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Numerical modelling of deformation and fracture in geomaterials
Yu.P. Stefanov, V.D. Evseev1, and R.A. Bakeev
Institute of Strength Physics and Materials Science, SB RAS, Tomsk, 634021, Russia 1 Tomsk Polytechnic University, Tomsk, 634050, Russia
The processes of localized deformation band formation and crack generation in brittle-plastic materials are simulated numerically in various loading conditions. The patterns of cracking and strain localized band for a sandstone specimen in compression are obtained.
To describe inelastic behavior of pressure-sensitive materials, the mathematical tools and concepts of plasticity theory are employed. In the computations the Nikolaevskii model with a non-associated flow rule is used. In the course of “plastic” deformation damages are accumulated. Macrocrack opening and material fracture are described explicitly. For this purpose the node-splitting technique is used and the free-surface conditions are given on newly formed boundaries. This provides self-acting account of crack generation for the entire calculation area in the course of deformation.
The use of the considered approaches in describing the behavior of a rock specimen as an ideal homogeneous medium allows a good agreement of fracture patterns as well as of various types of stress-strain dependences with the experimental findings. In order to describe in detail the behavior peculiarities of heterogeneous porous and cracked materials, it is necessary to consider their structural features and crack growth.
1. Introduction
Description of the behavior of geomaterials still remains to be an urgent problem in understanding the processes occurring in them and in predicting their behavior in various conditions. The macroscopic behavior of these materials can be brittle or near-plastic, depending on loading conditions. There is, however, a number of peculiarities to distinguish between their behavior and the plastic behavior of metals, where another physical mechanism is operative. These features are an influence of pressure on inelastic pseudoplastic deformation and pronounced changes in the specimen volume in the course of deformation.
Fracture of rocks is, first of all, associated with the presence of pores and cracks of different scales. Under any type of loading the stress-strain state is inhomogeneous. Near pores and cracks tensile stresses arise, which leads to tensile crack growth. The loading conditions have a pronounced effect, for instance, confining pressure and conditions on the end surfaces [1, 2]. An increase in the confining pressure limits the possibilities of crack growth, so that in certain conditions the material may behave like a plastic solid. It is known that with growth of friction on the end surfaces the rock demonstrates higher strength, and vice versa the presence of a lubricant causes a decrease in strength and often specimen cleavage. One of the reasons for such behavior is that friction results in the appearance of compression stresses that inhibit crack growth towards the ends, just like confining
pressure. Appropriate model of plasticity is best suited for describing its behavior.
For describing behavior of the brittle-plastic materials specified models in the difference forms are developed in [1-11]. Generalizations of the Coulomb-Mohr law in the Mises-Schleiher form are widely used to describe behavior of pressure sensitive materials beyond elastic limit. An example is the Drucker-Prager model [3]. The models with the independent dilatation coefficient were proposed by Rudnicki and Rice [4] and Nikolaevskii [2, 5].
However, for an adequate description of the complex medium behavior, the account for cracking and main crack formation remains an important constituent of theoretical constructs.
2. Governing equations
To describe deformation process, a set of continuum mechanics equations is numerically solved at given initial and boundary conditions. The set of equations involving the equations of motions the equation of continuity and the governing relations interrelating stress and strain increments. The problems are solved for plane-strain conditions. The equations are integrated by a well-known scheme [12] for solving dynamic equations.
Let us describe inelastic behavior of pressure-sensitive materials using the mathematical tools and terms of plasticity theory. Therefore, in the subsequent discussion by the term
© Yu.P. Stefanov, V.D. Evseev, and R.A. Bakeev, 2004
“plastic deformation” is meant inelastic material behavior regardless its nature. The Nikolaevskii model for a nonassociated flow rule is used for computations.
Prior to the onset of plastic deformation, consider medium behavior to be described by the hypoelastic law
Ds;; f
—гу- = 2u Dt
їу з ^kkSy-
Dt
' - siy ~ sik ® yk ~ syk °hk,
P = ~KV ’ a;y =~P8iy+ siy.
(i)
(2)
Here <5^ are the stress tensor components, Sj are the stress deviator components, P is the average pressure, K and ^ are the bulk and shear moduli;
hj = -jUj + uji), rnjj = j -i), (3)
where Ej- are the Cauchy strain rate tensor components, fflj- are the rotation velocity tensor components, ui are the displacement vector components.Total strains involves elastic and plastic ones
Є,-, = + es
(7)
ij ij ij The plasticity criterion
a Jj3 + J12 = Y, (8)
where Jj is the first invariant of stress tensor and J2 is the second invariant of stress deviator tensor, a is the internal friction coefficient, Y is the shear strength of a material or material cohesion. Plastic strain rate is assigned by the exp-
ression
. p , ЭФ Aep' ’
дву
where the plastic potential is
(9)
ФК ) = J2 +y Jl
a
2Y-J1 1 + const.
(10)
As a result, plastic strain rate is defined by the formula
Aep; =
2
s;; H-----Л
y 3
a
Ji |Sy
A
X.
(11)
Dilatancy or bulk inelastic strain is related to shear strain by the linear equation
i1p = 2Л^21/2>
(12)
where A is the dilatation coefficient. For A = a we have the Drucker and Prager model and for A = a = 0 the Prandtl-Reuss model.
Fracture of the material will be described using two approaches, namely, with an explicit description of crack formation and without it. In both approaches account will be
taken of damage accumulation and material softening. Assume that shear strength of the material depends on the accumulated inelastic strain and can be written in the form:
Y = Y,(1 + A(e) - De (e))
(13)
where A(e) is the function describing material hardening and De (e) describes softening or damaging, e = 2Ip ^2, h is the hardening coefficient, e* is the critical strain.
Take the linear dependence for hardening
A(e) = h 2ej' e* (14)
and the quadratic dependence for damage accumulation
De (e) = 2h(^e* )2 = Ae/e*. (15)
In the softening region, i.e. when e > e* of the function A and D can be different, continuity Y should only be allowed.
In describing crack growth explicitly, the fracture criterion is written as
eff
= an(1 - a) + aTa <a* (1 - DE) for an > 0, (16)
where on, oT are the normal and tangential components of the stress vector, respectively;
D,. =
P(J dif/ є*)
(17)
is damage, 0 < a < 1 governs the contribution of normal and tangential stresses, P and n are parameters, O, £ are the ultimate stress and ultimate value of bulk plastic strain, respectively,
Thus, it is assumed that fracture and crack opening occur in the presence and under the action of tensile stresses. In the course of plastic deformation the material accumulates damages, which govern bulk plastic deformation and result in material softening.
To describe cracking explicitly, the procedure of grid node separation is used in the numerical calculations. Such a method of fracture description was put forward and applied by different authors, for instance [13]. The technique suggests that cracks are generated on the boundaries of the calculation cells. The fracture condition is therefore verified for the boundaries between the cells. To do this, consideration is given to the stress state of pairs of cells with the common boundary and the average value of tensile stresses is calculated. When the local fracture criterion (16), which is checked for all the calculation cell pairs, is fulfilled, nodes of the calculation grid split and the free surface conditions are assigned on the newly formed boundaries.
With this in mind, in the calculation four calculation grids are used instead of one grid. For continuous material the calculation grid nodes coincide. In the case of crack nucleation the nodes split so to describe a particular crack shape. Stress state and accumulated damages in the vicinity of a given grid node determine a crack shape. As a result,
Fig. 1. The stress difference 0 = 01 -02 versus volumetric and axial strain of sandstone under different conditions on the specimen ends (a). Curves 1-3 are the respective calculation results for the specimen with fixed ends, with ideal sliding on the end surfaces and with corner irregularities and ideal sliding on the end surfaces. Curve 4 shows experimental data presented by Labuz et al. [15]
each cell of the calculation grid may have the nodes independent from nodes of surrounding cells. The use of this manner of cracking description makes it possible to take into account and consider slip of crack walls.
3. Calculation results
Consider the behavior of a vertically compressed homogeneous specimen. The material properties are: p0 = = 2.2 g/cm3, K = 12.28 GPa, ^ = 5.346 GPa, Y0 = 9.04 MPa, a0 = 0.546, A0 = 0.48. The material characteristics correspond to the macroscopic properties of sandstone.
The use of the considered approaches in describing the behavior of a rock specimen as an ideal homogeneous medium allows a good agreement of fracture patterns as well as of various types of stress-strain dependences with the experimental findings [14]. The numerical results obtained for the behavior of sandstone specimens in the plane strain conditions agree with experimental data presented by Labuz [15]. However, certain behavior features of such materials have been impossible to describe within the framework of the homogeneous medium. Moreover, the calculation results are in conflict with certain experimental findings.
A rectangular specimen being compressed, the end surface conditions influence strongly the maximum axial strain value as well as fracture pattern. When considering the behavior of homogeneous specimens in compression, tensile stresses do not arise. In this case, the sites of deformation localization are solely the singularities of geometry and loading conditions. As the friction coefficient increases at the specimen ends, deformation localization and fracture occur at a lower degree of the total axial strain and thus at a lower loading value. The localized deformation band pattern is somewhat different as well. For example, Figure 1 shows stress-strain curves for the homogeneous specimen under different end surface conditions.
Deformation in the conditions of ideal sliding on the specimen ends develops uniformly up to an ultimate degree. The latter being achieved, localization bands start to generate. The nucleation sites of localized deformation bands are defined by low stress and strain oscillations related to the wave background and to the calculation error. The deviations stem from the dynamic formulation of the problem. In the calculations performed their value does not exce-
Fig. 2. Deformation localization in sandstone under different conditions on the specimen ends. Heterogeneous specimen with a porosity of ~2 %: ideal sliding (a, b); fixed ends (c). Cracking of specimen with fixed ends (d)
a, GPa
0.2 0.4 0.6 0.8 1.0 s,%
Fig. 3. The stress difference versus axial strain of heterogeneous specimen under different conditions on the specimen ends. Curves 1-2 are the respective calculation results for the specimen with fixed ends and with ideal sliding on the end surfaces
ed 10-2 % of the maximum strain degree. Growth of friction on the end surfaces causes a more intensive development of deformation in the specimen corners and, consequently, more rapid formation of localized deformation bands. In this case, deformation localization and softening take place at an earlier stage of deformation. An explicit account of crack formation brings about no significant changes in the results, since cracks are generated in the zone of localized deformation.
In considering a heterogeneous specimen with a porosity of ~2 %, the specimen fails completely at an almost the same load. With the friction-free specimen ends the inelastic deformation development starts earlier. In this case, however, plastic deformation is more diffused in character than that in a specimen with the fixed ends (higher friction) where plastic deformation develops more rapidly in the central part. The highest increase in “strength” in the presence of friction has been attained in specimens with most distinctive features in the form of pores and cracks close to the end surfaces.
Figure 2 shows the calculation results of the strain intensity distribution and cracking in the heterogeneous specimen.
When cracking is taken into account explicitly, in a porous specimen cracks begin to generate near its ends, however, specimen failure into parts is observed at closer load values. Strength of specimens with the fixed ends turns out to be higher by 10-15 %, Fig. 3.
4. Conclusion
In the paper localized deformation band development and cracking are simulated numerically for elastic brittle-plastic materials under various loading conditions. The results obtained in this paper by using the Nikolaevskii dilatancy model agree with experimental findings of the
geological medium behavior under various loading conditions.
Using of models of elastic-brittle-plastic materials allows one to describe many of their behavior features. Such calculations can be helpful to verify the adequacy of a model, to select parameters and to explain certain behavior peculiarities. At the same time, the use of specific models to describe complex heterogeneous porous and cracked media as homogenous medium makes it impossible to allow for certain features of their behavior. In order to describe in detail the behavior peculiarities of heterogeneous porous and cracked materials, it is necessary to give due consideration for their structural features and crack growth.
This work is performed under the financial support of the Russian Foundation for Basic Research (Grant No. 0205-65346 and No. 04-05-64547a) and the Russian Academy of Sciences (Project Nos. 13.12 and 6.5.2).
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