Size Effect Law Research Articles

Fracture behavior and size effect of UHPFRC: Experimental and meso-scale numerical investigation

In this research, the fracture characteristics and size effect of ultra-high performance fiber reinforced concrete (UHPFRC) are investigated by notched three-point bending beam test and meso-scale numerical simulation. Four series of geometrically similar UHPFRC notched-beams with different sizes were tested at quasi-static condition. Crack patterns during the fracture process were obtained by digital image correlation and the phenomenon of crack initiation, multiple cracking and crack localization were well captured. The macro-scale fracture properties are analyzed by double-K fracture model (DKFM) from the perspective of composite toughening. Results show that the initial fracture toughness evaluates the toughness of matrix and the increment from initial fracture toughness to unstable fracture toughness evaluates the toughening due to fiber bridging. The initial fracture toughness and unstable fracture toughness have no size effect and can be treated as material inherent. Experimental results also show a slight decreasing tendency of strength with the increasing of specimen size, which can be approximately fitted by size effect law (SEL). Then a meso-scale numerical model was developed by explicit representation of fibers and directly considering bond-slip relationship of fiber–matrix. Simulation results show that the meso-scale numerical model developed in this research can well simulate the fracture of UHPFRC in terms of crack patterns, fracture process zone, mechanical response and size effect. Parametric study was also conducted based on the meso-scale numerical model and the effects of fiber content, fiber orientation and interfacial bond strength on fracture performance and size effect are analyzed.

Size effect on the FPZ length in notched beams: Mesoscopic method and theoretical analytical

A microscopic finite element model of three-point bending (TPB) notched lightweight concrete beams is established in this paper. The behavior of discrete crack initiation and propagation in notched beams is simulated based on the cohesive and interface frictional coupling model. The correctness of the microscopic model is verified by the experimental data from related literature, and the length evolution of the fracture process zone (FPZ) during the loading process of the microscopic notched beams is performed. Meanwhile, the evolution laws of macroscopic and mesoscopic characteristic parameters such as the crack-height ratio, porosity, lightweight aggregate strength, and the lightweight aggregate volume fraction on the FPZ length of the mesoscopic notched beams are studied. In addition, the size effect of the FPZ of the mesoscopic notched beams is analyzed. The size effect law based on the critical distance line criterion is derived and the effect of high-order terms of the stress field is considered. The functional relationship between the nominal strength, FPZ length, and notched beam height is obtained to explain the size effect phenomenon. The results show that the FPZ length obtained by the mesoscopic method is in good agreement with the FPZ length obtained by the critical distance line criterion.

Longitudinal failure mechanisms and crack resistance curves of unidirectional thermoplastic composites

This paper presents a study on the longitudinal fracture toughness, and associated crack resistance curves, of thermoplastic-based composite materials. Double-edge notched (DEN) specimens loaded in tension and compression are used to relate the size-effect laws with the crack resistance curves of unidirectional carbon-fibre reinforced PEKK. The results are corroborated with SEM and fractographic images to provide a better overview of the main failure mechanisms involved during the failure process. Moreover, tensile tests interrupted at 90% of the estimated failure load are CT-scanned to highlight the early stages of the crack propagation. This investigation confirms the suitability of the size-effect method for characterizing thermoplastic composites and it stands out as a first qualitative and quantitative analysis of their characteristic intra-laminar failure mechanisms.

Crack-parallel stress effect on fracture energy of plastic hardening polycrystalline metal identified from gap test scaling

The gap test is a new type of fracture test developed in 2020, in which the end supports of a notched beam are installed with gaps that close only after the elasto-plastic pads next to notch introduce a desired constant crack-parallel compression σxx (also called the T-stress). The test uses the size effect method to identify how such a compression alters the material fracture energy, Gf, and the characteristic size cf of the fracture process zone (FPZ). In 2020, experiments showed that a moderate σxx doubled the Gf of a quasibrittle material (concrete) and a high σxx reduced its Gf to almost zero. A preliminary study by Nguyen et al. (2021) showed that the gap test can be extended to plastic-hardening polycrystalline metals. A generalized scaling law with an intermediate asymptote for large-scale yielding in small structures was derived, and limited tests of aluminum alloy showed its applicability. In this study, geometrically scaled gap tests of notched three-point bend fracture specimens of aluminum are conducted at three different levels of σxx. An extended structural strength scaling law that captures the transition from the micrometer-scale FPZ through millimeter-scale yielding zone (YZ) to large-scale structures which follow linear elastic fracture mechanics (LEFM) is derived and then applied to analyze the effect of σxx. Presented here are the gap tests of aluminum alloy, in which three different levels of σxx are applied to scaled notched four-point-bend beams of depths D = 12, 24, 48 and 96 mm. Using an extended size effect law for plastic-hardening metals, it is found that, at crack-parallel stress σxx≈−40% of the yield strength, the critical J-integral value gets roughly quadrupled, not only because of the well-known enlargement of the hardening YZ whose width is of millimeter scale, but also because of the increase of the FPZ width of micrometer scale. These results can be reproduced neither by line crack models, including the LEFM, cohesive crack and phase-field models, nor by peridynamic and various nonlocal models that ignore the tensorial nature of the material stress at the crack tip. The crack band models, being able to represent an FPZ of finite width and a YZ whose size evolves depending on σxx, can capture the effect of crack-parallel stresses provided that a realistic 3D tensorial damage constitutive model is used. Here, Bai–Wierzbicki’s model is shown to capture the σxx effect on the Gf and Jcr qualitatively.

Type I size effect and failure behavior of woven composites under biaxial flexure

This study presents an experimental investigation into the failure and size effect exhibited by woven composites under biaxial flexure, induced via the ring-on-ring plate bending test. Square-shaped, epoxy/carbon twill woven composite plates of various sizes, scaled in three dimensions are considered. The biaxial flexural response of all specimens is found to be ductile and nonlinear with failure initiation at the top surface edges under combined in-plane compression and out of plane shear. These combined compression/shear cracks propagate towards the plate center, exhibiting a four-fold symmetry, and causing the plate to deform into a cone. Elastic stress analysis is presented to interpret and support the observations. The nominal stress at the onset of non-linearity is found to obey a type I size effect law, exhibiting a marked decrease with increased plate size but approaching a constant value. This size effect must be accounted for when designing woven composite parts to be subjected to biaxial flexure.

Role of CO[formula omitted] in geomechanical alteration of Morrow Sandstone across micro–meso scales

We study the effect of CO2 on the mechanical properties of Morrow Sandstone in Farnsworth Unit oilfield in Texas, where supercritical CO2 is being injected for sequestration and enhanced oil recovery (EOR) applications. We conduct mechanical compression tests with acoustic monitoring and microscratch tests on core samples taken from two wells in the Farnsworth field to quantify the role of CO2 on poroelastic moduli and fracture toughness across micro-to-meso scales. We quantify dynamic poroelastic moduli at different effective stresses and temperatures using argon and CO2 as saturating pore fluids and analyze the effect of the fluid type on property alteration. Variation in the effective stress is achieved by varying the confining stress from 5 MPa to 20 MPa and varying the pore pressure over 0–20 MPa range. The temperature is varied from 23 degC to 75 degC, which is relevant for sequestration-EOR applications. Microscratch tests are performed to investigate the nonlinear fracture behavior and estimate the change in the scratch toughness resulting from CO2 exposure. Scanning electron microscopy shows evidence of nonlinear fracture mechanisms. Using the Size Effect Law for fracture analysis, we find that scratch-induced deformation transitions from ductile to brittle regime as the penetration depth increases. We observe a 29%–45% reduction of the scratch toughness in the bedding-parallel direction and a 60%–82% reduction in the bedding-perpendicular direction, both of which are larger in magnitude compared to the 8%–10% increase in the dynamic elastic modulus at mesoscale due to injection-induced cooling. The grain orientation affects the fracture behavior and CO2-induced geomechanical alterations: a greater decrease in fracture toughness is observed in the bedding-perpendicular orientation.

Evaluation of equivalent crack propagation length and fracture energy of two commonly used rock fracture toughness test configurations based on Bažant’s size effect law

Purpose of this research is to apply the Bažant’s size effect law to investigate the effects of material properties, specimen size, normalized crack length and loading span on the equivalent crack propagation (ECP) length and fracture energy test results of cracked straight through Brazilian disc (CSTBD) specimens and notch semi-circular bend (NSCB) specimens, and to clarify the relationship between ECP length and fracture process zone (FPZ) length. The results indicate that the ECP length is positively correlated with the specimen radius, cf (the ECP length of the infinite-size specimen) and the loading span, and the influence of the normalized crack length on the ECP length has the dependence of specimen configuration. Both the apparent fracture energy and the equivalent fracture energy have size effect, and the fracture energy tested by the specimen with smaller cf are more reliable. Based on the analysis of the influence of normalized crack length and loading span on fracture energy test results, CSTBD specimen with normalized crack length of 0.4 is recommended for rock fracture toughness test. In addition, the FPZ length determined by digital image correlation technique is more accurate than that determined by stress intensity factor method and finite element model method, and the ratio of ECP length to FPZ length decreases with the increase of specimen size.

Mesoscale synergistic effect mechanism of aggregate grading and specimen size on compressive strength of concrete with large aggregate size

In this paper, a 2D finite element model with a mesoscale level was established. The effects of content, maximum particle size, and shape of aggregate on the strength of concrete were simulated. In addition, five mesoscopic models of aggregate gradation and specimen side length (100–450 mm) were established to investigate the influence law of aggregate grading and model size on the compressive strength of concrete. The simulation results were also compared and verified with four theoretical size-effect models. The results showed that the compressive strength shows a trend of decreasing and then increasing with the increase of aggregate content and falling with the growth of maximum aggregate size dmax. The peak stress of convex polygonal aggregates is higher than that of round and elliptical. In addition, when the ratio of model side length to the maximum aggregate particle size is about 3.5, the compressive strength gradually decreases with the increase of specimen size, up to 27.65 % decreased, showing a pronounced size effect. After comparative analysis, the simulated data in this paper fitted well with the Bažant’s Type-2, Kim’s modified, Jin’s modified, and Carpinteri’s size effect law (SEL). In addition, the data obtained from the simulation of this paper would better reflect the existing test conclusions. The mesoscale model established in this paper can significantly improve the effectiveness and efficiency of full-graded concrete modeling, and better simulate the strength difference between full-graded and wet-screened specimens. The difficulties in the mesoscale numerical simulation are solved to a certain extent.

Meso-scale modelling of size effect on shear behavior of Basalt fiber reinforced concrete deep beams

The work aims to examine the size effect on shear failure of deep concrete beams with Basalt Fiber Reinforced Polymer (BFRP) bars. A mesoscale numerical model for the deep concrete beam with BFRP bars was built to explicitly model the shear failure, considering concrete heterogeneity and concrete/bars interaction. The mesoscale simulation approach was firstly verified against the available test observations. It was then utilized to study the shear failure of geometrical-similar concrete deep beams having different structural sizes with BFRP bars and steel bars. The influence of beam depth, stirrup ratio and reinforcement type (i.e., steel bars and BFRP bars) on the shear failure and the corresponding size effect of concrete deep beams was explored. The simulation results indicate that: 1) the failure modes of concrete deep beams with BFRP bars and steel bars are basically similar, and they all exhibit noticeable size effect; 2) stirrups (i.e., steel bars and BFRP bars) can hinder the development of diagonal cracks within concrete, and thus weaken the size effect in shear strength; 3) the shear failure in beams with BFRP bars presents more substantial size effect than that of beams with steel bars; 4) the calculation models of shear capacity in the international codes do not apply to concrete deep beams with BFRP bars. Moreover, it is found that the size effect law developed previously can also be suitable for the shear failure of concrete deep beams with BFRP bars.

Discrete modeling of concrete failure and size-effect

Size-effect in concrete and other quasi-brittle materials defines the relation between the nominal strength and structural size when material fractures. The main cause of size-effect is the so-called energetic size-effect which results from the release of the stored energy in the structure into the fracture front. In quasi-brittle materials and in contrast to brittle materials, the size of the fracture process zone is non-negligible compared to the structural size. As a consequence, the resulting size-effect law is non-linear and deviates from the response predicted by linear elastic fracture mechanics. In order to simulate the size-effect, one needs to rely on numerical modeling to describe the formation, development and propagation of the fracture process zone. Although a number of models have been proposed over the years, it transpires that a correct description of the fracture and size-effect which accounts for boundary effects and varying structural geometry remains challenging. In this study, the Lattice Discrete Particle Model (LDPM) is proposed to investigate the effects of structural dimension and geometry on the nominal strength and fracturing process in concrete. LDPM simulates concrete at the aggregate level and has shown superior capabilities in simulating complex cracking mechanisms thanks to the inherent discrete nature of the model. In order to evaluate concrete size-effect and provide a solid validation of LDPM, one of the most complete experimental data set available in the literature was considered and includes three-point bending tests on notched and unnotched beams. The model parameters were first calibrated on a single size notched beam under three-point bending and on the mechanical response under unconfined compression. LDPM was then used to perform blind predictions on the load-crack mouth opening displacement curves of different beam sizes and notch lengths. Splitting test results on cylinders were also predicted. The results show a very good agreement with the experimental data. The quality of the predictions was quantitatively assessed. In addition, a discussion on the fracturing process and dissipated energy is provided. Last but not least, the Universal Size-Effect Law proposed by Bažant and coworkers was used to estimate concrete fracture parameters based on experimental and numerical data.

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.

Numerical study on shear capacity of geometrically similar RC interior beam‐column joints

Reinforced concrete (RC) beam-column joints often present brittle shear failure in the core area. The objective of the study is to study the effects of stirrup ratio and axial load ratio on the shear capacity of geometrically similar beam-column joints having different structural sizes. In this work, a series of 3D mesoscale simulations on the shear failure of RC beam-column joints were conducted and validated, in which the concrete heterogeneity was considered explicitly. The simulation results indicate that stirrups can improve the shear strength of joints, enhance the ductility, improve deformation ability, while weaken the size effect on the nominal shear strength. Moreover, the increasing axial load ratio would lead to the increase of shear strength, as well as the increase of size effect. Based on the modeling results, a novel semiempirical and semitheoretical size effect law considering the quantitative effect of stirrups and axial load ratio was proposed. In addition, the proposed size effect law was verified against with some available test data.

The Gap test – Effects of crack parallel compression on fracture in carbon fiber composites

This paper explores the global Mode I fracture energy of a carbon fiber composite subject to a biaxial stress state at the crack tip, specifically in which one stress component is compressive and parallel to the crack. Based on an experimental technique previously coined as The Gap Test and Bažant’s Type II Size Effect Law, it is found that there is a monotonic decrease in the Mode I fracture energy as the crack parallel compressive stress increases. Compared to the nominal value of fracture energy, where no crack parallel compression is applied, the fracture energy is observed to decrease by up to 37% for a compressive stress equal to 44% of the compressive failure limit of the composite. This weakening effect is attributed to splitting cracks that are induced at the crack tip due to the crack parallel compression, which are identified via crack tip photomicroscopy. This is a novel result that challenges the century old hypothesis of fracture energy being a constant material property and further, shows for the first time that crack parallel compression leads to a composite structure being dangerously weaker than expected.The experimental campaign is also buttressed with a computational campaign that provides a framework capable of capturing the effects of crack parallel compression. Through the use of the crack band model, which correctly characterizes the fracture process zone tensorially, coupled with a fully tensorial damage law, the simulated results provide satisfactory agreement with the experimental data. Conversely, when a reduced tensorial damage law defines the crack band it is shown that the structural strength and fracture energy are dangerously overpredicted. This emphasizes the importance of using a crack band model coupled with a fully tensorial damage law to accurately predict fracture in composites.

On interchangeability and selection of size effect and boundary effect experiments for characterization and prediction of quasi-brittle fracture of concrete

Size effect experiments using geometrically similar specimens of different sizes overlap boundary effect experiments using specimens of single size but different notches if sufficient sample designs are tested. This study shows the interchangeability and relevance of size and boundary effect experiments through in-depth analysis of comprehensive size effect experiments. Initially-designed five groups of geometrically similar concrete specimens with notch/size ratios of 0, 0.025, 0.075, 0.15 and 0.3 are rearranged into four groups with constant sizes of 40, 93, 215 and 500 mm. Thus, total nine groups of experimental results, containing abundant information on quasi-brittle fracture of concrete and associated size effect phenomenon, are analysed by three size effect laws (SELs) and a linearized boundary effect model (BEM). It is found that fracture behaviours revealed from comprehensive size effect experiments (using large and small geometrically similar specimens required by SEL) can also be confirmed by simple boundary effect experiments (using specimens of single convenient size). Discussion on SEL and BEM based on principles of mathematical fitting and data processing is provided for more objective quasi-brittle fracture experiments and modelling.

Meso-scale analysis of shear performance and size effect of CFRP sheets strengthened RC beams

In order to reflect and reveal the shear failure process and mechanism of carbon fiber reinforced plastic (CFRP) sheets strengthened reinforced concrete (RC) beam, a mechanical analysis model is established by using a 3D meso-scale simulation method. The model considers the heterogeneity of concrete materials, the interaction behaviors between reinforcements and concrete, and the complex interaction behaviors between CFRP sheets and concrete, simultaneously. Different types of meso-scale models of RC beam strengthened with CFRP sheets are designed to explore the influences of stirrup ratio and CFRP fiber ratio on the nominal shear strength and size effect behaviors. Test results indicate that: 1) The action mechanisms of stirrups and CFRP sheets are similar; 2) The nominal shear strength of CFRP sheets strengthened RC beams increases with the increase of stirrup ratio and CFRP fiber ratio, while the degree of the increasing range decreases; 3) The nominal shear strength of RC beams strengthened with CFRP sheets decreases with the increase of section size, nevertheless increasing the stirrup ratio and the CFRP fiber ratio will weaken the size effect; 4) Based on the numerical test results, a novel size effect law simultaneously considering the stirrup ratio, the CFRP fiber ratio and the mutual effects between them is proposed.

Effect of water-cement ratio and size on tensile damage in hardened cement paste: Insight from peridynamic simulations

This paper uses the improved bond-based peridynamics (BB PD) method to predict the crack evolution and macroscopic mechanical properties of the hardened cement paste at the micro-scale. Specifically, the prototype micro-elastic brittle (PMB) model is improved by considering an attenuation kernel function and the surface effect correction. Subsequently, the uniaxial tension simulations of hardened cement microstructures generated by μic software are systemically conducted to investigate the influences of the heterogeneity of microstructure, size effect, and water-cement ratio (w/c) on crack growth and mechanical properties, i.e., tensile strength and Young's modulus. The results show that the heterogeneity of the microstructure inevitably will cause the discreteness of the macroscopic properties of the numerical cement paste. Therefore, a statistical method is considered to analyze the mechanical properties of the hardened cement. Moreover, the tensile strength has a remarkable size effect, with the size increase of microstructure, the tensile strength decreases; the simulation results fit well with the existing analytical size effect model, i.e., Weibull size effect model, multifractal scaling law, and Bažant's size effect law. Besides, the higher ratio of w/c is likely to result in lower tensile strength, Young's modulus and premature damage to the cement paste. The predicted macroscopic mechanical properties are in good agreement with the previous simulation and experimental data, which demonstrates the reliability of the improved bond-based peridynamic method in studying the crack initiation and propagation of hardened cement paste.

An advanced experimental investigation of size effect on flexural fatigue behaviour of cement-bound granular materials

ABSTRACT This paper investigates the effect of specimen size on the flexural fatigue behaviour of two different locally-sourced pavement materials stabilised with 3% general purpose (GP) cement, viz., cement-stabilised Holcim road base (CSHRB) and cement-stabilised quartzite (CSQRT). Three different sizes of geometrically similar beams were tested under both static and dynamic four-point loading conditions. A suitable load pulse shape that simulates the critical stress response of the cement-treated base (CTB) under standard axe loading was generated for each beam size to perform stress-controlled flexural fatigue tests. The flexural fatigue test results showed the existence of a fatigue endurance limit in cement-stabilised pavement materials (CSPMs) at 28 days curing period. The analysis of experimental data revealed that the flexural properties of CSHRB (i.e. flexural strength and flexural fatigue performance) are appreciably affected by the size of the beam specimen employed in the flexural test while the flexural properties of CSQRT did not show a clear trend for increasing beam size. Bažant’s size effect law is in good agreement with the flexural strength results obtained in this study for CSHRB. For materials that exhibit size dependence on flexural strength, an empirical relation is proposed in this study to determine the flexural strength of any given beam size using the flexural strength of standard beam size or indirect tensile strength of the beam material. Furthermore, a stress-based fatigue model incorporating the size effect is proposed to predict the flexural fatigue life of CSPMs in service. The factors that give rise to erroneous strain-based fatigue models were critically discussed on the basis of the results of this study.

Crack face friction effects in the strength scaling of composites failing by kinked shear cracks under transverse compression

Recent research showed that the failure and strength scaling of composites under transverse compression exhibits a quasibrittle behavior, characterized by a non-negligible fracture process zone (FPZ). The present work is aimed at analyzing and quantifying the contributions of crack face friction towards this quasibrittleness, by focusing on fracture plane angles and the strength scaling. The failure, frictional effects, and strength scaling are numerically predicted via the fixed crack damage model, augmented with Puck’s friction-dependent damage initiation criterion and Ragueneau’s parallel coupling model for combined damage and friction evolution. The model is calibrated and verified using available test data and then used to analyze the transverse compressive failure in unnotched specimens and strength scaling in pre-cracked specimens, with and without friction. The analyses show that the friction-augmented model can accurately predict the fracture plane angle equal to 53 degrees in unnotched specimens, and can be tuned to reproduce other angles as well. Further, in all notched specimens failure occurs via a kinked shear crack propagating at an angle to the pre-crack, and having at its tip a large FPZ. The kink angle is observed to fall in the range of 44 to 47 degrees when friction is excluded, and 54 to 58 degrees when friction is included. The specimen strengths with or without friction, scale in accordance with the energetic type II size effect law. This finding is also supported analytically using energy balance-based arguments. The results reveal that the inclusion of friction can considerably increase the strength and enlarge the FPZ size (by nearly 90%), making the fracturing behavior significantly more quasibrittle. The strength increase is size dependent for smaller sizes, and appears to become roughly constant for larger specimen sizes. The FPZ of the kinked shear cracks is seen to embody a strongly multiaxial stress state. This makes the fracture energy associated with the kinked shear crack growth much higher than the pure mode II fracture energy. The frictional dissipation can further lead to an increase in this energy, which in the present case is found to be nearly 130%. These findings serve to emphasize the importance of considering and quantifying friction effects in the transverse compressive failure of composites.

A model scaling approach for fracture and size effect simulations in solids: Cohesive zone, smeared crack band and phase-field models

Despite many noteworthy contributions, the computational modeling of arbitrary fracture configurations and in particular, the quantification of size effect in quasi-brittle solids, still remains a challenging issue. In this work we proposed a model scaling approach for fracture and size effect simulations in solids. It employs only the geometric transformation relations and thus applies to any material model for fracture. Those popular models, i.e., the cohesive zone model, the smeared crack model and the phase-field cohesive zone model, are considered. With the geometric scaling factor intrinsically incorporated in the formulation, the resulting scaled models are able to predict the energetic size effect of a series of geometrically similar structures by parametric numerical simulations of the reference one. The 1-D analytical results of all the three scaled models are coincident with those from the underlying unscaled counterparts, validating the proposed model scaling approach. As it is independent of the mesh discretization and insensitive to the length scale, the scaled phase-field cohesive zone model (PF-CZM) is adopted for application to several representative benchmark tests of concrete and ice with mode-I and mixed I+II failure modes. The numerical results demonstrate that, owing to the strength-based crack nucleation criterion, the energy-based crack propagation criterion and the variational principle based path chooser, as well as the geometric scaling factor, are all incorporated into a standalone framework, the scaled PF-CZM is able to predict both the type II size effect law of pre-cracked quasi-brittle structures and the type I size effect law of intact ones. Moreover, those laborious manual manipulations in the numerical modeling of a large size range of geometrically similar structures, e.g., spatial discretization, enforcement of loading and boundary conditions, postprocessing of output data, etc., need to be done only once on the reference structure. These advantages make the scaled PF-CZM rather promising in the quantification of size effect in quasi-brittle structures.

Prediction of Mode I Fracture Toughness of Shale Specimens by Different Fracture Theories Considering Size Effect

In this study, mode I fracture tests on cracked straight-through Brazilian disc (CSTBD) and notched semi-circular bend (NSCB) shale specimens with different sizes were conducted to investigate the difference between maximum tangential stress fracture criterion and the size effect law (SEL) model in predicting apparent fracture toughness (Ka) of shale. In addition, the effects of specimen size and geometry on the Ka and the selection of fracture criterion on the prediction of the inherent fracture toughness (KIc) were also studied. The results show that the Ka increases with the increase of specimen size, and the difference between KIc of shale specimens with different sizes predicted by the fracture process zone length determined by the further improved maximum tangential stress (FIMTS) criterion is the smallest. For the prediction of Ka of NSCB specimen, the results predicted by the FIMTS criterion are the closest to the tested fracture toughness. However, the effect of SEL model applied to the prediction of Ka of NSCB specimens is poor. The effective establishment of SEL model requires high accuracy for test data, especially for the configuration with large variation of the dimensionless stress intensity factor (Y*) with normalized crack length (α).