Localization in Undrained Deformation 1

Transcription

1 Localization in Undrained Deformation 1 J. W. Rdnicki Dept. of Civil and Env. Engn. and Dept. of Mech. Engn. Northwestern University Evanston, IL John.Rdnicki@gmail.com Janary 7, To appear in the Proceedings of the 4 th Biot Conference on Poromechanics, Jne 8-10, 009. Colmbia University, New York, NY.

2 ABSTRACT Exploiting form invariance of a class of elastic plastic constittive relations for drained and ndrained deformation makes it possible to compare predictions for localization nder these limiting conditions. Reslts are shown to depend strongly on poroelastic constants reflecting the compressibility of solid and flid constitents and parameters describing inelastic volme strain and the dependence of the yield stress in shear on mean normal stress. INTRODUCTION Analysis of the onset of localization (shear banding, falting) as a bifrcation from homogeneos deformation [1,, 3] has proven to be a sefl framework when applied to materials modeled as rate-independent. For materials that are flid-satrated, as is often the case for geomaterials, flid diffsion in response to deformation introdces rate-dependence even when the drained (constant pore pressre) response is rateindependent. Becase only the instantaneos material properties enter the bifrcation condition, the application of this erion to rate-dependent materials, inclding those for which the rate-dependence is de to flid diffsion, is not prodctive and it is necessary to examine the evoltion of inhomogeneities. If, however, the drained response is rate-independent, then the alternative limit of ndrained deformation, corresponding to no flid mass change in material elements, is also rate-independent. If the ndrained response is rate-independent, the bifrcation analysis can be applied. Rice [4] showed, however, that for simple shear of a dilatant, pressresensitive material, homogeneos ndrained deformation is nstable in the sense that small spatial nonniformities grow exponentially in time when the localization condition is met in terms of drained response. In this case, this condition is met before the localization condition is met in terms of the ndrained response. Conseqently, it is nlikely that homogeneos ndrained deformation cold be sstained beyond the point at which the localization condition is met in terms of the drained response. Althogh Rice [4] focsed on the case of dilatant, frictional behavior, he noted that there were types of material behavior for which the condition for localization is met in terms of the ndrained response before it is met for the drained response. As discssed in more detail by Rdnicki [5], these can occr when the inelastic volme deformation John W. Rdnicki, Department of Civil and Environmental Engineering and Department of Mechanical Engineering, Northwestern University, 145 Sheridan Road, Evanston, IL , U. S. A.

3 is compactant and the material is on a frictional portion of the yield srface (yield stress in shear increases with mean compressive stress) or when the inelastic volme response is dilatant bt the stress state is on a cap of the yield srface (yield stress in shear decreases with increasing mean compressive stress). Sch conditions can occr near a transition in yield behavior from frictional to cap-like. Applications for high porosity rocks (porosity greater than abot 15%), which are typical of reservoir rocks and, hence, tend to be flid satrated, often enconter stress states in the vicinity of sch a transition. Rice [4] also noted that his analysis was for the special deformation state of simple shear combined with niaxial compression. For this special state the localization condition for both drained and ndrained deformation occrs at a peak in the stress strain crve. Analyses for more general deformation states [1,,3] show that, in general, localization for drained deformation is predicted to occr slightly before or well after peak. The analysis here takes advantage of a form invariance, noted in [5], for a wide class of rate-independent elastic plastic material models with inelastic volme change and pressre-sensitivity. In particlar, the form of the constittive relation for ndrained response is identical to that for drained response. Conseqently, ndrained response is obtained from the drained response by sbstittions that involve poroelastic material parameters. Using these same sbstittions in the predicted conditions for the onset of localization and the angle of the band for drained deformation yields reslts for the ndrained case. The reslts show that the predictions for ndrained response depend strongly on poroelastic parameters reflecting the compressibility of the solid and flid constitents. A limiting case is the soil mechanics approximation for which both solid and flid constitents are mch less compressible than the poros matrix. Althogh, as noted earlier, a more complete analysis reqires an examination of the time evoltion of heterogeneities. Becase of flid diffsion in response to heterogeneos deformation, local conditions will be neither drained nor ndrained regardless of the (global) conditions on the bondaries. Nevertheless, comparison of the reslts for the limiting cases drained and ndrained response can indicate when localization is first predicted and give gidance for a fller analysis. DRAINED RESPONSE The constittive relation is assmed to be the rate-independent, incrementally linear elastic plastic constittive relation sed in [1]. Strain increments are the sm and elastic and inelastic contribtions. Elastic strain increments are given by { } el dε = dσ ν /(1 + ν) δ dσ /G (1) kk where G is the shear modls, ν is Poisson s ratio and δ is the Kronecker delta. Inelastic strain increments have the form: in 1 s 1 s 1 kl dε = + βδ d + μδ σ kl kl h τ 3 τ 3 () This is the most general form for a smooth yield srface and plastic potential that

4 depend on stress only throgh the first and second stress invariants. In () 1/ ( ss /) s = σ σ /3 δ is the τ = is the Mises eqivalent shear stress, ( ) kk deviatoric stress, μ is a friction coefficient, and β is a dilatancy factor, eqal to in in 1/ dε / dγ, where in ( in in in dγ = de de ) and is the deviatoric portion of (). h is a kk in hardening modls eqal to the slope dτ / dγ for constantσ. The hardening modls is related to the tangent modls at constantσ, h = dτ / dγ by h = h/(1+ h / G) and, hence is approximately eqal to it for h/ G 1. Note, tan however, that for common testing configrations, axisymmetric compression and extension and plane strain, σ is not constant, and, conseqently, neither h nor h is tan directly related to the observed stress vs. strain crve. The analysis of [1] predicts a ical vale of the hardening modls at which localization occrs de tan h G 11+ ν 1 = β μ 1 + ν 3 + β + 91 ν 18 ( ) ( ) N ( μ) (3) and an angle between the band normal and the most compressive stress direction θ band π 1 = + arcsin α (4) 4 where ( / 3)( β + μ) N(1 ν) / 3 α = (5) 4 N In (3) and (5) N is eqal to the intermediate principal vale of the deviatoric stress, s, mltiplied by 3 and divided by τ. The N sed here is 3 times the N sed in [1] and the form of (4) is given by [6]. (These expressions neglect terms of order τ / G that wold enter from se of a proper co-rotational rate of stress.) If the material is flid satrated, then these reslts also apply for the limit of drained deformation. This limit pertains when the deformation occrs sfficiently slowly that flid mass diffsion can alleviate any alterations in pore flid pressre. UNDRAINED RESPONSE Rdnicki [5] (see, also [3, 7, 8]) has shown that the constittive relation for ndrained response has the same form as (1) and () with the following assmptions: (i) the elastic portion of strain increments can be described by linear, isotropic poroelasticity; (ii) the role of the pore pressre p in the inelastic strain increments is inclded by replacing the stress by the Terzaghi form of the effective stress, σ + pδ ; and (iii) the inelastic increment in the apparent void volme fraction is eqal to the inelastic volme strain increment. Rice [9] has arged on theoretical gronds that (ii) and (iii) are appropriate for geomaterials in which the primary mechanisms of inelastic deformation are microcracking from sharp-tipped flaws and frictional sliding on srfaces with small real areas of contact. In addition, (ii) seems to be spported by

5 observational evidence. With these assmptions, the constittive relation for the inelastic strain increments for ndrained response is identical to () with the following replacements: (i)( μ, β) (1 B )( μ, β) and (ii) h H = h+ ( μβkb / ς). B andζ are poroelastic parameters [10, 11]. B is Skempton s coefficient, eqal to the negative of the ratio of pore pressre change to mean normal stress change for ndrained elastic deformation, and ς = 1 K / K where K is the (drained) blk modls and K is another modls s s on the order of the blk modls of the solid constitents (ζ is the Biot parameter, often denoted α ). Skempton s coefficient B and the Biot parameterζ range from 0 to 1. In both each cases, the lower limit occrs for a very compressible pore flid and the pper when both the flid and solid constitents are incompressible (poros matrix is not necessarily incompressible). This pper limit is a good approximation for soils, thogh not generally for rocks. When B = 1(i) shows that the effective vales of μ and β become zero for ndrained deformation. Analogos to h for drained response, the modls H is the slope of the τ vs. in γ crve at constant σ for ndrained response. Note, however, that becase the pore pressre is changing for ndrained response, the effective compressive stress, σ + p is not constant. In the elastic strain increments (1), the drained Poisson s ratio ν and blk modls K are replaced by their ndrained conterparts 3 ν + (1 ν) Bς ν = 3 (1 ν ) Bς (6) and K [10, 11]. Since the elastic shear modls is the same for both drained and ndrained deformation (in an isotropic material), the expression for K is the sal one in terms of shear modls and ν. The ndrained Poisson s ratio ranges from ν ν 0.5 where the pper and lower limits correspond to the conditions mentioned above. The corresponding vale of K becomes nbonded and in this limit, ndrained response is elastically incompressible. Since the effective vale of β is zero for this case, both the elastic and inelastic strain increments are incompressible. Becase the ndrained response has the same form as the drained response, making the same replacements in (3) and (5) yields the predictions for the band angle α and the ical (ndrained) hardening modls for localization : n H α n ( B) ( / 3) 1 ( β + μ) 3 N(1 ν ) = (7) 4 N and H G 11+ ν 1 = ν 18 ( 1 B) ( β μ) ( 1 ν ) 3N ( 1 B)( β μ) (8)

6 The ical slope of the nderlying drained deformation at which localization occrs nd for ndrained response is h = H μβkb/ ς. Reslts For incompressible solid and flid constitents ( B = 1, ς = 1, ν = 0.5), α = 0 and, hence, the band angle is always 45 o, a featre noted previosly by [1]. In addition, the expressions for the ical hardening modli simplify: The terms involving β and μ vanish and the coefficient before the sqare bracket in (8) becomes 1/1. Figre 1 shows an example of how the compressibility of solid and flid constitents (reflected throgh vales of the poroelastic constants B and ζ ) affect predictions for localization for ndrained conditions for β = 0.3 and μ = 0.6. The horizontal axis in both panels is the deviatoric stress state parameter N which ranges from 1 for axisymmetric extension (left) to + 1for axisymmetric compression (right). Left panel shows the prediction for the band angle for drained deformation and for ndrained deformation for three sets of B and ζ. Right panel shows the drained nd ical hardening modls (3) and the ndrained ical modli, H (8) and h for cri t B = 0.8and ζ = 0.8. Band Angle drained B = 0.8, ζ = 0.8 B = 0.6, ζ = 0.8 B = 0.4, ζ = N hardening modls/g h /G, drained h nd /G, ndrained H /G, ndrained N Figre 1. The left panel shows that more compliant material constitents ( B, ζ < 1) redce the

7 predicted band angle from the drained vale, reaching 45 o in the limit of incompressible solid and flid constitents (soil mechanics approximation). The right panel compares the three ical hardening modli. Note that the ical vale for ndrained deformation H can be greater or less than the drained vale h. They are, however, approximately eqal and near zero slightly to the left of N = 0. This is consistent with both vales eqaling zero in the case of the simple shear case considered in [4]. Becase the term μβkb / ς is positive if β and μ have the same nd sign, h < h, bt the difference is smaller near axisymmetric compression (N = 1) becase the excess of H over h is larger here. ACKNOWLEDGEMENT Partial financial spport was provided by the US Dept. of Energy, Office of Science, Basic Energy Sciences, Geosciences Program throgh grant DE-FG0-93ER14344/A016 to Northwestern University. I am gratefl to Jose Andrade for enlightening discssions that motivated this work. REFERENCES 1. Rdnicki, J. W. and J. R. Rice Conditions for Localization of Deformation in Pressresensitive Dilatant materials, J. Mech. Phys. Solids, 3: Rice, J. R The Localization of Plastic Deformation in Theoretical and Applied Mechanics, Proc. of the 14th International Congress on Theoretical and Applied Mechanics, W. T. Koiter, ed. Delft: North Holland, Béselle, P. and J. W. Rdnicki Localization: Shear Bands and Compaction Bands, in Mechanics of Flid Satrated Rocks, Y. Gégen and M. Botéca, eds. London: Academic Press, Rice, J. R On the Stability of Dilatant Hardening for Satrated Rock Masses, J. Geophys. Res., 80: Rdnicki, J. W Diffsive Instabilities in Dilating and Compacting Geomaterials, in Mltiscale Deformation and Fractre in Materials and Strctres, T.-J. Chang and J. W. Rdnicki, eds. The Netherlands: Klwer, Rdnicki, J. W. and W. A. Olsson Reexamination of falt angles predicted by shear localization theory, in Proc. 3rd North American Rock mechanics Symposim (NARMS 98), Rock Mechanics in Mining, Petrolem and Civil Works, 3-5 Jne, 1998, Cancn, Mexico. Extended abstract in International Jornal of Rock Mechanics and Mining Sciences, 35(415), Rdnicki, J. W A Class of Elastic-Plastic Constittive Laws for Brittle Rock, J. of Rheology, 8: Rdnicki, J. W Effect of Pore Flid Diffsion on Deformation and Failre of Rock, in Mechanics of Geomaterials, Z. P. Bažant, ed. New York: John Wiley & Sons, Ltd., Rice, J. R Pore Pressre Effects in Inelastic Constittive Formlations for Fissred Rock Masses, in Advances in Civil Engineering throgh Engineering Mechanics. New York: American Society of Civil Engineers, Rice, J. R. and M. P. Cleary Some Basic Stress Diffsion Soltions for Flid-satrated Elastic Poros Media with Compressible Constitents, Rev. Geophys. Space Phys., 14: Wang, H. F Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrology. Princeton University Press. 1. Rnesson, K., R. Larsson, and S. Stre Localization in Hyperelasto-plastic Poros Solids Sbjected to Undrained Conditions, Int. J. Solids Strc., 35: