Bridging Law Research Articles

Experimental approach for mixed-mode fatigue delamination crack growth with large-scale bridging in polymer composites

An experimental apparatus utilizing double cantilever beam specimens loaded with uneven bending moments was developed to study the mixed-mode fatigue crack growth in composites. The approach is suitable when large-scale bridging of cracks is present. To illustrate the testing method, cyclic growth of delaminations in a typical fibre-reinforced polymer composite was investigated under a constant cyclic loading amplitude. Pure mode I, mode II and mixed-mode crack growth conditions were examined. The results, analysed using a J-integral approach, show that the double cantilever beam loaded with uneven bending moments configuration provides a robust approach to investigate the fatigue crack growth of composites for pure mode and mixed-mode cracking. A steady-state crack growth regime was observed for mode I and mixed-mode loading. For mode II loading, steady-state was absent, and a progressively decreasing crack growth rate observed. In addition to details concerning the equipment, a general discussion of the development of cyclic bridging laws for delamination growth in the presence of large-scale bridging is provided.

Application of the Shakedown Theory to Brittle-Matrix Fiber-Reinforced Cracked Composite Beams Under Combined Traction and Flexure

The cracking behavior of a composite beam with multiple reinforcing fibers under periodic traction-flexure is analyzed through a fracture mechanics-based model, where the edge-cracked beam section is exposed to external loads and crack bridging reactions due to the fibers. Assuming a rigid-perfectly plastic bridging law for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging forces are obtained from compatibility conditions. Under general load paths, shakedown conditions are explored by making use of the Melan's theorem, here reformulated for the discrete problem under consideration, where crack opening displacement at the fiber level plays the role of plastic strain in the counterpart problem of an elastic-plastic solid. The limit of shakedown is determined through an optimization procedure based on a linear programming technique.

Effect of initial crack length on the measured bridging law of unidirectional E-glass/epoxy double cantilever beam specimens

In this paper, the effect of initial delamination length is experimentally investigated on obtaining the mode I bridging law of unidirectional E-glass/epoxy double cantilever beam (DCB) specimens manufactured by hand layup method. To this end, an experimental test set-up is established for accurate measurement of crack tip opening displacement (CTOD) using digital image processing method. DCB tests are performed for three different delamination lengths and the corresponding bridging laws are calculated using J-integral approach. Results showed that the maximum bridging stress, the shape of bridging law and energy dissipation in bridging zone are slightly affected by changing initial crack length. In other words, the measured bridging law acts independent of initial delamination length. Therefore, the obtained bridging law can be used with the cohesive elements available in the commercial finite element software to simulate the delamination propagation behavior in unidirectional DCB specimens.

Brownian Gibbs property for Airy line ensembles

Consider a collection of N Brownian bridges \(B_{i}:[-N,N] \to \mathbb{R} \), B i (−N)=B i (N)=0, 1≤i≤N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak limit as N→∞ of the collection of curves scaled so that the point (0,21/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is added to each of the curves of this scaling limit, an x-translation invariant process sometimes called the multi-line Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which the curves are almost surely everywhere continuous and non-intersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with “wanderers” and “outliers”. We formulate our results to treat these relatives as well.Note that the law of the finite collection of Brownian bridges above has the property—called the Brownian Gibbs property—of being invariant under the following action. Select an index 1≤k≤N and erase B k on a fixed time interval (a,b)⊆(−N,N); then replace this erased curve with a new curve on (a,b) according to the law of a Brownian bridge between the two existing endpoints (a,B k (a)) and (b,B k (b)), conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edge-scaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property.An immediate consequence of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the long-standing conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi-line Airy process, and readily yields several other interesting properties of this process.

A Fiber Pull-Out Based Model for Synthetic Fiber Reinforced Concrete Beams under a Flexural Load

This work is intended to be a simple contribution to building a model able to implement theoretical results related to the random oriented fiber reinforced concrete in a procedure that could be used in structures analysis and design involving fiber reinforced elements. Here follows a short outline: In the introduction chapter the problem is presented together the work done. Section 2 develops some ancillary concepts of this material and its mechanical properties, while in Section 3, following the path of other researchers, the assumptions made to solve the problem are presented, together with the most relevant results related to presence of 3D randomly oriented fiber. In the following section a review of the mechanical process of fiber pull-out is done, and the results, mostly due to Victor Li researches, of a 3D randomly oriented synthetic fiber stress vs crack opening in a pull-out process from a cement matrix. In Section 5 the author, after making some assumptions about the configuration of the strain and crack geometry in the cross section where failure is assume to occur under flexural bending moment, the resultant stress is integrated to find the resultant internal moment vs increasing strain and crack width. In this analysis, the crack bridging law for synthetic fiber in FRC presented in the previous section is taken into account. In Section 6, a procedure to find a cross section configuration in equilibrium under external bending moment has been built. Under the assumption of a perfectly plastic collapse mechanism a numerical simulation is conducted on a specimen that undergoes a four-point bending test. A comparison with the trend of a similar test on a synthetic FRC sample has been done. The work is completed by the conclusions that could be inferred from this work.

Compressive strength of composite laminates with an open hole: Effect of ply blocking

This paper examines the application of a cohesive zone model to predict the open hole compressive strength of an IM7/8552 carbon fibre/epoxy quasi-isotropic multidirectional laminate and investigates the level-ply scaling or ply blocking effect on notch sensitivity. Cohesive zone models have been successfully applied to predict the damage from notches in engineering materials loaded in tension. They have also been used to determine the growth of fibre microbuckling from a hole in composite laminates under compression. The usual strategy is to replace the inelastic deformation associated with plasticity or microbuckling with a line-crack and to assume some form of stress-displacement (σ − δ) bridging law across the crack faces. Here a plastic fibre kinking analysis and a linear reduced (softening) σ − δ relationship are used for the prediction of the unnotched and open hole compressive strength; the theoretical results will be compared with experimental data in Part B of the 3rd World-Wide Failure Exercise.

A predictive multiscale computational framework for viscoelastic properties of linear polymers

A predictive multiscale computational framework has been proposed to study the viscoelastic properties of polymeric materials. Using the Inverse Boltzmann Method, both the static structures and dynamic behavior of all-atomistic models of polymers can be reproduced by a simple coarse-grained model, which bridges the scale from nano to meso. On this coarse-grained level, the entangled network of polymer chains is described via a primitive path analysis (Z1 code). This description allows extraction of the tube diameter and primitive chain length, quantities required to bridge the scale from meso to micro. Furthermore, by making the affine-deformation assumption, a continuum constitutive law for polymeric materials has been developed from the tube model of primitive paths, which bridges the scale from micro to macro. In this way, the different scales are crossed by using different bridging laws, which enable us to directly predict the viscoelastic properties of polymeric materials using a bottom-up approach. Our predicted dynamic moduli, zero-rate shear viscosities, and relaxation moduli of polyisoprene and polyethylene polymers are found to be in excellent agreement with experimental results. The proposed multiscale computational framework can also be naturally extended to the finite-deformation regime. Both the tube diameter a and primitive chain length L are found to increase with deformation, which enhances the viscous energy dissipation of polymers under extremely large deformations. To the authors' knowledge, this is the first work in which a multiscale computational framework has been proposed to predict the viscoelastic properties of entangled polymeric materials from the molecular level. Not only can the method put forth in this research be used to predict the viscoelastic properties of polymeric materials in a bottom-up fashion, it can also be applied to design the polymeric materials with targeted functions, within a top-down approach.

A generalized porosity formalism for isotropic and anisotropic effective opacity and its effects on X-ray line attenuation in clumped O star winds

We present a generalized formalism for treating the porosity-associated reduction in continuum opacity that occurs when individual clumps in a stochastic medium become optically thick. As in previous work, we concentrate on developing bridging laws between the limits of optically thin and thick clumps. We consider geometries resulting in either isotropic or anisotropic effective opacity, and, in addition to an idealized model in which all clumps have the same local overdensity and scale, we also treat an ensemble of clumps with optical depths set by Markovian statistics. This formalism is then applied to the specific case of bound–free absorption of X-rays in hot star winds, a process not directly affected by clumping in the optically thin limit. We find that the Markov model gives surprisingly similar results to those found previously for the single-clump model, suggesting that porous opacity is not very sensitive to details of the assumed clump distribution function. Further, an anisotropic effective opacity favours escape of X-rays emitted in the tangential direction (the ‘venetian blind’ effect), resulting in a ‘bump’ of higher flux close to line centre as compared to profiles computed from isotropic porosity models. We demonstrate how this characteristic line shape may be used to diagnose the clump geometry, and we confirm previous results that for optically thick clumping to significantly influence X-ray line profiles, very large porosity lengths, defined as the mean free path between clumps, are required. Moreover, we present the first X-ray line profiles computed directly from line-driven instability simulations using a 3D patch method, and find that porosity effects from such models also are very small. This further supports the view that porosity has, at most, a marginal effect on X-ray line diagnostics in O stars, and therefore that these diagnostics do indeed provide a good ‘clumping insensitive’ method for deriving O star mass-loss rates.

On the Role of Bridge Laws in Intertheoretic Relations

What is the role of bridge laws in intertheoretic relations? An assumption shared by many views about these relations is that bridge laws enable reductions. In this article, I acknowledge the naturalness of this assumption, but I question it by presenting a context within thermal physics (involving phase transitions) in which the bridge laws, puzzlingly, seem to contribute to blocking the reduction.

Numerical Study on Adhesive Interface Reinforcement of Piezoelectric Composite under Impact Load

The interfacial reinforcement with interlaminar chopped fibers of piezoelectric composite under impact electro-mechanical load was studied using nonlinear finite element method. A meso- mechanical model based on the main toughness reinforcement mechanism of single fiber bridging and pull out was adopted, and then a tri-linear bridging law was obtained, while the interface chopped fibers by defining nonlinear bidirectional spring elements between coincident nodes on the two crack surfaces within bridging zone and the energy release rate was calculated using the virtual crack closure technique. The numerical investigation indicates that the interlaminar chopped fiber can effectively reduce the crack tip energy release rate whether the applied voltage is positive or negative, which was an effective technique improve the interfacial toughness of the piezoelectric composite adhesive structure.

General alpha-Wiener bridges

An α-Wiener bridge is a one-parameter generalization of the usual Wiener bridge, where the parameter α > 0 represents a mean reversion force to zero. We generalize the notion of α-Wiener bridges to continuous functions α : [0, T ) → R. We show that if the limit limt↑T α(t) exists and is positive, then a general α-Wiener bridge is in fact a bridge in the sense that it converges to 0 at time T with probability one. Further, under the condition limt↑T α(t) 6= 1 we show that the law of the general α-Wiener bridge cannot coincide with the law of any non time-homogeneous Ornstein-Uhlenbeck type bridge. In case limt↑T α(t) = 1 we determine all the Ornstein-Uhlenbeck type processes from which one can derive the general α-Wiener bridge by conditioning the original Ornstein-Uhlenbeck type process to be in 0 at time T .

Determination of the fracture process zone under Mode I fracture in glass fiber composites

This study provides a simple yet effective procedure for the characterization of the fracture process zone (FPZ) developing in the interface of unidirectional laminates under Mode I delamination fracture. Double cantilever beam (DCB) coupons have been manufactured and tested. Three data reduction schemes available in the literature have been utilized for the calculation of the energy release rate (ERR) magnitude as a function of crack extension and the corresponding R-curves have been constructed. The R-curves were then reconstructed in terms of the experimentally registered pre-crack tip opening displacement ( δ*) and analytical functions have been used to describe their concatenate trend. The J-integral approach was then applied over the analytical functions to derive the corresponding bridging laws that describe the FPZ. The derived bridging laws were appropriately modified according to three different traction–separation models and implemented into user-developed interface finite elements (UEL) for the simulation of the fracture tests in ABAQUS® commercial software. Comparisons between numerical and experimental results have shown that the proposed straightforward procedure leads to an effective traction–separation law that can be used as a material property of the modeled interface.

Experimental Study of Damage Variable Law of Bridge and Tunnel with Underlying Goaf during Expressway

This paper takes geological conditions of the Li—Jun expressway by Kang JiaGou and Tong De coal mine goaf as prototype, using the simulation experiment of similar material, simulates the formation process and distribution of the mined rock fracture and the damage variable of the mined rock by using the simulation experiment of similar material, At the same time, It researches the evolvement law of rock’s fracture network in caving zone, fractured zone over goaf using fractal geometry theory. It analyses the damage variable law in caving zone, fractured zone using the damage mechanical theory, and found the relationship of damage variable and mined width. The conclusion of the experiment indicates: with the advance of the working surface, the fractal dimension of the mined rock crack goes through small → large →small, and stable process of change. In the identical push distance of the working surface, fractal dimension of caving zone is 1.0—1.2 times of fractured zone. With the advance of the working surface, damage variable increases gradually, under the identical mined width, damage variable of caving zone is greater than fracture zone, caving zone is about 1.04—1.11 times of fracture zone. Damage variable of goaf central zone is greater than edge.

Model for the Effect of Fiber Bridging on the Fracture Resistance of Reinforced-Carbon-Carbon

A micromechanical methodology has been developed for analyzing fiber bridging and resistance-curve behavior in reinforced-carbon-carbon (RCC) panels with a 3D composite architecture and a SiC surface coating. The methodology involves treating fiber bridging traction on the crack surfaces in terms of a weight function approach and a bridging law that relates the bridging stress to the crack opening displacement. A procedure has been developed to deduce material constants in the bridging law from the linear portion of the K-resistance curve. This approach has been applied to analyzing R-curves of RCC generated using double cantilever beam and single cantilever bend specimens to establish a bridging law for RCC. The bridging law has been implemented into a micromechanical code for computing the fracture response of a bridged crack in a structural analysis. The crack geometries considered in the structural analysis include the penetration of a craze crack in SiC into the RCC as a single-edge crack under bending and the deflection of a craze crack in SiC along the SiC/RCC interface as a T-shaped crack under bending. The proposed methodology has been validated by comparing the computed R-curves against experimental measurements. The analyses revealed substantial variations in the bridging stress (σo ranges from 11 kPa to 986 kPa, where σo is the limiting bridging stress) and the R-curve response for RCC due to the varying number of bridging ligaments in individual specimens. Furthermore, the R-curve response is predicted to depend on crack geometry. Thus, the initiation toughness at the onset of crack growth is recommended as a conservative estimate of the fracture resistance in RCC. If this bounding structural integrity analysis gives unacceptably conservative predictions, it would be possible to employ the current fiber bridging model to take credit for extra fracture resistance in the RCC. However, due to the large scatter of the inferred bridging stress in RCC, such an implementation would need to be probabilistically based.

Experimental and numerical investigation of Mode II fracture in fibrous reinforced composites

A straightforward procedure is described for utilizing experimentally evaluated bridging laws that characterize Mode II fracture growth of composite materials into numerical simulations. Unidirectional glass/epoxy end notch flexure (ENF) coupons have been fabricated and tested. Three data reduction schemes available in the literature were used for the construction of the R-curves together with the J-integral approach for the derivation of the bridging laws. Two traction— separation models have been utilized for the characterization of the fracture process zone (FPZ) developing during the delamination propagation process. The first model considers only the existence of a bridging zone behind the physical crack tip, whereas the proposed model considers both the existence of a bridging zone and of a cohesive zone in front of the physical crack tip. The traction—separation models were implemented into interface elements for the simulation of the ENF tests. Numerical results have shown that the proposed procedure together with the proposed traction— separation model is quite promising for simulations that involve Mode II fracture growth.

Factors influencing international fashion retailers' entry mode choice

PurposeThe purpose of this paper is to provide a theory‐based framework that informs a fashion retailer's entry mode choice into a foreign market.Design/methodology/approachAspects of transaction cost, bargaining, resource based, and internationalization theories were integrated to develop a conceptual framework for fashion retailers determining the best entry mode to foreign markets. Propositions were developed, which serve as bridge laws, bridging the gap between the theories and the investigation of fashion retailers' entry mode choice. A case study was used to demonstrate applicability of the developed propositions.FindingsThree groups of factors were identified that influence entry mode choice in the fashion retail market: firm‐specific factors of asset specificity, brand equity, financial capacity, and international experience; country‐specific factors of country risk, cultural distance, and government restrictions; and market‐specific factors of market potential and market competition. Nine propositions were generated, positing how each of the factors may influence a fashion retailer's entry mode choice.Research limitations/implicationsThe conceptual model and propositions require further empirical investigation. Future research also needs to systematically explore the interactions or trade‐offs between different determinate factors.Practical implicationsA fashion retailer can use the framework and propositions to systematically evaluate the company's case to justify an entry mode decision for a specific foreign market.Originality/valueThis is the first paper to describe the integration of theories to help explain factors affecting fashion retailers' entry mode choice.

The European Media Pluralism Monitor: Bridging Law, Economics and Media Studies as a First Step towards Risk-Based Regulation in Media Markets

The European Media Pluralism Monitor : Bridging Law, Economics, and Media Studies as a First Step towards Risk-Based Regulation in Media Markets

Chemistry and physics: no need for metaphysical glue

Using the notorious bridge law “water is H2O” and the relation between molecular structure and quantum mechanics as examples, I argue that it doesn’t make sense to aim for specific definition(s) of intertheoretical or interdiscourse relation(s) between chemistry and physics (reduction, supervenience, what have you). Proposed definitions of interdiscourse and part-whole relations are interesting only if they provide insight in the variegated interconnected patchwork of theories and beliefs. There is “automatically” some sort of interdiscourse relation if different discourses claim to have something to say about the same situation (event, system), which is the basis of (contingent) local supervenience relations, which, proper empirically support being provided, can be upgraded to ceteris paribus bridge laws. Because of the ceteris paribus feature, and the discourse dependence of event identification, there is at best only global supervenience of the “special sciences” on the physical (and of parts of physics on other parts of physics).

Crack Growth Resistance of Hybrid Fiber-Reinforced Cement Matrix Composites

The effect of hybrid fiber reinforcement on fracture energy and crack propagation in cement matrix composites is examined. The crack in cement matrix composites is allowed to fracture under mode-I loading with three-point bending beam specimens. The influence of fiber types and their combination is quantified by using the toughness index and fracture energy. A proper hybrid combination of steel fibers and polyvinyl alcohol microfibers enhances the resistance to both the nucleation and growth of the crack. The micromechanical model of hybrid composites by using a fiber bridging law is emphasized, and the numerical model prediction closely matches the behavior obtained from the experiment. The influencing role of the material parameters in the fracture tests (e.g., the fracture toughness index and fracture energy) becomes more apparent than ones used in some conventional strength-based or fiber pullout tests, and these fracture parameters could screen the effect of fiber/microfiber reinforcement in enhancing the crack growth resistance of cementitious composites. This study demonstrates that fundamental fracture tests are effective to characterize and develop high-performance hybrid fiber–reinforced cement matrix composites.

Experimental investigation of bridging law for single stitch fibre using Interlaminar tension test

In this study, a novel Interlaminar tension test (ITT) method was performed to experimentally investigate the bridging and fracture process of a single stitch fibre used to improve the delamination strength of composite laminates. Kevlar-29, of various thread thicknesses (44, 66, 88 and 132 tex), was used as the through-thickness stitch fibre in the ITT experiments. Key empirical force and displacement parameters, which governed the stitch fibre bridging law, were characterised and identified. Relationships of such parameters with thread thicknesses were determined. Fibre fracture load and fibre fracture energy are found to increase with increasing thread thickness. Frictional pull-out force greatly depends on the type of stitch fracture modes, which can be grouped into three categories. This paper aims to provide better physical understanding of the mechanics and mechanisms of stitch fibre fracture. By correlating critical stitch fracture parameters with stitch fibre thicknesses, the results expect to provide useful reference, which is essential and important for accurate stitch computational modeling and strength prediction of composites using stitching as the interlaminar reinforcement technique.