Size effect of fracture characteristics for anisotropic quasi-brittle geomaterials

Understanding the size effect exhibited by the fracture mechanism of anisotropic geomaterials is important for engineering practice. In this study, the anisotropic features of the nominal strength, apparent fracture toughness, effective fracture energy and fracture process zone(FPZ) size of geomaterials were first analyzed by systematic size effect fracture experiments. The results showed that the nominal strength and the apparent fracture toughness decreased with increasing bedding plane inclination angle.The larger the specimen size was, the smaller the nominal strength and the larger the apparent fracture toughness was. When the bedding inclination angle increased from 0° to 90°, the effective fracture energy and the effective FPZ size both first decreased and then increased within two complex variation stages that were bounded by the 45° bedding angle. Regardless of the inherent anisotropy of geomaterials,the nominal strength and apparent fracture toughness can be predicted by the energy-based size effect law, which demonstrates that geomaterials have obvious quasi-brittle characteristics. Theoretical analysis indicated that the true fracture toughness and energy dissipation can be calculated by linear elastic fracture mechanics only when the brittleness number is higher than 10;otherwise, size effect tests should be adopted to determine the fracture parameters.

Volume Change of Heterogeneous Quasi-brittle Materials in Uniaxial Compression

The volumetric strain was categorized into elastic and plastic parts. The farmer camposed of axial and lateral strains is uniform and determined by Hooke＇s law ; however, the latter consisting of axial and lateral strains is a fuaction af thickness af shear band determined by grndieat-dependeat plasticity by cansidering the heterngeneity of quasi- brittle materials. The non- uniform lateral strain due to the fact that shear band was farmed in the middle of specimen was averaged within specimen to precisely assess the volumetric strain. Then, the analytical expression for volumetric strain was verified by comparison with two earlier experimental results for concrete and rack. Finally, a detailed parametric study was carried out to investigate effects of constitutive parameters （ shear band thickness, elastic and softening rnoduli ） and geometrical size of specimen（ height and width of specimen ） on the volume dilatancy.

Investigation of mechanical behaviour of a quasi-brittle material using Karagozian and Case concrete (KCC) model

The mechanical behaviour of a quasi-brittle material,i.e.Pietra Serena sandstone,was investigated both numerically and experimentally in order to build a reliable numerical modelling system applicable to more complex cases.The Karagozian and Case concrete(KCC)model was exploited as the material constitutive law and a new method to utilise this model for efficient and accurate simulation of quasibrittle materials is discussed.The capability of this model is evaluated by comparing the results of the numerical simulations with the corresponding experimental results,and the method itself is critically assessed.

Multi-Scale Damage Model for Quasi-Brittle Composite Materials

In the present paper,a hierarchical multi-scale method is developed for the nonlinear analysis of composite materials undergoing heterogeneity and damage.Starting from the homogenization theory,the energy equivalence between scales is developed.Then accompanied with the energy based damage model,the multi-scale damage evolutions are resolved by homogenizing the energy scalar over the meso-cell.The macroscopic behaviors described by the multi-scale damage evolutions represent the mesoscopic heterogeneity and damage of the composites.A rather simple structure made from particle reinforced composite materials is developed as a numerical example.The agreement between the fullscale simulating results and the multi-scale simulating results demonstrates the capacity of the proposed model to simulate nonlinear behaviors of quasi-brittle composite materials within the multi-scale framework.

A Multiscale Method for Damage Analysis of Quasi-Brittle Heterogeneous Materials

A novel multiscale algorithm based on the higher-order continuum at both micro-and macrostructural level is proposed for the consideration of the quasi-brittle damage response of heterogeneous materials.Herein,the microlevel damage is modelled by the degradation of the homogenized stress and tangent stiffness tensors,which are then upscaled to govern the localization at the macrolevel.The C^1 continuity finite element employing a modified case of Mindlin’s form II strain energy density is derived for the softening analysis.To the authors’knowledge,the finite element discretization based on the strain gradient theory is applied for the modeling of damage evolution at the microstructural level for heterogeneous materials for the first time.The advantage of the novel C1 finite element formulation in comparison with the standard finite element discretization in terms of the regularization efficiency as well as the objectivity has been shown.An isotropic damage law is used for the reduction of the constitutive and nonlocal material behaviour,which is necessary for the physically correct description of the localization formation in quasi-brittle materials.The capabilities of the derived finite element to capture the fully developed localization zones are tested on a random representative volume element(RVE)for several different loading cases.By employing the conventional second-order computational homogenization,the microstructural material constitutive response is averaged over the whole RVE area.In order to model the loss of structural integrity when sharp localization is formed across RVE,the specific conditions which detect a completely formed localization zone are developed.A new failure criterion at the microstructural level has been proposed.The derived finite element formulation,as well as the multiscale damage algorithm,are implemented into the finite element program ABAQUS.The capabilities of the presented multiscale scheme to capture the effects of the deformation localization are demonstrated by few benchmark numerical examples.

Preliminary study on the determination of the Weibull modulus of strength distribution in quasi-brittle materials

In this paper,how to determine the Weibull modulus of a fracture strength distribution is discussed with its physical implications for quasi-brittle materials.Based on the Markov chain assumption,it is shown that the lifetime(i.e.,the time taken for formation of a critical defect)in a quasi-brittle material can be described by a gamma probabilistic distribution function.Prior to macroscopic failure,the effective number of energy barriers to be overcome is determined by the slope of the energy barrier spectrum,which is equivalent to the Weibull modulus.Based on a fracture mechanics model,the fracture energy barrier spectral slope and Weibull modulus can be calculated theoretically.Furthermore,such a model can be extended to take into account the crack interactions and defect-induced degradation.The predicted Weibull modulus is good agreement with that derived from available experimental results.

Ubiquitiformal Crack Extension in Quasi-Brittle Materials

Based on the concept of ubiquitiform,a ubiquitiformal crack extension model is developed for quasi-brittle materials.Numerical simulations are carried out using the ABAQUS software with the XFEM-based cohesive segments method to determine the ubiquitiformal crack extension path or fracture surface profile of the material under quasi-static loading.Such a ubiquitiformal crack model removes the singularity of a fractal crack;for the latter,the boundary value problem cannot be uniquely defined.In the simulation,the material properties,e.g.,the tensile strength,are assumed to obey the Weibull distribution.The meso-element equivalent method is used to determine the correlation between the Weibull distribution parameters and the aggregate gradation of concrete materials.The numerical results show that the complexities of the ubiquitiformal crack configurations are in good agreement with the previous experimental data.Through the numerical simulation,it is further demonstrated that the complexity of a ubiquitiformal crack is insensitive to the random spatial distribution of the aggregates,but more dependent on the Weibull distribution parameters which reflect the heterogeneity of the concrete.

A Novel Fractional Plastic Damage Model for Quasi-brittle Materials

In this work,a novel constitutive model is developed within the framework of fractional plasticity to delineate the coupling between inelastic deformation and damage of quasi-brittle materials.Faced with the common challenge of determining plastic flow direction,we resort herein to the Riemann–Liouville definition of fractional derivatives,instead of introducing an additional plastic potential.The pre-peak hardening behavior is described using an exponential function,while the post-peak softening response is viewed as the consequence of material damage.For describing damage evolution,a damage criterion is constructed in terms of plastic volume dilation related to micro-crack growth.This is conducive to supply a new insight for describing the complex influence of the non-orthogonality of plastic flow on damage evolution.For numerical applications,a semi-implicit return mapping algorithm is proposed.The predictive performance of the model is evaluated by comparing numerical simulations with experimental data under various loading paths.

A novel multi-level model for quasi-brittle cracking analysis with complex microstructure

The paper presents a novel multi-level model for quasi-brittle cracking analysis.Based on the partition of unity and information transmission technology,it provides a new non-re-meshing way to describe the cracking phenomenon in structures constructed from materials with complex microstructures.In the global model,the concept of the material particle is defined and the basic unknowns are the boundary displacements of these particles,which is different from the concept of the traditional displacement field.A series of enrichment functions with continuous steps is proposed,describing the boundary displacement affected by crack bands and allowing the intersections of crack bands with particle boundaries a priori unknown.Simultaneously,additional equations are introduced to determine element status and make the degrees of freedom of the global model remain at a stable level.Compared with previous research by our group,where the local description is equal to the global description on the boundary of a material particle,the introduced enrichment functions enable more accurate capture of the characteristics of the crack band.The model avoids the complex and dynamic model adjustments due to the activation and exit of representative volume elements(RVEs)and the accuracy of the description of the crack pattern can be ensured.The RVEs are activated at first,but then many of them exit the computation due to the unloading which reduces many of the degrees of freedom.Two examples of concrete specimens are analyzed,and the concrete fracture experiment and the digital image correlation(DIC)test are conducted.Compared with the reference solutions and the experimental data,even though the microstructure of concrete is very complex,the cracking process and crack pattern can be obtained accurately.

Phase-field modeling of fracture for quasi-brittle materials

This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topol-ogy.Within the computational mechanics community,several studies have treated the issue of modeling fracture using phase fields.Most of these studies have used an approach that implies the lack of a damage threshold.We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach.The formulation is systematically explained within a rigorous variational framework.Subsequently,we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation.Benchmark simulations in two dimensions and three dimensions are then presented.The results show that while an elastic stage and a damage threshold are ensured by the present model,good agreement with the results reported in the literature can be obtained,where such features are generally absent.

A FRACTURE-ENERGY-BASED ELASTO-SOFTENING-PLASTIC CONSTITUTIVE MODEL FOR JOINTS OF GEOMATERIALS

On the basis of plasticity and fracture mechanics for quasi-brittle materials, this article presented a constitutive model for gradual softening behavior of joints of geomaterials. Corresponding numerical tests are carried out at the local level. Characteristics of the model proposed are 1) plastic softening and dilatancy behavior are directly related to the fracture process of joint, and much less material and model parameters are required compared with those proposed by references; 2) the process of decohesion coupled with frictional sliding at both micro-scale and macro-scale is described.

ANALYSIS ON THE COHESIVE STRESS AT HALF INFINITE CRACK TIP

The nonlinear fracture behavior of quasi-brittle materials is closely related with the cohesive force distribution of fracture process zone at crack tip. Based on fracture character of quasi-brittle materials, a mechanical analysis model of half infinite crack with cohesive stress is presented. A pair of integral equations is established according to the superposition principle of crack opening displacement in solids, and the fictitious adhesive stress is unknown function . The properties of integral equations are analyzed, and the series function expression of cohesive stress is certified. By means of the data of actual crack opening displacement, two approaches to gain the cohesive stress distribution are proposed through resolving algebra equation. They are the integral transformation method for continuous displacement of actual crack opening, and the least square method for the discrete data of crack opening displacement. The calculation examples of two approaches and associated discussions are given.

Three-dimensional numerical solution and stress cage analysis of high conductive fractures pressure sealing

Aiming at the main problems in the design and analysis of well wall strengthening for fractured formations,a three-dimensional(3D)numerical simulation scheme is proposed.The three-dimensional finite element software is used to analyze the mechanical behavior of fractures and the pressure sealing process,and evaluate the stress cage effect.The main features of the model are as follows:(1)The equivalent fractures in the analytical model represent the function sum of the mechanical behavior of all fractures on the well wall,which is a functionally equivalent crack.(2)When evaluating the stress cage effect,the shape of the crack wedge filled with the plugging agent particles is formed by simulating the fractures opening process under the injection pressure,not a given regular shape.(3)In the model of calculating bull heading of block agent,the liquid pressure on the well wall is the injection pressure,which is a variation increased with time.The fluid pressure on the well wall in the stress cage calculation model is generated by the initial pore pressure.(4)The numerical evaluation of the stress cage effect is achieved by calculating the increase amplitude FX of the minimum hoop stress on the well wall.Using this model,several sets of injection pressure design values can be used for pressure plugging numerical simulation and stress cage effect evaluation calculation,and then the optimal and accurate quantitative value of“injection pressure,block agent particle size,safe mud window upper bound”are found through comparison.Finally,through an engineering example of horizontal well drilling pressure sealing in a shale gas reservoir developed by a fracture,we use the above theoretical tools to introduce the process and results of the numerical analysis of the extended mud window drilled in the shale gas fissure reservoir.