Power-Mohr Model
This viscoplastic model cpower combines the behavior of the viscoelastic two-component power model and the elasto-plastic Mohr-Coulomb model. In the model formulation, the total strain rate ˙ϵij is decomposed into elastic (˙ϵeij), viscous (˙ϵcij), and plastic (˙ϵpij) components:
The elastic strain rate ˙ϵeij is the only component contributing to the stress rate; the deviatoric behavior is visco-elasto-plastic and is expressed as
where ˙Sij and ˙eij are the deviatoric parts of the stress and strain rate tensors, ˙σij and ˙ϵij, and G is tangent shear modulus (see the Burgers model for a definition of the notation convention used in this section).
The volumetric behavior is elasto-plastic and has the form
where ˙σ0=(˙σ11+˙σ22+˙σ33)/3, ˙evol=˙e11+˙e22+˙e33, and K is tangent bulk modulus.
Creep is activated by the von Mises stress q=√3J2 in accordance with the Norton Power Law (J2=1/2SijSij is the second invariant of stress deviatoric tensor), and the creep rate is
The direction of creep flow is derived from the definition of q:
By definition, the creep intensity has two components (see the power model):
where
and σref1 and σref2 are two model parameters.
The plastic strain rate is defined using the Mohr-Coulomb flow rule,
where
The direction of plastic flow, ∂g/∂σij, is expressed using the definition of the Mohr-Coulomb potential function, g, and the plastic flow rate intensity, ˙ep, is derived from the Mohr-Coulomb yield criterion f = 0 (see the Mohr-Coulomb model). In the principal axes formulation, the yield and potential functions for shear yielding are
and for tension yielding, the functions are
where σ1 and σ3 are the minimum and maximum principal stresses (compression negative), C is the material cohesion, ϕ is the friction, ψ is the material dilation, σt is the tensile strength, Nϕ=(1+sinϕ)/(1−sinϕ), and Nψ=(1+sinψ)/(1−sinψ).
The model implementation closely follows the procedures for the power model and the Mohr-Coulomb model described in the manual. First, the viscoelastic response is calculated for the timestep Δt. Principal stresses and principal directions are computed, and the yield criterion is checked. If the criterion is not met, plastic strain increments are added for the step, and the increment intensity λ=˙epΔt is calculated to fulfil the yield condition f = 0. The procedure follows the lines presented in the Mohr-Coulomb model implementation, with the viscoelastic response replacing the “elastic guess” for the step.
power-mohr Model Properties
Use the following keywords with the zone property
(FLAC3D) or zone property
(3DEC) command to set these properties of the power-Mohr model.
- power-mohr
- bulk f
elastic bulk modulus, K
- cohesion f
cohesion, c
- constant-1 f
power-law constant, A1
- constant-2 f
power-law constant, A2
- dilation f
dilation angle, ψ
- exponent-1 f
power-law exponent, n1
- exponent-2 f
power-law exponent, n2
- friction f
angle of internal friction, ϕ
- poisson f
Poisson’s ratio, v
- shear f
elastic shear modulus, G
- stress-reference-1 f
reference stress, σref1
- stress-reference-2 f
reference stress, σref2
- young f
Young’s modulus, E
- flag-brittle b (a)
If true, the tension limit is set to 0 in the event of tensile failure. The default is false.
Key
- (a) Advanced property.
- This property has a default value; simpler applications of the model do not need to provide a value for it.
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 creep behavior is triggered by deviatoric stress, while the volumetric behavior does not consider creep.
- The tension cut-off is σt = min (σt, c/tanϕ).
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