Understanding the fracture mechanism of ring Brazilian disc specimens by the phase field method


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.

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Data availability

Data supporting this research article are available from the corresponding author via e-mail.


\(\Omega \) :

An arbitrary domain

\(\partial \Omega \) :

External boundary

\(\Gamma \) :

Internal discontinuity boundary

\({\bar{\mathrm{u}}}\) :

Preserved displacement

\(\partial \Omega _{\text {u}}\) :

Displacement boundary

\({\bar{t}}\) :

Preserved traction

\(\partial \Omega _{\text {t}}\) :

Traction boundary

d :

Phase field

\(\Pi \) :

Total potential energy

\(\Pi ^{\text {int}}\) :

Internal potential energy

\(\Pi ^{\text {ext}}\) :

External potential energy

\(E ({\varvec{\varepsilon }}, d)\) :

Elastic energy


Fracture energy

\(\psi ({\varvec{\varepsilon }})\) :

Elastic strain energy density function

\(g_{\text {c}}\) :

Critical energy release rate

\(l_{0}\) :

Length scale parameter

\(\gamma (d, \nabla d)\) :

Crack surface density

u :

Displacements vector

\({\varvec{\varepsilon }}\) :

Strain tensor

\({\varvec{\varepsilon }}^{+}\) :

Tensile strain tensor

\({\varvec{\varepsilon }}^{-}\) :

Compressive strain tensor

\(\varepsilon _{a}\) :

Principal strain

\({{\varvec{n}}}_{a}\) :

Principal strain direction

\(\psi ^{+}\) :

Elastic density caused by tension

\(\psi ^{-}\) :

Elastic density caused by compression

b :

Prescribed volume force

\({\varvec{\sigma }}\) :

Cauchy stress tensor

H :

History variable

\(Y_{I}\) :

Mode I dimensionless SIF

t :

Specimen thickness


Generalized finite element method


Discontinuous deformation analysis


Displacement discontinuity method


Particle flow code


Phase field method


Notched semi- circular bending

\({{\varvec{N}}}_{i}\) :

Shape function associated with the ith node

\({\text {B}}_i^u\) :

Strain gradient matrix of the node i

\({\text {B}}_i^d\) :

Cartesian derivative matrices of the node i

\({{\varvec{R}}}^{\text {u}}\) :

Displacement residue vector

\({{\varvec{R}}}^{\text {d}}\) :

Phase filed residue vector

\(K_n^u\) :

Displacement tangent matrix

\(K_n^d\) :

Phase field tangent matrix

\({{\mathbb {D}}}\) :

Fourth-order elasticity tensor

\({{\varvec{H}}}_{\upvarepsilon } (x)\) :

Heaviside function

\({{\mathbb {P}}}^{\pm }\) :

Fourth-order projection tensor

\({{\mathbb {J}}}\) :

Fourth-order tensor

R :

Radius of CCCD specimen

2a :

Crack length of CCCD specimen

k :

Numerical smoothing coefficient

\(\beta \) :

Crack inclination angles

E :

Young’s modulus

v :

Poisson’s ratio

P :

Diametric concentrated forces

\(R_{\text {i}}\) :

Inner diameter of ring Brazilian disc

\(R_{\text {o}}\) :

Outer diameter of ring Brazilian disc

\(\lambda \) :

Ratio of inner radius to outer radius

\(P^{*}\) :

Dimensionless peak load

a :

Fitting parameter

b :

Fitting parameter

c :

Fitting parameter

\(K_{\text {Ic}}\) :

Mode I fracture toughness

\(K_{\text {IIc}}\) :

Mode II fracture toughness


Stress intensity factors

\(Y_{II}\) :

Mode II dimensionless SIF


Extended finite element method


Discrete element method




General particle dynamics


Cohesive zone model


Centrally cracked circular disc


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The work is supported by the National Natural Science Foundation of China (Nos. 51839009 and 51679017).

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Correspondence to Xiaoping Zhou.

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Zhou, X., Wang, L. & Shou, Y. Understanding the fracture mechanism of ring Brazilian disc specimens by the phase field method. Int J Fract 226, 17–43 (2020). https://doi.org/10.1007/s10704-020-00476-w

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  • The phase field method
  • Ring Brazilian disc
  • Crack propagation
  • Fracture toughness