Maximum Hoop Stress Criterion Research Articles

An efficient moving-mesh strategy for predicting crack propagation in unidirectional composites: Application to materials reinforced with aligned CNTs

This paper presents an efficient numerical approach for reproducing the process of crack propagation inside unidirectional composites subjected to general loading conditions, with special reference to epoxy materials enhanced with embedded aligned CNTs. This approach involves a traditional FE framework improved by the Moving Mesh (MM) technique based on the Arbitrary Lagrangian-Eulerian (ALE) formulation and the Interaction Integral Method (M−integral). The MM serves as a powerful numerical tool to simulate the discrete crack advance with minimal remeshing, thus reducing the computational complexities. Instead, the M−Integral method, formulated for generally anisotropic materials, has been employed to extract the mixed-mode Stress Intensity Factors (SIFs), which are necessary to define the crack onset condition and propagation direction on the basis of the modified Maximum Hoop Stress Criterion. The proposed strategy includes the extended rule of mixtures to evaluate the homogenized elastic properties of nano-reinforced composites. The validity of the proposed methodology has been assessed through comparisons with experimental data and numerical results available in the literature.

Thermal and fracture analysis of collinear interface cracks in graded coating systems under ramp-type heating

Ceramic based thermal barrier coatings are extensively employed to shield critical engineering components from harsh temperature environments, where the typical Fourier's law may not be able to provide an accurate assessment of the physical process. This work adopts a more generalized heat conduction approach to reveal the thermal and fracture interaction mechanism of two collinear interface cracks in orthotropic functionally graded thermal coating systems by dual-phase-lag theory. The heating rate of the imposed thermal loading on the exterior bounding surface is modeled by ramp-type function, where the ramping period is comparable to the certain time spent for building the local thermal equilibrium within composites. The methods of Fourier transform and Laplace transform in conjunction with the singular integral equations are applied to solve the complex transient thermoelastic problem. Parametric investigations are conducted to investigate the impacts of ramping time, thermal lags, nonhomogeneous parameters, and crack spacing on the thermal and fracture characteristics. According to the maximum hoop stress criterion, higher fracture risk is discovered with the approaching of collinear interface cracks.

Dynamically propagating cracks in anisotropic plates subjected to hyperbolic thermal shock

ABSTRACT In this paper, dynamically propagating cracks in anisotropic plates exposed to a generalised thermal shock are investigated. Here, the governing equations are considered according to the Lord-Shulman (LS) model as a fully coupled generalised thermoelasticity theory. The crack is modelled in the context of the eXtended Finite Element Method (XFEM). To evaluate Stress Intensity Factors (SIFs), a technique based on the J integral and crack tip opening and sliding displacements, previously applied for stationary cracks, is developed for a dynamically propagating crack in orthotropic solids. Besides, a variety of functions for enriching the displacement field are derived relying on the behaviour near the tip of a dynamically propagating crack in anisotropic materials. The maximum hoop stress criterion is also modified by using the asymptotic stress field for a dynamically propagating crack. In addition, the effect of the angular variation of fracture toughness in orthotropic materials is accounted in computing the crack propagation speed. In some examples, dynamic crack growth in complex cracked orthotropic structures subjected to LS thermal shock is studied in detail. The impacts of fibre orientations and LS relaxation time are also investigated in these examples. In conclusion, applying non-Fourier thermal shock for modelling dynamically propagating cracks in orthotropic structures is recommended to obtain more realistic results. In addition, by increasing the LS relaxation time, the crack grows less oscillatory and is more aligned along the material fibres.

Investigation of the thermomechanical behavior of nonhomogeneous, layered materials with mixed-mode cracks subjected to rapid thermal shock loading

For mixed-mode, thermal shock crack problems with thermal lagging behaviors, it is difficult to obtain the transient thermal stress intensity factors (TSIFs). This paper aims to develop a method combining the Newmark method and the transient interaction energy integral method (IEIM) to extract the transient, mixed-mode TSIFs of nonhomogeneous, layered materials with mixed-mode cracks. The transient temperature field obtained by non-Fourier heat conduction theory is discretized in the spatial and temporal domains through the finite element method (FEM) and Newmark method, respectively. The effect of the continuity of thermomechanical parameters at the dual interface on the transient, mixed-mode TSIFs of nonhomogeneous, layered materials is discussed, then the fracture resistance of nonhomogeneous, layered materials is evaluated. Moreover, the maximum hoop stress criterion is utilized to obtain the crack growth angle and the equivalent, transient TSIFs of nonhomogeneous, layered materials. The influences of the interface on crack growth angle and equivalent transient TSIFs of nonhomogeneous layered materials are also studied. The present work is beneficial for the thermal design of nonhomogeneous, layered materials with mixed-mode cracks subjected to rapid thermal shock loading.

Criteria for crack path deviation in adhesive layer of bi-material DCB specimen

An alternative to traditional fracture mechanics methodology to predict direction for crack propagation in the adhesive layer of bonded stiff materials is demonstrated. The approach is based on the analysis of the location of maximum of the hoop stress in relation to the existing crack tip. Such method is very convenient and fast as it does not require a lot of computational resources and is easy to implement compared to other known numerical methods dealing with similar problems (e.g. X-FEM). The method is validated by fracture mechanics approach using energy release rate to predict crack propagation direction. The verification is done by using bi-material DCB specimen with relatively thick adhesive layer as an example.After proving the applicability of the maximum hoop stress criterion the parametric study on factors affecting crack propagation in the adhesive layer is carried out. Such parameters as bending stiffness of beams, thickness of the adhesive layer, distance to the bond-line, length of the initial pre-crack are analyzed.

Karışık Modlu Yorulma Çatlak Büyümesi Olayının Yeni Bir Hesaplamalı Yöntem ile İncelenmesi

Analysis of 3-D fatigue crack growth problems with mixed-mode loading has always been an interesting area in the field of fracture mechanics. Fracture failure under the influence of fatigue loading has been a common experience for various industries’ products, such as aerospace and automotive components. Any possible failure in these structures can result in high damage to these components or a serious risk for people’s health. The analysis of such failures may involve great challenges and complexities for obtaining the accurate solution. The complexities of the problem may not only be related to the loading type, but also to the specific geometry itself. Such problems are hard and costly to analyze with experimental methods. Therefore, it is important to establish the theoretical aspects of the process initially, and then having a computational procedure to solve the problem at hand. The crack growth law used in this procedure is NASGRO-type, and determination of the propagation angle is based on the maximum hoop stress criterion. Hypermesh and ANSYS APDL software are benefited during preprocessing of the propagation steps and application of submodeling procedure. Solution of the problem is performed with FRAC3D program, its enriched element methodology and newly implemented tools for crack growth. A specific example that includes cracking within an aircraft engine compressor blade is shown for demonstration purpose.

Effects of stop hole on crack turning, residual fatigue life and crack tip stress field

The aim of this paper is to research the effects of stop hole on crack turning, residual fatigue life and crack tip stress field, in which the location, size and shape of stop hole are considered. The numerical model is established according to M-integral technology, and crack turning angle is determined by maximum hoop stress criterion. The calculation results show that stop hole location has a significant effect on crack turning angle and residual fatigue life, and it should be drilled with a reasonable distance in vertical and horizontal direction around crack tip to achieve a better residual fatigue life retardation effect. The larger the stop hole size is, the wider effect range of stop hole on crack turning and residual fatigue life will be. The effects of elliptical stop hole on crack turning angle and residual fatigue life retardation are greater than that of round hole when the distance in vertical direction increases. Crack tip stress distribution is changed by increasing σy and decreasing σx, and the most dramatical changes occur at the circumferential direction of 90° and 270°.

Crack deflection in short fiber reinforced composites: Theory and experiment

AbstractIn the past decades different crack growth and crack deflection criteria have been proposed, e.g. the maximum hoop stress criterion [1], the minimum strain energy density criterion [2] or the maximum energy release rate criterion [3]. These criteria were extended concerning phenomena such as crack path instability [4,5] and the anisotropy of fracture parameters [4]. Another approach for the prediction of crack growth and deflection angle is based on the Jk‐integral [6], assuming the crack to grow into the direction of the J‐vector [7]. The resulting deflection angles coincide well with those from other criteria in the case of small mixed‐mode ratios KII/KI or deflection angles, respectively. One challenge is the accurate calculation of the second coordinate J2 [8] and the appropriateness of this approach has been shown [9]. Nevertheless, in the cases of anisotropic crack resistance parameters, large mode‐II loading or unstable crack paths, the classical J‐vector criterion fails.In this paper, an extended J‐vector criterion is introduced providing the crack growth direction in the case of anisotropic fracture parameters and elastic constants [9,10]. The theory is validated by experimental results for different anisotropy ratios and orientations.

Thermal shock resistance of functionally graded materials with mixed-mode cracks

The thermal shock resistance of functionally graded materials (FGMs) is studied in the presence of periodic mixed-mode cracks as characterization of a typical damaging effect of thermal shock loading on FGMs. A methodology of assessing the thermal shock resistance of FGMs is proposed featuring consideration of both strength and fracture theories. The thermal shock resistance is modeled through three distinct failure criteria: (i) the maximum local tensile stress criterion (ii) the maximum thermal stress intensity factor criterion and (iii) the maximum hoop stress criterion. The finite-difference approximation of the transient temperature field is obtained firstly according to Fourier's law of heat conduction. The corresponding thermal stress field is then obtained through the finite element analysis of the mechanical problem involving thermal strain. With such a coupling between the finite element method (FEM) and finite difference method (FDM) resolving the thermomechanical response, the mixed-mode thermal stress intensity factors (TSIFs) are accordingly extracted through the interaction energy integral method (IEIM). Some typical examples of periodic mixed-mode cracks are presented with the influences of crack spacing, crack length and crack angle on the thermal shock resistance of FGMs studied. Furthermore, the distribution patterns of material properties in FGMs are discussed in terms of their effects on the thermal shock resistance. The results indicate that the proposed methodology provides an effective pathway to assessment of thermal shock resistance of FGMs showing great significance in design and fracture evaluation of such materials.

On crack propagation in homogeneous and composite materials under mixed mode loading conditions

The direction of crack propagation under mixed mode loading conditions is investigated by FEM simulation in comparison with delamination experiments performed in the 4 point bending mode. Composites consisting of aluminum nitride attached to copper by active metal brazing were delaminated with use of a central notch for crack initiation. The main crack line propagated along the interface direction. However, SEM micrographs revealed that the crack line propagating along the interface is subjected to bifurcation. Small cracks digress from the main delamination line under kinking angles of 90° until they are stopped at the copper substrate. The observed behavior of crack propagation is studied by Linear Elastic Fracture Mechanics and by nonlinear Finite Element Analysis. In fact, the bifurcation angle of 90° can be considered as an indication for dominant influence of geometrical nonlinearities at the crack tip. Finite Element Analysis is performed in two stages starting from macroscopic level and then the analysis of crack tip behavior is continued on a microscopic level. On the macroscopic level, the whole experimental setup is simulated in order to derive an approximation of the stress distribution around the crack tip. Thereafter, the analysis is further extended on microscopic level, whereby local mesh refinement is continued until the element size approaches the dimension of the lattice constant. Thus, high stress values at the crack tip lead to considerable amounts of material rotation. Consequently, the observed direction of crack extension may be derived from the maximum hoop stress criterion.

Linear smoothed extended finite element method for fatigue crack growth simulations

In this paper, the recently proposed linear smoothed extended finite element method (LSmXFEM) is employed to simulate the fatigue crack growth. Unlike the conventional extended finite element method, the LSmXFEM does not require special numerical integration technique to integrate the terms in the stiffness matrix. The stress intensity factors (SIFs) are evaluated by using the domain form of the interaction integral technique. The fatigue crack growth rate is evaluated using the generalized Paris’ law in conjunction with the maximum hoop stress criterion. The robustness of the method is demonstrated with a few examples for which the results are available in the literature. Then, the fatigue crack growth from the numerical simulation is compared with the experimental investigations performed on CR5 grade cold formed steel. It is seen that the fatigue life and the crack path obtained from the proposed method is in close agreement with the experimental observation.

Symmetric Galerkin boundary element analysis of the interaction between multiple growing cracks in infinite domains

The use of the symmetric Galerkin boundary element method (SGBEM) for studying the quasi-static interaction between multiple growing micro-cracks is presented in this work. The micro-cracks can conveniently be modeled in infinite domains, and this type of analysis can be handled by the SGBEM in a straightforward manner. In fact, it reduces the size of the analysis due to the absence of a physical boundary. A quasi-static multi-crack growth model based upon the maximum hoop stress criterion (MHSC), and the SGBEM was developed in this work. An improved quarter-point crack-tip element and adjusted maximum crack increments were employed to enhance the accuracy and effectiveness of the crack growth prediction. The improved quarter-point element has been known for producing accurate stress intensity factors required by the MHSC, while the technique used to adjust the maximum crack increment at each iteration of crack growth simulations allows to achieve converged (accurate) crack extension paths even if a relatively large maximum crack increment is selected at the onset. Several numerical examples were presented to show the effectiveness of the proposed multi-crack growth model.

XEFGM for crack propagation analysis of functionally graded materials under mixed-mode and non-proportional loading

ABSTRACTIn the present study, a computational method based on the extended element free Galerkin method is adopted for crack propagation analysis of functionally graded materials under mixed-mode and non-proportional loading. The sub-triangulation technique for numerical integration, appropriate support domain and enrichment functions in the crack location are applied in the solution. The incompatible interaction integral method is used to evaluate the stress intensity factors. The maximum hoop stress criterion is adopted for crack propagation criterion. Good agreement and high convergence are observed in the simulation of crack initiation and crack growth direction along crack path between the proposed method results and the reference results available in the literature with lower degrees of freedom.

Simulation of dynamic and static thermoelastic fracture problems by extended nodal gradient finite elements

In this paper, the recent development of extended consecutive-interpolation 4-node quadrilateral element (XCQ4), which goes beyond certain limitations of conventional methods, is further employed to study dynamic and static thermoelastic fracture problems. In particular, static stress intensity factors (SIFs) and transient dynamic responses of two-dimensional (2-D) stationary cracks in elastic solids under thermal and thermal-mechanical loadings are investigated. In addition, simulation of quasi-static crack propagation under thermal-mechanical loading condition is also performed. To precisely capture the singularity and discontinuity of temperature, heat flux, and displacements due to the presence of crack, the unknown field variables of mechanical displacements and temperature are approximated in terms of the XCQ4 framework by different enrichment strategies, which include either the use of the conventional branch functions or the new ramp function associated with Heaviside function. Only adiabatic condition is considered on the crack surfaces. For quasi-static crack growth analysis, the maximum hoop stress criterion, which is also valid for thermo-elastic fracture mechanics, is adopted. For transient crack analyses, the inertial effect is taken into account for the interaction integral in the consideration of DSIFs. Through eight numerical examples for thermoelastic fracture problems, the validity of the XCQ4 is confirmed by a comparison of the obtained results to those derived from different approaches in previous studies. In addition, the accuracy, performance, and superior advantages of the XCQ4 over standard extended finite element approach are also demonstrated.

Fatigue crack propagation in heterogeneous materials under remote cyclic loading

A fatigue model for the analysis of crack propagation in heterogeneous materials is developed. The crack propagation is driven by cyclic loading and affected by the presence of neighboring inclusions which can have initial eigenstrains such as thermal strain and misfit strain. The static Semi-analytic solution obtained by Zhou et al. [2011a] is utilized to obtain the full stress filed and the stress intensity factors. In order to simulate the crack propagation in heterogeneous materials, the maximum hoop stress criterion is applied to predict crack propagation directions and the Paris-type law is employed to analyze the crack fatigue. Meanwhile, a zigzag crack path consisting of many small vertical and horizontal cracks models an arbitrary crack propagation path. The results show that material dissimilarity between the inclusion and matrix and the magnitude of cyclic loading could greatly influence the behaviors of crack propagation.

A Simple and Efficient X-FEM Approach for Non-planar Fatigue Crack Propagation

A simple and efficient extended finite element method (XFEM) approach has been presented to solve the 3-D fatigue crack propagation problems. In X-FEM, the crack is approximately described by local signed distances of the nodes around the crack face which makes it possible to simulate crack propagation on a fixed mesh without remeshing. In this work, a triangulation scheme is adopted to initialize and update the crack which enables an easy level-set representation for an arbitrary shaped or non-planar crack. The level-set functions are used to search the elements that have been fully or partly cut by crack face and will be enriched with either Heaviside function or singularity function. Furthermore, the level-set functions are used to create the local coordinate systems for crack front points which serve as the basis for denoting the singular field. The 3-D interaction integral method is adopted to calculate the stress intensity factors. The maximum principle hoop stress criterion is adopted to determine the crack propagation direction. The Paris law is used to perform fatigue crack propagation simulation. Some examples of planar and non-planar 3-D crack growth are solved to demonstrate the applicability and robustness of the proposed XFEM approach.

A local mesh replacement method for modeling near-interfacial crack growth in 2D composite structures

An extended-finite-element-based method is proposed to accommodate the arbitrary motion of a crack in a general two-dimensional domain containing different kinds of material interfaces. To obtain the accurate stress intensity factors (SIFs) when the crack tip approaches the interface, the interpolation method in the vicinity of the crack tip employed in the extended finite element method (XFEM) is replaced with one that is derived from a moving mesh patch. Mesh configurations in this patch are the same as that adopted in the finite element method (FEM) for crack problems. The boundary of the patch is required to be coincident with background-mesh element edges and only the patch mesh works during the computation. As a result, the major advantages of the XFEM for modeling crack growth are preserved. The simulations are accomplished using a new domain expression of an interaction integral for evaluating stress intensity factors, and the maximum hoop stress criterion for crack-growth direction prediction. Several numerical examples are presented to prove the capability and practicability of the proposed technique and the program.

Dynamic crack growth modeling technique based upon the SGBEM in the Laplace domain

This paper presents a new dynamic crack growth prediction tool based upon the symmetric Galerkin boundary element method (SGBEM) in the Laplace domain, combined with the maximum hoop stress (MHS) criterion. The fast Laplace inverse transform developed by Durbin is employed to obtain time-domain solutions of the stress intensity factors required by the MHS criterion. In this work, the proposed Laplace SGBEM tool is used to predict crack trajectories in homogeneous and non-homogeneous isotropic brittle materials under dynamic loading conditions. The effect of the elastic constant mismatches is investigated for the influence of crack–inclusion interaction on crack growth.

A DIC-based study of in-plane mechanical response and fracture of orthotropic carbon fiber reinforced composite

The in-plane elastic properties and quasi-static fracture response of an orthotropically woven carbon fiber reinforced composite are investigated using 3D digital image correlation. The elastic properties are determined as a function of fiber orientation, and the effect of fiber angles on the crack extension direction is investigated. The modified maximum hoop stress criterion is used to predict the crack extension angle of the examined material. The predicted angles are in very good agreement with those found experimentally. Optical observations and local strain data from the quasi-static fracture studies reveal that crack initiation occurs well before the maximum far-field load is reached.

An experimental/numerical investigation into the main driving force for crack propagation in uni-directional fibre-reinforced composite laminae

This paper presents an enriched finite element method to simulate the growth of cracks in linear elastic, aerospace composite materials. The model and its discretisation are also validated through a complete experimental test series. Stress intensity factors are calculated by means of an interaction integral. To enable this, we propose application of (1) a modified approach to the standard interaction integral for heterogeneous orthotropic materials where material interfaces are present; (2) a modified maximum hoop stress criterion is proposed for obtaining the crack propagation direction at each step, and we show that the “standard” maximum hoop stress criterion which had been frequently used to date in literature, is unable to reproduce experimental results. The influence of crack description, material orientation along with the presence of holes and multi-material structures are investigated. It is found, for aerospace composite materials with E1E2 ratios of approximately 10, that the material orientation is the driving factor in crack propagation. This is found even for specimens with a material orientation of 90°, which were previously found to cause difficulty in both damage mechanics and discrete crack models e.g. by the extended finite element method (XFEM). The results also show the crack will predominantly propagate along the fibre direction, regardless of the specimen geometry, loading conditions or presence of voids.