Understanding the fracture mechanism of ring Brazilian disc specimens by the phase field method
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ORIGINAL PAPER
Understanding the fracture mechanism of ring Brazilian disc specimens by the phase field method Xiaoping Zhou Yundong Shou
· Longfei Wang ·
Received: 30 April 2020 / Accepted: 7 August 2020 © Springer Nature B.V. 2020
Abstract Ring Brazilian disc specimens are favored for determining the tensile strength and mixed mode fracture toughness. To further understand the fracture mechanism of ring Brazilian disc specimens, the phase field method is used to investigate the cracking process and peak load of ring Brazilian disc specimens. First, the numerical validity and accuracy of the phase field method is verified by a benchmark example. Then, the effect of aperture ratio and crack inclination angle on the failure process and peak load of ring Brazilian disc specimens is studied. Finally, by combining the phase field method and J -integral method, the influence of prefabricated crack inclination angle and aperture ratios on mode I and II fracture toughness of cracked ring Brazilian disc specimens is discussed. Keywords The phase field method · Ring Brazilian disc · Crack propagation · Fracture toughness
Abbreviations ∂
An arbitrary domain External boundary
X. Zhou (B) · L. Wang School of Civil Engineering, Chongqing University, Chongqing 400045, People’s Republic of China e-mail: [email protected] Y. Shou School of Civil Engineering, Wuhan Univeristy, Wuhan 430072, P. R. China
u¯ ∂u t¯ ∂t d int ext E(ε, d) W (d) ψ(ε) gc l0 γ (d, ∇d) u ε ε+ ε− εa na ψ+ ψ− b σ H YI t GFEM DDA
Internal discontinuity boundary Preserved displacement Displacement boundary Preserved traction Traction boundary Phase field Total potential energy Internal potential energy External potential energy Elastic energy Fracture energy Elastic strain energy density function Critical energy release rate Length scale parameter Crack surface density Displacements vector Strain tensor Tensile strain tensor Compressive strain tensor Principal strain Principal strain direction Elastic density caused by tension Elastic density caused by compression Prescribed volume force Cauchy stress tensor History variable Mode I dimensionless SIF Specimen thickness Generalized finite element method Discontinuous deformation analysis
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X. Zhou et al.
DDM PFC PFM SCB Ni Biu Bid Ru Rd K nu K nd D H ε (x) P± J R 2a k β E v P Ri Ro λ P∗ a b c K Ic K IIc SIFs YI I XFEM DEM PD GPD CZM CCCD
Displacement discontinuity method Particle flow code Phase field method Notched semi- circular bending Shape function associated with the ith node Strain gradient matrix of the node i Cartesian derivative matrices of the node i Displacement residue vector Phase filed residue vector Displacement tangent matrix Phase field tangent matrix Fourth-order elasticity tensor Heaviside function Fourth-order projection tensor Fourth-order tensor Radius of CCCD specimen Crack length of CCCD specimen Numerical smoothing coefficient Crack inclination angles Young’s modulus Poisson’s ratio Diametric concentrated forces Inner diameter of ring Brazilian disc Outer di
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