofracmres are affected by boundary loads, the packing arrangement, and the range of tensile strengths in grains. I f the range of tensile strength distributions is small, that is the grains have uniform tensile strength, grain fracture occurs locally as shown in F]gure 4. If the range of tensile strengths in grains is large, grain fracture occurs randomly over the system, Stress concentrations due to these fractured grains do not hav~ a big effect in determining which grain is broken next, and the ten-Ale strength distribution is dominant ia determining microfracture propagation at the early age. For uniform grain strengths, stress concentration due to fractured grains results in macroscopic shear ftacaa~s
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