Mixed-mode Energy Release Rates Research Articles

Fracture research of adhesive-bonded joints for GFRP laminates under mixed-mode loading condition

Abstract The mixed-mode fracture characterization of adhesively bonded glass fiber-reinforced polymer (GFRP) plate joints is studied based on theoretical and experimental techniques. An improved beam model is used to estimate the compliance and the mixed-mode energy release rate (ERR) for GFRP four-point mixed-mode bending specimens. In this model, the deformation of the upper and lower GFRP plates is mutually independent, and the deformable adhesive is considered to satisfy the displacement compatibility conditions on the bonding interface. The explicit theoretical solutions of the compliance and ERR using the compliance method are reduced. The present theoretical solutions fit well with the finite element solutions by comparison with the rigid joint model. From the critical loads in experiment, the variation in fracture toughness on mixed-mode I/II has been determined by modifying the stiffness of the composite GFRP/GFRP substrate.

A Comparative Study of the Behavior of CFRP and GFRP Laminates in a Plate Specimen Using Modified Virtual Crack Closure Technique (MVCCT)

The applications of polymer composites in aircraft industry have exponentially increased in the recent years due to their high strength to weight ratio. Presence of delaminations in composites is inevitable which affects the structural stability due to reduction in structural stiffness and strength. The degradation of a structural component depends on the geometric characteristics of delamination, nature of loading and material characteristics. Damage tolerance study is thus essential to determine the extent of degradation of the structure due to the presence of delamination. The present paper brings about a comparision between the behaviour of a standard plate specimen made up of Carbon Fibre Reinforced Polymer (CFRP) and Glass Fibre Reinforced Polymer (GFRP) laminates with circular delaminations of varying diameters and subject to compressive load.A constant compressive load in terms of initial displacements was applied on a quasi-isotropic square plate specimen of dimensions 200 mm x 200 mm with a thickness of 2.88 mm of both CFRP and GFRP configurations. A circular delamination was introduced at the centre of the plate and its diameter and position along the thickness direction were varied and studied. Using ABAQUS codes of practice, Strain Energy Release Rate (SERR) was computed. The principles of Modified Virtual Crack Closure Technique (MVCCT) was used to compute SERR. Delamination propagates when the computed mixed mode energy release rate exceeds the critical value, GC. Depending on the external loading and material properties, Total strain energy release rate, GT (GT=GI+GII+GIII) was used to predict growth of delamination. The onset of delamination growth was determined by plotting the values of (GT/GC) across various delamination sizes along the thickness of the plate and reported.

Mixed mode delamination of asymmetric beam-like geometries with cohesive stresses

The partition of the energy release rate of beam-like geometries is analyzed here. Several methods for partitioning mode I and II energy release rates are presented and critically compared. A Timoshenko beam theory with Dugdale cohesive stress is applied and results show a mixed mode energy release rate dependence on the interface fracture toughness. A geometry is proposed that eliminates this dependence, offering a method to characterize the fracture properties of interfaces between dissimilar materials.

Fracture Toughness Analysis of Delamineted Composites under Variable Temperatures

Abstract The main goal of the present paper is to approach the modeling of one of the most important and critical failure modes for composite laminates which is known as interlaminar delamination in the aeronautical structures. The analytical model is based on a fracture mechanics approach; it’s used to estimate the total mixed mode energy release rate for composite laminates. A finite element simulation has been achieved in combination with the virtual crack closure technique (VCCT) to analyze the effect of temperature on the interlaminar fracture toughness growth of a delaminated carbon/epoxy material, namely IM7/8552 subjected to mechanical loading at variable temperatures. The developed model may serve as the basis for treating different types of thermal and mechanical loading, different stacking sequences and thickness of lamina in order to build safe working conditions for composite laminates.

Creep crack analysis of viscoelastic material by numerical manifold method

In order to gain an insight into creep induced cracking in viscoelastic materials, the creep crack phenomenon in viscoelastic materials are numerically investigated in this paper based on the numerical manifold method (NMM). To more realistically capture the mechanical behaviour in the vicinity of crack tips, the local displacements around crack tips are enriched with the viscoelastic crack-tip asymptotic displacement field. To better represent the viscoelastic materials, an incremental viscoelastic constitutive law based on the generalized Kelvin-Voigt model is incorporated into the NMM. To achieve these treatments in the NMM, an incremental viscoelastic NMM formulation is derived. In addition, an incremental formulation of the mixed mode viscoelastic energy release rate (ERR), which can be tackled in the same process with the incremental viscoelastic constitutive law, is derived and incorporated into the NMM for investigating the variations of the ERR in viscoelastic materials. Based on the developed method, three creep crack problems with different loading conditions and pre-existing crack configurations are simulated, and the crack opening and sliding displacements as well as corresponding mixed mode ERR are analysed. Simulation results show good agreement with the theoretical solutions, which indicate that the proposed NMM is suitable for complex creep crack problems.

Particle- and crack-size dependency of lithium-ion battery materials LiFePO<sub>4</sub>

Lithium-ion batteries have become a widely-used commodity for satisfying the world’s mobile power needs. However, the mechanical degradation of lithium-ion batteries initiated by micro cracks is considered to be a bottleneck for advancing the current technology. This study utilizes a finite element method-based virtual crack closure technique to obtain particle- and crack-size-dependent estimates of mixed-mode energy release rates and stress intensity factors. Interfacial cracks in orthotropic bi-materials are considered in the current study, whereas the crack extension along the interface is assumed. The results show that energy release rate, stress intensity factor, and the propensity of crack extension are particle- and crack-size- dependent. In particular, our results show that for smaller plate-like LiFePO4 particles (100 nm × 45 nm), a crack has lesser tendency to extend if crack-to-particle size is less than 0.2, and for 200 nm × 90 nm particles, similar results are obtained for crack-to-particle sizes of less than 0.15. However, for larger particles (500 nm × 225 nm), it requires an almost flawless particle to have no crack extension. Therefore, the current study provides insight into the fracture mechanics of LiFePO4 and the associated crack-to-particle size dependency to prevent crack extensions.

Decomposition of 3-D mixed-mode energy release rates using the virtual crack extension method

A technique was implemented for decomposing 3-D mixed-mode energy release rates using the Virtual Crack Extension (VCE) method. The technique uses a symmetric/anti-symmetric approach to decompose local crack-front displacements that are substituted into the global VCE energy release rate form. The subsequent expansion leads to the mixed-mode energy release rate expressions. As a result of the expansion, previously unaddressed modal–interaction coupling terms are found to impact the mixed-mode energy release rates. This development expands the VCE method’s advantages over similar procedures when simulating arbitrary crack growth by providing the means to calculate both mixed-mode energy release rates and their variations.

A High-order extended finite element method for extraction of mixed-mode strain energy release rates in arbitrary crack settings based on Irwin's integral

SUMMARY An analytical formulation based on Irwin's integral and combined with the extended finite element method is proposed to extract mixed-mode components of strain energy release rates in linear elastic fracture mechanics. The proposed formulation extends our previous work to cracks in arbitrary orientations and is therefore suited for crack propagation problems. In essence, the approach employs high-order enrichment functions and evaluates Irwin's integral in closed form, once the linear system is solved and the algebraic degrees of freedom are determined. Several benchmark examples are investigated including off-center cracks, inclined cracks, and two crack growth problems. On all these problems, the method is shown to work well, giving accurate results. Moreover, because of its analytical nature, no special post-processing is required. Thus, we conclude that this method may provide a good and simple alternative to the popular J-integral method. In addition, it may circumvent some of the limitations of the J-integral in 3D modeling and in problems involving branching and coalescence of cracks. Copyright © 2013 John Wiley & Sons, Ltd.

On the Through-the-Width Multiple Delamination, and Buckling and Postbuckling Behaviors of Symmetric and Unsymmetric Composite Laminates

Multiple delamination causes severe degradation of the stiffness and strength of composites. Interactions between multiple delamination, and buckling and postbuckling under compressive loads add the complexity of mechanical properties of composites. In this paper, the buckling, postbuckling and through-the-width multiple delamination of symmetric and unsymmetric composite laminates are studied using 3D FEA, and the virtual crack closure technique with two delamination failure criteria: B-K law and power law is used to predict the delamination growth and to calculate the mixed-mode energy release rate. The compressive load-strain curves, load-central deflection curves and multiple delamination process for eight composite specimens with different initial delamination sizes and their distributions as well as two angle-ply configurations 04//(±θ)6//04 (θ = 0° and 45°, and “//” denotes the delaminated interface) are comparatively studied. From numerical results, the unsymmetry decreases the local buckling load and initial delamination load, but does not affect the global buckling load compared with the symmetric laminates. Besides, the unsymmetry affects the unstable delamination and buckling behaviors of composite laminates largely when the initial multiple delamination sizes are relatively small.

Mixed mode fracture properties characterization for wood by Digital Images Correlation and Finite Element Method coupling

This paper seeks to characterize the mixed-mode energy release rate for orthotropic media subjected to a mixed-mode complex loading under service conditions. Experimental full-field information on Digital Image Correlation along with numerical modeling by the Finite Element Method have been considered as complementary approaches to determining fracture behavior in wood. An optimization technique allows calculating the kinematic state from Crack Relative Displacement Factor. A finite element analysis is performed in the local stress field characterization based on Stress Intensity Factor. This hybrid method has proven that the mixed-mode energy release rate uncoupling for fracture mode separation is achieved without taking into account local elastic mechanical properties; these properties can then be deduced by computing reduced elastic compliances from the additional information provided by Digital Image Correlation and Finite Element Method coupling. Such a hybrid method has been employed for cracked Douglas fir specimens undergoing complex loading in tension for various mixed-mode ratio values.

Analysis of mixed-mode interlaminar fracture and damage behavior of GFRP woven laminates at cryogenic temperatures

The objective of this work is to investigate the interlaminar fracture and damage behavior of glass fiber reinforced polymer (GFRP) woven laminates loaded in a mixed-mode bending (MMB) apparatus at cryogenic temperatures. The finite element analysis (FEA) is used to determine the mixed-mode interlaminar fracture toughness of MMB specimen at room temperature (RT), liquid nitrogen temperature (77 K) and liquid helium temperature (4 K). A FEA coupled with damage is also employed to study the damage distributions within the MMB specimen and to examine the effect of damage on the mixed-mode energy release rate. The technique presented can be efficiently used for characterization of mixed-mode interlaminar fracture and damage behavior of woven laminate specimens at cryogenic temperatures.

Analysis and Testing of Mixed-Mode Interlaminar Fracture Behavior of Glass-Cloth∕Epoxy Laminates at Cryogenic Temperatures

This paper presents results from an analytical and experimental study of the effect of temperature and mixed-mode ratio on the interlaminar fracture toughness in glass-cloth∕epoxy laminates. Mode I, mode II, and mixed-mode tests were conducted by the double-cantilever beam, end-notched flexure, and mixed-mode bending test methods at room temperature, liquid nitrogen temperature (77 K), and liquid helium temperature (4 K). A finite element model was used to perform the delamination crack analysis. Mode I, mode II, and mixed-mode energy release rates at the onset of delamination crack propagation were computed using the virtual crack closure technique. The fracture surfaces were examined by scanning electron microscopy to correlate with the interlaminar fracture properties.

Effects of mean stresses on multiaxial fatigue life prediction based on fracture mechanics

A linear elastic fracture mechanics analysis is presented to explore the effects of mean stress on fatigue life prediction under multiaxial loading conditions based on uniaxial fatigue data. Material damage in the form of small cracks is assumed to exist in a material element of interest. The mixed-mode energy release rate or the J integral is used as the primary fatigue damage governing parameter. The uniaxial Goodman relation is generalized to demonstrate the effects of mean stresses on fatigue life under multiaxial loading conditions. The generalization is also applicable for nonlinear constant fatigue life relations in the Haigh diagram.

Analysis of the driving force for a generally oriented crack in a functionally graded strip sandwiched between two homogeneous half planes

The driving forces for a generally oriented crack problem embedded in a Functionally Graded strip sandwiched between two half plane are analyzed using singular integral equations with Cauchy kernels, and integrated using Lobatto-Chebyshev collocation. Mixed-mode Stress Intensity Factors (SIF) and Strain Energy Release Rates (SERR) are calculated. The Stress Intensity Factors are compared for accuracy with previously published results. Parametric studies are conducted for various non-homogeneity ratios, crack lengths, crack orientation and thickness of the strip. It is shown that the SERR is more complete and should be used for crack propagation analysis.

An efficient procedure for determining mixed-mode energy release rates in practical problems of delamination

A computationally efficient procedure is presented for the prediction of mixed-mode strain energy release rates in practical problems of delamination. In this procedure, an analytical crack tip element analysis is used for the determination of all singular field quantities. By comparison with two- and three-dimensional finite element results, the procedure is shown to be accurate for mixed-mode problems where mode I, mode II and/or mode III crack tip singularities are present. The procedure is applicable for those cases where a near-tip inverse-square-root singularity exists, as well as those where an oscillatory singularity exists. For these latter cases, an alternative approach to using oscillatory field quantities to characterize crack advance is suggested.

The Characterization of Mixed-Mode Energy Release Rates in Orthotropic Materials

The aim of this work was to predict, from the material constants, mixed-mode energy release rates in orthotropic materials, in particular the general cases in which the crack is aligned at a random angle to the principal material direction, normal to the plane of orthotropy. Two-dimensional finite element models with various fibre orientations were generated. The finite element models were validated by comparing two sets of contour plots of deformation, one resulting from the finite element analysis and the other from moiré interferograms of the experimental work. On comparison there was shown to be a strict similarity between experimentally determined and computational deformation fields. Variations of the energy release rates were investigated for both rapid and stable crack growth. This was accomplished by generating two-dimensional stable crack growth finite element models. In general, energy release rates were found to be strongly affected by the fibre orientation. An increase of the angle of the crack growth direction caused a decrease of the mode I energy release rate and, by contrast, an increase of the mode II energy release rate, but the mode II energy release rate was always a small fraction of the mode I value. Crack extension caused a gradual increase of the mode I energy release rate both for coplanar and non-coplanar crack growth. However, there was no significant effect found on the mode II energy release rate.

Interfacial crack growth in thermomechanically loaded bimaterial joints

This paper considers quasi-static and dynamic propagation of straight and curvilinear interface cracks in brittle bimaterials subjected to mechanical crack surface loads and/or superimposed thermal strains acting along the ligament. Explicit stress intensity vector formulae are introduced and discussed in view of interface mechanics features: applied loading, interface, crack-tip velocity and curvature of the interface. In addition, the role of a bulk interphase layer between the adherent bimaterial components is addressed by assessing numerically calculated mixed-mode energy release rates. Finally, crack kinking out of the bulk interlayer (interphase) and the prospective kinking angle are investigated.

Finite element calculation of stress intensity factors for interfacial crack using virtual crack closure integral

This paper presents a successful implementation of the virtual crack closure integral method to calculate the stress intensity factors of an interfacial crack. The present method would compute the mixed-mode stress intensity factors from the mixed-mode energy release rates of the interfacial crack, which are easily obtained from the crack opening displacements and the nodal forces at and ahead of the crack tip, in a finite element model. The simple formulae which relate the stress intensity factors to the energy release rates are given in three separate categories: an isotropic bimaterial continuum, an orthotropic bimaterial continuum, and an anisotropic bimaterial continuum. In the example of a central crack in a bimaterial block under the plane strain condition, comparisons are made with the exact solution to determine the accuracy and efficiency of the numerical method. It was found that the virtual crack closure integral method does lead to very accurate results with a relatively coarse finite element mesh. It has also been shown that for an anisotropic interfacial crack under the generalized plane strain condition, the computed stress intensity factors using the virtual crack closure method compared favorably with the results using the J integral method applied to two interacting crack tip solutions. In order for the stress intensity factors to be used as physical variables, the characteristic length for the stress intensity factors must be properly defined. A study was carried out to determine the effects of the characteristic length on the fracture criterion based the mixed-mode stress intensity factors. It was found that the fracture criterion based on the quadratic mixture of the normalized stress intensity factors is less sensitive to the changes in characteristic length than the fracture criterion based on the total energy release rate along with the phase angle.

Mixed Mode Energy Release Rates for Adhesively Bonded Beam Specimens

This paper presents closed-form solutions for the energy release rate and the mode ratio of an equal adherend adhesively bonded beam specimen subject to a mixed mode bending load. The developed expressions explicitly account for the thickness and the material properties of the adhesive layer using a beam on an elastic foundation model. The accuracy of the expressions is verified using dimensional analysis and comparison with finite element results, and it is shown that they correlate well with experimental fracture data from adhesively bonded beam specimens with varying crack lengths.

An Investigation of the Existence of a Work Potential for Fiber-Reinforced Plastic

Experimental data from axial and torsional deformation tests on inelastic, angle-ply fiber-reinforced plastic laminates and related theory are used to show that mechanical work is independent of path for suitably limited deformation paths, and thus is a potential. First, results from proportional deformation tests are shown to support the existence of a work potential. Domains of path-independence are then established for non- proportional deformation tests of laminates strained well into the range of nonlinear in elastic behavior. Lastly, a study is described in which a work potential-based theory is used to determine critical mixed-mode energy release rates for laminates subjected to axial and torsional deformations. This relatively simple analysis is shown to provide qualitative correlation between fracture surface morphology and calculated energy release rates.