WIPP-Drucker Model

Viscoplasticity is also modeled in FLAC3D or 3DEC by combining the viscoelastic WIPP model with the elasto-plastic Drucker-Prager model. Of the plasticity models currently embodied in FLAC3D/3DEC, the Drucker-Prager model is the most compatible with the WIPP model, because both models are formulated in terms of the second invariant of the deviatoric stress tensor. Viewed in the π-plane, both models exhibit responses that depend only on radial distance from the isotropic-stress locus. The response of the Mohr-Coulomb model, on the other hand, is not isotropic, because the intermediate principal stress does not enter into its formulation.

The following development is slightly different from that presented in the Drucker-Prager model, and is provided to demonstrate the compatibility of the Drucker-Prager formulation with the creep formulation given in the WIPP-reference creep model.

The shear yield function for the Drucker-Prager model is

(1)fs=τ+qϕσkϕ

where fs = 0 at yield, σ=σkk/3, and τ=J2, where J2 is the second invariant of the deviatoric stress tensor. Parameters qϕ and kϕ are material properties.

Because

(2)J2=σdijσdij/2

(e.g., see Malvern 1969, p. 93) τ may be related to the stress magnitude, ˉσ, defined by ˉσ=3σdijσdij/2:

(3)ˉσ=3τ

The plastic potential function in shear, gs, is similar to the yield function, with the substitution of qψ for qϕ as a material property that controls dilation (similar to the Drucker-Prager model).

(4)gs=τ+qψσ

If the yield condition (fs = 0) is met, the following flow rules apply:

(5)˙ϵdpij=λgsσdij
(6)˙ϵp=λgsσ

where λ is a multiplier (not a material property) to be determined from the requirement that the final stress tensor must satisfy the yield condition. Superscript p denotes “plastic,” and d denotes “deviatoric.” By differentiating Equations (2) and (6), we obtain

(7)˙ϵdpij=λσdij2τ
(8)˙ϵp=λqψ

In this elastic/plastic formulation, these equations are solved simultaneously with the condition fs = 0 and the condition that the sum of elastic and plastic strain rates must equal the applied strain rate.

This implementation of the Drucker-Prager model also contains a tensile yield surface, with a composite decision function used near the intersection of the shear and tensile yield functions. The tensile yield surface is

(9)ft=σσt

where σt is the tensile yield strength. The associated plastic potential function is

(10)gt=σ

Using a similar approach to that for shear yield, the strain rates for tensile yield are

(11)˙ϵdpij=0
(12)˙ϵp=λ

where λ is determined from the condition that ft = 0. Note that the tensile strength cannot be greater than the value of mean stress at which fs becomes zero (i.e., σt<kϕ/qϕ).

When both creep and plastic flow occur, we assume that the associated strain rates act “in series”:

(13)˙ϵdij=˙ϵdeij+˙ϵdvij+˙ϵdpij

where the terms represent elastic, viscous, and plastic strain rates, respectively. We first treat the case of shear yield, fs > 0. Extending Equation (13), we obtain

(14)˙ϵdij=˙σdij2G+σdij2ˉσ{3˙ϵ+3λ}

In contrast to the creep-only model, the volumetric response of the viscoplastic model is not uncoupled from the deviatoric behavior unless qψ = 0, so

(15)˙ϵkk=3˙ϵ=˙σkk3K+λqψ

The iteration procedure embodied in the creep solution scheme can be extended to include plastic strain increments. We have

(16)σdij=σdij+2GΔt[˙ϵdijσdij2ˉσ(3˙ϵ+3λ)]

And Equation (15) becomes

(17)σ=σ+(˙ϵkkλqψ)KΔt

The value of λ can be adjusted in each iteration so that the solution converges to fs = 0. Using Newton’s method for roots,

(18)λ=λfs(fsλ)

Note that fs is evaluated with “new” stress components, σdij. The derivative in Equation (18) can be evaluated:

(19)fsλ=fsσdijσdijλ+fsσσλ

Hence,

(20)fsλ=GΔtKqϕqψΔt

assuming that the mean stress components (σdij and ˉσ) are constant.

For tensile yield, σ>σt. Further, if the shear stress is nonzero, the following function is used to decide if shear or tensile yield is occurring:

h=ττpαp(σσt)

where

τp=kϕqϕσt
αp=1+qϕ2qϕ

Tensile yield is declared if h < 0; otherwise, shear yield occurs. In the former case, the last (plastic) term of Equation (13) is zero, and the value of λ is such that Equation (17) reduces to

σ=σt

In order to include softening behavior, an accumulated plastic strain, ϵdp, is computed, based on the second invariant of the deviatoric strain-increment tensor, as follows:

(21)ϵdp:=ϵdp+Δt˙ϵdpij˙ϵdpij/2

There is no built-in support for softening tables, but a FISH function that scans the grid every few steps and recomputes properties based on the current value of ϵdp may be written (see WIPP-Drucker Model: Compression Test Showing Localization for an illustration of this technique).



wipp-drucker model properties

Use the following keywords with the zone property (FLAC3D) or zone property (3DEC) command to set these properties of the WIPP-Drucker model.

wipp-drucker
activation-energy f

activation energy, Q

bulk f

bulk modulus, K

cohesion-drucker f

material parameter, kϕ

constant-a f

WIPP model constant, A

constant-b f

WIPP model constant, B

constant-d f

WIPP model constant, D

constant-gas f

gas constant, R

creep-rate-critical f

critical steady-state creep rate, ε̇ss

dilation-drucker f

material parameter, qk

exponent f

WIPP model exponent, n

friction-drucker f

material parameter, qϕ

temperature f

zone temperature, T

poisson f

Poisson’s ratio, v

shear f

shear modulus, G

tension f

tension limit, σt. The default is 0.0.

young f

Young’s modulus, E

creep-strain-primary f (r)

primary creep strain, ε̇s

creep-rate-primary f (r)

primary creep rate, εs

strain-shear-plastic f (r)

accumulated plastic shear strain

strain-tension-plastic f (r)

accumulated plastic tensile strain

Key

(r) Read-only property.
This property cannot be set by the user. Instead, it can be listed, plotted, or accessed through FISH.

Notes:

  • Only one of the two options is required to define the elasticity: bulk modulus K and shear modulus G, or Young’s modulus E and Poisson’s ratio v.
  • The tension cut-off is σt = min (σt, kϕ/qϕ)
  • The creep behavior is triggered by deviatoric stress, while the volumetric behavior does not consider creep.