Total Energy Release Rate Research Articles

Interlaminar fracture of multidirectional glass/epoxy laminates under mixed-mode I + II loading

The mixed-mode I + II interlaminar fracture of multidirectional glass/epoxy laminates is investigated. Mixed-mode bending (MMB) tests were performed on specimens with delaminations in 0/θ-type interfaces, with θ varying from 0 to 90°. Preliminary three-dimensional finite-element analyses validated the beam theory model (BTM) used for analysing experimental data. The compliances measured are in a good agreement with BTM predictions. The total critical energy release rate Gc varies linearly with the mode II ratio GII/G, although some discrepancies are observed in the high-mode II results for the 0/45 and 0/90 specimens.

Analysis of composite skin–stiffener debond specimens using a shell/3D modeling technique

The application of a shell/3D modeling technique for the simulation of skin/stringer debond in a specimen subjected to tension and three-point bending was studied. The global structure was modeled with shell elements. A local three-dimensional model, extending to about three specimen thicknesses on either side of the delamination front was used to model the details of the damaged section. Computed total strain energy release rates and mixed-mode ratios obtained from shell/3D simulations were in good agreement with results obtained from full solid models. The good correlation of the results demonstrated the effectiveness of the shell/3D modeling technique for the investigation of skin/stiffener separation due to delamination in the adherents.

Delamination in elastic shear deformable circular layers

This work extends the analytical solution of an interface crack in straight layered structures to circular layered structures. A small segment at the vicinity of an interface crack tip in a circular laminated beam is analyzed by a novel shear deformable bi-layered circular beam theory. Two concentrated forces are found existing at the crack tip due to the requirement of the equilibrium condition. Closed-form solution of the total energy release rate of the interface crack is obtained as the half of the product of the concentrated forces and the corresponding displacement gradient discontinuities at the crack tip. Closed-form expressions of the mode I and II components of the energy release rate are also obtained by global and local methods. Numerical verifications are conducted by analyzing the interlaminar delamination of a circular beam with an edged crack and comparing with the baseline results obtained through finite element analysis. Excellent agreements between the present method and finite element analysis on the predictions of total energy release rate and mode partition verify the accuracy and efficiency of the present solution.

Measurement of the Total Energy Release Rate for Cracks in PZT Under Combined Mechanical and Electrical Loading

Four-point-bending V-notched specimens of lead zirconate titanate (PZT) poled parallel to the long axis are fractured under conditions of controlled crack growth in a custom-made device. In addition to the mechanical loading electric fields, up to 500V∕mm are applied parallel and anti-parallel to the poling direction, i.e., perpendicular to the crack surface. To determine the different contributions to the total energy release rate, the mechanical and the piezoelectric compliance, as well as the electrical capacitance of the sample, are recorded continuously using small signal modulation/demodulation techniques. This allows for the calculation of the mechanical, the piezoelectric, and the electrical part of the total energy release rate due to linear processes. The sum of these linear contributions during controlled crack growth is attributed to the intrinsic toughness of the material. The nonlinear part of the total energy release rate is mostly associated to domain switching leading to a switching zone around the crack tip. The measured force-displacement curve, together with the modulation technique, enables us to determine this mechanical nonlinear contribution to the overall toughness of PZT. The intrinsic material toughness is only slightly dependent on the applied electric field (10% effect), which can be explained by screening charges or electrical breakdown in the crack interior. The part of the toughness due to inelastic processes increases from negative to positive electric fields by up to 100%. For the corresponding nonlinear electric energy change during crack growth, only a rough estimate is performed.

Energy release characteristics of the nanoscale aluminum-tungsten oxide hydrate metastable intermolecular composite

Tungsten oxides are of interest as an oxidant for metals in metastable intermolecular composites (MICs), a reactive nanoscale powder useful for such applications as electric matches and gun primers. Smaller particles typically lead to fast reaction rates in this class of energetic material, and we have synthesized nanoscale WO3∙H2O using wet chemistry. Analysis by electron microscopy and small angle x-ray scattering revealed an approximately 100-nm-wide by7-nm-thick platelet morphology. X-ray diffraction verified the orthorhombic structure and composition of the hydrate. A MIC material was formulated using 44nm Al as the fuel. Performance was measured using a pressure cell where total enthalpy change and energy release rate was measured. This report includes the thermodynamic analysis of the pressure cell (calorimetry) that allows the determination of these metrics. Accuracy of the technique is discussed. Performance of the hydrate was found to significantly exceed that of MIC formulated with dehydrated tungsten oxide for one formulation, having an energy release of approximately 1.8MJ∕kg at a rate of approximately 215GW∕m2, compared to around 1.1MJ∕kg at a rate of around 130GW∕m2 for the dehydrated formulation. The data show that the enhanced behavior of the hydrated MIC formulation resulted from the reaction of aluminum with the interstitially bound water, which had additional energy release and generated hydrogen gas.

Application of prestressed transparent composite beams in fracture mechanics

This work presents the mixed-mode I/II/III version of the prestressed end-notched flexure system for the general delamination characterization of composite materials. The novel fracture mechanical configuration combines the traditional mode-I double-cantilever beam and the mode-II end-notched flexure specimens with the mode-III modified split-cantilever beam. First, a steel roller with a given diameter - which should not exceed the critical crack opening - is inserted to the delamination front, this fixes the mode-I part of the total energy release rate. Second, the prestressed specimen is put into the special rig of the MSCB specimen, and with the help of a prestressing screw the mode-III part of the total energy release rate is also fixed. Third, the prestressed specimen is put into a simple three-point bending setup and the mode-II part of the total energy release rate is increased up to fracture initiation. Using this method, i.e. varying the crack opening displacement by the steel roller and the crack te aring displacement by the MSCB rig the fracture surface in the GI-GII-GIII space can be obtained. To demonstrate the applicability and limitations of the novel system experiments on glass-fiber reinforced polyester specimens were performed including separated mode-I, mode-II, mode-III, mixed-mode I/II, II/III, I/III and I/II/III fracture tests. To reduce the measured data a previously validated improved beam theory scheme is applied. Finally, the surface of the fracture criterion is constructed by the generalization of the traditional criterion by Williams.

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.

Stiffness and fracture analysis of laminated composites with off-axis ply matrix cracking

Matrix cracks parallel to the fibres in the off-axis plies is the first intralaminar damage mode observed in laminated composites subjected to static or fatigue in-plane tensile loading. They reduce laminate stiffness and strength and trigger development of other damage modes, such as delaminations. This paper is concerned with theoretical modelling of unbalanced symmetric laminates with off-axis ply cracks. Closed-form analytical expressions are derived for Mode I, Mode II and the total strain energy release rates associated with off-axis ply cracking in [0/θ]s laminates. Stiffness reduction due to matrix cracking is also predicted analytically using the Equivalent Constraint Model (ECM) of the damaged laminate. Dependence of the degraded stiffness properties and strain energy release rates on the crack density and ply orientation angle is examined for glass/epoxy and carbon/epoxy laminates. Suitability of a mixed mode fracture criterion to predict the cracking onset strain is also discussed.

Modelling off-axis ply matrix cracking in continuous fibre-reinforced polymer matrix composite laminates

The fracture process of composite laminates subjected to static or fatigue tensile loading involves sequential accumulation of intra- and interlaminar damage, in the form of transverse cracking, splitting and delamination, prior to catastrophic failure. Matrix cracking parallel to the fibres in the off-axis plies is the first damage mode observed. Since a damaged lamina within the laminate retains certain amount of its load-carrying capacity, it is important to predict accurately the stiffness properties of the laminate as a function of damage as well as progression of damage with the strain state. In this paper, theoretical modelling of matrix cracking in the off-axis plies of unbalanced symmetric composite laminates subjected to in-plane tensile loading is presented and discussed. A 2-D shear-lag analysis is used to determine ply stresses in a representative segment and the equivalent laminate concept is applied to derive expressions for Mode I, Mode II and the total strain energy release rate associated with off-axis ply cracking. Dependence of the degraded stiffness properties and strain energy release rates on the crack density and ply orientation angle is examined for glass/epoxy laminates. Suitability of a mixed mode fracture criterion to predict the cracking onset strain is also discussed.

Mixed mode I + II interlaminar fracture of glass/epoxy multidirectional laminates – Part 1: Analysis

An analytical study was conducted on mixed mode bending (MMB) glass/epoxy multidirectional specimens with delaminations in 0/ θ and θ/− θ type interfaces. Three-dimensional finite element models and the virtual crack closure technique (VCCT) were used to evaluate a modified beam theory (MBT) data reduction scheme. The specimen compliance and total strain energy release rate G were accurately predicted by MBT, and there were only small mode partitioning differences relative to VCCT for 0/ θ specimens. Moreover, VCCT mode partitioning for 0/45 and 0/90 specimens rapidly converged to MBT predictions for increasing crack closure increments of the order of the fracture process zone.

Investigation of the M -integral in crack-damaged piezoelectric ceramics

The physical interpretation of the M-integral is investigated in the analysis of crack-damaged piezoelectric problems. The relation between the M-integral and the change of the total electric enthalpy (CTEE), i.e., M = 2CTEE, is derived with a theoretical derivation procedure for two-dimensional piezoelectric problems. It is shown that the M-integral may provide a more natural description of electric enthalpy release due to the formation of the pre-existing microcracks associated with the damaged body, rather than the description of the total potential energy release rate as interpreted for conventional brittle solids. For crack-damaged piezoelectric ceramics, numerical calculation of the M-integral is discussed. Based on the pseudo-traction electric displacement method, M = 2CTEE has also been proved by the numerical results.

On the calculation of energy release rates for cracked laminates with residual stresses

Prior methods for calculating energy release rate in cracked laminates were extended to account for heterogeneous laminates and residual stresses. The method is to partition the crack tip stresses into local bending moments and normal forces. A general equation is then given for the total energy release rate in terms of the crack-tip moments and forces and the temperature difference experienced by the laminate. The analysis method is illustrated by several example test geometries. The examples were verified by comparison to numerical calculations. The residual stress term in the total energy release rate equation was found to be essentially exact in all example calculations.

Interface fracture of polymer films: blister test experiments and modelling

The interface fracture between a rigid substrate and polymer film is investigated in this work using pressurised blister test experiments and modelling. The interface crack growth is studied for two different types of polymer films: stiff and compliant ones. The pressurised blister test is used to provide critical pressure-crack length curves for different loading media (water and electrolyte solutions) and loading rates. Two different analytical approaches and a numerical modelling concept are used to determine the critical total energy release rate as a function of the crack length (crack resistance curve or R-curve). A relatively flat R-curve is observed for the system with the stiff polymer film, whilst R-curve for the compliant film system exhibits an increasing tendency. The mixed-mode fracture behaviour occurs for both investigated polymer film systems, as shown by the value of the mixed-mode angle that is constant for all investigated crack lengths. R-curves are nearly unaffected by different loading media, whereas the loading rate has a strong influence on the interface fracture of the compliant file system. Finite element method-based prediction of the total energy release rate is in good agreement with that obtained from analytical expressions.

Effect of inclusion size on mechanical properties of polymeric composites with micro and nano particles

The effect of the particle size on the mechanical properties of polymeric composites reinforced with spherical particles was investigated. The size of particles varied from macro (0.5 mm) to nano (15 nm) scale. It was found that particle sizes at micro scale have little influence on the Young’s modulus of the composite and that Young’s modulus increases as the size of particles decreases at nano scale. It was also observed that tensile strength of the composite is significantly dependent on particle size. At 1 vol% loading, the tensile strength increased as the particle size decreased. However, the trend for the composite with alumina nanoparticles of 3% volume fraction was found to be opposite. TEM and SEM micrographs showed higher likelihood of poor dispersions in the composite with 3 vol% nanoparticles than that with 1 vol%. To understand the effect of the particle size in micron scale on the failure process, finite element analyses showed that total strain energy release rate for particle/matrix debonding growth decreases as particle size decreases and that sliding fracture mode becomes dominant as the debonding grows. It was found that interfacial fracture toughness does not depend on particle size but increases substantially when the sliding fracture mode prevails.

Fracture Analysis of Shear Deformable Bi-Material Interface

Based on a novel split bi-layer shear deformable beam model capable of capturing the local deformation at the crack tip, the explicit closed-form solutions of bi-material interface fracture are presented in this paper. A recently developed novel shear deformable bi-layer beam theory is briefly reviewed, from which the deformation at the crack tip is explicitly derived. A new expression for the energy release rate is then obtained using the J integral, in which several new terms associated with the transverse shear force are present; this represents an improved solution compared to the one from the classical beam model. By exploiting the two concentrated crack tip forces, the general loadings acting at the crack tip are decomposed into two groups which produce only the mode I and mode II energy release rates, respectively; the total energy release rate is thus decomposed into the mode I and II components in a global sense. The stress intensity factor referred to as local decomposition is also obtained including the transverse shear effect. The difference between the global and local mode decompositions is clarified, and a simple relationship between them is provided. The effect of the existence of a thin layer of adhesive on the stress intensity factor is further studied by an asymptotic analysis. A simple and improved expression for the T stress, the nonsingular term of stress at the crack tip, is also given. The fracture parameters of several commonly used interface fracture specimens are summarized. The present fracture analysis including the transverse shear effect is in better agreement with finite element analyses and shows advantages and improved accuracy over the available classical solutions.

Theoretical modelling of a conducting crack in a magnetoelectroelastic solid

A theoretical study of a magnetoelectroelastic composite plane with a single conducting crack is presented. Crack is assumed to be arbitrarily oriented with respect to the poling direction of the medium. General solutions for fully coupled magnetoelectroelasticity expressed in term of complex potential functions are used to formulate the problem. The formulation of a conducting crack is reduced to a classical Riemann-Hilbert problem and explicit analytical solutions are derived for the field intensity factors, total and mechanical energy release rates and crack tip hoop stress. The formulation is then extended to obtain common fracture parameters of an arbitrarily oriented permeable crack. Selected numerical results are presented for a conducting and a permeable crack in magnetoelectroelastic composite BaTiO3-CoFe2O4. It is found that crack orientation, volume fraction of piezoelectric phase and magnitude of electro-magnetic and mechanical loading have a significant influence on fracture parameters.

Energy Release Rate Prediction in Stiffened-skin Structure Using a Three-dimensional Crack Tip Element Analysis

Total energy release rates and energy release rate components are predicted for a stiffened-skin structure that is loaded in bending and which contains a delamination between its skin and a hat-shaped stiffener. This is done using threedimensional crack tip element and three-dimensional finite element analyses. An isotropic configuration and a variety of graphite/epoxy layups are considered, for which the two analyses are shown to be in good agreement. The results indicate the practicality of, and considerable computational savings that may be achieved by, employing crack tip element analyses for energy release rate assessments in realistic structural geometries. They are also used to qualitatively examine the effects of layup on energy release rate distributions in this geometry.

Study of Interfacial Delamination by Considering the Effect of Temperature and Moisture during Solder Reflow for QFN Package

Experiment has been conducted to measure the solubility Csat of mold compound and to obtain the moisture weight loss curves at various temperature. Moisture desorption modeling has been conducted to calculate the moisture diffusivity for desorption D(T) by matching with the experimental results. Finite Element Analysis (FEA) has also been conducted to calculate the transient development of strain energy release rate (ERR) at the interfacial crack tip due to thermal stress only (Gt), hygrostress only (Gh), and the combined effect (Gtotal) during solder reflow. ERR is computed based on the Virtual Crack Closure Technique (VCCT). It is found that Gh is significantly smaller than Gt, however the effect of hygrostress significantly increases the total strain energy release rate when combined with the thermal effect. The maximum Gtotal occurs at the peak of the solder reflow profile. The effects of crack size and geometrical parameters have been studied. The result imply that the interfacial crack is unstable and has a high tendency of growing to a significant extent. ERR is strongly influenced by the thickness of the package while length between the edge of die and pad has a moderate affect on the ERR.

Computation of Energy Release Rate and Mode Separation in Delaminated Composite Plates by Using Plate and Interface Variables

A model for analyzing mixed-mode delamination problems in laminated composite plates under general loading conditions is studied. The first-order shear deformable laminated plate theory and the interface methodology, which in turn is based on fracture mechanics, are adopted. The laminate is modeled as an assembly of laminated plate and interface layers in the thickness direction. When the limit case of interface stiffness coefficients approaching infinity is considered, a perfect adhesion between plate models is simulated. On the other hand, delamination between sublaminates is taken into account by assuming zero values for interface stiffnesses. Lagrange and penalty methods are adopted to simulate connections between plate elements. By using a variational approach and the virtual crack closure concept, expressions for total energy release rate and its mode components along the delamination front are obtained, in terms of both interface variables and plate stress resultant discontinuities. These formulas shed light on the effect of transverse shear in interface fracture analysis and establish the improvement and the accuracy of the proposed formulation compared to other plate-based delamination models existing in the literature. The method is implemented in a two-dimensional finite element analysis, which makes use of shear deformable plate elements and interface elements. To illustrate the present method, some typical delamination problems involving mode I, mode II, and mode III are examined and the numerical energy release rate distributions are compared to highly accurate three-dimensional (3D) finite element solutions. These results show that the procedure is accurate when a reasonable number of plate models are included in the analysis and more computationally efficient than 3D finite element models, for which determining fracture energies may lead to a remarkable increase in model complexity.

Modelling damage of multiple delaminations and transverse matrix cracking in laminated composites due to low velocity lateral impact

This paper presents the latest development by the authors in modelling damage due to low velocity impact, i.e., negligible role of inertia to laminated composites. The most common modes of damage are transverse cracking and delamination. The problem has been addressed in a series of publications by the authors. Upon a successful partition of the total energy release rate into individual modes, a mixed mode delamination propagation criterion has been employed for modelling delamination propagation, which is also applicable to multiple delamination problems. A simplistic ply-discount technique has been employed to incorporate the effects of transverse matrix cracking. The model has been applied to a number of problems from idealistic DCB to practical problems, such as quasi-static indentation of filament-wound pipes as widely used in offshore and many other industries. An attempt has been made to employ a continuum model to simulate delamination so that it can be implemented using a commercial FE code for wider engineering applications.