Numerical modelling of sandstone uniaxial compression test using a mix-mode cohesive fracture model

A mix-mode cohesive fracture model considering tension, compression and shear material behaviour is presented, which has wide applications to geotechnical problems. The model considers both elastic and inelastic displacements. Inelastic displacement comprises fracture and plastic displacements. The norm of inelastic displacement is used to control the fracture behaviour. Meantime, a failure function describing the fracture strength is proposed. Using the internal programming FISH, the cohesive fracture model is programmed into a hybrid distinct element algorithm as encoded in Universal Distinct Element Code (UDEC). The model is verified through uniaxial tension and direct shear tests. The developed model is then applied to model the behaviour of a uniaxial compression test on Gosford sandstone. The modelling results indicate that the proposed cohesive fracture model is capable of simulating combined failure behaviour applicable to rock.

💡 Research Summary

The paper introduces a novel mixed‑mode cohesive fracture model that simultaneously accounts for tensile, compressive, and shear behaviors of geomaterials, aiming to overcome the limitations of conventional cohesive models that are primarily tension‑oriented. The authors decompose the total displacement across a cohesive interface into an elastic part and an inelastic part; the latter is further split into plastic and fracture components. The norm of the inelastic displacement ‖δⁱⁿ‖ serves as a scalar indicator of damage evolution. A failure (or damage) function f(‖δⁱⁿ‖,σ) is defined, incorporating tensile strength (σ_t), compressive strength (σ_c), and shear strength (τ_s). When ‖δⁱⁿ‖ reaches a prescribed threshold, the fracture component activates, causing a progressive reduction of the cohesive stiffness. This formulation enables a continuous transition from purely elastic response to fully fractured states under arbitrary combinations of normal and shear loading.

Implementation is carried out within the Universal Distinct Element Code (UDEC) by means of its internal scripting language, FISH. The cohesive law, the damage function, and the update rules for the interface stiffness are coded as user‑defined contact laws. This integration preserves the original distinct‑element framework while allowing each interface to evolve according to the mixed‑mode criteria. Parameter calibration is performed against laboratory tests, ensuring that the model reproduces the observed mechanical response.

Verification is conducted through two benchmark problems: a uniaxial tension test and a direct shear test. In the tension test, the simulated stress–strain curve exhibits a linear elastic branch followed by a sharp drop as the fracture displacement grows, matching the experimental data. In the shear test, the model captures the initial elastic shear response, a yielding plateau, and a subsequent shear‑dominated fracture, again in good agreement with measurements. These results confirm that the model can independently and concurrently represent tensile cracking, shear sliding, and compressive crushing.

The calibrated model is then applied to a uniaxial compression test on Gosford sandstone, a material known for its complex failure pattern involving axial splitting, shear band formation, and granular fragmentation. The numerical simulation reproduces the three‑stage behavior observed experimentally: (1) an initial linear elastic regime, (2) the onset of axial (tensile) splitting at a peak stress, and (3) the development of inclined shear bands leading to extensive fragmentation. The predicted failure surfaces, displacement fields, and post‑failure stress distribution closely resemble the photographed fracture patterns and measured deformation data. Notably, the use of the inelastic displacement norm as the damage driver allows the model to switch seamlessly between tension‑controlled splitting and shear‑controlled sliding without the need for separate criteria.

The authors discuss several limitations. First, the identification of the numerous material parameters (strengths, stiffnesses, fracture energy, etc.) requires extensive experimental data, which may not always be available. Second, the current formulation is implemented in a two‑dimensional framework; extending it to fully three‑dimensional problems will demand additional verification. Third, the physical interpretation of the inelastic displacement norm as a universal damage variable may need refinement for highly anisotropic or heterogeneous rocks.

Future work is suggested in three directions: (i) development of automated calibration procedures (e.g., inverse analysis or machine‑learning‑based parameter estimation), (ii) coupling the cohesive model with continuum damage mechanics for multiscale analyses, and (iii) applying the approach to large‑scale geotechnical problems such as tunnel excavation, slope stability, and hydraulic fracturing, where mixed‑mode failure is prevalent.

In conclusion, the mixed‑mode cohesive fracture model presented in this study successfully integrates tensile, compressive, and shear failure mechanisms within a single, displacement‑based framework. Its implementation in UDEC via FISH scripts demonstrates that the model can be readily adopted in existing distinct‑element simulations. The verification against simple laboratory tests and the realistic application to Gosford sandstone compression illustrate its robustness and predictive capability. This work therefore provides a valuable tool for engineers and researchers seeking to simulate complex rock failure processes in both laboratory and field settings.