Energy Release Rate Components Research Articles

Crack propagation in filled elastomers: 3D study of mechanisms involving the filler agglomerates

This paper presents the impact of Carbon Blacks agglomerates, at different concentrations, on crack propagation mechanisms in a Carbon black (CB) filled Ethylene Propylene Diene Monomer (EPDM) elastomer. As shown by Transmission electron microscopy, these CB agglomerates (CBaggl) consist of aggregates clusters with interpenetrating elastomer, and for this reason, are soft and deformable. Crack tip observation using X-ray tomography demonstrates that these CBaggl can either undergo fracture or arrest/ deviate a crack during its propagation. This causes higher energy dissipation at the crack tip, which contributes to the dissipative component of the strain energy release rate G. For this reason, it is found that among the two materials tested with a significant amount of CBaggl (more than 3%), the material with the highest concentration has a slower crack propagation speed at high G.

Novel bi-layer beam elements for elastic fracture analysis of delaminated composite beams

A general finite element method is proposed for linear elastic fracture analysis of delaminated bilayer composite beams. An interface deformable bilayer beam model is adopted in modeling the virgin portions of bilayer beams, and it is capable of guaranteeing the continuity condition of internal forces at the tip of delamination because the bilayers are modeled as two separate shear deformable sub-beams and both the interface normal and shear compliances are introduced to account for interface deformation. The stiffness matrices of bilayer beams using the linear and quadratic interpolation polynomials are both derived using the principle of minimum potential energy. Then, a finite element program is developed by assembling the element stiffness matrices of virgin and delaminated portions of bilayer beams and applying the corresponding equivalent nodal forces. Two typical fracture specimens (i.e., single leg bending and end-notched flexure bilayer beams) are analyzed using the proposed bilayer beam elements. The computed displacement and energy release rate are compared well with those of analytical solutions, and the mode components of energy release rate for single leg bending specimen are also obtained using the virtual crack closure technique. It demonstrates that the proposed novel bilayer beam elements are capable of effectively and efficiently predicting the displacement and energy release rate (including its mode components) of delaminated bilayer beams. The developed bilayer beam element is general in nature and accurate and efficient in particular, because it can achieve the same accuracy as the high order beam theory and be easily implemented in the existing commercial finite element software.

Experimental evaluation of cohesive laws components of mixed-mode I + II fracture characterization of cortical bone

Mixed-mode I + II fracture characterization of cortical bone tissue is addressed in this work. The mixed-mode bending test was used to impose different mode ratios. An equivalent crack length data reduction method was considered to obtain the strain energy release rate components. Crack opening and shear displacements were measured by means of digital image correlation. These quantities were then integrated to propose a direct evaluation of cohesive laws. The components of the cohesive laws for each mixed-mode loading were obtained by the uncoupled and Högberg’s methods. The later provided consistent evolution of strain energy release rate and peak stresses components in function of mode-ratio, revealing its appropriateness regarding the fracture characterization of cortical bone under mixed-mode I + II loading.

Thermoelastic effect on inter-laminar embedded delamination characteristics in Spar Wingskin Joints made with laminated FRP composites

This paper presents two sets of full three-dimensional thermoelastic finite element analyses of superimposed thermo-mechanically loaded Spar Wingskin Joints made with laminated Graphite Fiber Reinforced Plastic composites. The study emphasizes the influence of residual thermal stresses and material anisotropy on the inter-laminar delamination behavior of the joint structure. The delamination has been pre-embedded at the most likely location, i.e., in resin layer between the top and next ply of the fiber reinforced plastic laminated wingskin and near the spar overlap end. Multi-Point Constraint finite elements have been made use of at the vicinity of the delamination fronts. This helps in simulating the growth of the embedded delamination at both ends. The inter-laminar thermoelastic peel and shear stresses responsible for causing delamination damage due to a combined thermal and a static loading have been evaluated. Strain energy release rate components corresponding to the Mode I (opening), Mode II (sliding) and Mode III (tearing) of delamination are determined using the principle of Virtual Crack Closure Technique. These are seen to be different and non-self-similar at the two fronts of the embedded delamination. Residual stresses developed due to the thermoelastic anisotropy of the laminae are found to strongly influence the delamination onset and propagation characteristics, which have been reflected by the asymmetries in the nature of energy release rate plots and their significant variation along the delamination front.

Hygrothermal effect on the strain energy release rates and mode mixity of asymmetric delaminations in generally layered beams

An analytical framework is presented to calculate the mode I and II components of the strain energy release rate (SERR) of a delamination between two generally layered sub-laminates, considering the residual hygrothermal stresses effect. First, the sub-laminates are modeled as Timoshenko beams, the crack tip is assumed rotationally flexible and the mode partitioning is based on the global method. Next, SERR and mode mixity are obtained for an arbitrarily loaded delaminated beam and reduced for some typical delamination fracture tests. An asymmetrically delaminated fiber metal laminate (FML) is analyzed under double cantilever beam (DCB) and end-notched flexure (ENF) configurations. For the ENF configuration, the inherent contact problem was analytically approached. We demonstrate that the thermal effect on the SERR and mode mixity of the FML, rather ignored so far in the literature, can be non-negligible. Finally, the analytical results are validated against numerical predictions using the finite element method.

The effects of shear and near tip deformations on interface fracture of symmetric sandwich beams

The effects of shear on energy release rate and mode mixity in a symmetric sandwich beam with isotropic layers and a debond crack at the face-sheet/core interface are investigated through a semi-analytic approach based on two-dimensional elasticity and linear elastic fracture mechanics. The semi-analytic expressions for the shear components of energy release rate and mode mixity phase angle which have been derived in Li et al. (2004) for bi-material beams are extended to sandwich beams and the necessary numerical coefficients derived through accurate finite element analyses. The expressions are combined with earlier results for sandwich beams subjected to bending moments and axial forces in order to obtain solutions for general loading conditions and for an extensive range of geometrical and material properties. The physical and mechanical significance of the terms of the energy release rate which depend on the shear forces are explained using structural mechanics concepts and introducing crack tip root rotations to account for the main effects of the near tip deformations. The results are applicable to laboratory specimens used for the characterization of the fracture properties of sandwich composites for civil, marine, energy and aeronautical applications, provided the lengths of the crack and the ligament ahead of the crack tip are above minimum lengths which are defined in the paper.

Adhesion failure propagation analyses of Spar Wingskin Joints made with curved laminated FRP composite and FGM panels

Spar Wingskin Joint (SWJ) comprises of a thin spar which overlaps and is co-cured to a thick wingskin by suitable adhesive materials. These joints are mostly used in integral modular fabrication of aircraft and fuselage wing structures usually made with curved laminated Fibre Reinforced Polymeric (FRP) composite panels. This paper deals with the analyses of initiation of adhesion failure and its propagation in SWJs subjected to transverse loadings using Finite Element Analyses (FEA) and fracture mechanics approach. Normal and shear stress distributions at different interfacial locations between the adherends and the adhesive have been evaluated. Tsai-Wu coupled stress failure criteria have been used to identify the location of onset of adhesion failure. The critical location for onset of adhesion failure has been found to be at the toe-end of the spar wingskin overlap region. Pre-embedded adhesion failure has been simulated at this location and Strain Energy Release Rate (SERR) components corresponding to Mode I, Mode II and Mode III of adhesion failure have been determined using the principle of Virtual Crack Closure Technique (VCCT). Rate of propagation of adhesion failure has been evaluated by sequential release of Multi-Point Constraint (MPC) and contact finite elements provided at the pre-embedded failure front. It has been observed that Mode I component of SERR primarily governs the process of adhesion failure propagation. The appropriate spar overlap length on the wingskin has been determined in order to prevent adhesion failure. Further, an attempt has been made to improve the structural integrity of the SWJ by reducing the stress concentration at the ends of spar and wingskin overlap. This has been achieved by obtaining a favourable stress distribution over the entire joint structure by using Functionally Graded Materials (FGM) in the spar and wingskin adherend instead of FRP composite panels. Exponential gradation indices have been used for material grading of the spar and wingskin adherends. It has been seen that peel and shear stresses and values of SERR components as well as total SERR reduce significantly by use of FGMs with suitable exponential material grading index.

Mixed-mode fatigue crack growth and life prediction of an automotive adhesive bonding system

Mixed-mode fatigue crack growth behavior of an automotive adhesive bonding system (AA5754O-A951-BM4601) has been investigated using a compound compact mixed-mode specimen. The test was performed under load control with 0.1 R ratio and 3 Hz frequency. A long-distance moving microscope was employed to record the real-time length of the fatigue crack in the adhesive layer during testing. The strain energy release rate components of the crack were analyzed by finite element method. The pure-mode fatigue crack growth test results show that a given value of GI results in a fatigue crack growth rate over an order of magnitude higher than for an equal value of GII. The mixed-mode tests results indicate that under a given loading angle, the fatigue crack growth behavior is also influenced by the test load due to the changing shear mode ratio when the crack grows. The fatigue crack growth data were then processed by a generalized Paris relation formula and employed in the direct fatigue life prediction of single lap shear joints with the same bonding system. The life prediction shows good results in the low-cycle region but slightly conservative results in the high-cycle region when compared to coupon fatigue tests.

Strain based investigation of mode I crack near strong-weak interface in SSY regime

The paper presents an investigation on mode I cracked bi-material comprising elastic-plastic materials that are elastically identical but have different yield strengths and plasticity properties. The interface between the materials is considered to be thin and perfect. A strain based theoretical model is used to include the effect of non-linearity in plasticity characteristics of materials for estimation of the magnitude of plastic energy transfer across the interface when the crack in stronger material is near the interface of weaker material in SSY regime. Energy transfer is quantified by the component of energy release rate of the crack due to inhomogeneity caused by property mismatch between the materials. The behaviour of the model is demonstrated at discrete positions of crack near the interface of strength and plasticity mismatched steels under the action of monotonic tension in plane stress condition. Amplification effect due to energy transfer towards stronger steel is found to result in higher values of energy release rate and stress intensity parameter at the crack tip vis-à-vis the far field values. The effect is maximum when the crack tip is at the interface of steels. The results are compared with those of an available model based on Dugdale's criterion that uses stress parameter and is applicable to elastic-perfectly plastic materials. Difference is noticed between the two. Finite element analysis of the bi-material is undertaken to validate the present model. Numerical and theoretical results are found to be in close agreement.

A damage mechanics based failure criterion for fiber reinforced polymers

In this paper an energy-based failure envelope for unidirectional fiber reinforced polymers (FRP) has been developed. It is based on the stress and the damage energy release rate (DERR) components, which makes it suitable for being used as a damage surface in continuum damage mechanics (CDM), when a thermodynamically consistent evolution rule is intended to be used. The proposed failure surface is an interactive quadratic function of the DERR components accompanying with linear stress components in the form of an invariant. The mathematical formulation of the criterion is originated from the fully interactive Tsai-Wu criterion. The failure locus of the two criteria were compared with each other for several unidirectional FRP composites under several combined stress states. The two criteria were found to be in a reasonably good agreement for investigated composites with Eglass and carbon fibers, especially where the experimental data were available.

Finite-Layer Method: Evaluation of Stresses and the Modal Components of Energy Release Rate on the Midplane of Edge-Cracked Composite Specimens

On the basis of the finite-layer method, a technique for calculating the two-dimensional stress-strain state and interlaminar stresses in edge-cracked specimens in tests on determination of the Mode I, II, and III fracture toughnesses of composites is proposed. Taking into account the symmetry and antisymmetry of specimens, the calculation is reduced to a boundary-value problem for a system of partial differential equations describing the deformation of one of the layers. Using the method of straight lines, the two-dimensional problem is reduced to a one-dimensional boundary-value problem, which is solved by the stable Godunov–Grigorenko numerical method. Calculations for specimens subjected to three types of the tests are performed, and the distributions of normal and tangential stresses on the specimen midplane and the distributions of modal components of the fracture toughnesses GI, GII, and GIII along the delamination front line are presented. Comparisons with results of the one-dimensional solution and with known results obtained by the virtual crack closure technique are presented.

Bone fracture characterization under mixed-mode I + II loading using the MMB test

Fracture characterization of bovine cortical bone under mixed-mode I+II loading was performed in this work. With this aim, a miniaturized version of the mixed-mode bending testing setup was constructed. Three different mode ratios were analysed to cover a wide range of mode mixities, thus leading to an appropriate definition of fracture criterion of bovine cortical bone under mixed-mode I+II loading. A suitable data reduction scheme based on crack equivalent concept was used to get the Resistance-curves of the strain energy release rate components. The proposed procedure was validated by means of a mixed-mode I+II cohesive zone model using a bilinear softening law. In general, it was verified that the numerical load-displacement as well as the Resistance-curves are in agreement with the global trend obtained experimentally, thus validating the proposed procedure concerning mixed-mode I+II fracture characterization.

Inter-laminar delamination analyses of Spar Wingskin Joints made with flat FRP composite laminates

3D Finite Element Analyses have been carried out for Spar Wingskin Joints made with laminated Graphite Fibre Reinforced Plastic composites subjected to uniformly distributed transverse load. The delamination has been pre-embedded at the most likely location, i.e. in resin layer between the top and next ply of the fibre reinforced plastic laminated wingskin and near the spar overlap end. Multi-Point Constraint finite elements have been made use of at the vicinity of the delamination fronts. This helps in simulating the growth of the embedded delamination at both ends. The inter-laminar peel and shear stresses responsible for causing delamination damage have been evaluated. Strain energy release rate components corresponding to the Mode I (opening), Mode II (sliding) and Mode III (tearing) of delamination are determined using the principle of Virtual Crack Closure Technique. These are seen to be different and non-self-similar at the two fronts of the embedded delamination. Mode I Strain energy release rate (GI) primarily governs the process of delamination propagation.

Free Edge Mixed Mode Delamination Analysis of Laminated Composites with Wrap-Around Configuration: A Finite Element Study

Finite element analyses of laminated composites were done in the present study with respect to suppression of free edge delamination by an innovative technique. Wrap-around configuration was considered to determine its effectiveness over the wrapper-less laminated configuration on laminated composites. Nodal stresses were generated ahead of the crack tip through finite element analysis. This was used for determining interlaminar normal stress and inter laminar shear stress distributions at the critical interface. Later virtual crack closure technique was used to estimate the strain energy release rate components for several sizes of virtual crack extensions through a single finite element analysis. Computational analysis predicts Mode-I delamination as dominant mode of failure. This mode of delamination was significantly suppressed with wrap-around configuration on laminated composites.

Failure analysis of carbon nanotube/epoxy composites having a broken carbon nanotube

In the present work, multi-scale finite element analysis of carbon nanotube/epoxy composites having a broken carbon nanotube has been performed to assess the severity of such a fiber (i.e. carbon nanotube) break. A square representative volume element having nine-carbon nanotubes has been considered for the analysis. Considering a small debonding around the broken fiber as a crack front, stress redistribution around the break is studied. Using the concept of linear elastic fracture mechanics, strain energy release rates around the debonding have been calculated using virtual crack closure integral method. Quadratic stress criterion has been used to assess the delamination initiation at the interface of broken carbon nanotube and the matrix. Virtual crack closure integral along with quadratic stress criterion has been used to determine the critical strain energy release rate. Results from the present analysis show that the fiber volume fraction does not have significant influence on the ineffective length of the broken carbon nanotube. A high magnitude of interfacial shear stress is observed to have developed in the vicinity of the fiber break indicating the chances of debonding. All the three components of strain energy release rate have been determined and the influences of important parameters on the value of strain energy release rate have also been studied.

Brittle interfacial cracking between two dissimilar elastic layers: Part 2—Numerical verification

A thorough program of 2D finite element method (FEM) simulations is carried out parametrically on a bimaterial double cantilever beam (DCB) model in MSC/NASTRAN. The Young’s modulus ratio, the Poisson’s ratio, the thickness ratio, and the DCB tip loads are varied over their entire practically useful domains for different values of the crack extension size. Extensive comparisons are made between the results of the analytical theory that was developed in Part 1 by Harvey et al. (2015) and FEM results. This paper reports the outcome of these comparisons. The present analytical theory and the supporting mathematical techniques are thoroughly verified. Overall, excellent agreement is observed between the present analytical theory and the FEM results for the crack extension size-dependent energy release rate (ERR) components and the stress intensity factors (SIFs).

Flexural fracture response of a novel iron carbonate matrix – Glass fiber composite and its comparison to Portland cement-based composites

This paper explores the fracture properties of a novel and sustainable glass-fiber reinforced composite, the matrix for which is formed through the aqueous, anoxic, room-temperature carbonation of (waste) metallic iron powder along with other minor ingredients. A comparison of the properties of this binder with Ordinary Portland Cement pastes, which constitutes one of the most common and economic ceramic matrices is also provided. The iron-based binder system exhibits fracture parameters (fracture toughness, KICS and critical crack tip opening displacement, CTODC, determined using two parameter fracture model, TPFM) that are significantly higher when compared to those of the OPC systems in both the unreinforced and glass fiber reinforced states. The beneficial influence of the unreacted metallic iron particles of large aspect ratio, on the fracture parameters of iron-based binders are elucidated. The strain energy release rates show trends that are in line with the fracture parameters from TPFM. The elastic and inelastic components of strain energy release rate are separated in an effort to capture the fundamental toughening mechanisms in these systems. The fracture parameters determined using a non-contact, digital image correlation technique are found to relate well to those obtained from TPFM.

A refined theoretical model for mode I crack near the interface of strength and plasticity mismatched materials in K dominant regime

The paper investigates a cracked bi-material, comprising elastic-plastic materials that are elastically identical but strength and plasticity mismatched, under the action of monotonic and uniform far field tension in small scale yielding (SSY) or K dominant regime. A through, mode I, edge crack in weaker material is assumed to advance perpendicularly toward the interface of stronger material. The interface between the materials is considered to be thin, strong and perfect. A theoretical model is developed with the help of basic mathematical principles to account for the influence of non-linearity in plastic behavior of constituent materials during estimation of plastic energy transfer toward stronger interface material when the crack tip is near the interface. The criterion of the onset of yielding of interface material is included in the model. Energy transfer is represented by the component of energy release rate of the crack due to inhomogeneity caused by property mismatched interface. Shielding effect over the crack is quantified by energy release rate and stress intensity parameter at the crack tip. Two elastically similar steels with different yield strengths and plasticity properties are chosen for computational purpose under plane stress condition. Differences between fracture results of the proposed model and those of an available model based on Dugdale’s criterion that ignores material non-linearity and is valid for elastic-perfectly plastic materials are presented. The proposed model is validated.

Finite Element Analysis of Damage Modeling of Fiber Reinforced Laminate Plate

The paper deals with identifying of the damage model for a bundle of T300 and AS4D fibers under tensile load. The damage model is implemented in ANSYS for a one-dimensional bar element and obtained the strain-stress response of a bundle of fibers. The delamination of laminate plate, which consists of unidirectional fiber reinforced layers, is investigated. The methodology adopts the first-order shear laminate plate theory and fracture and contact mechanics. Within the interface modeling there are calculated the individual components of energy release rate along the lamination front. Numerical results are given for mixed mode delamination problems. Numerical example is done by the commercial ANSYS code.

Damage growth behavior and interlaminar fracture resistance of CFRP laminates under shear fracture mode

Interfacial failure problem of composite laminates was experimentally studied to know the dependence of interlaminar fracture resistance on various conditions as well as conditions of migration of the delamination into lamina to understand three-dimensional complex damage growths. A test method was proposed to realize a stable damage growth without influence of free edge under varying shear fracture mode in laminated composites. We could numerically evaluate critical energy release rate (ERR) for the delamination growth using a finite element method based on the measured shapes of the delamination and the applied load. Not only total ERR but also the ERR components in two directions parallel and normal to the fiber were calculated. The two components of the energy release were more appropriate to discuss the damage growth problem than the mode II and III components. When the normal component to the fiber was relatively small, the total ERR controlled the instability of the delamination. The delamination migrated into the lamina (that is, jumped to the other interface) where the ERR components normal to the fiber reached some level. The migration caused an anchoring effect on the delamination growth.