Two types of laminated composite plates reinforcing pure copper with carbon steel were prepared. One was the steel-copper-steel (SCS) hot-rolled clad plates, and the other the copper-steel-copper (CSC) diffusion-bonded plates. As a result of fatigue testing using these specimens, the fatigue crack showed a zigzag crack front with a large roughness on the fracture surface of the inner copper sandwiched between both carbon steels in the SCS type plate with large thickness, while it showed a relatively smooth crack front on the fracture surface of the steel part. An influence of such a fracture surface roughness on the fatigue crack propagation was discussed in terms of fracture mechanics parameter. The fatigue crack propagation rate, da/dN, for all the laminated composite plates and the carbon steel base plates gathered in a small band expressed by the power law of ΔK_<est>/E, using the stress intensity factor range Δ K_<est> evaluated from the measured crack opening displacement and Young's modulus E. If examined in detail, however, the da/dN was lower in the thicker SCS type plate than in the CSC and the base plates, due to the large roughness on the fracture surface of the inner copper part.
The A15052/AFRP laminates often suffer from the cyclic bending moment of variable amplitude during the service. Thus the fatigue crack initiation would occur and propagate in the aluminum layer of laminate by the cyclic bending moment. The fatigue crack propagation in aluminum layer of A15052/AFRP laminate is commensurated with the delamination between aluminum layer and fiber layer. The delamination deteriorates the fiber bridging behind the crack tip of laminate. Therefore the influence of variable cyclic bending moment on the delamination and the fatigue crack propagation behavior in A15052/AFRP laminate need to be investigated. In this study, A15052/AFRP laminate composite of three thin sheets of A15052 and two layers of unidirectional aramid fibers was prepared and tested. The shape and size of the delamination zone formed along the fatigue crack between A15052 sheet and aramid fiber-adhesive layer behind the crack tip were measured by an ultrasonic C-scan. The relationships among da/dN, ΔK, the cyclic bending moment and the delamination zone size were studied.
Fatigue behavior of SiC fiber reinforced SiC matrix (SiC/SiC) composites was carried out under bending conditions to investigate the fatigue damage and failure mechanisms. It was found that fatigue cracks initiated at several sites and then grew or coalesced each other. Finally, a crack grew significantly as a main crack to cause the specimen failure. The change of rigidity during fatigue tests was measured. The rigidity decreased with increasing fatigue damage in the specimen. Although the relationship between rigidity and number of cycles depended on applied stress, fatigue failure occurred when the rigidity reduced to a critical value. The change of the rigidity was considered to be a useful parameter to evaluate the fatigue damage of SiC/SiC composite material.
Smooth specimens of an aluminum alloy 2024-T6 reinforced with 20 volume percent of silicon carbide particles (SiCp) were fatigued under four-point bending. The X-ray diffraction method was applied to measure the loading and residual stresses in each constituent phase. The phase stresses were determined from the diffractions of Al 222 and SiC 116. The compressive residual stress in the aluminum alloy phase increased with increasing stress cycles. For the SiC phase, when the ratio of the number of stress cycles to fatigue life, N/N_f, was larger than about 0.15, the compressive residual stress increased with stress cycles. Difference between the phase stresses measured at the maximum and zero loads was examined during fatigue. When N/N_f was larger than about 0.15, the change of the phase stress of both aluminum alloy and SiC particles decreased with the number of stress cycles. A lot of cracks in the matrix and decohesions at the interface were observed on the specimen surface. The decrease of the phase stress was caused by the initiation and propagation of fatigue cracks. The fatigue damage was successfully evaluated by the X-ray method.
The dynamic and fatigue properties of short-fiber reinforced Chloroprene rubber have been studied as functions of interphase conditions and fiber content. The loss factor and dynamic ratio generally decreased with fiber content and showed different patterns according to interphase conditions. The better interphase condition showed the higher storage modulus(E_1). Therefore, the short-fiber reinforced rubber could have the better isolation in frequency ratio of √<2> min. compared to frequency ratio of √<2> max. The spring constant of rubber decreased about 21% after the fatigue test. On the contrary, that of reinforced rubber increased in all cases. The temperature of matrix increased about 2.5 times and one of reinforced rubber showed 1.7 〜 2 times up after the test. It is found that the better interphase condition, the smaller changing rate of spring constant and the lower specimen temperature at the same fiber content. Double coatings of rubber solution and bonding agent seems to be better interphase condition than a single coating of rubber solution or bonding agent.
Continuous fiber reinforced metal matrix composites (MMCs) have been researched and developed because of both high specific strength and high temperature capability and various trial high temperature structural components have been successfully fabricated in the field of aeronautics, aerospace and power generation industries. However, there is the limited number of research for long-term durability performance at practical use temperature ranges in comparison with the static strength characteristics. The final goal of this research is to establish the fatigue damage tolerance design concepts for continuous fiber partially reinforced Ti-alloy matrix composite (TMCs) rotating parts in aircraft engines. This paper summarizes the elevated temperature low cycle fatigue behavior for Ti-alloy matrix composites.
One of critical technological issues for the realization of the supersonic transportation(SST) is to ensure the long-term high temperature structural integrity of advanced polymer matrix composite system. This research has been conducted as a part of long-term durability performance of composites and structures in Japan Supersonic Research Program for 2^<nd> Generation Supersonic Civil Transport. The final goal is to achieve through long-term and short-term tests under conditions simulating SST flight, development of associated predictive and accelerated test methods, and assessment of durability performances for design. This paper summarizes open-hole fatigue behavior for the interesting and candidate high temperature polymer matrix composites.
The present paper treats an identification method of bending stiffness components of symmetric laminates using vibration data that consist of the low natural frequencies and the corresponding vibration modes. The present method is based on the virtual work principle and the least squares method, and it can give a nondestructive evaluation of bending stiffness components of laminated composites within a large-scale structure such as a wing structure. The method is applied to symmetrically laminated carbon/epoxy plates. The five bending stiffness components of laminated composites are identified using the vibration data obtained by a holographic apparatus. It is also numerically shown that the present method gives a precise evaluation of bending stiffness components insensitive to measurement errors.
The applicability of the CED, that has the meaning of strain energy area density without any restriction on constitutive equation, to the resistance evaluation of fiber pulling out in fiber reinforced composites was studied, for an axisymmetric model with a cylindrical crack between a fiber and matrix, by using the discontinuous model. The path-independent expression of CED with no restriction on constitutive equation was derived first and the finite element analyses of the model were carried out to investigate the properties of CED for fiber debonding problem. The CED was evaluated not only by the derived path-independent expression but also by other methods and it was demonstrated that the evaluation of CED can be conducted easily taking the effects of friction in debonded interface and singularity at the edge of debonding, that are difficult to evaluate when the conventional methods are employed, into consideration.
Composite materials technology is gradually accepted as a basis for studying the behaviour of concrete. To that end, finite element methods (FEM) are employed on composite section images. A random generator-based (RG) system is conventionally used to simulate a particulate structure of spherical grains. Neither does this allow generation of the dense-random structure of aggregate grains in concrete on macro-level (with a volume fraction of about 0.75), nor of the meso-structure of binder particles between the aggregate particles at lower water to cement ratios. Particularly, the Interfacial Transition Zone (ITZ) between bulk paste and the aggregate particles is of paramount importance for strength and durability of the material. The paper presents an analytical method relating major characteristics of the spatial structure in a randomly generated spherical particle mixture to relevant features of this structure, such as the areal fraction and the grain section size distribution function, displayed in a randomly selected cross-section. The presented example concerns a Fuller mixture, but the solution is also available for arbitrary sieve curves. In doing so, the section image for FEM application of the material can directly be derived from design data such as the sieve curve and the volume fraction of the aggregate. Studies of structure-insensitive to moderately structure-sensitive properties, depending on material composition, can be performed in this way. For structure-sensitive properties, like debonding and crack initiation, the mutual arrangement of particles (configuration) should be 'realistically' simulated. For that purpose, SPACE is introduced. It is based on a dynamic mixing of grains (reflecting the production process) of which the volume density is gradually increased to the required level. The paper shows the dramatic differences in geometrical statistical properties of particulate structures simulated by the classical RG system and by SPACE.
In this work, the homogenized elastic-viscoplastic behavior of long fiber-reinforced laminates under in-plane loading is predicted by taking directly into account the micro-scopic structure and stacking sequence of laminae. To this end, a homogenization theory of nonlinear time-dependent composites is applied to such laminates, leading to the macro-scopic rate-type constitutive equation of laminates and the evolution equations of micro-scopic and average stresses in each lamina. The macroscopic constitutive equation has a stiffness tensor and a stress relaxation function which are evaluated explicitly in terms of the microscopic structure and stacking sequence of laminae. To verify the present theory, uniaxial tensile tests are performed on carbon fiber/epoxy laminates. It is thus shown that the present theory is successful in predicting the anisotropic viscoplasticity in in-plane tension of unidirectional and cross-ply laminates and the negligible viscoplasticity exhibited by quasi-isotropic laminates.
A methodology to constitute three-dimensional finite element model of heterogeneous material, such as textile composite, is investigated on the basis of the X-ray CT image of the cross-section. A new breed of experimental system is devised to obtain the CT images of a specimen successively along the loading axis with short intervals. We discretize the specimen by means of the fixed grid/mesh elements, and identify the material property of elements with relation to the voxel value contained in the element. An inverse problem for the identification is posed in the context of the damage mechanics in consideration of the analogy between the voxel value and effective resisting area. The posed inverse problem is formulated so as to determine the damage variables in the damage effect tensor to realize the displacement field of the specimen observed through the experimental system.
In this paper, the displacement field for a semi-infinite wedge anisotropic plate under transverse loading was first derived. The corresponding stress field was then determined. Stress singularities at the tip of the wedge were also obtained. To prove the validity of the derived theory, a clamped zero-degree twelve-ply carbon-fiber-reinforced-plastic (CFRP) plate with a wedge of 270 degrees was prepared as the test specimen. The electronic speckle pattern interferometry (ESPI) technique was employed to find the whole-field displacement. Well-matched condition between the theory derived and ESPI results was obtained. In contrast to an isotropic plate, the derived theory shows that stress singularities can be found not only above but also below 180 degrees of the wedge angle.
This paper deals with progressive ply-cracking damage and nonlinear deformation of CFRP cross-ply laminate. Tensile tests in 0°-axis direction and in off-axis direction are carried out on five kinds of CFRP cross-ply laminates which are different in stacking lay-up. Under 0°-axis tension, initiation and evolution of the ply-cracking damage in 90° plies are observed, and corresponding stress-strain relation is slightly nonlinear after the damage. On the other hand, under off-axis tension, stress-strain response shows nonlinear deformation, while the ply-cracking damage is not remarkable except for the specimen with large fracture strain. Therefore, it is concluded that the nonlinear deformation of the laminates under off-axis tension is mainly attributed to the nonelastic deformation of the matrix resin. In order to describe the progressive ply-cracking damage and the nonlinear deformation of the CFRP cross-ply laminates, a ply-cracking damage theory and a nonlinear lamination theory are developed, respectively.
The propagation path of matrix cracks parallel to the fiber direction in continuous fiber-reinforced plastics (FRP) under remote mode I loading was predicted by the stress intensity factors and the nonsingular T-stress, T_m, in a matrix layer. These parameters for two FRP models were calculated by using the two-dimensional boundary element method. One model is the inhomogeneous FRP model which has a laminated structure composed of matrix and fiber, and the other is three-layered FRP model in which a matrix layer is sandwiched between homogenized FRP plates. When the residual stress in the matrix layer is negligible, T_m is negative for typical FRP composites. For long cracks, the absolute value of T_m for the inhomogeneous FRP model is larger than that for the three-layered FRP model, but the sign of T_m for both models is the same. A crack in typical FRP composites is predicted to run along the centerline of the matrix layer because of a negative T_m value and the sign of the mode II matrix stress intensity factor. For long cracks, the propagation path of matrix cracks in inhomogeneous FRP was predictable from two parameters of three-layered FRP model. The effects of the residual stress, material properties, the height of the cracked matrix layer, and the crack length on the crack propagation path were discussed on the basis of the results of the present analysis.
The most effective material in shape memory alloy (SMA) is the TiNi alloy, because its shape recovery characteristics are excellent. The composite material was molded with shape memory function, which the fiber is Ti_<50>-Ni_<50> shape memory alloy and the matrix is epoxy resin having good adhesive and optical sensitivity. It has been assured that this composite material could be used as the model material of photoelastic experiment for intelligent material or structures. In this research, the composite material with shape memory function is used as model material of photoelastic experiment. Photoelastic experimental hybrid method is developed in this research. It is confirmed that it is valuable on the obtaining stress intensity factor and the separation of stress components from only isochromatic data. The measuring method of stress intensity factor of intelligent material by photoelectric experiment is introduced.
Acrylic resin matrix composites with different contents of iron powder were fabricated under a magnetic field. Iron powder was arrayed parallel to the magnetic flux, and a distribution of powder was dependent on the magnetic field intensity. Mechanical properties of the composites were examined by four-point bending test. The bending strength and the deflection at fracture were improved in the direction of powder arrangement, and increased and then decreased with increasing powder content. The change of the strength with the powder content was explained by strengthening and weakening mechanisms. Young's modulus did not change with the powder content nor the magnetic filed intensity, but took a constant value in a range of the experiment.
The purpose of this study is to investigate static mechanical properties of Ti-alloy matrix composite fabricated by HIP treatment. Ti-6Al-4V alloy and W short fiber were used for matrix and reinforcement, respectively. This MMC has been processed two kinds of secondary working. Secondary works are rotary-swaging or shape-forging. Ti-6Al-4V alloy is known as one of representative Ti-alloys and W short fiber possesses excellent mechanical properties, such as high tensile strength, hardness number and high melting temperature. According to these experimental results, it is found that hardness and tensile strength of this material increases with the increase in volume fraction of W short fiber. Tensile strength of shape forging material is lower than rotary-swaging one. In addition, the highest value of tensile strength achieved among the metal matrix composites for this material is 1520MPa under the condition of W fiber volume fraction of V_f=9% (secondary working is rotary-swaging).
Stainless steel fiber (SSF) reinforced aluminum composites were fabricated by hot isostatic pressing (HIP) and subsequently elongating in its axial direction by rotary swaging treatment. The density and hardness were measured and microstructures of the specimens were observed with a scanning electron microscope and an optical microscope. The tensile test and fracture surface analysis were conducted. The hardness and the tensile strength of the composites increase significantly by only a small amount addition of SSFs (0.78vol%) in comparison with those of the non-reinforced aluminum material. The SSFs distributes homogeneously after HIP processing and prefer to align along the longitudinal direction in the aluminum matrix after rotary swaging treatment. The comparison of orientation coefficient shows that this treatment is effective to rearrange the orientation of fibers. The main fracture mode of the SSF/A1 MMCs is interfacial debonding between SSFs and the Al matrix.
AC4CH-based metal matrix composites with 9Al_4O_3・2B_2O_3 reinforcement have been produced using squeeze casting method, after T6 heat treatment, we evaluated fatigue life characteristics of matrix and MMC composite and investigated fracture mechanism. the results are summarized as follows. 1. T6 heat treated MMC decreased mechanical properties because of MgO・Al_2O_3 of spinal type by the reactions between whisker and small amount of Mg in the matrix. 2. Fatigue limit of MMC was a 71% higher than that of T6 non-heat treatment, fatigue strength of T6 heat treated MMC lower general. 3. In the fatigue fracture of whisker, surface of whisker before T6 heat treatment had no reacive products and damage was very small but whisker after T6 heat treatment had a lot of reactive products.
We have observed in our previous work that the tensile strength along the longitudinal direction of unidirectional CFRP under constant strain-rate loading is highly time-temperature dependent and obeys the same time-temperature superposition principle with that for matrix resin. We present in this paper a prediction scheme for the tensile strength by making adjustments pertinent to polymer matrix to Rosen's formula for fiber reinforced composite with elastic matrix. The prediction scheme requires the CFRP strength in the glassy region and the shear relaxation modulus of matrix resin. The prediction curve based on the proposed method captured the test data adequately over the wide range of temperatures and loading rates.
The effects of fiber/matrix interface on the inplane notched strength, the interlaminar shear strength, and the mode I and II interlaminar fracture toughness were investigated for 5H satin woven C/C composite materials by coating bismaleimide-triazine co-polymer (BT-resin). Carbonized C/C composites were used in this study with the heat-treatment temperature of 1600℃. While the notched strength of C/C composites without BT-resin coating indicated clear notch sensitivity, that with BT resin coating (weakened interface) was well estimated by the net section stress criterion. The initial fracture toughness under mode I loading, G_<Ic>, was insensitive to BT-resin coating. Moreover, the propagation values of the fracture toughness, G_<IR>, increased by coating BT-resin. On the other hand, the mode II interlaminar fracture toughness, G_<Ilc>, decreased by coating BT-resin. The change of the fracture mechanism in microscopic scale was well correlated to the change of the inplane and interlaminar fracture behavior. Comparison between inplane notched strength and interlaminar fracture toughness suggested the possibility to cope with both inplane and interlaminar properties by optimizing the interface control.
The damage which is caused by the low velocity impact in composite laminates is the form of the failure which occurrs frequently in application structures. As the results of impact loading in composite laminates, matrix cracking, delamination and eventually fiber breakage for higher impact energies can occur. Even though no visible impact damage is observed, damage can exist inside composite laminates. This damage can reduce considerably the strength and stiffness of the composite laminates. And it can influence the structure stability and can be the initiation point of structure failure. The objective of this study is to assess impact properties, to determine the inside damage of composite laminates by impact loading and to evaluate residual compressive strength after impact. For this purpose a series of impact and compression-after-impact tests are carried out on composite laminates made of carbon fiber reinforced epoxy resin matrix with lay up patterns of [(±45)(0/90)_2]s and [(±45)(0)_3(90)(0)_3(±45)]. UT-C scan is used to determine impact damage characteristics and CAI(Compression After Impact) tests are carried out to evaluate quantitatively the reduction of compressive strength by impact loading.
Composite patch repair of aircraft structural panels has recently received wide attention. When this repair technique is applied to the aircraft structural panels, it is imperative to design composite patches which are reliable enough to operate in severe environmental conditions. In designing composite patches, the accurate stress fields in the repaired structural panels and the bonded composite patches should be analyzed since highly complicated three-dimensional stress fields are developed. To obtain the accurate stress fields in repaired structural panels and bonded composite patches, usually the conventional three-dimensional finite element method is used. However, the conventional three-dimensional finite element method involves with a long computational time for iterative stress analysis. In this paper, we develop an effective three-dimensional finite element method based on the concept of domain decomposition with independent interfaces. Using this method, we carry out several parametric studies to examine the effect of patch shape and size on the stress fields in a repaired structural panel and bonded composite patches. The method is also used to determine the optimum patch shape and size so as to endure maximum tensile applied stress by a mathematical programming.
Crack closure in an epoxy resin plate with an embedded linear shape-memory alloy (SMA) wire as an actuator in smart structure members was investigated as a function of the duration of the supply of electric current to the SMA wire under a tensile load using finite-element analysis (FEA). The recovery force of SMA was considered in the FEA. The simulated isochromatics obtained by the FEA were in good with those obtained by the experiments. Therefore, the simulated isochromatics suggested the possibility of applications in the prediction of crack closure in epoxy resin plates with embedded SMA wires heated by supplying electric current.
Transverse load tensile tests were carried out to evaluate mechanical properties and investigate the failure behavior of the high-strength fabric yarns Zylon, Kevlar, and Spectra. Tests with blunt loaders produced stress-strain curves that were similar to uniaxial tensile test results. However, loaders with sharpened contact edges reduced the properties drastically. The Zylon yarn showed the highest modulus, tensile strength, and failure strain in all tests. The trends in the results for the three yarns were in accord with previous results generated under fragment impact load conditions, suggesting that simpler quasistatic tests can be used to assist the design of ballistic barriers.
Interaction of multiple axial pre-cracks in a pressurized fuselage was investigated using a small-scale model of an idealized fuselage. The strain between multiple cracks under rapidly crack propagation was measured. The recorded crack velocities justified the use of successive finite element analysis with large shell deformation. The experimental results of a rupturing model fuselage represented the variations of stress intensity factors K_I and K_<II> in the mixed mode and the remote stress components σ_<0X> with crack extension. The crack kinking location, kinking angle and the off-axis crack trajectory are predicted by numerical analysis and the validity of the calculated results is discussed in comparison with the experimental data.
An investigation was undertaken into the interfacial crack kinking phenomena in a bimaterial specimen of epoxy and aluminum alloy using a combination of experimental method and numerical simulation. It was found that all kinked fractures occurred at loading angles equal to or larger than 120°, and this was attributed to the direction of the global shearing mode. Three categories of fracture pattern were identified (A, B and C). In the case of type A fracture, the (J_1^0)_<kink> integrals were generally higher than the homogeneous epoxy J_<ic> at loading angles of 120° and above. In contrast, for types B and C fracture, the J_1^0 integrals were consistently lower than the homogeneous epoxy J_<ic>. Predictions of crack kinking behavior made using the Maximum Energy Release Rate Criterion (MG-criterion) were found to agree well with the observed experimental results.
Cyclic loading tests were performed for the through-thickness center cracked specimen of annealed copper under load-and displacement-controlled conditions. The crack opening/closing behaviors were experimentally examined, and then calculated by means of elastic-plastic finite element method, employing the constitutive equation, which was proposed by the authors in their earlier papers. The calculated load-displacement responses were in quite good agreement with the measured ones under both the test conditions. The large crack opening generated on the first tensile load excursion decreases on the following unloading and compressive loading excursions under the load-controlled condition at the load ratio R_L=-1.0. The crack opening displacement reaches the smaller value at the second maximum tensile load point than at the first one. Meanwhile, the greatly opened crack on the first tensile load excursion reduces its opening displacement at the successive unloaded point, and the crack opening displacements at the tensile peak load and the unloaded point tend to reach the corresponding stable states under the displacement-controlled cyclic loading at the displacement ratio R_D= 0. The calculation describes quite well such crack opening/closing behaviors under both the cyclic loading conditions.
Numerical prediction of three-dimensional dynamic fracture is very important for the integrity of complicated machine components and vital structures, including manufacturing tools. For the purpose of the present subject, it is important to understand the responses of time varying fracture parameter such as the dynamic J integral or the dynamic stress intensity factor along the curved-crack front in 3-D bodies. However, the mechanism of three-dimensional dynamic fracture has not been clarified. To elucidate it, the authors carried out experimental and computational studies. First, in the experimental study, high-speed photographs of dynamically propagating crack fronts were recorded. In the computational studies, in order to ease the three-dimensional numerical evaluation of the dynamic J integral, an expression in terms of equivalent domain integral (EDI) is used. Furthermore, to overcome the difficulties in three-dimensional dynamic fracture simulation, a three-dimensional moving finite element method together with an automatic element control method is developed. The moving finite element simulation together with the dynamic J integral evaluation procedure makes it possible to calculate accurate distribution of the dynamic energy release rate along the propagating crack-front at each time step. Using the experimentally recorded history of three-dimensional dynamic fracture, the generation-phase simulation is carried out. Based on the dynamic J integral distribution along the dynamically propagating crack fronts, the mechanism of three-dimensional dynamic fracture is discussed.
In establishing a long life fatigue design criterion and estimating the reliability of the spot-welded thin sheet structure such as the automobile and train, it is necessary to estimate the fatigue strength of spot-welded joint. So far, many investigators have numerically and experimentally studied on the method of systematic fatigue strength estimation for various spot-welded joint. Also considerable amount of data on the fatigue strength of spot-welded joint has been accumulated. But, it is difficult to determine the fatigue design criterion based on the existing data when the fatigue design is modified. Therefore, many test were required to determine a new criterion for modified design. It takes a lot of money and time. Thus, an expert system was developed to predict the fatigue design criterion including the fatigue strength and to predict fatigue life of spot-welded joint. The expert system consists of material destination database, spot-welding condition database and fatigue design module. And this expert system was developed based on the knowledge base, numerical calculation module, interface engine and user interface. It is expected that the expert system can economically provide a designer and manufacture with useful information on the material properties and predicting the fatigue design criterion for modified design.
This study examines an elastic layer with a penny-shaped crack situated in its middle plane. The crack surfaces are subjected to a uniform pressure. The layer surfaces are free from stress for case I while smooth-clamp conditions are prescribed on the layer surfaces for case II. By assuming series expansions for the normal displacement on the crack surfaces, both cases are reduced to analytical and exact solutions through infinite system of simultaneous equations. Numerical results are obtained to examine the effects of layer thickness and boundary conditions of the layer surfaces on stress distributions.
By employing the Stroh formalism, a general solution satisfying the basic laws of two-dimensional linear anisotropic elasticity has been written in a complex variable formulation. To study the stress singularity, suitable stress functions have been assumed. The singular order near the multi-bonded wedge apex can then be found by satisfying the boundary conditions. Since there are many material constants and boundary conditions involved, the characteristic equation for the singular order usually becomes cumbersome. It is therefore difficult to get any important parameters to study the failure initiation of the multi-bonded wedges. Through a careful mathematical manipulation, a key matrix N^^^ that contains the information of material properties and wedge geometries has been found to be a dominant matrix for the determination of the singular order. A closed-form solution for the order of stress singularity is thus written in a simple form. Special cases such as the wedge corners, cracks, interfacial joints or cracks, a crack terminating at the interface, free edges of multiplayer media, etc. can all be studied in a unified manner.
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