Fictitious Crack Model Research Articles

An analytical method for predicting mode-I crack propagation process and resistance curve of rock and concrete materials

In this paper, taking into account the fictitious crack model and the closure effect of the crack opening displacement (COD) induced by the cohesive force, an analytical method with the crack mouth opening displacement (CMOD) as the control parameter is proposed to predict the mode-I crack propagation process of rock and concrete materials by using stress intensity factor (SIF)-based criteria. With this method, the load-crack mouth opening displacement (P-CMOD) curve, crack extension resistance curve (KR-curve), and fracture process zone (FPZ) length during crack propagation are analytically obtained. The validity of the proposed analytical method is then verified with the experimental results of three-point bending tests on rock and concrete beams. Therefore, given the tensile strength ft, Young’s modulus E, initial fracture toughness KICini, and fracture energy GF, the complete mode-I crack propagation process of rock and concrete beams under three-point bending can be predicted accurately using the proposed analytical method without iteration in calculating the cohesive stress. Finally, the influence of the closure effect of the COD induced by the cohesive force on the P-CMOD curve, KR-curve, and FPZ length is discussed. The results show that although ignoring the closure effect of the COD induced by the cohesive force has little effect on the shape of the P-CMOD curve, the value of crack extension resistance and the maximum FPZ length are significantly underestimated, and the calculation error increases with the crack propagation and then decreases gradually when the FPZ length reaches the maximum.

Analytic solution for the bond stress-slip relationship between rebar and concrete

This paper presents a three stage bond stress-slip relationship between rebar and concrete under static and dynamic loads, with coordinate values at key points determined by an analytic solution. Based on the elastic mechanic solutions of a thick-walled cylinder under uniform pressure, radial stress and radial slip at the end of the elastic stage is proposed. When concrete cracks, the fictitious crack model of fracture mechanics is used to consider the softening behavior of concrete under tension. The radial stress and radial slip at the partial cracked stage and full cracked stage is presented, and the bond stress is deduced from the radial stress based on the equilibrium of forces. By assuming the rebar between adjacent ribs is a circular truncated cone, the relationship between slip and radial strain is established. Meanwhile, as an additional external constraint in the thick-walled cylinder model, the restraint of stirrups is taken into account. For the dynamic effect, the dynamic tensile strength and elastic modulus of concrete is introduced to consider the effect of the loading rate. The bond strength calculated by the presented model is compared with the experimental results, and the errors are generally within 15%. The presented bond stress-slip relationship is then used to simulate pull-out test specimens, including the effect of stirrups and loading rates by ABAQUS software. The comparison between simulated and experimental results shows that the proposed bond stress-slip relationship can successfully predict the bond stress-slip between rebar and concrete.

The analysis of failure in concrete and reinforced concrete beams with different reinforcement ratio

In this paper the analysis of failure and crack development in beams made of concrete is presented. The analysis was carried out on the basis of the performed experimental investigation and numerical simulations. A fictitious crack model based on nonlinear fracture mechanics was applied to investigate the development of strain softening of tensile concrete in plain concrete and slightly reinforced concrete beams. The role of strain softening was also discussed according to the inclined crack propagation in highly reinforced concrete beams. The analysis has brought the evidence that the mode of failure in flexural beams varies according to a longitudinal reinforcement ratio. A brittle failure due to the formation of a flexural crack takes place in plain and slightly reinforced concrete beams, and strain softening of tensile concrete is of paramount importance at failure crack initiation and propagation. A stable growth of numerous flexural cracks is possible in moderately reinforced concrete beams, and then the load carrying capacity is connected with reaching the yield stress of reinforcing steel or concrete crushing in the compression zone. In higher reinforced concrete beams without transverse reinforcement, brittle failure can take place due to shear forces and the development of diagonal cracks. However, strain softening of tensile concrete is not the only mechanism influencing the propagation of an inclined crack. Such mechanisms as aggregate interlock and dowel action of steel bars contribute more importantly to the development of failure crack.

Tensile fracture of a single crack in first-year sea ice.

The break-up of sea ice in the Arctic and Antarctic has been studied during three field trips in the spring of 1993 at Resolute, NWT, and the fall of 2001 and 2004 on McMurdo Sound via in situ cyclic loading and fracture experiments. In this paper, the back-calculated fracture information necessary to the specification of an accurate viscoelastic fictitious (cohesive) crack model is presented. In particular, the changing shape of the stress separation curve with varying conditions and loading scenarios is revealed.This article is part of the theme issue ‘Modelling of sea-ice phenomena’.

The fracture extreme theory for determining the effective fracture toughness and tensile strength of concrete

A theoretical method for determining the effective fracture toughness KICe and tensile strength ft of concrete was proposed based on the fictitious crack model. With the fracture extreme theory (FET), only the experimental peak load of one specimen in fracture test is required for the calculation. The application of the proposed method was illustrated by the three-point bending notched concrete beam. The KICe, ft, critical crack tip opening displacement CTODc, and critical crack propagation length Δac of beams with different initial notch ratio a0/h were obtained. The results obtained by the proposed method were in good agreement with those by the experimental data and existing analytical method, which confirmed its applicability. And the calculated values of KICe are insensitive to a0/h using the proposed method.

Predicting the Shear Behavior of Reinforced Concrete Beams Using Non-linear Fracture Mechanics

This paper deals with the challenging problem of predicting the load carrying capacity of reinforced concrete shear-critical beams. To simulate the cracking behavior of concrete, the discrete crack approach based upon non-linear fracture mechanics is used. An algorithm with two pathways of implementation has been proposed so as to implement fictitious crack model to analyze reinforced concrete beams using finite-element method. The merit of the proposed algorithm is its capability to recognize and to incorporate them in the analysis procedure. The proposed method is capable of predicting simultaneous multiple shear cracks, load-deformation behavior, and ultimate shear capacity of reinforced concrete beam. The obtained results show a good compliance with the available experimental benchmark studies on reinforced concrete beams failed in shear. As one of the important aspects of shear capacity is the size effect issue, some large-scaled test beams have been numerically simulated using the proposed algorithm and the results have been compared with that of ACI 318-11 building code as well as the well-known modified compression field theory (MCFT). The comparison corroborated the robustness of the proposed algorithm in detecting the well-known shear-scaling phenomenon. In the same time, the comparison emphasized on the weakness of the current codes of practice to overestimate the shear capacity of large-scaled beams owing to not taking the size effect into account.

A comparative study on stress intensity factor-based criteria for the prediction of mixed mode I-II crack propagation in concrete

Combined with the fictitious crack model, the stress intensity factor (SIF)-based criteria are widely adopted to determine the crack propagation of mixed mode I-II fracture in normal strength concrete. However, less research is reported on the applicability of the different SIF-based criteria when they are used to analyze the crack propagation process of concrete with different strength grades. With this objective in mind, three-point bending and four-point shear tests were conducted in this study on C20, C50 and C80 grade concrete to measure the initial fracture toughness, fracture energy, load-crack mouth opening/sliding displacement (CMOD/CMSD). Four SIF-based criteria, including two initial fracture toughness-based (with/without mode II component of SIF KII) and two nil SIF-based (with/without KII), were introduced to determine crack propagation and predict the P-CMOD/CMSD curves for the notched concrete beams under four-point shear loading. The results indicated that the difference between the peak loads from experiment and from the analysis based on the nil SIF criterion with KII approximately increases with the increase of the concrete strength. By contrast, the predicted peak load and P-CMOD/CMSD curves adopting the initial fracture toughness-based criterion with KII showed better agreement with experimental results for the different concrete strength. Meanwhile, in the case of the initial fracture toughness-based criteria, the predicted initial load was underestimated if the component of KII was not considered. However, the fracture mode transformed from mixed mode I-II to mode I after the crack initiation, meaning the KII component in the criterion had a less significant effect on the crack propagation process.

Simplified equations for determining double‐K fracture parameters of concrete for 3‐point bending test

AbstractThe paper presents simplified polynomial equations for determining the double‐K fracture parameters of concrete for 3‐point bending beams with variable strengths and material properties of concrete. The derived equations avoid complexities involved in computations of fracture parameters using existing analytical methods. The input data required for systematic computation in the study for deriving the nondimensional fracture parameters are obtained using a fictitious crack model. It is inferred that for a relative size of initial crack length, critical load and corresponding crack opening displacement maintain a linear relationship in their nondimensional forms. The value of critical mouth opening displacement can also be determined for known value of peak load using the derived nondimensional equation, thus avoiding the measurement of the crack mouth opening displacement in the experiment. Further, the derived polynomial equations predict the double‐K fracture parameters of concrete with negligible error as compared to those obtained based on experimental results.

An Introduction of Boundary Element Method.to the Analysis of Crack Propagation in Concrete

The dual boundary element method (DBEM) incorporated with fictitious crack model (FCM) is used to analyse and simulatethe crack propagation .in concrete. The fracture in concrete is represented by the FCM in which the fracture zone is replaced by applying closing forces on both crack surfaces. When the force at the ficti tious crack tip exceeds the maximum tensile strength of the concrete, the fictitious crack will propagate perpendicular to the maximum principal stress. Three-point bending specimen is used to check the numerical analysis and the res ults is compared to the analysis by finite element method and experimental results.

Experimental Study on the Tensile Properties of Rock‐Mortar Interface under Different Strain Rates

In this paper, an experimental study was carried out on a rock‐mortar interface specimen under three different strain rates (10−6, 10−5, and 10−4 s−1) using the MTS322 electrohydraulic servo loading system, and a new constitutive relation function of fictitious crack model (FCM) according to the axial‐stress‐crack‐width curves of the rock‐mortar interface is established, because the traditional nonlinear softening function easily distorts, shakes, and so cannot describe the damage‐evolution process of the rock‐mortar interface accurately. Through the use of a precise servo actuator system and three extensometers measured axial‐stress‐crack‐width curves, it is shown that the rock‐mortar interface is very sensitive to the strain rate. The tensile strength increases with strain rate, the crack width decreases at the same time, and the axial‐stress‐crack‐width curves gradually evolve from a concave‐downward trend to a linear decreasing trend. At the same time, the new constitutive relation function can reflect the tensile strength, crack width, and the downward trend of the rock‐mortar interface more accurately.

The fracture mechanism of circular/elliptical concrete rings under restrained shrinkage and drying from top and bottom surfaces

Due to uniform shrinkage along the radial direction, drying from both top and bottom surfaces has been recommended to replace drying from outer circumference surface in the restrained circular ring test to assess cracking tendency of concrete. However, non-uniform shrinkage along the height direction under drying conditions is significant, and its effect on crack initiation and propagation in a concrete ring is not clearly understood. To investigate the fracture mechanism of the restrained ring test under drying from top and bottom surfaces, three series of circular and elliptical ring specimens with heights of 30 mm, 50 mm and 75 mm are tested to measure the cracking ages. A fracture mechanics based numerical method is proposed by introducing fictitious crack model to simulate the fracture process and predict the cracking age of a concrete ring under restraint. The effects of ring geometric profile, specimen height and moisture gradient on crack development are discussed. The results indicate that, under drying from both top and bottom surfaces, crack initiates partly along the height direction at the inner circumference of a concrete ring, and propagates along the radial direction, one by one, until the crack propagated throughout the whole cross-section. The moisture gradient along the height direction has significant effect on the crack driving force, which is dominated by the moisture gradient and steel ring restraint near the exposed surface, whose proportion increases with the increase in distance from the exposed surface.

Vector form Intrinsic Finite Element Based Approach to Simulate Crack Propagation

AbstractThis paper presents a new approach to simulate the propagation of elastic and cohesive cracks under mode-I loading based on the vector form intrinsic finite element method. The proposed approach can handle crack propagation without requiring global stiffness matrices and extra weak stiffness elements. The structure is simulated by mass particles whose motions are governed by the Newton's second law. Elastic and cohesive crack propagation are simulated by proposed VFIFE-J-integral and VFIFE-FCM methods, respectively. The VFIFE-J-integral method is based on vector form intrinsic finite element (VFIFE) andJ-integral methods to calculate the stress intensity factors at the crack tips, and the VFIFE-FCM method combines VFIFE and fictitious crack models (FCM). When the stress state at the crack tip meets the fracture criterion, the mass particle at the crack tip is separated into two particles. The crack then extends in the plate until the plate splits into two parts. The proposed VFIFE-J-integral method was validated by elastic crack simulation of a notched plate, and the VFIFE-FCM method by cohesive crack propagation of a three point bending beam. As assembly of the global stiffness matrix is avoided and each mass particle motion is calculated independently, the proposed method is easy and efficient. Numerical comparisons demonstrate that the present results predicted by the VFIFE method are in agreement with previous analytical, numerical and experimental works.

A simple model for fatigue crack growth in concrete applied to a hinge beam model

In concrete structures, fatigue is one of the major causes of material deterioration. Repeated loads result in formation of cracks. Propagation of these cracks cause internal progressive damage within the concrete material which ultimately leads to failure. This paper presents a simplified general concept for non-linear analysis of concrete subjected to cyclic loading. The model is based on the fracture mechanics concepts of the fictitious crack model, considering a fiber of concrete material, and a simple energy based approach for estimating the bridging stress under cyclic loading. Further, the uni-axial fiber response is incorporated in a numerical hinge model for beam analysis. Finally, the hinge model is implemented into a finite element beam element on a constitutive level. The proposed model is compared to experimental results on both fiber- and beam level. The proposed model shows good performance and seems well suited for the description of fatigue crack growth in concrete.

Cohesive cracked-hinge model for simulation of fracture in one-way slabs on grade

Numerical analysis of slab on grade structures subjected to mechanical loads is a complex matter often requiring computationally expensive models. In order to develop a simplified and general concept for non-linear analysis of slab on grade structures, this paper presents a cohesive cracked-hinge model aimed at the analysis of the bending fracture of the cemented material. The model is based on the fracture mechanics concepts of the fictitious crack model with a linear stress–crack opening relationship. Moreover, the paper presents a two-parameter spring foundation model applied to realistically capture the continuity in the supporting medium. The functionality of the proposed model is compared to numerical analysis with application of the more conventional cohesive zone model. The results obtained show that the methodology is a attractive and powerful one well-suited for practical use and further development.

General cracked-hinge model for simulation of low-cycle damage in cemented beams on soil

The need for mechanistic constitutive models to evaluate the complex interaction between concrete crack propagation, geometry and soil foundation in concrete- and composite pavement systems has been recognized. Several models developed are either too complex or designed to solve relatively simple problems, e.g. limited to one type of load configuration or test set-up. In order to develop a general and mechanistic modeling framework for non-linear analysis of low-cycle damage in cemented materials, this paper presents a cracked-hinge model aimed at the analysis of the bending fracture of the cemented material. The model is based on the fracture mechanics concepts of the fictitious crack model. The proposed hinge is described in a general and consistent format, allowing for any type of stress-crack opening relationship and unloading- reloading formulation. The functionality of the proposed hinge model is compared to numerical- and experimental results. The proposed hinge shows good performance and seems promising for the description of low-cycle fracture behavior in cemented materials.

Fracture Properties of Polystyrene Aggregate Concrete after Exposure to High Temperatures.

This paper mainly reports an experimental investigation on the residual mechanical and fracture properties of polystyrene aggregate concrete (PAC) after exposure to high temperatures up to 800 degrees Celsius. The fracture properties namely, the critical stress intensity factor (), the critical crack tip opening displacement (CTODC) for the Two-Parameter Model, and the fracture energy (GF) for the Fictitious Crack Model were examined using the three-point bending notched beam test, according to the RILEM recommendations. The effects of polystyrene aggregate (PA) content and temperature levels on the fracture and mechanical properties of concrete were investigated. The results showed that the mechanical properties of PAC significantly decreased with increase in temperature level and the extent of which depended on the PA content in the mixture. However, at a very high temperature of 800 °C, all samples showed 80 percent reduction in modulus of elasticity compared to room temperature, regardless of the level of PA content. Fracture properties of control concrete (C) and PAC were influenced by temperature in a similar manner. Increasing temperature from 25 °C to 500 °C caused almost 50% reduction of the fracture energy for all samples while 30% increase in fracture energy was occurred when the temperature increased from 500 °C to 800 °C. It was found that adding more PA content in the mixture lead to a more ductile behaviour of concrete.

Determination of Double-K Fracture Parameters of Concrete Using Split-Tension Cube: A Revised Procedure

This paper presents a revised procedure for computation of double-K fracture parameters of concrete split-tension cube specimen using weight function of the centrally cracked plate of finite strip with a finite width. This is an improvement over the previous work of the authors in which the determination of double-K fracture parameters of concrete for split-tension cube test using weight function of the centrally cracked plate of infinite strip with a finite width was presented. In a recent research, it was pointed out that there are great differences between a finite strip and an infinite strip regarding their weight function and the solution of infinite strip can be utilized in the split-tension specimens when the notch size is very small. In the present work, improved version of LEFM formulas for stress intensity factor, crack mouth opening displacement and crack opening displacement profile presented in the recent research work are incorporated. The results of the double-K fracture parameters obtained using revised procedure and the previous work of the authors is compared. The double-K fracture parameters of split-tension cube specimen are also compared with those obtained for standard three point bend test specimen. The input data required for determining double-K fracture parameters for both the specimen geometries for laboratory size specimens are obtained using well known version of the Fictitious Crack Model.

Enhanced fictitious crack model accounting for material drawn into the cohesive zone: physically based crack closure criterion

According to the fictitious crack model, the crack width is infinitely thin at crack initiation and with further loading the crack opens, but no material exists in the cohesive zone. Despite that, the crack is assumed to able to sustain a load. While this is possible at the atomic level, it is not a realistic assumption for the macroscopic behavior we are considering. Invoking the mass conservation law and by postulating a relation between the porosity f of the cohesive zone and the damage variable $$\kappa _{n}$$ , which is that part of the damage variable $$(\kappa )$$ that is related to opening of the crack, it is shown that the fictitious crack model can be enhanced so that it accounts for material that is drawn into the cohesive zone. In turn, the amount of material present in the cohesive zone implies a consistent and physically based crack closure criterion. The concepts are first discussed in detail for uniaxial loading and then generalized to arbitrary loadings.

Thermomechanical failure modeling and investigation into lining optimization for a Ruhrstahl Heraeus snorkel

Abstract Snorkels used in Ruhrstahl Heraeus (RH) degassers in the steel industry undergo severe thermomechanical loads due to normal cyclic operations including preheating, submerging, and idle time. A finite element model was developed to simulate the first operational process cycle experienced by a RH snorkel. The presented analysis investigates the thermomechanical failure mechanisms of the magnesia chromite bricks in the wear lining. The Drucker–Prager, Drucker–Prager creep and fictitious crack models were applied to model creep and failure. The wedge-splitting test using an inverse analysis method, and the modified shear and compressive creep tests were utilized to characterize the mechanical properties of the refractories at elevated temperatures. The results show that at the beginning of the submerging process, tensile failure occurs closest to the hot face of the lowest refractory section prior to shear failure or creep. Because of refractory creep, the hot face of the wear lining does not experience shear failure during submerging. Furthermore, shear failure and creep contribute to the process of joint opening at the hot face. The concept of using less brittle material for the wear lining and a relative stiff monolith indicated that the tensile failure occurring from the hot thermal shock can be mitigated.

Experimental validation and fracture properties analysis on wedge splitting concrete specimens with different initial seam-height ratios

In order to study the fracture properties of non-standard concrete wedge splitting tensile specimens with different initial seam-height ratios and the impact on concrete fracture toughness based on aggregates cohesive force, five groups non-standard concrete wedge splitting specimens with different initial seam height ratios as 0.2, 0.25, 0.3, 0.35 and 0.4 were designed and constructed for the experiment. Based on double-K fracture model and fictitious crack model, a new fracture toughness calculation model for non-standard concrete wedge splitting tensile specimen with different initial seam height ratios was built. The results showed that: the value of initial cracking load Pini and unstable fracture load Pun decreased gradually with the increase of specimens’ initial seam height ratio. With the increase of seam height ratios, the value of the critical effective crack length ac showed a linearly increasing trend, but the subcritical crack propagation length Δac showed a decrease trend. The unstable fracture toughness KunIC and the cohesive toughness KcIC were affected by seam height ratios, decreased with initial seam height ratios increasing, and the initial fracture toughness KiniIC was independent of it that could be treated as material constant. When calculating the initial fracture toughness KiniIC, the model in this paper could eliminate the errors caused at the moment of determining initial cracking load Pini, which proved its advantage. For the specimens with smaller initial seam height ratio, its fracture features was influenced by maximum tension stress and showed some shearing failure.