Convergence in non‐associated plasticity and fracture propagation for standard, rate‐dependent, and Cosserat continua

Hageman, Tim; Sabet, Sepideh Alizadeh; Borst, René de

doi:10.1002/nme.6561

Cited by 8 publications

(12 citation statements)

References 50 publications

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“…The second case which we consider is a fracture propagating due to shear stresses, while being limited by inertial and visco-plastic effects in the domain. 35,36 This case consists of a 500 × 250m domain on which external forces are applied (𝜏 xx = 8.55MPa, 𝜏 yy = 10MPa, 𝜏 xy = 1.8MPa). To prevent stress waves from reflecting from the domain boundaries, absorbing boundary conditions are employed.…”

Section: Resultsmentioning

confidence: 99%

A time‐based arc‐length like method to remove step size effects during fracture propagation

Hageman

Borst

2021

Numerical Meth Engineering

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An arc-length like method is presented which alters the size of the time increment when simulating crack propagation problems. By allowing the time increment to change during the time step a constraint can be imposed, which is used to enforce the fracture to propagate a single element length per time step. This removes the effect of the (interface) element size on propagating fractures, and therefore allows smooth fracture propagation during the simulation.The benefits of the scheme are demonstrated for three cases: mode-I crack propagation in a double cantilever beam, a shear fracture including inertial and viscoplastic effects in the surrounding material, and a pressurized fracture inside a poroelastic material. These cases highlight the ability of this scheme to obtain more accurate and nonoscillatory results for the force-displacement relation, to remove numerically induced stepwise fracture propagation, and to allow for arbitrary propagation velocities. An added benefit is that plastic strains surrounding a fracture are no longer affected by the (interface) element size.

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Section: Resultsmentioning

confidence: 99%

A time‐based arc‐length like method to remove step size effects during fracture propagation

Hageman

Borst

2021

Numerical Meth Engineering

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“…A Cosserat continuum introduces an internal length scale

ℓ_{c}

. It removes the ill‐posedness and the ensuing the mesh dependence caused by the nonassociated plasticity, 46–49 and maintains the quadratic convergence of Newton–Raphson schemes used to solve nonlinear boundary value problems 50 …”

Section: Governing Equationsmentioning

confidence: 99%

“…A nonassociated Drucker–Prager plasticity model is used to model the plastic deformations, with the yield function and plastic potential defined as:

f = \sqrt{3 J_{2}} + α_{s} p_{s} - k_{s} g = \sqrt{3 J_{2}} + β_{s} p_{s}

with

p_{s}

the solid pressure (tension positive),

J_{2}

the second invariant of the deviatoric stresses in a Cosserat continuum 45,54 and

α_{s}

β_{s}

, and

k_{s}

related to the angle of internal friction

ϕ

, dilatancy angle

ψ

, and cohesion

c

through:

α_{s} = \frac{6 \sin ϕ}{3 - \sin ϕ} β_{s} = \frac{6 \sin ψ}{3 - \sin ψ} k_{s} = c \frac{6 \cos ϕ}{3 - \sin ϕ} .

More implementation details related to the integration of the plastic strain and the used tangential stiffness matrix are given in Ref. 50.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Using the discretization from Equations (21)–(26), the momentum balance from Equation (10) is discretized as:

{bold-italicf}_{e x t} - {bold-italicf}_{i n t} - {bold-italicf}_{d} = bold0

with the external force vector defined in a standard manner as:

{bold-italicf}_{e x t} = \int_{Γ_{t}} {bold-italicN}_{s}^{T} \bar{t} + {bold-italicN}_{s}^{T} {bold-italicR}^{T} S R {bold-italicN}_{s} (\frac{γ}{β normalΔ t} (() {bold-italicu}^{t + Δ t} - {bold-italicu}_{t}) - (() \frac{γ}{β} - 1) {\dot{bold-italicu}}^{t} - (() \frac{Δ t γ}{2 β} - Δ t) {\ddot{bold-italicu}}^{t}) d Γ,

where

\bar{boldt}

is the traction imposed on the boundary of the domain, and

bold-italicS

is the damping matrix used for the absorbing boundary conditions: 50,71

S = {bold-italicρ}_{s} [\begin{matrix} c_{p} & 0 & 0 \\ 0 & c_{s} & 0 \\ 0 & 0 & 0 \end{matrix}],

using the pressure and shear stress wave speeds

c_{p}

and

c_{s}

. The internal force vector is given by:

\begin{matrix} true \\ right {bold-italicf}_{i n t} & left = \int_{Ω} B^{T} <...> \end{matrix}

…”

Section: Governing Equationsmentioning

confidence: 99%

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Stick‐slip like behavior in shear fracture propagation including the effect of fluid flow

Hageman

Borst

2021

Num Anal Meth Geomechanics

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Shear‐based fracture propagation in fluid‐saturated porous materials is investigated using a displacement–pressure formulation that includes acceleration and inertial effects of the fluid. Pressure‐dependent plasticity with a nonassociated flow rule is adopted to realistically represent the stresses in the porous bulk material. The domain is discretized using unequal order T‐splines and cast into a finite element method using Bézier extraction. An implicit scheme is used for the temporal integration. The solid acceleration‐driven fluid flow reacts to stress waves, but it results in pressure oscillations. Adding fluid acceleration terms dampens these oscillations and increases the fluid pressure near the fracture tips. By simulating a typical shear fracture case, it is shown that stick‐slip like, or stepwise, fracture propagation occurs for a high permeability, also upon mesh refinement. The acceleration driven fluid flow results in a build‐up of pressure near the fracture tip. Once this pressure region encompasses the fracture tip, propagation arrests until the pressure has diffused away from the crack tip, after which propagation is resumed and the build‐up of pressure begins anew. This results in a stick‐slip like behavior, with large arrests in the fracture propagation. Stepwise propagation related to the initial conditions has also been observed, but disappears once the fracture length exceeds the size of the region influenced by the initial conditions.

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“…While these works have mainly focused on material softening, surprisingly, little attention has been given to the regularization of nonassociative models. Recent works [26,38,52] on this topic employ rate-dependent and/or Cosserat contin-uum models to obtain mesh-objectivity in non-associative plasticity. Another instance can be found in Ulloa et al [53], where gradient-enhanced plasticity is considered for the same purpose.…”

Section: Introductionmentioning

confidence: 99%

A micromechanics-based variational phase-field model for fracture in geomaterials with brittle-tensile and compressive-ductile behavior

Ulloa,

Wambacq,

Alessi

et al. 2021

Preprint

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This paper presents a framework for modeling failure in quasi-brittle geomaterials under different loading conditions. A micromechanics-based model is proposed in which the field variables are linked to physical mechanisms at the microcrack level: damage is related to the growth of microcracks, while plasticity is related to the frictional sliding of closed microcracks. Consequently, the hardening/softening functions and parameters entering the free energy follow from the definition of a single degradation function and the elastic material properties. The evolution of opening microcracks in tension leads to brittle behavior and mode I fracture, while the evolution of closed microcracks under frictional sliding in compression/shear leads to ductile behavior and mode II fracture. Frictional sliding is endowed with a non-associative law, a crucial aspect of the model that considers the effect of dilation and allows for realistic material responses with non-vanishing frictional energy dissipation. Despite the non-associative law, a variationally consistent formulation is presented using notions of energy balance and stability, following the energetic formulation for rate-independent systems. The material response of the model is first described, followed by several benchmark finite element simulations. The results highlight the ability of the model to describe tensile, shear, and mixed-mode fracture, as well as responses with brittle-to-ductile transition. A key result is that, by virtue of the micromechanical arguments, realistic failure modes can be captured, without resorting to the usual heuristic modifications considered in the phase-field literature. The numerical results are thoroughly discussed with reference to previous numerical studies, experimental evidence, and analytical fracture criteria.

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Convergence in non‐associated plasticity and fracture propagation for standard, rate‐dependent, and Cosserat continua

Cited by 8 publications

References 50 publications

A time‐based arc‐length like method to remove step size effects during fracture propagation

A time‐based arc‐length like method to remove step size effects during fracture propagation

Stick‐slip like behavior in shear fracture propagation including the effect of fluid flow

A micromechanics-based variational phase-field model for fracture in geomaterials with brittle-tensile and compressive-ductile behavior

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