Multi-parameter Fracture Mechanics Research Articles

Fatigue crack propagation under corrosion of high-strength steel

Corrosion is nearly unavoidable phenomenon in most of the metal materials. In this paper, its effect on the sharp edge crack propagation under tensile loading is investigated. A rectangular plate with a perpendicular crack and an elliptical corrosion pit nearby is modelled via finite elements and fracture behavior of the crack is analyzed. The multi-parameter fracture mechanics concept is applied, i.e. the higher-order terms of the Williams expansion are calculated by means of the over-deterministic method and utilized for tangential stress approximation in the vicinity of the crack tip. Thus, the generalized MTS criterion could be used for estimation of the crack deflection angle. The calculations were performed for a selected corrosion pit size and location considering various crack lengths. The results are discussed and a crack length with the highest probability to deflect from its original perpendicular propagation direction is found.

Crack Deflection Under Mixed-Mode Loading Conditions in Fine-Grained Composites Based on Water Glass-Activated Slag

This paper is devoted to an analysis of the crack deflection angle in a semi-circular disc made of a fine-grained composite based on water glass-activated slag. Three-point bending together with an inclined crack ensures the crack propagation in I + II mixed mode. Generally, concrete material exhibits quasi-brittle fracture behavior, which is difficult to understand. Multi-parameter fracture mechanics (MPFMs) concept shall help us to describe the crack propagation and the influence of choice of a suitable distance for application of generalized fracture criteria is investigated. In this work, the well-known maximum tangential stress (MTS) criterion as well as strain energy density (SED) criterion are applied in their multi-parameter (MP) form to find the initial kink angle. Williams power expansion is used to approximate the crack-tip stress field. Its coefficients need to be calculated numerically via combination of the finite-element (FE) solution and over-deterministic method. Results are discussed and recommendations are stated regarding the critical length parameter as well as the number of the Williams expansion (WE) coefficients.

A computational study of the mixed–mode crack behavior by molecular dynamics method and the multi – Parameter crack field description of classical fracture mechanics

The overarching objective of the paper is to analyze the mixed-mode crack propagation direction angles by molecular dynamics method and to investigate the validity of continuum-based linear elastic fracture mechanics crack growth criteria. For this purpose, an embedded atom potential (EAM) available in LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) molecular dynamics (MD) software is utilized to accurately pinpoint mixed-mode crack growth. The study is focused on the application of the different approaches for determination of the initial crack propagation angle. Copper and aluminum plates with the central crack under complex mechanical stresses (Mode I and Mode II loading) are studied by extensive MD simulations. Williams’ expansion for the crack tip fields containing the higher-order terms is used. The crack propagation direction angles for combinations of Mode I and Mode II loadings are obtained by 1) multi-parameter fracture mechanics approach based on three fracture mechanics criteria: maximum tangential stress (MTS), maximum tangential strain and strain energy density (SED); 2) atomistic modeling for the mixed-mode loading of the plane medium with the central crack. The temperature effects during fracture processes in MD simulations are considered and temperature field distributions for mixed-mode crack propagation are obtained.

Propagation conditions of an eccentric crack in a semi-circular disk loaded in I+II mixed mode

Multi-parameter fracture mechanics concept is applied to investigate crack propagation conditions during a three-point bending (3PB) test performed on an asymmetrical semi-circular disk under I+II mixedmode loading. The asymmetry is represented by an eccentricity of the angled crack as well as by different distances of both supports during the 3PB test from the middle of the specimen. The Williams series expansion is used for the crack-tip stress field approximation and fracture response of the crack is assessed by means of the combination of the numerical and analytical solution. Comparison between the individual configurations is carried out and discussion on the importance of the higher-order terms considered in the Williams expansion for various cases of asymmetry is presented.

Utilization of Williams’ Power Series for Estimation of Crack Behavior under Mixed-Mode Loading

Multi-parameter fracture mechanics concept has been applied to investigate crack behavior under mixed-mode loading, particularly in a semi-circular bending disc. The so-called Williams’ series expansion is used for the crack-tip stress field approximation. It has been shown that application of the generalized fracture mechanics concept can be crucial for materials with specific fracture behavior, such as elastic-plastic or quasi-brittle one, when fracture occurs not only in the very vicinity of the crack tip, but also in a more distant surrounding. Then, considering the higher-order terms of the Williams’ expansion in fracture criteria can be helpful. The attention is devoted to the analysis of the influence of various distances between the supports during the three-point bending test on the results of the further crack propagation direction.

Molecular Dynamics Modeling of Crack Propagation

The study is focused on the application of different approaches for initial crack propagation angle determination. Copper plate with the central crack under complex mechanical stresses (Mode I and Mode II loading) is studied by extensive molecular dynamics simulations using EAM potential. In addition, the complete Williams expansion for the crack tip fields containing higher-order terms is used. Crack propagation angle is obtained by 1) multi-parameter fracture mechanics approach based on three fracture mechanics criteria, MTS, maximum tangential strain and SED; 2) atomistic modeling for mixed-mode loading of plane medium with the central crack. From our simulations we can derive crack propagation directions and crack angles. MD calculations were run for three different values of the mixity parameter: 0.4, 0.5 and 0.6. Calculated values of crack angles were -51:5 deg, -46:6 deg and -42:2 deg accordingly. All the fracture criteria tested give similar values of crack growth angle for different values of the mixity parameter. It is shown that initial crack propagation angles given by both approaches are very close, especially when higher order terms of Williams series expansion for stress/displacement field description are taken into account. Thus, one can conclude that the criteria of classical continuum mechanics MTS and SED can give satisfactory predictions for crack initiation direction. Crack propagation direction angles given by conventional fracture mechanics reasonably agree with the angles obtained from molecular dynamics simulations.

Crack propagation in mixed-mode specimens described via multi-parameter fracture mechanics

The main objective of the paper is to employ a multi-parameter fracture mechanics concept to describe crack propagation through a specimen loaded in mixed-mode. This concept in particular was used because it has been shown that application of the generalized fracture mechanics concept can play a key role for materials with specific fracture behaviour, i.e. when fracture processes occur not only in the very vicinity of the crack tip, but also at larger distances from it. Two mixed-mode (I+II) geometries for the investigation of crack behaviour are presented here. The Williams series expansion is used for crack-tip stress field approximation. Then, considering the higher-order terms of the Williams expansion with regard to maximum tangential stress criterion can provide better estimates of the crack deflection angle. The coefficients of the Williams expansion were determined by means of the over-deterministic method for the purposes of this work. This analysis was performed for each cracked configuration, which is very time-consuming and makes the analysis very extensive. The crack propagation angle obtained by means of the generalized fracture criterion is discussed in detail. It was found that single-parameter fracture mechanics is sufficient when applied close to the crack tip and when mode I of loading prevails, while multi-parameter fracture mechanics can be recommended at larger distances from the crack tip and for configurations where mode II becomes dominant.

Multi-parameter fracture mechanics

A mixed-mode geometry has been chosen to investigate a crack propagation using the multi-parameter fracture mechanics concept. The so-called Williams’ series expansion is used for the crack-tip stress field approximation. It has been shown that application of the generalized fracture mechanics concept can be crucial for materials with specific fracture behaviour, such as elastic-plastic or quasi-brittle one, when fracture occurs not only in the very vicinity of the crack tip, but also in a more distant surrounding. Then, considering the higher-order terms of the Williams’ expansion in fracture criteria (describing the crack stability and/or crack propagation direction) can bring more precise results. The coefficients of the Williams’ expansion must be calculated numerically (for instance by means of the over-deterministic method in this work) for each cracked configuration, which is very time-consuming, and the analysis is very extensive even for a few basic cracked specimen configurations. On the other hand, a suitable choice of the geometrical configuration of the cracked disc enables performing experiments only on the specimens that could prove the theory about the importance of using the higher-order terms.

Multi-parameter failure assessment of a bi-material V-notch – Crack initiation from a free-edge singularity

Multi-parameter fracture mechanics criteria are used to describe crack initiation conditions in general singular stress concentrators. In the cases of stress concentrators described by only one singular stress term, multi-parameter description provides a qualitative and quantitative increase of accuracy. In this paper multi-parameter criteria, based on an average tangential stress and average strain energy density factor, are described and used to determine a crack initiation direction and the critical load in the case of a free-edge singularity. The critical load expresses a failure/stability criterion for the studied general singular stress concentrator. Contrary to criteria employing singular stress terms only, a criterion of this kind is able to cover the influence of the direction of the external load.

Wedge Splitting Test: Displacement Field Analysis by Multi-parameter Fracture Mechanics

Wedge Splitting Test: Displacement Field Analysis by Multi-parameter Fracture Mechanics

A computational study of the mixed-mode crack behavior by molecular dynamics method and the multi-parameter crack field description of classical fracture mechanics

The overarching objective of the paper is to analysis the mixed-mode crack propagation direction angles by molecular dynamics method and to investigate the validity of continuum-based linear elastic fracture mechanics crack growth criteria. For this purpose, an embedded atom potential (EAM) available in LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) molecular dynamics (MD) software is utilized to accurately pinpoint mixed-mode crack growth. The study is focused on the application of the different approaches for the determination of the initial crack propagation angle. Copper and aluminum plates with the central crack under complex mechanical stresses (Mode I and Mode II loading) are studied by extensive MD simulations. Williams’ expansion for the crack tip fields containing the higher-order terms is used. The crack propagation direction angles for combinations of Mode I and Mode II loadings are obtained by 1) the multi-parameter fracture mechanics approach based on three fracture mechanics criteria: maximum tangential stress (MTS), maximum tangential strain and strain energy density (SED); 2) atomistic modeling for the mixed-mode loading of the plane medium with the central crack. The temperature effects during fracture processes in MD simulations are considered and the temperature field distributions for mixed-mode crack propagation are obtained.

Multi-parameter description of the crack-tip stress field: Analytic determination of coefficients of crack-tip stress expansions in the vicinity of the crack tips of two finite cracks in an infinite plane medium

The study is aimed at analytical determination of coefficients in crack tip stress expansions for two collinear finite cracks of equal lengths in an infinite plane medium. The study is based on the solutions of the complex variable theory in plane elasticity theory. Multiparametric presentation of the stress field near the crack tips in the infinite plate with two collinear cracks of finite lengths is obtained and analyzed for a full range of mixed mode loading from pure tension to pure shear. The method of analytical determination of coefficients of the complete asymptotic expansion of the stress field near the crack tip is presented. The influence of consideration of various numbers of terms of the series expansion on the stress distribution is discussed, and the significance of the multi-parameter fracture mechanics approach is emphasized.

Crack propagation direction in a mixed mode geometry estimated via multi-parameter fracture criteria

A numerical parametric study on the crack growth direction under mixed mode conditions (I+II) is introduced. An eccentric asymmetric four-point bending of a single edge notched beam specimen is considered as an example, since various levels of mode mixity can be thus obtained. The crack growth direction is estimated using the maximum tangential stress (MTS) criterion, the strain energy density (SED) criterion and the crack tip displacement (CTD) criterion. Multi-parameter fracture mechanics based on description of the stress and deformation fields in the cracked body by means of their approximation using several initial terms of the Williams power series is employed. It was found out that whereas the use of the CTD and SED criteria should be accurate enough even in their classical (or two-parameter) form for most geometries and distances of their application from the crack tip, the generalized form of the MTS criterion with at least four terms of the series should be applied, if the distance of its application is not less than 0.01 of the specimen width. Moreover, it is able to describe the typical dependence of the initial kink angle on the distance from the crack tip where the criterion is applied.

Complete Williams Asymptotic Expansion Near The Crack Tips of Collinear Cracks of Equal Lengths in an Infinite Plane Medium

The study is aimed at analytical determination of coefficients in crack tip stress expansions for two collinear finite cracks of equal lengths in an infinite plane medium. The study is based on the solutions of the complex variable theory in plane elasticity theory. Multiparametric presentation of the stress filed near the crack tips in the infinite plate with two collinear cracks of finite lengths is obtained and analyzed for a full range of mixed mode loading from pure tension to pure shear. The method of analytical determination of coefficients of the complete Williams asymptotic expansion of the stress field near the crack tip is presented. The influence of consideration of various numbers of terms of the series expansion on the stress distribution is discussed, and the significance of the multi-parameter fracture mechanics approach is emphasized.

Estimation of the crack propagation direction in a mixed-mode geometry via multi-parameter fracture criteria

The presented work introduces a numerical parametric study on the crack propagation direction under mixed-mode conditions (mode I + II). It is conducted for the geometry of an eccentric asymmetric fourpoint bending of a single edge notched beam specimen; various levels of mode-mixity are ensured by modifications in the crack length and crack eccentricity. The direction of crack propagation is estimated semianalytically using both the maximum tangential stress criterion and the strain energy density criterion (implemented as a procedure within the used finite element computational code) as well as numerically (from verification reasons). Multi-parameter fracture mechanics is employed in the presented work for precise analytical evaluation of the stress field in the cracked specimen. This theory is based on description of the stress and deformation fields in the cracked body by means of their approximation using several initial terms of the Williams power series. Recent studies show that utilization of only first term of the series, which corresponds to the stress intensity factor (SIF), the single controlling parameter for the crack initiation and propagation assessment in brittle materials, is insufficient in many crack problems. It appears also in this study that the higher-order terms of the asymptotic crack-tip field are of great relevance for the conducted analysis, similarly to a number of other fracture phenomena (near-crack-tip stress field approximation, non-linear zone extent estimation, etc.).

Application of Multi-Parameter Fracture Mechanics to Study of Crack Propagation Angle in Selected Mixed-Mode Geometry

The multi-parameter fracture mechanics becomes more and more significant, because it is shown that it can help to describe fracture processes occurring in cracked specimens more precisely than conventional linear elastic fracture mechanics. In this paper, the concept based on the Williams expansion derived for approximation of stress/displacement crack-tip fields is presented and applied on a mixed-mode configuration. Two fracture criteria for estimation of the initial crack propagation angle are introduced. A parametric study is performed in order to investigate the dependence of the crack propagation angle on the stress intensity factors ratio. Influence and importance of taking into account the so-called higher-order terms of the Williams expansion are discussed and some recommendations are stated.

Estimation of the Zone of Failure Extent in Quasi-Brittle Specimens with Different Crack-Tip Constraint Conditions from Stress Field

A multi-parameter fracture mechanics concept based on the Williams power series is applied on novel cracked specimen geometries utilizing combined boundary conditions of the wedge splitting and the three-point bending test. Crack tip stress fields for various configurations (causing different constraint conditions at the crack tip and thus also different fracture process zone extents) are numerically investigated and subsequently analytically reconstructed using developed procedure. An importance of using higher order terms of the Williams series is demonstrated.

Crack Path Investigation Using the Generalized Maximum Tangential Stress Criterion: Antisymmetrical Four-Point Bending Specimen

A mixed-mode geometry has been chosen in order to investigate crack propagation through the specimen. There exist several criteria for estimation of the crack path, and this work enriches the concept by application of multi-parameter fracture mechanics. Particularly, the classical MTS criterion is presented in its generalized form when several initial terms of the so-called Williams series expansion are used for the tangential stress approximation. The results obtained show that the generalized approach can be important in some special cases, when the criterion shall be applied in a larger distance from the crack tip.

Detailed Crack-Tip Stress Field Description in a Specimen Subjected to Mixed-Mode Loading

A mixed-mode cracked configuration is investigated using the multi-parameter fracture mechanics concept based on the analytical description of the stress/displacement field near a crack tip by means of the Williams series expansion. It is shown that using only one (singular) parameter as it is usual for brittle materials is not sufficient if the accurate stress distribution also further from the crack tip shall be known. Tangential stress distribution in various distances from the crack tip is presented and importance of the higher-order terms of the Williams expansion emphasized. Moreover, initial crack propagation direction is investigated by means of the MTS criterion and utilization of its generalized form is discussed.

Using the Multi-Parameter Fracture Mechanics for more Accurate Description of Stress/Displacement Crack Tip Fields

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.