Expression For Energy Release Rate Research Articles

Strain energy release rate formulae for 3D finite element

Three expressions for the strain energy release rate for three-dimensional singular and non-singular finite elements are derived based on Irwin's virtual crack-closure method. The strain energy release rate for the three modes of fracture mechanics can be expressed in terms of the nodal forces ahead of the crack front and the opening displacements behind it. The material is assumed to be isotropic. The formulae are derived for three different element types, namely, 20-node singular element, eight-node and 20-node non-singular elements. The validity of the derived formulae is checked by comparing their results to those from the literature for mode I and mixed mode I and II problems.

Energy release rate for a dynamically growing crack at the interface of dissimilar isotropic materials

The strain energy release rate for a crack at the interface of dissimilar isotropic materials was derived from an existing crack tip field. In the derivation, the J-integral along a special near tip contour was used. A single expression of the strain energy release rate corresponding to different definitions of stress intensity factors was obtained. The validity of the general expression was discussed.

Bimaterial plane with elliptic hole under uniform tension normal to the interface

A bimaterial plane subject to uniform tensile traction normal to the interface is analyzed. The plane consists of two dissimilar elastic half-planes each with a semi-elliptic notch. The stress analysis is carried out by using a rational mapping function and complex stress ones. Stress Intensity of Debonding is defined as an index for the stress intensity at the tip of debonding and is investigated. The expression of energy release rate is derived by applying the residue theorem to a contour integral expressed by the complex stress function. Stress distributions around a circular hole and on the interface are shown. Stress concentration factors are studied for various oval shapes, material constants, and debonding lengths. Expressions of stress concentration factor are given.

Mixed Mode II-Mode III Fracture of Adhesive Joints

This paper presents approximate closed-form expressions for the local energy release rate in Mode I, Mode II, and Mode III along the idealized straight crack front of a cracked generalized adhesive joint. The development is based on beam theory, the J-integral, and a beam-on-an-elastic-foundation model. Fracture experiments were performed with adhesively bonded split-cantilever-beam (SCB) specimens of different adherend widths and thickness ratios using two different adhesives, and it was seen that the fracture surface had a peculiar stepped arrowhead shape. The analysis and the experiments showed that both the energy release rate and the mode mix vary along the crack front of an SCB specimen, making it unsuitable for the determination of fundamental fracture properties. However, the SCB specimen provided a useful means to test an engineering approach to predict the out-of-plane fracture loads of adhesive joints.

Delamination of Unidirectional Glass Fibre-Epoxy with Cut Plies Loaded in Four Point Bending

Constant thickness unidirectional specimens with plies cut across the complete width have been tested in four point bending. With the cut plies on both the tension and compression side, specimens failed by delamination initiating above and below the ends of the cut plies, and propagating along the interfaces between continuous and discontinuous plies. Finite element analysis has been carried out to determine the stress distributions and strain energy release rates, and an analytical expression for the strain energy release rate has also been derived. Predictions of the delamination stresses correlate well with measured values.

A double-plate finite-element model for the impact-induced delamination problem

A double-plate model is proposed to simulate a composite laminate with impact-induced delamination. Mindlin plate finite elements with large deflection have been employed to calculate stress and strain fields. On the basis of the modified crack closure method, a general expression for the strain energy release rate at the delamination front is derived. The proposed model was first evaluated by using an isotropic double cantilever beam. It is shown that this model is simple and computationally efficient. Subsequently, the present model was employed to predict impact-induced delamination in composite laminates and was compared with experimental results. The effect of shear cracks in the lamina adjacent to the delamination is discussed. The similarities between delaminations produced by different impact loads is identified and investigated by the use of the double-plate model.

The effect of residual stresses on the debonding of coatings—I. A model for delamination at a bimaterial interface

An analysis is presented of how a general set of residual stresses is predicted to influence the debonding behaviour of a coating. This is oriented towards relatively thick coatings, in which significant through-thickness variations of residual stresses may occur. A through-thickness deposit crack is assumed to exist and the external loading is pure bending. An expression for the interfacial strain energy release rate, G i, comprised of three terms, due to the applied loading, to the residual stresses and to an interaction term dependent on both, is derived. An experimental procedure to characterise the residual stress distribution is described. During interfacial fracture, the crack path, and therefore the loading state at the crack tip, are pre-determined by the plane of weakness. Since the critical value of G 1 for interfacial failure has been observed to be a function of this loading state, a procedure is described for determining the phase angle characterising it for a given applied load and residual stress state. The results of these analyses are presented for various generalised residual stress states. In the second part of this pair of papers, the model is applied to experimental data for thermally sprayed coatings.

ANALYTICAL SOLUTIONS AND NUMERICAL EXAMPLES FOR THERMOELASTIC INTERFACE CRACK PROBLEMS IN DISSIMILAR ANISOTROPIC MEDIA

The problem associated with an interface crack between anisotropic dissimilar media subjected to a remote heat flux as welt as traction on the crack surface is studied in this article. Based on the complex variable representations of the plane elastic problem developed by Lekhnitskii and a special technique of analytical continuation, the full field solutions of the displacements and thermal stresses are obtained in closed form. It is indicated that the nature of singularities in the stress field at the crack tips is unaltered by the presence of an applied heat flux if the heat conductivity coefficients obey the Onsager reciprocal relations. The explicit expressions for the strain energy release rate and crack opening displacements are also provided. The former is treated as the fracture parameter to characterize the occurrence of interface cracking. Numerical examples for graphite / epoxy composite bonded to metal and ceramic materials bonded to metal substrates are examined and detailed results are given.

On the analysis of the end-notched flexure specimen for measuring mode II fracture toughness of composite materials

This paper deals with the correction of mode II strain energy release rate, G II, of composite laminates measured with the end-notched flexure (ENF) specimen. A derivation is given of the expressions for compliance and strain energy release rate, in which the asymmetric flexure of the specimen and the shear deformations are considered. The results are compared with finite element results. It is shown that the expression for compliance based on the shear deformation beam theory may result in some errors when it is used to calculate G II even though it gives an accurate prediction of compliance. By considering the deformation at the crack root, an improved expression for strain energy release rate is obtained. The values of G II obtained from the improved expression are almost the same as those obtained from the finite element method. A test procedure for determining G II by using the ENF specimen is suggested.

Analysis of Toughening Mechanism of Ceramic-Matrix Composites in Consideration of Interface Debonding. 1st Report. Energy Release Rate for Interface Debonding Crack and Evaluation of Critical Debonding Length.

In this study, the energy release rate is derived for interface debonding cracks in ceramic-matrix composites, using approximate solutions of an axisymmetric cylindrical fiber-matrix model. The energy release rate expression includes the important effects of compressive residual stresses and frictional sliding stress on the debonded interface in a closed-form equation. A parametric study of the energy release rate is carried out and the effects of various parameters on the energy release rate are clarified and compared with results of other studies. The method to determine the critical debonding length is proposed based on the energy release rate and the fracture criterion of the interface and fiber. The predicted debonding length gives good agreement with published experimental results.

The modelling and control of failure in bi-material ceramic laminates

Recent experimental and theoretical work on simple, single phase, laminated systems has indicated that failure resistant ceramics can be produced using an elegant method that avoids many of the problems and limitations of comparable fibrous ceramic composites. Theoretical work on these laminated systems has shown good agreement with experiment and simulated the effects of material properties and laminate structure on the composite performance. This work has provided guidelines for optimised laminate performance. In the current study, theoretical work has been simply extended to predict the behaviour of bi-material laminates with alternating layers of weak and strong material with different stiffnesses. Expressions for the strain energy release rates of internal advancing cracks are derived and combined with existing criteria to predict the failure behaviour of these laminates during bending. The modelling indicates three modes of failure dictated by the relative proportions, thicknesses and interfacial properties of the weak and strong phases. A critical percentage of strong phase is necessary to improve failure behaviour, in an identical argument to that for fibre composites. Incorporation of compliant layers is also investigated and implications for laminate design discussed

A Plate Finite Element for Modeling Delaminations

Laminate constitutive relations are derived for a composite plate, the midplane of which is offset from the reference x-y plane. The equations are similar to the conventional laminates except that the definition of the [ A], [ B], and [ D] matrices are dif ferent. This offsetting procedure is convenient for finite element modeling of delamina tions. Starting with the three-dimensional J-integral, an expression for strain energy release rate for delaminations is derived. Unlike the crack closure method, the G can be derived with one single computation; further, the distribution of G along the delamination front can also be obtained. The method is illustrated for an anisotropic DCB specimen.

Micromechanical Analysis of Fiber-Reinforced Composites with Interfacial Phenomena. 4th Report, Evaluation of Mechanical Properties of Unidirectional Fiber-Reinforced Composites.

In the 3rd report of the sequential papers which dealt with analysis of the composite showing interfacial phenomena, micromechanical analysis was performed on a unidirectional fiber-reinforced composite producing interfacial debonding between a fiber and a matrix, and analytical expressions for energy release rates of a matrix crack and a debonding one were derived. In the present paper, evaluation of mechanical properties of a composite with such an interfacial debonding like a resin-based composite, is carried out based on the theoretical analysis of the 3rd paper. As a result, fracture toughness of such a composite can be evaluated in terms of the critical size of the crack at which the unstable propagation of the matrix crack will be arrested and then the debonding between a matrix and a fiber will occur. The effect of material parameters such as the volume fraction of a fiber, and the length of the debonding, etc., on the toughness can be explained by the interfacial phenomenon. The results obtained are consistent with the common experimantal facts. It should be particularly emphasized that there exists an optimal value of the surface energy of the interface for obtaining a tougher composite.

Analysis of multiple matrix cracking in [± θm/90 n] s composite laminates. Part 2: Development of transverse ply cracks

In this paper the progressive transverse ply cracking in [± θ m/90 n] s composite laminates is investigated theoretically. A general and simple expression for the energy release rate due to transverse matrix cracking is obtained using the potential energy approach in classical fracture mechanics and the assumption of a through-the-thickness flaw; thermal residual stresses are taken into account. The laminate resistance to crack multiplication is examined for both uniform and nonuniform crack spacing by substituting the measured applied stress/crack density data into corresponding energy release rate expressions. The resistance curve concept is employed to predict crack growth with increasing applied load. The results indicate that the assumption of uniform cracking and a resistance curve concept is a convenient and acceptable method for modelling crack initiation and multiplication in composite laminates.

The Formation and Propagation of Matrix Microcracks in Cross-Ply Laminates during Static Loading

Recently, a variational mechanics analysis approach has been used to de termine the thermoelastic stress state in cracked, cross-ply laminates (Nairn, J. A. 1989. J. Comp. Mat. , 23:1106). The analysis included a calculation of the energy release rate due to the formation of a microcrack in the 90° plies. This paper describes a series of ex perimental results on a wide variety of composite material systems and of cross-ply layups of generic type [0m,/90n]s. The variational mechanics energy release rate analysis can be used to predict all features of the experimental results and to draw some new conclusions about the progression of damage in cross-ply laminates. The recommended experiments are to measure the density of microcracks as a function of applied stress. Such results can be fit with the energy release rate expression and used to measure the microcracking or intralaminar fracture toughness. The measured microcracking fracture toughnesses are: Hercules AS4/3501-6-240 J/m2, Fiberite 934/T300—690 J/m2, DuPont Avimid® K Polymer/IM6-960 J/m2, Fiberite 977-2/T300-1800 to 2400 J/m2, and ICI PEEK/AS4- 3000 J/m2. Experiments that measure only the stress to initiate microcracking are specifically not recommended because they do not give an accurate measure of the micro cracking fracture toughness.

Micromechanical Analysis of Fiber-Reinforced Composites with Interfacial Phenomena. 2nd Report. Evaluation of Mechanical Properties of Discontinuous Fiber-Reinforced Composites.

In the previous paper, micromechanical analysis was performed on a discontinuous fiber-reinfor-ced composite material showing interfacial sliding behavior between a fiber and a matrix, and analytical expressions for energy release rate and elastic modulus, etc., were derived. In the present paper, evaluation of mechanical properties of a composite with such an interfacial sliding characteristics, like a ceramic-based composite, is conducted based on the theoretical analysis of the previous paper. As a result, the calculated elastic modulus is in good agreement with the experimental one. Furthermore, fracture toughness of such a composite can be evaluated in terms of the critical size of the crack which travels across the material in an unstable manner. The effect of material parameters such as the volume fraction of the fiber, friction stress between the fiber and matrix, and the magnitude of sliding, on toughness can be explained by the interfacial phenomenon, whose results are consistent with common experimental facts. It should also be particularly emphasized that there exists an optimal value of friction stress between a fiber and a matrix for obtaining a tougher composite

Strain energy release rates and the fatigue growth of matrix cracks in model arrays in composite laminates

Crack growth in the transverse plies of cross-ply composite laminates has been investigated both experimentally and theoretically. Expressions for the strain energy release rate associated with the growth of cracks in model arrays have been obtained using both the compliance approach and the energy method. Measurements of compliance change with crack length were obtained using glass-epoxy laminates and compared with various predictions. Correlations between the crack growth rate and the strain energy release rate range indicate that a Paris law is applicable.

Fracture Mechanics of Unidirectional Composites

A new approach to the fracture mechanics analysis of unidirectional com posites has been formulated. First, using the shear-lag model, we obtained an approximate stress-analysis for composites containing cracks and longitudinal splits. From this new stress analysis, we derived expressions for the energy release rate due to both self-similar crack propagation and to longitudinal splitting. The new expressions are easy to use. Be cause material heterogeneity, material anisotropy, finite width effects, and various possible failure modes are all specifically taken into account, we suggest that these new expressions should replace conventional fracture mechanics for analyzing fracture data in unidirec tional composites. Experimental work has verified that the new expressions fit experimen tal data significantly better than conventional fracture mechanics expressions.

The Strain Energy Release Rate of Composite Microcracking: A Variational Approach

A variational analysis approach has been used to determine the two- dimensional thermoelastic stress state in cross-ply laminates of type [0 m/90 n] s and [90 m/ 0 n] s. The stress analysis was used to calculate the energy release rate due to formation of a new microcrack. The analysis accurately includes the effect of residual thermal stresses. When compared with experiments, the new energy release rate expressions are found to predict typical data using a single value for the critical energy release rate for microcrack ing. This critical energy release rate has a physical interpretation as a microcracking fracture toughness or as an intralaminar fracture toughness. We also used the variational solution to get an analytical expression for the longitudinal expansion coefficient of the microcracked cross-ply laminate as a function of microcrack density. Finally, variational theorems are used to show that this new microcracking fracture analysis is rigorously more accurate than previous attempts at the same problem.

Analysis and Experimental Characterization of the Center Notch Flexural Test Specimen for Mode II Interlaminar Fracture

The interlaminar Mode II critical strain energy release rates of unidirec tional graphite/epoxy, graphite/polyetheretherketone (PEEK), and graphite/epoxy with interleaf material were characterized using the End Notch Flexural (ENF) and Center Notch Flexural (CNF) test specimens. The CNF test method was developed specifically for this study. Compliance and strain energy release rate expressions of the CNF are derived and are in good agreement with finite element results for a wide range of crack lengths. Mode II critical energy release rates for both test methods were found to be in quite good agreement.