Strain Energy Density Criterion Research Articles

Mixed-mode I/II fracture criterion for crack initiation assessment of composite materials

This paper presents a mixed-mode I/II fracture criterion to investigate the crack initiation in orthotropic materials in which the crack is directed along fibers. The minimum strain energy density criterion is extended to investigate the cracked orthotropic materials. The reinforced isotropic solid model based on collinear crack propagation along fibers is proposed as an advantageous model to study the fracture behavior of composites. This model introduces fibers as reinforcements of the isotropic matrix in orthotropic materials in which their effects are qualified by defining reinforcement factors at tension and shear modes. The proposed criterion can predict the crack initiation phenomenon. Therefore, in this study a new concept of linear fracture toughness for orthotropic materials is proposed. Experimental data are used to validate the output of the proposed criterion. The coincidence of fracture limit curves and experimental data indicates the ability of the new criterion to predict crack initiation in orthotropic materials.

On predicting mode II fracture toughness (KIIc) of hot mix asphalt mixtures using the strain energy density criterion

Mode II or in-plane sliding of crack faces is one of the possible fracture modes for asphalt pavements which can often take place due to traffic loads passing over the crack at low temperature conditions. Hence pavement engineers need to know the value of mode II fracture toughness (KIIc) of asphalt mixtures as an important design parameter. However, despite a large number of mode I fracture toughness (KIc) values available in the literature for different asphalt concrete materials, only limited KIIc values have been reported for such construction materials. In this paper to fill this research gap, a number of pure mode I and pure mode II fracture toughness experiments are conducted at low temperature using symmetric and asymmetric semi-circular bend specimens made of different asphalt mixtures. It was found that the mode II fracture toughness of each mixture is approximately 0.9–1.1 times the corresponding value of KIc. It was demonstrated that the experimentally obtained fracture toughness ratio (KIIc/KIc) for the investigated asphalt concretes are in good agreement with theoretical predictions of the minimum strain energy density criterion. Consequently, as an engineering application of this research, mode II fracture toughness of investigated asphalt mixtures was estimated in terms of their more common mechanical parameters such as KIc and Poisson’s ratio values.

A new mixed mode I/II failure criterion for laminated composites considering fracture process zone

In this paper, by considering the absorbed energy in the fracture process zone and extension of the minimum strain energy density theory for orthotropic materials, a new mixed mode I/II failure criterion was proposed. The applicability of the new criterion, to predict the crack growth in both laminated composites and wood species, was investigated. By defining a suitable damage factor and using the mixed mode I/II micromechanical bridging model, the absorbed energy in the fracture process zone was considered. It caused the new criterion to be more compatible with the nature of the failure phenomena in orthotropic materials, unlike available ones that were conservative. A good agreement was obtained between the fracture limit curves extracted by the present criterion and the available experimental data. The theoretical results were also compared with those of the minimum strain energy density criterion to show the superiority of the newly proposed criterion.

Energy-based approach for fracture assessment of several rocks containing U-shaped notches through the application of the SED criterion

This work presents an energetic continuum approach for the fracture assessment of rocks containing U-shaped notches and subjected to Mode I loading conditions. Three different methodologies are proposed, all based on the premise that brittle failure will occur when the average strain energy density over a certain control area reaches a critical value that only depends on the material, as stated by the Strain Energy Density (SED) criterion.The first method proposed (A) deals with the application of the SED criterion through an expression with a series of already tabulated parameters, which are particularised for the analysed rocks by rational extrapolation. By contrast, the second method (B) aims to obtain numerically the previously extrapolated parameters, and the third method (C) directly relates the strain energy density with the applied load, without the use of those parameters.The research is based on the results obtained from an exhaustive experimental programme comprising 300 fracture specimens tested in four-point bending conditions. These tests combine parallelepiped samples made of six different types of rocks (two marbles, two limestones, a sandstone and a granite) and containing eight different notch radii (varying from 0.15 mm up to 15 mm).Thus, this work aims to show the potential, capacity and limitations of the SED criterion in rock fracture analyses, by comparing the experimentally obtained fracture loads to those predicted by the three proposed methodologies.

Estimation of crack propagation direction angles under mixed mode loading in linear elastic isotropic materials by generalized fracture mechanics criteria and by molecular dynamics method

The crack growth directional angles in the isotropic linear elastic plane with the central crack under mixed-mode loading conditions for the full range of the mixity parameter are found. The traditional strain energy density, maximum tangential stress and maximum tangential strain criteria are introduced and analyzed. The accuracy of each model is investigated and discussed by comparing its results to the molecular dynamics modelling. Three fracture criteria of traditional linear fracture mechanics (maximum tangential stress, and minimum strain energy density criteria) are generalized by mean the multi-parameter stress expansion in the vicinity of the crack tip. Atomistic simulations of the central crack growth process in an infinite plane medium under mixed-mode loading using Large-scale Molecular Massively Parallel Simulator (LAMMPS), a classical molecular dynamics code, are performed. The inter-atomic potential used in this investigation is Embedded Atom Method (EAM) potential. The plane specimens with initial central crack were subjected to Mixed-Mode loadings. The simulation cell contains 400000 atoms. The crack propagation direction angles under different values of the mixity parameter in a wide range of values from pure tensile loading to pure shear loading in a wide diapason of temperatures (from 0.1 К to 800 К) are obtained and analyzed. It is shown that the crack propagation direction angles obtained by molecular dynamics method coincide with the crack propagation direction angles given by the multi-parameter fracture criteria based on the strain energy density and the multi-parameter description of the crack-tip fields.

Crack Growth Direction in Mixed Mode Loading: A Strain Energy Density Approach

The crack growth in metallic materials using fast and reliable simulations of 2-D and linear elastic finite element models is investigated. The effect of the stress intensity factor in mode I and II (KI, KII) on the fracture behavior of stainless steel and the associated strain energy density factor in mixed mode crack propagation were studied numerically to determine crack propagation angle θ in linear elastic fracture investigation. In order to implement the determination of the crack propagation direction using the strain energy density criterion S, the numerical finite element program ANSYS was used. ANSYS APDL macros were developed to generate the geometry, material properties, boundary conditions and mesh size of the model for the conducted analyses. To demonstrate the capability of crack propagation trajectories using the proposed method under mixed mode situation, an edge crack specimen was considered with initial crack having the same length but at different inclination angles under a uniaxial tension load. Results obtained from the developed models had a good agreement (average deviation of 4.63%) with the results available in the literatures.

Frictional crack initiation and propagation in rocks under compressive loading

Crack initiation and propagation in a brittle material such as a rock are affected by the friction between crack surfaces. Choosing an appropriate fracture criterion for frictional crack surfaces plays an important role in determining fracture parameters. To this end, a two-dimensional crack propagation code (TDCPC) was developed in this study based on the displacement discontinuity method (DDM) to predict the crack propagation path under the effect of friction. In this study, three classes of fracture criteria were defined: (1) classical fracture criteria (maximum tangential stress criterion, minimum strain energy density criterion, and maximum strain energy release rate criterion) under mixed mode loading without the effect of friction (CFC-I-II), (2) classical fracture criteria under the pure mode II (shear fracturing) without the effect of friction (CFC-II), and (3) the Swedlow criterion which takes into account the effect of friction between crack surfaces (SW-II). A special element (Swedlow-element) was developed and used in the displacement discontinuity method to predict crack trajectories under the effect of friction. The Swedlow criterion was developed to study the fracture process under bi-axial loading. Results indicated that CFC-I-II are not valid for frictional crack surfaces. Crack propagation paths predicted by CFC-II and SW-II under uniaxial loading proved that friction coefficient has a significant effect on the crack initiation angle (the first stages of the crack propagation path), and away from the crack tip, CFC-II and SW-II almost have the same path. For bi-axial loading, crack propagation paths predicted by SW-II deviate away from the original crack as friction coefficient is increased. The stress ratio (Ko) and friction coefficient have opposite effects on the crack propagation path. Crack trajectory deviates away from the original crack under the effect of friction coefficient and moves toward the original crack under the effect of the stress ratio. The present study indicated that the results predicted using TDCPC are in good agreement with the experimental and numerical results of other studies.

Rupture analysis of rubber in the presence of a sharp V-shape notch under pure mode-I loading

The rupture behavior of styrene-butadiene rubbers (SBR) in the presence of a V-shape notch is investigated for the first time both experimentally and theoretically. In the experiments, V-notched samples of SBR are tested under tensile loading and their rupture displacements are determined. Afterwards and in the analytical field, the rupture loads of tested rubbers are predicted using the averaged strain energy density (ASED) criterion. The key idea of this criterion (i.e. the almost uniaxial state of stress field near the notch tip) is verified through non-linear finite element modeling. It is shown that good agreement exists between the predictions of the ASED criterion and the experimental results obtained for SBR. Moreover, the microscopic study of the ruptured surfaces of the notched SBR demonstrates its high roughness which can be attributed to the resistance of the rubber chains against the crack growth.

Modelling of the ductile-brittle fracture transition in steel structures with large shell elements: A numerical study

In this paper, the strain energy density (SED) criterion is proposed for predicting the ductile-brittle fracture transition (DBFT) in ships and offshore structures. In finite element simulations, these structures are discretized by relatively large shell elements which precludes the modelling of the local stress and strain states in the vicinity of a crack. Critical values of the SED are determined based on local simulations of fracture for a range of temperatures and plane stress states. The local simulations of fracture are based on combined use of the Gurson model for ductile damage and fracture and the Richie-Knott-Rice (RKR) criterion for brittle fracture. After calibrating the Gurson model and the RKR criterion to existing experiments on offshore steel, critical values of the SED are found by analysing a representative plate element with a generic through-thickness crack using a refined solid element mesh. The proposed failure model is evaluated by simulating drop tests on steel-plated structures found in the literature. The present study indicates that the SED criterion is a useful concept for practical design of ships and offshore structures at sub-zero temperatures, but further assessment against experimental data is necessary to fully establish its credibility.

Energy-based assessment of brittle fracture in VO-notched polymer specimens under combined compression-shear loading conditions

This research presents some experimental, numerical, and theoretical results on brittle fracture of disk-type test specimens weakened by V-notches with end-holes under mixed mode I/II loading with negative mode I contributions. First, 54 fracture tests are conducted on VO-notched Brazilian disk specimens made of the general-purpose polystyrene under mixed mode I/II loading with negative mode I contributions. Then, two energy-based brittle fracture criteria, namely the averaged strain energy density and averaged strain energy density based on the equivalent factor concept are proposed to predict the experimentally obtained fracture loads of the tested general-purpose polystyrene specimens. Additionally, the fracture initiation angles of the tested VO-notched Brazilian disk specimens are predicted by using averaged strain energy density criterion. The finite element analyses, as well as the experimental observations, show that although brittle fracture in the specimens under mixed mode I/II loading takes place from the applied load side of the notch border by local tensile stresses, the notch bisector line and the other sides of the notch border sustain compressive stresses. In fact, this phenomenon states the concept of mixed mode I/II loading with negative mode I contributions. Finally, it is shown that good agreement exists between the experimental results and the theoretical predictions of the two energy-based fracture criteria.

Towards a physics-based relationship for crack growth under different loading modes

In an attempt to understand quasi-static delamination growth under mixed mode loading conditions from a physics-based perspective, this work first evaluated cracking in isotropic materials. The critical Strain Energy Density (SED) approach is adopted, because physically the onset of crack growth is expected to occur when the energy available near the crack tip reaches a critical value.The main hypothesis of the present paper is that the critical SED for onset of crack growth is constant for a given material, and independent of the loading mode. The relationship derived from this hypothesis therefore relates the physical onset of crack growth and the angle at which that occurs for any opening mode through the SED.To test this hypothesis, results from literature were taken and shear fracture tests on foam specimens were performed, which both were compared with the derived relationship. The excellent correlation demonstrated the validity of the physics-based relationship, which explains the observed differences between mode I and mode II fracture toughnesses and illustrates why concepts like the Stress Intensity Factor (SIF) alone are insufficient to explain the observations. The developed relationship allows to derive the mode II fracture toughness from mode I fracture toughness tests and the material’s mechanical properties.

Residual static strength and the fracture initiation path in adhesively bonded joints weakened with interfacial edge pre-crack

This paper deals with the application of fracture mechanics approaches for predicting the residual static strength and the crack kinking angle of adhesively bonded joints containing interfacial edge pre-cracks. The interfacial cracks are created due to different factors such as inappropriate surface preparation which cause a significant reduction of the joint strength. To investigate the residual strength of interfacial cracked adhesive joints and predict the crack kinking angle, three different approaches including the maximum tangential stress (MTS), the minimum strain energy density (SED) and the maximum tangential strain energy density (MTSED) were assessed. To this end, single lap joints (SLJs) containing a brittle adhesive material and with different pre-crack sizes and various substrate thicknesses were manufactured and tested. The results were also verified by applying fracture mechanics approaches on previously published experimental data. According to the results, it was concluded that in mode II dominant cases, the predictions of kinking angle using the MTS method was in good agreement with the experimental observations, while in mode I dominant cases the mentioned approach provided poor predictions. It was also found that the SED criterion could be a precise model for predicting the crack extension angle in mode I dominant conditions. The results also showed that the MTS criterion predicts the residual static strength of interfacial cracked adhesive joints very well.

Static assessment of nanoscale notched silicon beams using the averaged strain energy density method

This paper extends the averaged Strain Energy Density (SED) method to the static assessment of notched components at the nanoscale. First, in situ micromechanical testing of notched nano-cantilever beams made of single-crystal silicon is briefly reviewed. Then, an alternative strategy based on the Theory of Critical Distances is employed to evaluate the control volume and the critical SED. The method is later verified against experiments and FE analyses. The SED method successfully estimates the load at fracture of nanoscale notched specimens with a maximum discrepancy of 4.7%. Moreover, the method is mesh-independent, and therefore very coarse meshes can be employed in numerical analyses. Finally, the results are discussed on the basis of the breakdown of continuum fracture mechanics at the nanoscale. The extension of the SED approach to the micro- and nanoscales provides a fast and simple tool for the design of micro- and nanodevices.

Fracture Assessment of Inclined Double Keyhole Notches in Isostatic Graphite

In the present contribution, the static strength of isostatic graphite using keyhole notch specimens under mixed mode loading is investigated. An experimental program was performed and in total, 18 new experimental data are provided. In addition, different loading mode ratios are considered by varying the inclination angle of the notch with respect to the direction of the applied load. The criterion based on the averaged value of the strain energy density over a control volume at the notch edge is applied to assess the static strength of specimens. A sound agreement is found between experimental data and the results obtained from strain energy density criterion.

Compressive Brittle Fracture Prediction in Blunt V-Notched PMMA Specimens by Means of the Strain Energy Density Approach

The paper aims to examine the suitability of the strain energy density criterion in predicting the fracture behavior of blunt V-notched specimens under compression load. Recent studies on local stress fields have shown that the strain energy density, averaged over a specific control volume which embraces the notch round border, could be a robust parameter in the brittle fracture assessment of several materials. A set of experimental results recently published in the literature on compressive brittle fracture of V-notched specimens made out of polymethyl methacrylate has here been considered. Finite element analyses have been performed on plane strain condition and experimental data have been summarized by means of the SED criterion. It has been shown that the proposed criterion permits a satisfactory evaluation of the fracture load of polymethyl methacrylate specimens weakened by notches having different opening angles and radii.

Fracture Loads Prediction on Notched Short Glass Fibre Reinforced Polyamide 6 Using the Strain Energy Density

This paper provides an energetic approach useful for the prediction of critical loads on U-notched components without an ideally linear elastic behaviour. The methodology has been applied to 100 fracture specimens of short glass fibre reinforced polyamide 6 (SGFR-PA6), combining four different fibre contents (5, 10, 30 and 50 wt %) and five different notch radii (0.00, 0.25, 0.50, 1.00 and 2.00 mm). The proposal combines the application of the strain energy density criterion with the use of the whole absorbed energy in the tensile test (elastic-plastic area under the stress-strain curve). With all of this, the fracture loads have been well estimated in this type of material.

Mathematical modeling of the crack growth in linear elastic isotropic materials by conventional fracture mechanics approaches and by molecular dynamics method: crack propagation direction angle under mixed mode loading

The crack growth directional angles in the isotropic linear elastic plane with the central crack under mixed-mode loading conditions for the full range of the mixity parameter are found. Two fracture criteria of traditional linear fracture mechanics (maximum tangential stress and minimum strain energy density criteria) are used. Atomistic simulations of the central crack growth process in an infinite plane medium under mixed-mode loading using Large-scale Molecular Massively Parallel Simulator (LAMMPS), a classical molecular dynamics code, are performed. The inter-atomic potential used in this investigation is Embedded Atom Method (EAM) potential. The plane specimens with initial central crack were subjected to Mixed-Mode loadings. The simulation cell contains 400000 atoms. The crack propagation direction angles under different values of the mixity parameter in a wide range of values from pure tensile loading to pure shear loading in a wide diapason of temperatures (from 0.1 К to 800 К) are obtained and analyzed. It is shown that the crack propagation direction angles obtained by molecular dynamics method coincide with the crack propagation direction angles given by the multi-parameter fracture criteria based on the strain energy density and the multi-parameter description of the crack-tip fields.

Estimation of Fracture Loads in AL7075-T651 Notched Specimens Using the Equivalent Material Concept Combined with the Strain Energy Density Criterion and with the Theory of Critical Distances

The main goal of this paper is the application of the Strain Energy Density (SED) criterion and the Theory of Critical Distances (TCD), both of them in combination with the Equivalent Material Concept (EMC), to predict the fracture loads of aluminum alloy Al7075-T651 Compact Tension (CT) specimens containing U-shaped notches. For this purpose, 45 fracture tests were performed combining two rolling orientations (transverse and longitudinal) and 6 notch radii, which cover from crack-type defects (0 mm) up to 2 mm-notch radius. Crack-type specimens are used to define the fracture properties of the material and the rest of the tests are used to check and compare the experimental fracture loads with the loads predicted using the different aforementioned criteria: SED, EMC-SED and EMC-TCD. The theoretical results of the fracture load predictions for the virtual brittle material obtained employing the EMC are in good agreement with the experimental results reported for real samples.

Composite cylindrical shells under combined loading

The structural design of a cylindrical shell made of a unidirectional glass/epoxy fiber composite and subjected to torsional moment and internal pressure is studied. An anisotropic model for analysis of composite material is applied. The minimum wall thickness of the shell, with the use of the Tsai-Hill failure criterion, is derived. In this connection, the strain energy density criterion (SED) is applied, by which the critical crack length is determined. In this paper, we illustrate the fracture methodology towards the design of a cylindrical shell structure for determining safe operation. Finite element analysis and analytical method were also established.

尖锐V型切口混凝土梁的应力强度因子研究

The failure behavior of the concrete beams with V notches is usually predicted by the notch stress intensity factor (NSIF), which quantify the intensities of the asymptotic linear elastic stress distributions around the notches. For a V-notched beam, the NSIF is determined by the notch angle. The strain energy density fracture criterion is used to judge the fracture failure of a member according to whether the strain energy density in a certain volume reaches the critical value. If the volume is small enough to neglect the higher-order solutions of the Williams equation, the strain energy density criterion can be used to calculate the NSIF. In view of the type-Ⅰ load condition, the theoretical NSIFs of the V notches obtained with the mean strain energy density fracture criterion and Carpinteri’s finite fracture mechanics method respectively, agree well with each other. Moreover, both the theoretical NSIFs given by the 2 criteria are fairly consistent with the experimental results from various V-notched concrete beam specimens.