Combined Viscoplasticity-Embedded Discontinuity Model for 3D Description of Rock Failure Under Dynamic Loading
- PDF / 2,803,850 Bytes
- 16 Pages / 595.276 x 790.866 pts Page_size
- 83 Downloads / 199 Views
ORIGINAL PAPER
Combined Viscoplasticity‑Embedded Discontinuity Model for 3D Description of Rock Failure Under Dynamic Loading Timo Saksala1 Received: 27 April 2020 / Accepted: 10 July 2020 © The Author(s) 2020
Abstract This paper presents a combined viscoplasticity-embedded discontinuity model for 3D analyses of rock failure processes under dynamic loading. Capabilities of a rate-dependent embedded discontinuity model, implemented with the linear tetrahedral element, for mode I (tension) loading induced fractures is extended to compressive (shear) failure description by viscoplastic softening model with the Drucker–Prager yield criterion. The return mapping update formulas are derived for the corner plasticity case exploiting the consistency conditions for both models simultaneously. The model performance is demonstrated in 3D numerical simulations of uniaxial tension and compression test on a heterogeneous rock at various loading rates. These simulations corroborate the conception that the rate sensitivity of rock is a genuine material property in tension while structural (inertia) effects play the major role in compression at high loading rates (up to 1000 s−1). Finally, the model is validated with predicting the experiments of dynamic Brazilian disc test on granite. Keywords Rock fracture · Rate sensitivity · Embedded discontinuity FEM · Drucker–Prager viscoplasticity · Dynamic Brazilian disc test List of Symbols DEM Discrete-element method DIF Dynamic increase factor DP Drucker–Prager criterion EAS Enhanced assumed strains concept FEM Finite-element method c0 Cohesion E Elasticity tensor E Young’s modulus gDP Plastic potential gd, gc Softening slope control parameters GIc, GIIc Mode I and II fracture energy hd, hc Softening moduli for tensile and shear failure HΓd Heaviside function at discontinuity I1, J2 First and the second invariants of the stress tensor kDP Drucker–Prager parameter nd Discontinuity normal vector Ni Nodal interpolation function m1, m2 Discontinuity tangent vectors * Timo Saksala [email protected] 1
Civil Engineering, Tampere University, POB 600, 33101 Tampere, Finland
MΓd Auxiliary function for discontinuity qd, qc Softening function for discontinuity and plastic flow sd, sc Viscosity moduli for tensile and shear failure tΓd Traction vector for discontinuity 𝐮ei Nodal displacement vector x Placement vector αDP Drucker–Prager parameter αd Crack opening vector β Shear control parameter Γd Discontinuity plane 𝛿Γd Dirac’s delta at discontinuity 𝛆, 𝛆e , 𝛆vp Total, elastic and viscoplastic strain tensor 𝜅d , 𝜅̇ d Internal variable and its rate for discontinuity 𝜅c , 𝜅̇ c Internal variable and its rate for shear 𝜆̇ d , 𝜆̇ c Crack opening and plastic increment ν Poisson’s ratio ρ Material density 𝛔 Stress tensor σt, σc Tensile and compressive strength 𝜑 Friction angle 𝜑Γd Ramp function for discontinuity 𝜙d , 𝜙DP Yield criteria for discontinuity and shear ψ Dilation angle Ω±e Positive and negative sides of discontinuity
13
Vol.:(0123456789)
T.
Data Loading...
