A computational strategy for damage‐tolerant design of hollow shafts under mixed‐mode loading condition

Three‐dimensional numerical analyses, using the finite element method (FEM), have been adopted to simulate fatigue crack propagation in a hollow cylindrical specimen, under pure axial or combined axial‐torsion loading conditions. Specimens, made of Al alloys B95AT and D16T, have been experimentally tested under pure axial load and combined in‐phase constant amplitude axial and torsional loadings. The stress intensity factors (SIFs) have been calculated, according to the J‐integral approach, along the front of a part through crack, initiated in correspondence of the outer surface of a hollow cylindrical specimen. The crack path is evaluated by using the maximum energy release rate (MERR) criterion, whereas the Paris law is used to calculate crack growth rates. A numerical and experimental comparison of the results is presented, showing a good agreement in terms of crack growth rates and paths.     

A hybrid approach to determine fracture resistance for mode I and mixed‐mode I and II fracture specimens

This paper proposes a hybrid approach to determine the fracture resistance for mode I and mixed‐mode I and II fracture specimens, combining both numerically computed and experimentally measured load (P) versus load‐line displacement (LLD or Δ) relationships for metallic fracture specimens. The hybrid approach predicates on the same principle as the conventional, multiple‐specimen experimental method in determining the energy release rate. The hybrid method computes the P–Δ curves from multiple finite element (FE) models, each with a different crack depth. The experimental procedure measures the P–Δ curve from a standard fracture specimen with a growing crack. The intersections between the experimental P–Δ curve and the numerical P–Δ curves from multiple FE models dictate the LLD levels to compute the strain energy (U) using the area under the numerical P–Δ curves. This method provides accurate estimates of the J resistance data for both SE(B) specimen under mode I loading and single‐edge notched specimens under mixed‐mode I and II loading.     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

Experimental studies on mixed‐mode dynamic fracture behaviour of aluminium alloy plates with narrow U‐notch

Mixed‐mode dynamic fracture behaviour of cast aluminium alloy ZL205A thin plates with narrow U‐notch was studied by split Hopkinson tensile bar apparatus. Specimens with different loading angles were designed to realize different fracture modes. The same loading condition was maintained during the tests. Recovery specimens show that crack propagates along the notch direction. Force–elongation relations show that with the loading angle increasing, the fracture force increases while the final elongation decreases. Deformation and fracture process was observed by a high‐speed camera. Displacement distribution around the crack was calculated through digital image correlation technique. Based on the photos and displacement results, initiation time of the crack was derived. Besides, two stress components (normal stress and shear stress) applied on the fracture surface were investigated. Results show that crack initiation stresses at different loading angles satisfy the ellipse equation. Pure mode I and II fracture stresses are 425.3 and 236.7 MPa, respectively. Furthermore, specific fracture energy of different specimens was calculated. The energy data vary with loading angle and located on an approximate upward parabolic curve. From the curve, the minimum specific fracture energy of the thin plate specimen is 42.0 kJ/m² under loading angle of 76.3°.     

Numerical and experimental investigation of mixed‐mode fracture parameters on silicon nitride using the Brazilian disc test

Engineering applications of ceramics can often involve mixed‐mode conditions involving both tensile and shear loading. Mixed‐mode fracture toughness parameters are evaluated for applicability to ceramics using the Brazilian disc test on silicon nitride. Semi‐elliptical centrally located surface flaws are induced on the disc specimens using Vickers indentation and compression loaded to fracture with varying levels of mode mixity. The disc specimens are modelled via 3D finite element analysis and all three modes of stress intensity factors computed along the crack front, at failure load. We present a numerical and experimental investigation of four widely used mixed‐mode fracture criteria and conclude that the critical strain energy release rate criterion is simple to implement and effective for silicon nitride under mixed‐mode conditions.     

Error estimation of stress intensity factors for mixed‐mode cracks

This paper presents an a posteriori error estimator for mixed‐mode stress intensity factors in plane linear elasticity. A surface integral over an arbitrary crown is used for the separate calculation of the combined mode's stress intensity factors. The error in the quantity of interest is based on goal‐oriented error measures and estimated through an error in the constitutive relation. Copyright © 2006 John Wiley & Sons, Ltd.     

A direct analytical method to extract mixed‐mode components of strain energy release rates from Irwin's integral using extended finite element method

A new analytical approach, within the extended finite element framework, is proposed to compute mixed‐mode components of strain energy release rates directly from Irwin's integral. Crack tip enrichment functions in extended FEM allow for evaluation of integral quantities in closed form (for some crack configurations studied) and therefore resulting in a simple and accurate method. Several benchmark examples on pure and mixed‐mode problems are studied. In particular, we analyze the effects of high‐order enrichments, mesh refinement, and the integration limits of Irwin's integral. The results indicate that high‐order enrichment functions have significant effect on the convergence, in particular when the integral limits are finite. When the integral limits tend to zero, simpler strain energy release rate expressions are obtained, and high‐order terms vanish. Nonetheless, these terms contribute indirectly via coefficients of first‐order terms. The numerical results show that high accuracy can be achieved with high‐order enrichment terms and mesh refinement. However, the effect of the integral limits remains an open question, with finite integration intervals chosen as h ∕ 2 tending to give more accurate results. Copyright © 2013 John Wiley & Sons, Ltd.     

Influence of shear parameters on mixed–mode fracture of concrete

The influence of the mode II fracture parameters on the mixed mode fracture experimental tests of quasibrittle materials is studied. The study is based on experimental results and numerical analyses. For the numerical study, a procedure for mixed mode fracture of quasibrittle materials is presented. The numerical procedure is based on the cohesive crack approach, and extends it to mixed mode fracture. Four experimental sets of mixed mode fracture were modelled, one from Arrea and Ingraffea and another from a nonproportional loading by the authors, both with bending concrete beams. Two other sets of experimental fracture were modelled, based on double-edge notched testing; in these tests an important mode II is beforehand expected. The numerical results agree quite well with experimental records. The influence of the main parameters for mode II fracture on the mixed mode fracture is studied for the four experimental set of tests and compared with these results. In all them, large changes in the mode II fracture energy hardly modify the numerical results. The tangential and normal stresses along the crack path during the loading proccess are obtained, also with different values of the mode II fracture energy. For the studied experimental tests it is concluded that the crack is initiated under mixed mode but propagated under predominant mode I. This allows a development of mixed mode fracture models, mainly based on standard properties of the material measured by standard methods, avoiding the problems associated with the measurement of mode II fracture parameters, such as mode II fracture energy and cohesion.     

Experimental and numerical investigation of the effect of temperature on mixed‐mode fracture behaviour of AM60 Mg alloy

The aim of present paper is to experimentally investigate mixed‐mode fracture behaviour of AM60 Mg alloy at low and elevated temperatures. For this purpose, mode I, 45° mixed‐mode, and mode II tests were conducted using a modified version of Arcan device at three different temperatures. An elastic‐plastic finite element model was used to extract necessary geometric parameters. Crack resistance curves (J‐R) and critical J‐integral of the material were extracted. The results indicated that, for all loading modes, maximum critical J‐integral value was observed at ambient temperature and decreased by either increasing or decreasing the temperature. It was observed that effect of temperature on fracture behaviour is much larger at temperatures above 0°C rather than sub‐zero temperatures. By changing the loading angle to go from mode I to mode II, a decreasing trend was observed in the values of critical fracture parameters at all temperatures. Finally, the surfaces were examined using scanning electron microscopy (SEM).     

A High‐order extended finite element method for extraction of mixed‐mode strain energy release rates in arbitrary crack settings based on Irwin's integral

An analytical formulation based on Irwin's integral and combined with the extended finite element method is proposed to extract mixed‐mode components of strain energy release rates in linear elastic fracture mechanics. The proposed formulation extends our previous work to cracks in arbitrary orientations and is therefore suited for crack propagation problems. In essence, the approach employs high‐order enrichment functions and evaluates Irwin's integral in closed form, once the linear system is solved and the algebraic degrees of freedom are determined. Several benchmark examples are investigated including off‐center cracks, inclined cracks, and two crack growth problems. On all these problems, the method is shown to work well, giving accurate results. Moreover, because of its analytical nature, no special post‐processing is required. Thus, we conclude that this method may provide a good and simple alternative to the popular J‐integral method. In addition, it may circumvent some of the limitations of the J‐integral in 3D modeling and in problems involving branching and coalescence of cracks. Copyright © 2013 John Wiley & Sons, Ltd.     

Phase‐field material point method for brittle fracture

The material point method for the analysis of deformable bodies is revisited and originally upgraded to simulate crack propagation in brittle media. In this setting, phase‐field modelling is introduced to resolve the crack path geometry. Following a particle in cell approach, the coupled continuum/phase‐field governing equations are defined at a set of material points and interpolated at the nodal points of an Eulerian, ie, non‐evolving, mesh. The accuracy of the simulated crack path is thus decoupled from the quality of the underlying finite element mesh and relieved from corresponding mesh‐distortion errors. A staggered incremental procedure is implemented for the solution of the discrete coupled governing equations of the phase‐field brittle fracture problem. The proposed method is verified through a series of benchmark tests while comparisons are made between the proposed scheme, the corresponding finite element implementation, and experimental results.     

Tangential strain‐based criteria for mixed‐mode I/II fracture toughness of cement concrete

Experimental and theoretical works are performed on the mixed‐mode I/II brittle fracture of cement concrete tested by edge cracked semicircular bend specimens. Theoretical background of the traditional fracture criteria including strain energy density, maximum tangential stress, and maximum tangential strain (MTSN) are introduced. The ability of each fracture criterion in prediction of the fracture test data is investigated. The comparison between the evaluations by the traditional criteria and the experimental data shows that none of them are capable of successfully estimating the fracture resistance of cement concrete. An enhanced version of the MTSN criterion is then employed to predict the test data. It is demonstrated that the extended MTSN criterion can successfully predict the test data in a higher accuracy than traditional criteria.     

An accurate scheme for mixed‐mode fracture analysis of functionally graded materials using the interaction integral and micromechanics models

The interaction integral is a conservation integral that relies on two admissible mechanical states for evaluating mixed‐mode stress intensity factors (SIFs). The present paper extends this integral to functionally graded materials in which the material properties are determined by means of either continuum functions (e.g. exponentially graded materials) or micromechanics models (e.g. self‐consistent, Mori–Tanaka, or three‐phase model). In the latter case, there is no closed‐form expression for the material‐property variation, and thus several quantities, such as the explicit derivative of the strain energy density, need to be evaluated numerically (this leads to several implications in the numerical implementation). The SIFs are determined using conservation integrals involving known auxiliary solutions. The choice of such auxiliary fields and their implications on the solution procedure are discussed in detail. The computational implementation is done using the finite element method and thus the interaction energy contour integral is converted to an equivalent domain integral over a finite region surrounding the crack tip. Several examples are given which show that the proposed method is convenient, accurate, and computationally efficient. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary element analysis of three‐dimensional mixed‐mode thermoelastic crack problems using the interaction and energy domain integrals

A three‐dimensional boundary element method (BEM) implementation of the interaction integral methodology for the numerical analysis of mixed‐mode three‐dimensional thermoelastic crack problems is presented in this paper. The interaction integral is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains, temperatures and derivatives of displacements and temperatures at internal points can be evaluated using the appropriate boundary integral equations. Several examples are analysed and the results compared with those available in the literature to demonstrate the efficiency and accuracy of the implementation to solve straight and curved crack‐front problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Modelling the fracture of concrete under mixed loading

A simple and efficient numerical procedure for mixed mode fracture of quasibrittle materials is shown: This technique predicts crack trajectories as well as load-displacement or load-CMOD responses. The model is based on the cohesive crack concept and uses the local mode I approach. Numerical results agree quite well with three experimental sets of mixed mode fracture of concrete beams; one from Arrea and Ingraffea, another from García, Gettu and Carol and from a nonproportional loading by the authors. In constrast to more sophisticated models, this method offers two major advantages: it requires only material properties measured by standardized methods and it can easily be implemented with general multipurpose finite element codes.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

On the use of an anti‐symmetric four‐point bend specimen for mode II fracture experiments

The edge‐cracked beam specimen subjected to anti‐symmetric four‐point bend (ASFPB) loading has been conventionally used in the past for investigating the pure mode II fracture experiments in many engineering materials. However, it is shown through finite element analysis that the ASFPB specimen sometimes fails to produce pure mode II conditions. For anti‐symmetric loads applied close to the crack line, there are considerable effects from K_I and T‐stress in the ASFPB specimen. Pure mode II is provided only when the applied loads are sufficiently far from the crack plane.     

The maximum principle violations of the mixed‐hybrid finite‐element method applied to diffusion equations

The abundant literature of finite‐element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial–boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time‐dependent solutions for such problems during small time duration obtained by using a non‐conforming mixed‐hybrid finite‐element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite‐difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non‐physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial–boundary value problem will be presented. One of the propositions given to avoid any non‐physical oscillations is to use the mass‐lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd.