Crack Tip Displacement Fields Research Articles

A Discontinuous Cellular Automaton Method for Modeling Rock Fracture Propagation and Coalescence Under Fluid Pressurization Without Remeshing

We present a formulation of a discontinuous cellular automaton method for modeling of rock fluid pressure induced fracture propagation and coalescence without the need for remeshing. Using this method, modelers discretize a discontinuous rock-mass domain into a system composed of cell elements in which the numerical grid and crack geometry are independent of each other. The level set method, which defines the relationship between cracks and the numerical grid, is used for tracking the crack location and its propagation path. As a result, no explicit meshing for crack surfaces and no remeshing for crack growth are needed. Discontinuous displacement functions, i.e., the Heaviside functions for crack surfaces and asymptotic crack-tip displacement fields, are introduced to represent complex discontinuities. When two cracks intersect, the tip enrichment of the approaching crack is annihilated and is replaced by a Heaviside enrichment. We use the “partition of unity” concept to improve the integral precision for elements, including crack surfaces and crack tips. From this, we develop a cellular automaton updating rule to calculate the stress field induced by fluid pressure. Then, the stress is substituted into a mixed-mode fracture criterion. The cracking direction is determined from the stress analysis around the crack tips, where fracture fluid is assumed to penetrate into the newly developed crack, leading to a continuous crack propagation. Finally, we performed verification against independent numerical models and analytic solutions and conducted a number of simulations with different crack geometries and crack arrangements to show the robustness and applicability of this method.

The Research on Oblique Crack Tip Deformation Field of Airship Envelope Composites Based on the Digital Speckle Correlation Method

The tearing mechanical properties and the deformation field distribution of 45 degree oblique direction of the crack tip was studied on the stratospheric airship envelope reinforced by plain woven Kevlar-fiber composite. Here, the material samples were prefabricated with the initial length of 20mm unilateral incision, and the digital speckle correlation experimental method was used to obtain the perpendicular crack tip displacement and strain fields with artificial spots under mono-axial tear test. The results show the deformation field changes with the characteristic of discrete step and the strain concentration phenomenon, and the strain concentration zone expands as increasing tensile load. The warp and weft step maximum displacement reaches 6.00mm and 1.7mm, and deformation in every step exhibits a diminishing step law, the warp deformation in the rear area of the crack tip are evenly, while the shear and weft strain volatile in a wide range

Effect of overload on crack closure in thick and thin specimens via digital image correlation

The displacement field of compact tension (CT) specimens have been mapped by digital image correlation (DIC) local to growing fatigue cracks to study overload effects for plane stress and plane strain. We have extracted crack opening displacement (ΔCOD) and stress intensity (K) determined by a Muskhelishvili fit to the crack tip displacement field to infer the closure load. In both cases a classical knee was observed upon unloading consistent with closure which disappeared during the accelerated growth following OL, before increasing during retardation. In both cases following OL the crack growth rate is perturbed for a distance similar to the plastic zone.

EXTENDED FINITE ELEMENT METHOD WITH NONLINEAR GEOMETRY AND ITS APPLICATION IN FRACTURE MECHANICS

A extended finite element (X-FEM) method and corresponding Fortran code are developed in the modeling and simulation for nonlinear geometry and fracture mechanics problems. X-FEM can model a domain without explicitly meshing the crack surface. This method can treat an arbitrary crack independent of the mesh and crack growth. The X-FEM formulas with nonlinear geometry are deduced. In order to model the crack discontinuity, a Heaviside step function and a two-dimensional asymptotic crack-tip displacement field are added to the traditional finite element approximation for the local enrichment based on the theory of partition of unity. The crack is described by two level set functions. The X-FEM computational algorithm is presented in the framework of Lagrangian description in order to model the arbitrary discontinuities in large deformations. The stress intensifying the factors of a crack are calculated by using the multi-point displacement extrapolation method and least square fitting. Finally, a numerical example is presented to demonstrate the accuracy and efficiency of the X-FEM and the FORTRAN code in large deformation crack problems. It is found that X-FEM is superior to the traditional FEM in the modeling and simulation of crack being and crack growth program.

The fracture behavior of birch and spruce in the radial-tangential crack propagation direction at the scale of the growth ring

Abstract Crack-tip displacement fields have been computed based on digital image correlation for spruce (Picea abies [L.] Karst.) and birch (Betula pendula Roth.) wood, which were submitted to pure mode I loading in the RT-direction under both green and air-dried conditions. Moreover, crack propagation was modeled based on both linear elastic fracture mechanics (LEFM) and nonlinear fracture mechanics, relying on the fictitious crack model (FCM). The measured and modeled load versus the crack-mouth opening displacement (CMOD) curves and displacement fields were compared. In the case of spruce, the load-CMOD curves simulated by the FCM coincide well with the measured ones. On the contrary, measured near crack-tip displacement fields in both green and air-dried spruce are better comparable with the LEFM predictions than with the FCM predictions. In the case of green birch, the simulated FCM curve follows the measured curve quite well, but in air-dried birch the simulated FCM curve has a better fit than the LEFM-curve only before maximum load. In birch, the FCM predicts the displacement fields better than the LEFM. In both species, moisture content has a big effect on the softening behavior. In both spruce and birch, the FCM overestimates the displacements ahead of crack tip, whereas the LEFM model underestimates the displacements.

Identification of a cohesive zone model from digital images at the micron-scale

We present a new methodology for the identification of a zone cohesive model that describes material failure. The material under consideration fails by crazing. The study is conducted at the micron scale in order to capture and analyze the fracture mechanism. The crack tip displacement fields are measured optically by Digital Image Correlation. The local stress intensity factors (mode I and II) and the location of the equivalent elastic crack tip are calculated during the loading. The variation of the location of the equivalent crack tip is used to track the initiation and growth of the process zone, up to the onset of crack propagation. These experimental measurements are used to define the appropriate parameters in a cohesive zone model. The methodology addresses the onset of crazing, the traction–separation profile and the maximum opening corresponding to the local nucleation of a crack. The cohesive parameters that are derived from the experimental data are consistent with results available in the literature. In addition, the model enables the characterization of the normal and tangential mode of the cohesive model.

Hygrothermal Fracture Analysis of Orthotropic Functionally Graded Materials UsingJk-Integral-Based Methods

This paper puts forward two differentJk-integral-based methods, which can be used to perform mixed-mode fracture analysis of orthotropic functionally graded materials subjected to hygrothermal stresses. The first method requires the evaluation of both components ofJk-integral, whereas the second method employs the first componentJ1and the asymptotic crack tip displacement fields. Plane orthotropic hygrothermoelasticity is the basic theory behind theJk-integral formulation, which is carried out by assuming that all material properties are functions of the spatial coordinates. Developed procedures are implemented by means of the finite element method and integrated into a general purpose finite element analysis software. Temperature and specific moisture concentration fields needed in the fracture analyses are also computed through finite element analysis. Each of the developed methods is utilized in conjunction with the superposition technique to calculate the hygrothermal fracture parameters. An inclined crack located in a hygrothermally loaded orthotropic functionally graded layer is examined in parametric analyses. Comparisons of the results generated by the proposed methods do indicate that both methods lead to numerical results of high accuracy and that the developed form of theJk-integral is domain independent. Further results are presented so as to illustrate the influences of crack inclination angle, crack length, and crack location upon the modes I and II stress intensity factors.

Analysis of Multi-Crack Growth in Asphalt Pavement Based on Extended Finite Element Method

An extended finite element method (XFEM) for multiple crack growth in asphalt pavement is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite element with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Finally, the propagation path of the cracks in asphalt pavement under different load conditions is presented.

Characteristic of crack tip fields for dynamic fracture in anisotropic materials

Abstract The constant length crack steadily propagating in an anisotropic body, acted on at infinity by uniform biaxial and shear loading, is investigated. The analytical solutions to the dynamic crack tip stress fields and displacement fields are derived. The obtained crack tip fields contain second order term, and are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. Crack propagation characteristics are represented by crack propagation velocity M and fiber direction α . The faster the crack velocity is, the greater the dynamic stress components and dynamic displacement components around crack tip is. Distributions of the stress fields and displacement fields vary with M and α but the inclinations of stress and displacement distribution are almost the same. The effects of biaxial loading coefficient on crack tip fields are also considered.

In-Situ SEM J-Testing and Crack-Tip Micromechanics of Zircaloy-4 by Three-Point Bending

A three-point bend fixture has been designed, fabricated, and utilized to demonstrate the feasibility of performing in-situ J-testing at ambient and elevated temperatures inside a scanning electron microscope (SEM). Using the three-point bend test technique, in-situ SEM J-testing has been performed to measure the crack mouth opening displacement and crack extension as a function of the applied load in order to generate J-R curves for Zircaloy-4 at 25°C and 316°C. Once the J-R curve is determined, an equivalent KJ-resistance (KJ-R) curve is computed on the basis of a relationship between the J-integral (J) and the stress intensity factor (K). The J-R and KJ-R curves of Zircaloy-4 exhibit a rising R-curve behavior, while the elastic K-R curve underestimates the fracture resistance of Zircaloy-4 once substantial crack extension has occurred. For the specimen dimensions considered, the J-R curves generated by in-situ SEM J-tests are not sensitive to the specimen geometry and measure the actual fracture resistance of the material. Furthermore, the onset of crack extension is dictated by the emission of one or more slipbands from the crack tip, and a change in the crack-tip displacement field, followed by void formation along the slipband, and linkage of the voids with the main crack.

3D/4D Strain Mapping Using <i>In Situ</i> X-Ray Microtomography

X-ray microtomography (XMT) has been utilized for the in-situ observation of various structural materials under external disturbance such as loading. In-situ XMT provides a unique possibility to access the three-dimensional (3D) character of internal microstructure and its time evolution behaviours non-destructively, thereby enabling advanced techniques for measuring local strain distribution. Local strain mapping is readily enabled by processing such high-resolution tomographic images either by the particle tracking technique or the digital image correlation technique. Procedures for tracking microstructural features which have been developed by the authors, have been applied to analyse localised deformation and damage evolution in a material. Typically several tens of thousands of microstructural features, such as particles and pores, are tracked in a tomographic specimen (0.2 - 0.3 mm3 in volume). When a sufficient number of microstructural features is dispersed in 3D space, the Delaunay tessellation algorithm is used to obtain local strain distribution. With these techniques, 3D strain fields can be measured with reasonable accuracy. Even local crack driving forces, such as local variations in the stress intensity factor, crack tip opening displacement and J integral along a crack front line, can be measured from discrete crack tip displacement fields.

A new analytical method for stress intensity factors based on insitu measurement of crack deformation under biaxial tension

A new approach for the calculation of stress intensity factors (SIF) for isotropic and orthotropic materials under biaxial tension loading was proposed in this paper. In order to determine SIF from the full-field displacement data, an asymptotic expansion of the crack tip displacement field was performed. The deforming shape and surface residual stress of the crack tip was obtained at the early extended stage of the loading process by using optical microscope and X-ray diffraction measurement. During this stage, a modified Dugdale Model, which takes into account the coupled effect at the crack tip, was proposed for the open displacement of the crack tip. In this paper, the SIFs of two types of silicon steel sheet with isotropic and orthotropic properties were calculated using the modified Dugdale Model based on the biaxial tension experimental data. From the results, it was found that analysis using the modified Dugdale Model is an effective way to evaluate SIF under biaxial stress.

Quantifying crack tip displacement fields with DIC

Crack paths under both fatigue and fracture conditions are governed by the crack tip displacement field and the material deformation characteristics, including those influenced by metallurgical anisotropy. Experimental techniques such as thermoelasticity and photoelasticity have been successfully used to characterise the elastic stress fields around cracks but they do not take into account either plasticity or anisotropy. Considerable work has been carried out to characterise crack tip stress fields from displacement measurements. The current method of choice for obtaining displacement field data is digital image correlation (DIC) which has undergone significant advances in the recent years. The ease of use and capabilities of the technique for full field displacements has led to improved methods for characterising crack tip displacement fields based on data obtained from DIC. This paper gives an overview of some of the applications of DIC for crack tip characterisation such as K, T-stress and crack tip opening angle (CTOA) measurements as well as data obtained from 3D measurements of a propagating crack.

The Higher Order Asymptotic Fields for a Anti-Plane Crack in a Linear Functionally Gradient Material

The exponential and power material functions are often applied to functionally gradient materials (FGM). Obviously, it is of fundamental significance to study FGM with arbitrary material function. Because an arbitrary function can be treated as finite linear segments approximately, it is essential to research FGM with a linear material function. Crack-tip higher order stress and displacement fields for an anti-plane crack perpendicular to the direction of property variation in a FGM with a linear shear modulus along the gradient direction are obtained through the asymptotic analysis. The asymptotic expansions of crack tip stress fields bring out explicitly the influence of non-homogeneity on the structure of the stress field. The analysis reveals that only the higher order terms in the expansion are influenced by the material non-homogeneity. Moreover, it can be seen from expressions of higher order stress fields that at least three terms must be considered in the case of FGM in order to explicitly and theoretically account for non-homogeneity effects on crack tip stress fields.

Analysis of a transiently propagating crack in functionally graded materials under mode I and II

Crack tip stress and displacement fields for a transiently propagating crack along gradient in functionally graded materials (FGMs) with a linear variation of shear modulus are developed. The higher order terms of the transient stress and displacement fields at crack tip were obtained by transforming the general partial differential equations of the dynamic equilibrium into Laplace’s equations whose solutions have harmonic functions. Thus, the fields can be expressed very simply. Using these stress components, isochromatics and the first invariant at crack tip are generated. The results show that the isochromatics (constant maximum shear stress) for mode I crack tilt backward around the crack tip with an increase of crack tip acceleration c ˙ ( d c / d t ) , and tilt forward around the crack tip with an increase of rate of change of dynamic mode I stress intensity factor K ˙ I ( d K I / d t ) . The isochromatics for mixed mode crack move to upper direction with an increases of K ˙ I and K ˙ II , and lower direction with an increase of c ˙ . Contours of the first stress invariant for mode I crack enlarge around the crack tip with an increase of c ˙ , and decrease around the crack tip with an increase of K ˙ I . As K ˙ I ( II ) decreases at crack initiation, the predicted kinking angles increase. As c ˙ increases, the predicted kinking angles also increase.

Characteristics of a transiently propagating crack in functionally graded materials

When a crack propagates with acceleration, deceleration and time rates of change of stress intensity factors, it is very important for us to understand the effects of acceleration, deceleration and time rates of change of stress intensity factors on the individual stresses and displacements at the crack tip. Therefore, the crack tip stress and displacement fields for a transiently propagating crack along gradient in functionally graded materials (FGMs) with an exponential variation of shear modulus and density are developed and the characteristics of a transiently propagating crack from the fields are analyzed. The effects of the rate of change of the stress intensity factor and the crack tip acceleration on the individual stresses at the crack tip are opposite each other. Specially, the isochromatics (constant maximum shear stress) of Mode I tilt backward around the crack tip with an increase of crack tip acceleration, and tilt forward around the crack tip with an increase of the rate of change of the dynamic mode I stress intensity factor.

Cohesive-zone laws for void growth — I. Experimental field projection of crack-tip crazing in glassy polymers

A hybrid framework for inverse analysis of crack-tip cohesive-zone model is developed in this two-part paper to measure cohesive-zone laws of void growth in polymers by combining analytical, experimental, and numerical approaches. This paper focuses on experimental measurements of the cohesive-zone laws for two nonlinear fracture processes in glassy polymers, namely multiple crazing in crack-growth toughening of rubber-toughened high-impact polystyrene (HIPS) and crazing of steady-state crack growth in polymethylmethacrylate (PMMA) under a methanol environment. To this end, electronic speckle pattern interferometry (ESPI) is first applied to measure the crack-tip displacement fields surrounding the fracture process zones in these polymers. These fields are subsequently equilibrium smoothed and used in the extraction of the cohesive-zone laws via an analytical solution method of the inverse problem, the planar field projection method (P-FPM) [Hong, S., Kim, K.-S., 2003. Extraction of cohesive-zone laws from elastic far-fields of a cohesive crack tip: a field projection method. Journal of the Mechanics and Physics of Solids 51, 1267–1286]. Results show that the proposed framework of the P-FPM could provide a systematic way of finding the shape of the cohesive-zone laws governed by the different micro-mechanisms in the fracture processes. In HIPS, inter-particle multiple crazing develops and the craze zone broadens ahead of a crack-tip under mechanical loading. The corresponding cohesive-zone relationship of the multiple-craze zone is found to be highly convex, which indicates effectiveness of rubber particle toughening. It is also observed that the effective peak traction, 7 MPa, in the crack-tip cohesive zone of HIPS (30% rubber content) is lower than the uniaxial yield stress of 9 MPa, presumably due to stress multi-axiality effects. In contrast, in PMMA, methanol localizes the crack-tip craze, weakening the craze traction for craze-void initiation to about 9 MPa and the fibril pull-out stress to less than 6 MPa. This reduction in cohesive traction, coupled with a strongly concave traction–separation cohesive-zone relationship, signifies environmental embrittlement of PMMA. These experimentally determined cohesive-zone laws are compared with detailed numerical analyses of effective microscale-void growth ahead of a crack tip in Part II.

Experimental and numerical investigation of a cracked transversely graded plate subjected to in plane bending

The variation of stress intensity factor along the thickness in a cracked transversely graded plate subjected to in plane bending is investigated in this study. A transversely graded plate having elastic modulus varying continuously along the thickness was prepared by embedding glass beads in epoxy resin. An edge crack in this plate was subjected to in plane bending and the crack tip displacement field on the surfaces of the plate was measured using digital image correlation (DIC). Using the recently reported asymptotic displacement fields for cracked transversely graded plates (Wadgaonkar, S.C., Parameswaran, V., 2009. Structure of near tip stress field and variation of stress intensity factor for a crack in a transversely graded material, Journal of Applied Mechanics 76 (1), 011014), the stress intensity factor (SIF) on the surfaces of the plate was calculated from the experimental data. The results of this part of the study indicated that the extent of variation of the SIF across the plate thickness is nearly the same as that of the elastic modulus. An expression to calculate the variation of the SIF through the plate thickness was developed assuming simple bending of the plate. The predicted variation of SIF was verified through finite element calculations. Further, the behavior of the SIF near the intersection of the crack front and the plate surfaces, the extent of dominance of the corner singular field and the exponent of the corner singularity were also investigated in detail. Finally, the effect of gradation strength and gradation type on the SIF was also studied.

Real-time acquisition of fatigue crack images for monitoring crack-tip stress intensity variations within fatigue cycles

Digital images of fatigue crack behaviour have been acquired in real time at high cycle fatigue rates (77 Hz) using a high-speed camera at 1000 frames/s. Digital image correlation has then been used to determine the crack-tip position and stress intensity variations ( KI and KII) within selected cycles. This has been achieved for a pre-cracked aluminium compact tension (CT) specimen subjected to constant load amplitude fatigue crack cycling. The crack-tip displacement field has been determined at 14 points within each cycle. In this proof of concept study, despite noise in the inferred displacement fields, by least squares fitting the displacement field rather than the strain field to the Muskhelishvili's form crack-tip stress field, the crack-tip stress intensity inferred from the measured crack-tip displacement field was found to good accuracy (around 0.2 MPa m1/2). Furthermore, the observed sinusoidal variation was in excellent agreement with the nominal Δ K obtained from the applied fatigue amplitude confirming the accuracy of the method; a drift of around 0.42 MPa m1/2 was observed in Kmean from cycle to cycle. No evidence of closure was observed at low K. In principle this method permits the identification of crack-tip closure and crack growth, as well as the application of constant Δ Keff cycling, through intermittent monitoring throughout the duration of a fatigue test. The current method is well suited to more complex cases where crack closure, residual stresses, or more complex geometries mean that it is difficult to predict Δ Keff a priori and a number of potential applications of the method are identified.

Determination of Stress Intensity Factor for Cracks in Orthotropic Composite Materials using Digital Image Correlation

Abstract: This paper focuses on the application of the digital image correlation (DIC) technique to determine the stress intensity factor (SIF) for cracks in orthotropic composites. DIC is a full‐field technique for measuring the surface displacements of a deforming object and can be applied to any type of material. To determine the SIF from full‐field displacement data, the asymptotic expansion of the crack‐tip displacement field is required. In this paper the expansion of the crack tip displacement field is derived from an existing solution for strain fields. Unidirectional fibre composite panels with an edge crack aligned along the fibre were tested under remote tensile loading and the displacements were recorded using DIC. The SIF was calculated from the experimental data by fitting the theoretical displacement field using the least squares method. The SIF thus determined was in good agreement with theoretical results and therefore demonstrates the applicability of the derived displacement field and DIC technique for studying fracture in composites.