Localization in Undrained Deformation

Transcription

1 Localization in Undrained Deformation J. W. Rudnicki Dept. of Civil and Env. Engn. and Dept. of Mech. Engn. Northwestern University Evanston, IL Fourth Biot Conference on Poromechanics, Columbia University, June 8 1, 29

2 Background (Rice, JGR, 1975) σ σ=constant τ y h/2 h/2 x dτ 1 dγ = + [ dτ μd( σ p) ] G H 1 β dε = d( σ p) + dτ μd( σ p) M H p p d ε = βd γ [ ] Equilibrium: τ / y = ε Fluid mass conservation: ( q / ρ ) + = y t and Darcy's Law: q p = ρκ y p ε κ = y y t

3 Geometric Interpretation of Constitutive Parameters τ dε p p = βdγ > (dilation) τ p dγ 1 μ > p dγ ( ) H = H /1 + H / G tan yield surface elastic response p - dε (compaction) 1 μ, μ< G σ = constant σ (compression) γ

4 undrained (βμ > ) response curve τ / τ inc σ p = const, μ >, β > dec σ p = const, μ<, β < γ / γ drained (σ p = const) response curve

5 τ / τ 4 3 Localization predicted to occur here for undrained response BUT γ / γ Localization predicted to occur here for drained response. Rice showed that small spatial perturbations grow exponentially in time beginning here

6 βμ < (compaction on frictional y.s. or dilation on cap y. s.) τ / τ 2 drained response curve 1 inc σ p = const, μ <, β > dec σ p = const, μ>, β < undrained response curve γ / γ

7 Layer Problem is special because Localization predicted to occur at peak of both drained and undrained response curves Band formation is parallel to the layer boundaries More generally, there is a strong dependence on deviatoric stress state (Lode angle).

8 Generalize: Constitutive Relation (drained, p = constant) { } el dε = dσ ν /(1 + ν) δ dσ /2G ij ij ij kk in 1 sij 1 skl 1 dε = ij + βδ μδ dσ ij kl kl h 2τ 3 + 2τ 3 ( ) τ = ( ss /2), s = σ σ / 3 δ 1/2 ij ij ij ij kk ij μ is a friction coefficient = dτ tan ( /3) yield ( σ ) / dσ, σ = /3 in in in in in β is a dilatancy factor = dε / dγ, dγ = (2 de de ) de = dε dε δ in in in ij ij kk ij in h= dτ / dγ, σ = constant h = dτ / dγ = h/(1 + h/ G) kk ij ij kk 1/2

9 Localization condition yields h crit and θ band τ G σ = ( ) h = h /1 + h / G tan crit crit constant γ σ 3 θ band σ 1 h / crit G axisymmetric extension sinθLode θ band axisymmetric compression sinθLode β =.3, μ =. 6 β =. 3, μ =.6 β =.3, μ =. 6 β =.3, μ =.6

10 Questions How does band angle predicted for undrained deformation differ from prediction for drained? How it depend on deviatoric stress state (Lode angle)? How do critical hardening moduli for drained and undrained deformation depend on stress state? Will also turn out that predictions for undrained deformation depend strongly on poroelastic parameters.

11 Undrained Response has the same form as drained if (Rudnicki, 1985) (i) the elastic portion of strain increments can be described by linear, isotropic poroelasticity; (ii) the role of the pore pressure in the inelastic strain increments is included by replacing the stress by the Terzaghi form of the effective stress, σ σ p (iii) the inelastic increment in the apparent void volume fraction is equal to the inelastic volume strain increment. dv in = dε in Rice [1977] has argued on theoretical grounds 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 surfaces with small real areas of contact. In addition, (ii) seems to be supported by experiments.

12 To get the undrained response from the drained, make the following substitutions in expression for inelastic strain increments: ( μβ, ) (1 B) ( μβ, ) h H = h+ ( μβ KB/ ς ) where ( Δσ ) = Δp/ /3 kk ς = 1 K / K, Biot's coefficient (often α) s where K is drained elastic bulk modulus for porous solid K is related to bulk modulus of solid constituents s B = Skempton's coefficient, For elastic strain increments replace Poisson s ratio by its undrained value: Note: for B = 1 (incompressible solid and fluid constituents, soil mechanics approx), effective value of μ and β vanish; i.e., inelastic pressure dependence and volume change vanish (Runesson et al., 1998) ν u 3 ν + (1 2 ν) Bς 1 =,, for B = 1, ς = 1 3 (1 2 ν) Bς 2

13 Dependence on Lode angle and poroelastic parameters for μ =.6, β = Band Angle axisymmetric extension drained B =.8, ζ =.8 B =.6, ζ =.8 B =.4, ζ = N = 2 sin(θ Lode ) crit hardening modulus/g axisymmetric compression μβkb / ς h crit /G, drained h crit und /G, undrained H crit /G, undrained N = 2 sin(θ Lode ) Note: localization condition is met for undrained response before drained response h crit /G = critical hardening modulus for drained deformation H crit /G = critical hardening modulus for undrained deformation h crit und /G = critical hardening modulus of underlying drained response curve at undrained response satisfies localization condition.

14 Dependence on Lode angle and poroelastic parameters for μ =.6, β = -.3 Band Angle axisymmetric extension drained B=.4, ζ=.6 B=.6, ζ=.8 B=.8, ζ= N=2 sin(θ Lode ) crit hardening modulus/g axisymmetric compression μβkb / ς h crit /G, drained h crit und /G, undrained H crit /G, undrained N=2 sin(θ Lode ) h crit /G = critical hardening modulus for drained deformation H crit /G = critical hardening modulus for undrained deformation h crit und /G = critical hardening modulus of underlying drained response curve at undrained response satisfies localization condition.

15 Conclusions and Caveats Relation between localization conditions and band angle for drained and undrained deformation depend strongly on deviatoric stress state (Lode angle) and poroelastic parameters. Band angle decreases as solid and fluid constituents become less compressible. For a Drucker Prager (Rudnicki Rice) constitutive relation undrained response (and hence localization conditions) can be obtained directly by substitution (no need to do separate analysis) Relevance of undrained localization condition is unclear when it occurs after the drained condition. Analysis should really include possibility of spatial perturbations (as in Rice, 1975) for which local and global drainage conditions will differ.