Study on crack growth behaviour in rocks having pre-existing narrow flaws under biaxial compression

Abstract

The analysis of crack growth on a rock with pre-existing flaws is mostly carried out in open flaw configurations whereas limited study can be found for narrow flaw configurations. Unlike open flaw, narrow flaws are initially open and subsequently gets closed during loading. It also experiences friction which needed to be accounted for. Very few studies are available, to understand the effect of friction between the flaw surfaces. Additionally, the lateral confinement due to the biaxial stress state would influence the crack initiation and their subsequent growth. In the present study, an experimental investigation is conducted to understand the crack growth behaviour of narrow flaws under biaxial stress environments. A novel biaxial compression test setup is developed using conventional triaxial cells. The tests were conducted using gypsum having uniaxial compressive strength of 15.5 MPa and Young’s modulus of 1.037 GPa, with a single flaw. The specimen’s size effect due to lateral confinement is investigated by varying the ratio of the specimen width to flaw length. The change in the behaviour of crack growth and its crack stresses with respect to initiation and peak level was studied for different flaw angles. Subsequently, the numerical analysis was performed using the Extended Finite Element Method (XFEM) in conjunction with the Cohesive Zone Model (CZM) systematically validated with laboratory experiments. Analysis was further extended for higher confinement levels to know the effect of crack patterns and their stress behaviour. The study observed a change in critical flaw angle at which sooner initiation happens when the confinement increases. Hence, there was a change in crack initiation stress and their corresponding peak stress behaviour with respect to flaw angles when the lateral confinement increases. Also, the crack pattern was found to be influenced by various factors such as the ratio specimen width to flaw length, frictional coefficient between the flaw surface and the confinement stress level.

Appendix: Numerical validation of biaxial plane stress condition

The rubber foam membrane is composed of closed pores embedded in the Polyurethane (PU) matrix, which can undergo full compressibility with significantly lesser expansion in the lateral direction (Kim et al. 2019). When the specimen is subjected to loading in one direction, it experiences deformation in the other two directions, where the lateral deformation is restrained using acrylic plated in one of the directions. Normally, the current setup without rubber foam experiences a stress concentration at the interface between the gypsum. When the rubber foam is placed between the acrylic and gypsum, strain undergone by the specimen gets accumulated on the membrane, hence no stress is transferred in that direction. Hence, it is necessary to monitor how much stress is observed on the acrylic sheet for achieving the plane stress condition. However, there is a complication in monitoring the setup under high confined air pressure within the cell. Therefore, an additional numerical analysis is performed to check the effect of plane stress state conditions using a rubber foam membrane.

The numerical model is simulated as a two-dimensional problem for the biaxial setup of cross-sectional view as shown in Fig. 

Fig. 25

a Cross-section of the biaxial setup b Mesh geometry c Boundary condition adopted and d Interaction used between gypsum, rubber foam and PMMA

25. The model consists of a gypsum specimen with acrylic plates fixed which is restrained in the lateral direction and rubber foam is inserted between the acrylic and gypsum on both sides. The length of Acrylic and Rubber foam is 180 mm and their respective thickness is 26 mm and 5 mm. For gypsum, the length of the material is taken as 150 mm and its thickness 28 mm. A 4-node bilinear plane strain quadrilateral element is adopted for all three materials. The analysis is solved in quasi-static loading by a reduced integration method with an hourglass control technique. Based on mesh optimization, a coarser mesh is chosen for PMMA and a finer mesh for gypsum material and rubber foam. The boundary condition of the model is shown in Fig. 25c, where the direction of ‘x’ and ‘y’ movement is restricted in PMMA. For gypsum material, the top end of the specimen is fixed at the y-direction and at the bottom end where loading is applied in the y-direction. No boundary condition is given for rubber foam material since the material is allowed for compression during loading.

The material input parameter used in the numerical model is listed in Table 3. The constitutive model assigned for gypsum is CZM with XFEM and for PMMA it is assumed as a linear elastic material. In the case of rubber foam, the hyper foam material is adopted which simulates highly compressible material having large deformation characteristics (Briody et al. 2012; Kim et al. 2019). The material property for rubber foam is calibrated with uniaxial test data from the experiment study, where their stress–strain responses are compared (Fig. 26). The test is performed on rubber foam membrane at a strain rate of 10^–3 s⁻¹ (Kim et al. 2019) upto 0.8% strain of axial thickness of the material (Fig. 26). A tie constraint (‘○’ symbol) is given at the interaction between the PMMA and rubber foam replicating the glueing performance. A surface to surface contact type (‘□’ symbol) is assumed between gypsum and rubber foam with a friction coefficient value of 0.68 based on literature (Fig. 25d).

Table 3 Material parameters adopted in the numerical model

Fig. 26

a Open cell rubber foam used for performing uniaxial loading test and b Experimentally and numerically obtained stress versus strain plot obtained for uniaxial loading test

The result of the failure pattern obtained at numerical analysis is shown in Fig. 27(a and b). In both the analyses, the rubber foam membrane was observed to be deforming on the interaction with gypsum and variation in stress was observed in the gypsum specimen. The stress distribution of PMMA is monitored on both left and right of the specimen and the variation of stress along with the height of PMMA is in Fig. 28.

Fig. 27

Final failure pattern obtained under two different confinement loadings a 0.65 MPa and b 1.25 MPa

Fig. 28

a Stress observed in ‘x direction’ on acrylic from rubber foam membrane for 0.65 MPa and 1.25 MPa on b left side and c right side of PMMA

The plot shows the stress in the x-direction that is exhibited at the acrylic sheet for both the adopted confinements. The stress is plotted with respect to nodal points observed along the vertical direction of the PMMA material. From the plot, it can be noted that the maximum stress value of 1.37 kPa acting at acrylic is significantly lower than the confinement applied 1.24 MPa. Hence, it is evident that the adopted rubber foam can absorb the stress that is exhibited in the third direction of the specimen during biaxial loading. Therefore the proposed setup relatively achieves its plane stress state condition in the present study.

See Figs. 25, 26, 27 and 28 and Table 3.