Bridging Law Research Articles

Assessing fracture behavior in Composite-Metal bonded joints under opening and sliding Modes: Insights from Experiments, CZM, and FEA

The fracture behavior of AA 7050-T6/CFRP adhesive joints was investigated under mode I and II loading. Digital Image Correlation (DIC) provided precise measurements of crack length and crack tip opening/shear displacement (CTOD/CTSD), while the Extended Global Method (EGM) was used to determine the Energy Release Rate (ERR) and resistance curves (R-curves) for each fracture mode. The direct method was applied to establish the traction-separation law (TSL) and bridging law. Based on observed failure mechanisms, a novel cohesive law with five linear components was proposed, with parameters derived from experimentally obtained TSL and bridging law. Experimental determination of cohesive zone lengths for both fracture modes was achieved using the DIC technique, which was then compared with analytical predictions. Finally, the five-linear and trapezoidal cohesive zone model (CZM) was used to simulate the mode I and mode II fracture processes, respectively, within a finite element framework.

Bridging Behavior of Palm Fiber in Cementitious Composite

This study addresses the growing need for sustainable construction materials by investigating the mechanical properties and behavior of palm fiber-reinforced cementitious composite (FRCC), a potential eco-friendly alternative to synthetic fiber reinforcements. Despite the promise of natural fibers in enhancing the mechanical performance of composites, challenges remain in optimizing fiber distribution, fiber–composite bonding mechanism, and its balance to matrix strength. To address these challenges, this study conducted extensive experimental programs using palm fiber as reinforcement, focusing on understanding the fiber–matrix interaction, determining the pullout load–slip relationship, and modeling fiber bridging behavior. The experimental program included density calculations and scanning electron microscope (SEM) analysis to examine the surface morphology and diameter of the fibers. Single fiber pullout tests were performed under varying conditions to assess the pullout load, slip behavior, and failure modes of the palm fiber, and a relationship between the pullout load and slip with the embedded length of the palm fiber was constructed. A trilinear model was developed to describe the pullout load–slip behavior of single fibers, and a corresponding palm-FRCC bridging model was constructed using the results from these tests. Section analysis was conducted to assess the adaptability of the modeled bridging law calculations, and the analysis result of the bending moment–curvature relationship shows a good agreement with the experimental results obtained from the four-point bending test of palm-FRCC. These findings demonstrate the potential of palm fibers in improving the mechanical performance of FRCC and contribute to the broader understanding of natural fiber reinforcement in cementitious composites.

A finite element analysis method for random fiber-aggregate 3D mesoscale concrete based on the crack bridging law

Mesoscale modeling is an emerging analytical method that can be used to study fiber-reinforced concrete. However, fine fibers with micrometer diameters are distributed in millions in a standard specimen, so it is impossible to establish a mesoscale model and further analyze it, which severely limits the investigation of the reinforcement mechanism of fine fibers on concrete. This paper efficiently established a highly applicable mesoscale analytical model for random fiber-polyhedral aggregate concrete in Abaqus software based on a comprehensive examination of the crack bridging law and the supplementation of the fiber spacing theory, combined with the improvement of the algorithms of aggregate generation, fiber arrangement, and the judgment of the fiber-aggregate contact by the secondary development method of Python, as well as the matrix-dual arrangement method. Additionally, basalt fiber concrete and mortar specimens were prepared and tested, and the parameters of the model were calibrated to verify the model's authenticity. The result shows that the method proposed in this paper can accurately and efficiently perform mesoscale concrete analysis with fine fibers. In the peak load stage, the fiber has the strongest alleviation effect (Δ = 0.033) on the matrix damage, 10 times that of the crack initiation stage (Δ = 0.003). With the development of damage, fiber stress increases gradually from the crack initiation stage (0.68 σfu), and the growth rate (41.53 %) is the largest at the peak load stage (σfu), which reveals the complex dynamic bridging effect of fiber on the matrix. This investigation proposes a novel and broad applicability methodology for the finite element analysis of fiber-reinforced aggregate mesoscale concrete.

A new method for determining fracture toughness and bridging law of asymmetric double cantilever beam

This work presents a new approach for determining the fracture toughness and bridging law of asymmetric double cantilever beam (ADCB). A key advantage of this method is that it eliminates the need for real-time monitoring of crack length and extra experiments to obtain elastic parameters, thus reducing potential errors from different test operators. Through force analysis, formulations are derived for compliance, energy release rate and relative sliding displacement at the pre-crack tip. The corresponding bridging stress is determined by the J-integral method. To validate the proposed method, the obtained results are compared with experimental data from various data reduction methods, including the modified beam theory (MBT) and modified elastic beam theory (MEBT). Notably, the results from the proposed method show strong alignment in fracture toughness values and bridging laws with those obtained through alternative methodologies. Furthermore, numerical modeling for the delamination using a tri-linear cohesive constitutive law is conducted to provide additional validation.

Influence of Fiber Dimensions on Bridging Performance of Polyvinyl Alcohol Fiber-Reinforced Cementitious Composite (PVA-FRCC)

This study investigates the influence of fiber dimensions on the bridging performance of polyvinyl alcohol fiber-reinforced cementitious composite (PVA-FRCC) through an experimental and analytical program. Bending tests, bridging law calculations, and section analysis are conducted. Bending tests of notched specimens of PVA-FRCC with six different PVA fiber dimensions are performed to determine the load–deflection (LPD) and bending moment–crack mouth opening displacement (CMOD) relationships. The fiber volume fraction for all PVA-FRCCs is set to 2%. It is found that the load capacity of PVA-FRCC with a 27 μm diameter fiber is much higher than that of the other fibers, and the load capacity decreases as the fiber diameter increases. The study proposes parameters for the characteristic points of the tri-linear model for the single-fiber pullout model as functions of diameter, bond fracture energy, elastic modulus, cross-sectional area, and perimeter of the fiber. These findings provide valuable insights into the behavior of PVA-FRCC under different fiber dimensions. Bridging law calculations are conducted to obtain tensile stress–crack width relationships using the developed single-fiber pullout models. The Popovics model for the complete tensile stress–crack width relationship is adopted to obtain a better fit with the bridging law calculation, and then section analysis is conducted. The bridging law calculation results show that the maximum tensile stress decreases as the fiber diameter increases. It is also determined that most of the smaller-diameter fibers ruptured, whereas the larger fiber diameters pulled out from the matrix. The section analysis results show good agreement with the maximum bending moments obtained from the bending test.

Influence of Z‐pin material on the mechanical properties of fine Z‐pin reinforced composites

AbstractThe insertion of finer Z‐pins is an effective method to reduce the microstructure damage and maintain the in‐plane performance of composite laminates. In this paper, an ultrasound guided insertion process is proposed to achieve the insertion of fine Z‐pins and the influence of Z‐pin material on the mechanical properties of Ø0.11 mm Z‐pin reinforced composite laminates is revealed based on experiments and simulations. For in‐plane mechanical properties, stainless steel Z‐pins help to reduce the loss of compression modulus and compression strength. For the interlaminar mechanical properties, stainless steel Z‐pins and carbon fiber Z‐pins exhibit bilinear and trilinear bridging laws respectively, resulting in the mode‐I maximum load and fracture toughness of carbon fiber Z‐pin reinforced composites being greater than that of stainless steel Z‐pin reinforced composites. By adjusting the insertion spacing, it is found that for unidirectional composite laminates reinforced by Ø0.11 mm carbon fiber Z‐pins with an insertion spacing of 2 mm, the compression strength reduction is only 3.39% while the mode‐I fracture toughness can be improved by 14.4 times in comparison with the unpinned specimens, achieving a better balance between in‐plane performance loss and interlaminar reinforcement effect.Highlights The effect of Z‐pin material on fine Z‐pin reinforced composites is studied. The compression properties are simulated by the RVE model. The mode‐I fracture properties are simulated by the DCB unit strip model.

The over-notched flexure test method for determining the fracture toughness and bridging law of laminates

This work introduces a novel approach for analyzing the over-notched flexure (ONF) test, aiming to calculate the fracture toughness and bridging law. The important feature of this new methodology lies in obviating the necessity for continuous observation of crack length and relative sliding displacement at the tip of pre-crack, and extra tests for determining flexural modulus. Formulations are derived for compliance, fracture toughness, sliding displacement at the tip of pre-crack, and the corresponding bridging stress. To validate the accuracy of this approach, a comprehensive assessment is conducted based on experimental results from T800/epoxy and E-glass/polyester laminates with various stacking sequences. Comparisons between the calculated fracture toughness and bridging law from the proposed approach and the experiment results present a satisfactory consistence between each other. Furthermore, finite element analysis of the ONF test is carried out. The obtained bridging law is incorporated into a tri-linear cohesive law for simulating delamination. The good agreements between the load–displacement response from tests and predictions provide further validation of the proposed method. This method can simplify the testing process and reduce experimental errors, which makes it easy for application.

Study on bending failure and crack characteristics in ductile fiber‐reinforced concrete beams

AbstractIn this study, we designed six polypropylene fiber‐reinforced concrete (PP‐FRC) beams and three reinforced concrete (RC) beams for experimental studies. Then, we analyzed the effects of incorporating PP fibers on the cracking performance of PP‐FRC beams from the perspectives of load‐displacement response, crack distribution, crack depth, crack width, crack spacing, and deformation coordination. The research results showed that the cracking load and number of cracks were positively correlated with the volume ratio of PP fibers, while the maximum crack width, average crack spacing, and average crack depth were negatively correlated with the volume ratio of PP fibers. Due to the bridging effect of fibers, the PP‐FRC beam body and reinforcement could deform in a coordinated manner within a specific loading range after cracking, which increased the deformation coordination ability between the body and reinforcement. In this study, the cracking mechanism of PP‐FRC beams was investigated. Firstly, the formula for calculating the cracking moment of PP‐FRC beams was derived from the perspective of equivalent bending tensile strength. Then, based on the fiber bridging law and the bond strength between PP‐FRC and reinforcement, theoretical formulas were proposed to predict the average crack spacing and maximum crack width of PP‐FRC beams. Finally, through comparison, it was found that the proposed formulas could be used for characteristic crack prediction.

Bond and Cracking Characteristics of PVA-Fiber-Reinforced Cementitious Composite Reinforced with Braided AFRP Bars

Easy maintenance and high durability are expected in structures made with fiber-reinforced cementitious composite (FRCC) reinforced with fiber-reinforced polymer (FRP) bars. In this study, we focused on the bond and cracking characteristics of polyvinyl alcohol (PVA)-FRCC reinforced with braided AFRP bars (AFRP/PVA-FRCC). Pullout tests on specimens with varying bond lengths were conducted. Beam specimens were also subjected to four-point bending tests. In the pullout tests, experimental parameters included the cross-sectional dimensions and the fiber volume fractions of PVA-FRCC. A trilinear model for the bond constitutive law (bond stress–loaded-end slip relationship) was proposed. In the pullout bond test with specimens of long bond length, bond strength was found to increase with increases in both the fiber volume fraction and the cross-sectional dimension of the specimens. Bond behavior in specimens of long bond length was analyzed numerically using the proposed bond constitutive law. The calculated average bond stress–loaded-end slip relationships favorably fitted the test results. In bending tests with AFRP/PVA-FRCC beam specimens, high ductility was indicated by the bridging effect of fibers. The number of cracks increased with increases in the fiber volume fraction of PVA-FRCC. In specimens with a fiber volume fraction of 2%, the load reached its maximum value due to compression fracture of the FRCC. The crack width in PVA-FRCC calculated by the predicted formula, considering the bond constitutive law and the fiber bridging law, showed good agreement with the reinforcement strain–crack width relationship obtained from the tests.

Risk assessments for possible school shooters

Bridging law enforcement and mental health care is tricky, and in the case of possible violence in a student, this relationship can be especially strained. The chilling phrase “anyone could have stopped me” was the title of a session on this topic at the American Psychiatric Association meeting last spring.

Determination of mixed-mode I/II fracture toughness and bridging law of composite laminates

Accurate determination of the fracture toughness and the bridging law in a simple way is meaningful for the composite structural design. This study proposes a semi-analytical approach to calculate the mixed-mode I/II fracture toughness, and the crack opening/sliding displacements. Based on the relation between them, the mixed-mode I/II bridging law is obtained by using the J-integral theory. The obtained fracture toughness and distribution of the bridging stress show good agreements with test results, which illustrates the applicability of the semi-analytical approach. The obtained bridging law is further integrated into a tri-linear cohesive zone model for simulating the delamination behavior. Numerical load–displacement responses are consistent with experimental ones, further verifying the semi-analytical approach. Only experimental load and displacement data are necessary in this semi-analytical approach while specific equipment and tedious measurements on the delamination length and the relative crack opening/sliding displacement during the delamination tests are avoided, which makes the proposed approach simple and applicable for engineering application.

Thermal cycling effects on the mode-I delamination of multilayer laminated composites: An experimental-numerical approach

In this research, the influence of thermal cycling on the fracture toughness and maximum load in the mode-I delamination of polymer matrix composites (PMCs) was investigated. To reduce the effect of the remote ply orientation on the fracture toughness during initiation and propagation delamination, double cantilever beam (DCB) specimens with a stacking sequence of unidirectional [0]16 were considered. The specimens were subjected to thermal-cycled loading between 15°C and 65°C for 50-150 number cycles. One group of un-cycled specimens was examined at the commencement of the research as a control group to determine the effects of thermal cycling loading on mechanical characteristics of the second group specimens. During the DCB experiment, the specimens were inspected by 2 real-time cameras to record the delamination length and initial crack tip opening displacement (ICTOD). The strain energy release rate (SERR) approach was used for obtaining the critical fracture toughness from the experimental data. By employing the digital image correlation (DIC) method, the initial crack tip separation profile was obtained. The measured bridging laws with the cohesive zone elements model are used to accurately simulate the delamination propagation in DCB specimens. Investigation of load variations revealed that critical fracture toughness in the mode-I of delamination is firmly affected by the thermal cycling process.

Bridging Law Application to Fracture of Fiber Concrete Containing Oil Shale Ash

Concrete is a widely used material in various industries, including hazardous waste management. At the same time, its production creates a significant carbon footprint. Therefore, intensive research is being conducted to create more eco-friendly concrete, for example, partially replacing cement with by-products such as oil shale ash (OSA) or improving properties by adding dispersed fibers such as basalt fibers (BFs). The article consists of experimental testing of nine types of concrete and the modeling of crack propagation in bending. The basic trends of crack propagation in samples of concrete with OSA and BFs are simulated using a two-dimensional Finite Element (FE) model considering only material degradation on the opening crack surface and experimental data of three- and four-point bending tests. Crack propagation is modeled using the bridging law approach. A surrogate model for predicting the peak loading as a function of tensile strength and fracture work was created. An examination of the results of the FE model shows that the bilinear and nonlinear bridging law functions best describe the crack growth in the analyzed material. A comparison of experimental and modeled results showed that the length of the composite BF strongly affects the accuracy of the numerical model.

Flexural Characteristics of Functionally Layered Fiber-Reinforced Cementitious Composite with Polyvinyl Alcohol Fibers

The main purpose of this study is to investigate the flexural characteristics of a functionally layered fiber-reinforced cementitious composite (FL-FRCC) with polyvinyl alcohol fibers and to verify the adaptability of the proposed tri-linear stress-strain model based on the bridging law for large fiber orientation intensity, which shows the fiber orientation distribution as almost 2-D. The average maximum bending moment of FL-FRCC specimens is almost twice that of homogeneous (Hmg-FRCC) specimens, which indicates that the FL-FRCC specimens lead to larger bending capacity. The proposed wide-range stress-strain model based on the bridging law was verified and showed good adaptability with the experimental results through a comparison with the conducted section analysis.

Bridge laws in the Philosophy of Mind and in the Physical Sciences

Resumo: No debate sobre a redutibilidade da mente ao corpo, argumenta-se que não é plausível supor que tal redução possa se dar apenas a partir das condições físicas basais, mas que é necessário levar em conta também as leis de ponte psicofisiológicas. Essa posição é geralmente considerada antirreducionista, na Filosofia da Mente, mas se prefere aqui chamá-la de “reducionismo indutivo”, devido à analogia com duas outras formas de determinação nas Ciências Físicas: o determinismo causal e o reducionismo escalar espacial. A discussão é feita com base em “sondas epistemológicas” abstratas, como o demônio de Laplace, o demônio escalar e o demônio psicofisiológico. Critica-se também a noção de causalidade sincrônica usada por Searle.

Study on the dynamic response of offshore bridge under earthquake action and tsunami impact

This paper takes an offshore continuous rigid bridge as the background bridge to reveal the dynamic response law of an offshore bridge under the action of earthquake and tsunami. A tsunami wave is simulated by a dam-break wave. A numerical model of a dam-break impact of a bridge pier is established in Fluent software, and k−ε turbulence model is chosen to solve the Navier–Stokes equation. Meanwhile, ABAQUS software is used to establish a three-dimensional nonlinear full bridge model. The dam-break wave working condition is proposed according to the Stoker theory. The bridge dynamic response index under a single earthquake action and a continuous earthquake tsunami action is calculated, and the dynamic response rules under the two actions are summarized. The results show that compared with the tsunami action alone, the sequential action of the earthquake and tsunami will cause a greater dynamic response of the bridge structure. With the increase in the height of the incoming tsunami wave, the waveform trend difference of the response index will be more significant. Moreover, the influence coefficient of the dynamic response index of the bridge tends to increase as the probability level of the earthquake occurrence decreases.

Contesting the “Secular” via the Judiciary

This article looks at some cases of legal contestation in three African countries (Mali, Senegal, and Nigeria) where Muslim religious authorities/institutions brought challenges to constitutional-legislative proposals in the national arena, seen by religious leaders as unacceptable due to lack of resonance with religious values. These are cases where laws/directives aiming to regulate social/religious life vs. the state are contested on grounds of religious conviction: with objections put forward in the name of an entire faith community despite internal differences within it, and despite the fact that in all countries the religious pluriformity necessitates bridging laws. The article makes the argument that there are limits on the religious contestation of secular state laws, to a point that the effort tends to become counter-productive in securing religious freedom or influencing politics in a positive sense, and instead might endanger a pragmatic modus vivendi of the religious and the political spheres in nominally secularist orders. Some conclusions are drawn on the resulting phenomenon of growing tensions between particularist narratives and state narratives in Africa.

A semi-analytical method for the determination of fracture toughness and bridging law in ELS test

Fracture toughness and bridging law are necessary for characterizing the delamination behavior with the effect of fiber bridging. Accurate determination of them in a simple way is particularly important for the design of composite structures. The end-loaded split (ELS) method, which can obtain stable delamination growth, has been applied in the ISO 15,114 standard to measure the mode II fracture toughness. However, geometric nonlinearity and difficulty on the crack observation are unavoidable. To deal with those problems, a semi-analytical method is proposed in this study for determining the fracture toughness and the bridging law. Predictions from the semi-analytical method are consistent with experimental results of CFRP and GFRP laminates. Furthermore, numerical modelling is performed based on a tri-linear cohesive zone model which integrates the determined bridging law. Numerical load–displacement responses have good agreements with the experimental ones, which illustrates the applicability and accuracy of the proposed method. The proposed method overcomes the problem of real-time monitoring of the crack growth length and the relative sliding displacement at the pre-crack tip, and considers the correction factors to solve the nonlinear problem caused by the large displacement, which is simple and efficient for practical application.

Fiber bridging mechanism in moisture‐induced mode I delamination in carbon/epoxy composites: Finite element analysis and experimental investigation

AbstractProlonged exposure of fiber reinforced composite structures to high moisture and temperature in outdoor environment could lead to the degradation of mechanical properties of the materials. To provides reliable prediction of the delamination behavior as the moisture progressively ingresses into the composites, we proposed a Bilinear‐Exponential Traction‐Separation (BETS) law‐which can account for the fiber bridging mechanism‐for investigating the mode I delamination of unidirectional carbon fiber reinforced epoxy composite laminates in wet states. Finite element analysis (FEA) of the delamination model was conducted to evaluate the effects of moisture content on the global force‐displacement curves. A cohesive zone model (CZM) was used to describe the delamination behavior at the interface. With regard to the force‐displacement curves, the BETS law agrees well with results from experimental study (using the double cantilever beam testing on the wet specimens) at both the elastic and failure regions. The delamination model governed by the BETS law also showed good agreement with the crack length versus the crosshead displacement. The FE model that considered the inputs from the BETS law yielded prediction about the evolution of the damage parameter and crack growth profile. In particular, the predicted crack extension increases linearly with increasing crosshead displacement. The proposed BETS law has the advantage of not requiring crack growth monitoring during experiment, and only one fitting parameter was needed to describe the bridging law at different moisture content levels.

Multiscale modelling of curing-induced z-pin/composite interfacial cracks

The curing process generates thermal residual stress in z-pinned composites, leading to partial or complete cracking along z-pin/composite interfaces. In this study, we applied a novel multiscale framework to evaluate the delamination behaviors of z-pinned composites considering the effects of z-pin/composite interfacial cracks. Unit cell models were used to represent the typical microstructures and micromechanisms of z-pinned composites with diverse interfacial conditions and to predict z-pin pullout behaviors. The derived z-pin bridging laws are then used to simulate the behaviors of individual z-pins in macroscale models. The load-versus-displacement relationships of z-pinned composites under mode-Ⅰ loading conditions are predicted using macroscale models and then verified experimentally. A parametric study reveals that interfacial cracks can affect the failure behaviors of z-pinned composites by changing the microscale bridging mechanisms of individual z-pins. It is theoretically feasible to customise z-pin/composite interfacial bonding properties, z-pin surface roughness, and curing temperature to optimise the delamination properties of z-pinned composites.