Delamination of layered structures on elastic foundation

This study presents an interface fracture mechanics analysis of delamination of a layered beam resting on a Winkler elastic foundation subject to general mechanical loads. A crack tip element on elastic foundation model is established first, through which, two concentrated forces existing at the crack tip are determined in closed-form. Then total energy release rate of the crack can be expressed in term of these two forces. By using available numerical results in the literature, the phase angle of the total energy release rate is also obtained. To verify the validity and accuracy of the solutions, debonding of a bonded overlay from the base structure resting on a Winkler elastic foundation is analyzed using the present solution. Comparisons with the baseline results obtained by finite element analysis suggest that the present analytical solution provides an excellent estimation of the total energy release rate and its phase angle for interface cracks in layered structure on elastic foundation. This study provides an approximated analysis of the debonding of a thin overlay debonding from the concrete pavement, where the effect of the base structure is simplified by a Winkler elastic foundation. This solution can also be used to analyze other similar delamination problems, such as local delamination in laminated composites, and face sheet delimitation in sandwich beams.     

Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation

In this article, an exact analytical solution for buckling analysis of moderately thick functionally graded (FG) sector plates resting on Winkler elastic foundation is presented. The equilibrium equations are derived according to the first order shear deformation plate theory. Because of the coupling between the bending and stretching equilibrium equations of FG plates, these plates have deflection under in-plane loads lower than the critical buckling load acting on the mid-plane. The conditions under which FG plates remain flat in the pre-buckling configuration are investigated and the stability equations are obtained based on the flat plate assumption in the pre-buckling state. The stability equations are simplified into decoupled equations and solved analytically for plates having simply supported boundary condition on the straight edges. The critical buckling load is obtained and the effects of geometrical parameters and power law index on the stability of functionally graded sector plates are studded. The results for the critical buckling load of moderately thick functionally graded sector plates resting on elastic foundation are reported for the first time.     

Effect of mode III fracture on composite laminates

The digital speckle correlation method (DSCM) is preformed to study the inplane displacements along the x and y direction of laminated composite panels containing preset elliptical damage. The different principle axis of the ellipse and the different sequence of laminates are considered in the experiments. It is shown that the method is very useful to get the displacements on the laminate surface and between the adjacent plies. According to the experiment results, the deformations of x and y direction can be obtained. The conclusion that the mode III fracture may exist and may have an effect on the crack growth is formed. The total strain energy release rate is calculated by the finite element method. Using Mindlin plate theory and the virtual crack closure technique, the energy release rate of mode III fracture can be calculated by FEM. The results show that mode III fracture has an influence on the total energy release rate and also on the delamination growth. The energy release rate of mode III fracture cannot be ignored. The delamination growth also is influenced by the stacking sequence.     

Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation

The main purpose of this study is to investigate buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. In-plane compressive loads may be applied to either radial, circumferential, or all edges of circular/annular sector plates. Based on the classical plate theory (CPT), critical buckling loads and fundamental frequencies of the circular/annular sector plates under simply-supported and clamped boundary conditions are obtained by using differential quadrature method (DQM). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is considered to be constant. Convergence study is carried out to demonstrate the stability of the present method. To confirm the excellent accuracy of the present approach, a few comparisons are made for limited cases between the present results and those available in literature. Critical buckling load and fundamental frequency parameters of the circular/annular sector thin plates are computed for different boundary conditions, various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Influence of bending rotations on three and four-point bend end notched flexure tests

It has been experimentally observed that mode II critical energy release rate (G_IIC) values determined by four-point end notched flexure test and three-point end notched flexure test are different for the same material. At the present work correction factors related to bending rotations are introduced to explain the differences between values of G_IIC obtained by three point and four point end notched flexure tests. The bending angle leads to the contact zones between specimen and supports and specimen and load rollers changing in both test configurations. The present analysis has been carried out by the classical beam theory, neglecting shear effects and assuming the hypothesis of small rotated angles. Results show that the relative differences between corrected and uncorrected values of G_IIC are greater in the case of four-point end notched flexure than in the case of three-point end notched flexure test.     

Numerical evaluation of the effects of friction and geometric nonlinearities on the energy release rate in three- and four-point bend end-notched flexure tests

Nonlinear finite element analyses are used to examine the effects of friction and geometric nonlinearities on the energy release rate in three- and four-point bend end-notched flexure tests. Energy release rates are first determined by a recently developed direct energy balance approach. It is shown that the finite diameter loading rollers that are typically used in practical test set-ups cause both tests to be inherently nonlinear. The effect of these nonlinearities on the energy release rate is shown to be larger in the four point than the three point test and to increase with increasing roller diameter, increasing coefficient of friction along the crack plane, and decreasing supporting span length. For the four point test, the effect of these nonlinearities is also shown to increase with increasing ratio of inner to outer span length. Next, energy release rates at the onset of crack advance are determined by a simulated compliance calibration technique. This “perceived toughness” is compared with predictions of the “true toughness” given by the direct energy balance approach at the same load. It is shown that perceived toughnesses from this simulated compliance calibration procedure are larger than previously reported results that were obtained in a similar fashion using linear theory. In addition, the perceived toughness is shown to strongly depend upon the range used for fitting the load versus deflection data to obtain compliance. These findings are used to make some general recommendations regarding use of the two test methods and their associated data reduction techniques.     

Free vibration analysis of continuous grading fiber reinforced plates on elastic foundation

The free vibration characteristics of rectangular continuous grading fiber reinforced (CGFR) plates resting on elastic foundations have been studied, based on the three-dimensional, linear and small strain elasticity theory. The foundation is described by the Pasternak or two-parameter model. The CGFR plate is simply supported at the edges and is assumed to have an arbitrary variation of fiber volume fraction in the thickness direction. Suitable displacement functions that identically satisfy the simply supported boundary conditions are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method (DQM) to obtain natural frequencies. Convergence studies have been performed on CGFR plates on the elastic foundations. It is shown that the present method has a rapid convergent rate, stable numerical operation and very high accuracy. Besides results for CGFR plate with arbitrary variation of fiber volume fraction in the plate’s thickness are compared with discrete laminated composite plate. The main contribution of this work is to present useful results for continuous grading of fiber reinforcement in the thickness direction of a plate on elastic foundation and comparison with similar discrete laminate composite plate. Results indicate the advantages of using CGFR plate with graded fiber volume fractions over traditional discretely laminated plates.     

Effect of the loading rate on mode I interlaminar fracture toughness of laminated composites

The objective of the present study is to determine the influence of the loading rate on the critical energy release rate G_Ic of fibre-reinforced epoxy laminates. In order to perform pure mode I loading at higher opening velocities, a new test device is developed. The approach is based on a symmetrical opening displacement applied to a DCB specimen. In the data reduction, the influence of the kinetic energy has to be taken into account. The results obtained on the unidirectional carbon-epoxy laminate T300/914 at crack opening rates up to 1.6 m/s show a slight effect of the loading rate on G_Ic.     

Measurement of mode I and mode II rock dynamic fracture toughness with cracked straight through flattened Brazilian disc impacted by split Hopkinson pressure bar

In order to find an effective and convincing method to measure rock dynamic fracture toughness for mode I and mode II, cracked straight through flattened Brazilian disc specimens of marble, which were geometrically similar for three size, were diametrically impacted by split Hopkinson pressure bar on the flat end of the specimen with three load angle respectively. History of stress intensity factors (K_I(t) for opening mode I, and K_II(t) for sliding mode II), mode mixture ratio (K_I(t)/K_II(t)), as well as mode I and mode II dynamic fracture toughness at crack initiation (K_Id and K_IId) were determined with the experimental–numerical method. It is found that there is a unique size effect for dynamic fracture test with the specimens, the mode mixture ratio is not solely determined by load angle (the angle between load direction and crack line) as in the static loading; the pure mode II load angle is 19° for the ?50 mm specimen, however it is 10° for the ?130 mm and ?200 mm specimens; the mode II load angle decreases with increment of specimen size. Realization of pure mode II is justified by the mode mixture ratio approaching zero, it can be realized under certain load angle and loading rate for the specimen of specified size. K_IId is generally greater than K_Id. Both K_Id and K_IId increase with increment of specimen size, and this trend for K_IId is more remarkable than that for K_Id.     

Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation

Vibration analysis of a functionally graded rectangular plate resting on two parameter elastic foundation is presented here. The displacement filed based on the third order shear deformation plate theory is used. By considering the in-plane displacement components of an arbitrary material point on the mid-plane of the plate and using Hamilton’s principle, the governing equations of motion are obtained which are five highly coupled partial differential equations. An analytical approach is employed to decouple these partial differential equations. The decoupled equations of functionally graded rectangular plate resting on elastic foundation are solved analytically for levy type of boundary conditions. The numerical results are presented and discussed for a wide range of plate and foundation parameters. The results show that the Pasternak (shear) elastic foundation drastically changes the natural frequency. It is also observed that in some boundary conditions, the in-plane displacements have significant effects on natural frequency of thick functionally graded plates and they cannot be ignored.     

Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations

Nonlinear vibration, nonlinear bending and postbuckling analyses are presented for a sandwich plate with FGM face sheets resting on an elastic foundation in thermal environments. The material properties of FGM face sheets are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equation of the plate that includes plate-foundation interaction is solved by a two-step perturbation technique. The thermal effects are also included and the material properties of both FGM face sheets and homogeneous core layer are assumed to be temperature-dependent. The numerical results reveal that the foundation stiffness and temperature rise have a significant effect on the natural frequency, buckling load, postbuckling and nonlinear bending behaviors of sandwich plates. The results also reveal that the core-to-face sheet thickness ratio and the volume fraction distribution of FGM face sheets have a significant effect on the natural frequency, buckling load and postbuckling behavior of the sandwich plate, whereas this effect is less pronounced for the nonlinear bending, and is marginal for the nonlinear to linear frequency ratios of the same sandwich plate.     

Tapered beam on elastic foundation model for compliance rate change of TDCB specimen

A modified beam theory is developed to predict compliance rate change of tapered double cantilever beam (TDCB) specimens for mode-I fracture of hybrid interface bonds, such as polymer composites bonded to wood. The analytical model treats the uncracked region of the specimen as a tapered beam on generalized elastic foundation (TBEF), and the effect of crack tip deformation is incorporated in the formulation. A closed-form solution is obtained to compute the compliance and compliance vs. crack length rate change. The present TBEF model is verified with finite element analyses and experimental calibration data of compliance for wood-wood and wood-composite bonded interfaces. The compliance rate change can be used with experimental critical fracture loads to determine the respective critical strain energy release rates or fracture toughness of interface bonds. The present analytical model, which accounts for the crack tip deformation, can be efficiently and accurately used for compliance and compliance rate-change predictions of TDCB specimens and reduce the experimental calibration effort that is often necessary in fracture studies. Moreover, the constant compliance rate change obtained for linear-slope TDCB specimens can be applied with confidence in mode-I fracture tests of hybrid material interface bonds.     

A numerical fracture analysis of indentation into thin hard films on soft substrates

In this paper, finite element simulations of spherical indentation of a thin hard film deposited on a soft substrate are carried out. The primary objective of this work is to understand the mechanics of fracture of the film due to formation of cylindrical or circumferential cracks extending inwards from the film surface. Also, the role of plastic yielding in the substrate on the above mechanics is studied. To this end, the plastic zone development in the substrate and its influence on the load versus indentation depth characteristics and the stress distribution in the film are first examined. Next, the energy release rate J associated with cylindrical cracks is computed. The variation of J with indentation depth and crack length is investigated. The results show that for cracks located near the indenter axis and at small indentation depth, J decreases over a range of crack lengths, which implies stability of crack growth. This regime vanishes as the location of the crack from the axis increases, particularly for a substrate with low yield strength. Finally, a method for combining experimental load versus indentation depth data with simulation results in order to obtain the fracture energy of the film is proposed.     

Effects of test procedures and lithology on estimating the mode I fracture toughness of rocks using empirical relations

Empirical estimation is a common method for getting mode I fracture toughness K_IC of rock. By collecting data from tests in this study and literature, 204 sets of K_IC and tensile strength σ_t test data are obtained for new empirical K_IC‐σ_t relations regression. The empirical relations make the estimation of K_IC values from σ_t conveniently, but test procedures and lithology will influence its reasonability and reliability. Results indicate that the empirical K_IC‐σ_t relations obtained from the four different suggested K_IC test methods are all in good but obviously different linear relationship. The analyses show that cracked chevron notch Brazilian disc specimen (CCNBD) test‐based empirical relation is more accurate for estimating K_IC than the other three test‐based empirical relations. As to different lithology, isotropic rocks such as sandstone and carbonatite may be more appropriate for the application of empirical estimation method. However, for coarse grained or anisotropic rocks such as granite and marble, estimation method should be applied carefully because of possibly weak K_IC‐σ_t relations.     

Three-dimensional temperature dependent free vibration analysis of functionally graded material curved panels resting on two-parameter elastic foundation using a hybrid semi-analytic,differential quadrature method

In this study, free vibration analysis of initially stressed thick simply supported functionally graded curved panel resting on two-parameter elastic foundation (Pasternak model), subjected in thermal environment is studied using the three-dimensional elasticity formulation. The material properties are temperature dependent and the temperature is assumed to have uniform and non-uniform distributions through the thickness direction of the curved panel. In order to discretize the governing equations, the differential quadrature method in the thickness direction and the trigonometric functions in longitudinal and tangential directions in conjunction of the three-dimensional form of the Hamilton’s principle are used. The convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for both isotropic and functionally graded material (FGM) curved panels. By examining the results of thick FGM curved panels for various geometrical parameters and temperature distribution models with the inclusion of supporting elastic foundation, the influence of these parameters and in particular, those due to functionally graded material (FGM) parameters are studied.     

Fracture mechanics approach to facesheet delamination in honeycomb: measurement of energy release rate of the adhesive bond

Two types of experiments were designed and performed to evaluate the adhesive bond in honeycomb sandwich panels. The tensile bond strength between the facesheet and the core was determined through the flatwise tension test. The fracture toughness of the bond line was measured through the double cantilever beam test. Fracture toughness values varied for different facesheet thicknesses and core materials. Toughness was also different for the bag and tool sides of the panels for all specimen types.     

Fracture assessment of U-notches under mixed mode loading: two procedures based on the ‘equivalent local mode I’ concept

Two fracture criteria are proposed and applied to blunt-notched components made of brittle materials loaded under mixed mode; the former is based on the averaged strain energy density over a given control volume, the latter on the cohesive crack zone model. In both instances use of the equivalent local mode I hypothesis is made. Only two material properties are needed: the ultimate tensile strength and the fracture toughness. Numerical predictions of rupture loads from the two criteria are compared with experimental measurements from more than 160 static tests with notched beams. The samples are made of PMMA and tested at − 60°C to assure a bulk behaviour almost linear elastic up to rupture. Notch root radii range from 0.2 to 4.0 mm and load mixicity varies from pure mode I to a prevailing mode II. The good agreement between theory and experimental results adds further confidence to the proposed fracture criteria.     

Vibration analysis of nonhomogeneous moderately thick plates with point supports resting on Pasternak elastic foundation using element free Galerkin method

In this paper shear deformable plate theory in combination with Element-Free Galerkin Method (EFGM) is used for free vibration analysis of nonhomogeneous moderately thick plates with point supports resting on a two-parameter elastic foundation. It is shown that the vibration results obtained by this method are in a very good agreement with the available literatures in spite of using low numbers of nodes which can be considered as an inconvenience in some other methods to reach a satisfactory accuracy. Also, applicability of the method is demonstrated by solving numerical examples for different values of homogeneity variation parameter, aspect ratio, thickness to length ratio, foundation parameters, various types of boundary conditions and different numbers of point support. The numerical results present valuable information for engineers and designers in various structural applications and also prove useful to use as benchmarks for further references.     

Multifractal detection of overlaps based on a stereometric analysis of fracture surfaces: Application to fractures of sintered carbides

A methodology is presented of a multifractal image analysis which was applied to real stereometric files from profilographometric examinations of the fractures of WC-Co sintered carbides. The analysis was conducted based on the research scheme and theoretical basis discussed in [Stach S, Cybo J. Multifractal detection of overlaps based on a stereometric analysis of fracture surface: assumptions, Materials Characterization (in this issue).[1]]. The solution presented enables the detection of the locations of overlaps on a fracture surface and their graphic presentation.