Vibration analysis of rotating systems with open and breathing cracks

This paper is concerned with vibration analysis of rotating systems containing cracks. The flexibility matrix of cracked element is calculated with modified integration limits which is more accurate than conventional methods. The effect of this modification on the coefficients of flexibility matrix is presented for a simple rotor system containing open crack. To model the crack breathing behavior, a new finite element approach is introduced and implemented. Then, the dynamic response of a rotor with a breathing crack is evaluated by using the frequency/time domain approach (short time Fourier transform). The ability of short time Fourier transform to detect small cracks is investigated and compared with the transient response. The results provide a possible basis for an on-line monitoring system.     

A strong discontinuity based adaptive refinement approach for the modeling of crack branching

In the current work, the physical phenomena of dynamic fracture of brittle materials involving crack growth, acceleration and consequent branching is simulated. The numerical modeling is based on the approach where the failure in the form of cracks or shear bands is modeled by a jump in the displacement field, the so called ‘strong discontinuity’. The finite element method is employed with this strong discontinuity approach where each finite element is capable of developing a strong discontinuity locally embedded into it. The focus in this work is on branching phenomena which is modeled by an adaptive refinement method by solving a new sub-boundary value problem represented by a finite element at the growing crack tip. The sub-boundary value problem is subjected to a certain kinematic constraint on the boundary in the form of a linear deformation constraint. An accurate resolution of the state of material at the branching crack tip is achieved which results in realistic dynamic fracture simulations. A comparison of resulting numerical simulations is provided with the experiment of dynamic fracture from the literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monitoring fatigue cracks and SIFs in plate structures solving inverse boundary value problems

In this work we present methods for the detection of cracks in plate and shell structures. In contrast to most of the common monitoring methods taking advantage of the reflection of surface waves at crack faces, the presented approach is based on the strain measured at different locations on the surface of the structure. This allows both the identification of crack position parameters, such as length, location and angles with respect to a reference coordinate system and the calculation of stress intensity factors (SIF). The solution of the direct problem is performed on the basis of the BFM (body force method) and the method of assembled point dislocations. The inverse problem is solved applying the PSO (particle swarm optimization) algorithm. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A boundary element analysis for stress intensity factors of multiple circular arc cracks in a plane elasticity plate

In this paper, a numerical approach for analyzing interacting multiple cracks in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by considering the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are given to illustrate that the numerical approach is very accurate for analyzing interacting multiple cracks in an infinite linear elastic media under remote uniform stresses. In addition, the displacement discontinuity method with crack-tip elements is used to analyze a multiple crack problem in a finite plate. It is found that the boundary element method is also very accurate for investigating interacting multiple cracks in a finite plate. Specially, a generalization of Bueckner’s principle and the displacement discontinuity method with crack-tip elements are used to analyze multiple circular arc crack problems in infinite plate in tension (including: Two Collinear Circular Arc Cracks, Three Collinear Circular Arc Cracks, Two Parallel Circular Arc Cracks, Three Parallel Circular Arc Cracks and Two Circular Arc Cracks) in a plane elasticity plate. Many results are given.     

A novel size independent symplectic analytical singular element for inclined crack terminating at bimaterial interface

Cracks often exist in composite structures, especially at the interface of two different materials. These cracks can significantly affect the load bearing capacity of the structure and lead to premature failure of the structure. In this paper, a novel element for modeling the singular stress state around the inclined interface crack which terminates at the interface is developed. This new singular element is derived based on the explicit form of the high order eigen solution which is, for the first time, determined by using a symplectic approach. The developed singular element is then applied in finite element analysis and the stress intensity factors (SIFs) for a number of crack configurations are derived. It has been concluded that composites with complex geometric configurations of inclined interface cracks can be accurately simulated by the developed method, according to comparison of the results against benchmarks. It has been found that the stiffness matrix of the proposed singular element is independent of the element size and the SIFs of the crack can be solved directly without any post-processing.     

Multiple crack fatigue growth modeling by displacement discontinuity method with crack-tip elements

This paper presents a numerical approach for modeling multiple crack fatigue growth in a plane elastic infinite plate. It involves a generation of Bueckner’s principle, a displacement discontinuity method with crack-tip elements (a boundary element method) proposed recently by the author and an extension of Paris’ law to a multiple crack problem under mixed-mode loading. Because of an intrinsic feature of the boundary element method, a general multiple crack growth problem can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is conveniently modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. Fatigue growth modeling of an inclined crack in an infinite plate under biaxial cyclic loads is taken into account to illustrate the effectiveness of the present numerical approach. As an example, the present numerical approach is used to study the fatigue growth of three parallel cracks with same length under uniaxial cyclic load. Many numerical results are given.     

Shape and topological sensitivity analysis in domains with cracks

The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.     

Singular integral equation method for 2D fracture analysis of orthotropic solids containing doubly periodic strip-like cracks on rectangular lattice arrays under longitudinal shear loading

The doubly periodic arrays of cracks represent an important mesoscopic model for analysis of the damage and fracture mechanics behaviors of materials. Here, in the framework of a continuously distributed dislocation model and singular integral equation approach, a highly accurate solution is proposed to describe the fracture behavior of orthotropic solids weakened by doubly periodic strip-like cracks on rectangular lattice arrays under a far-field longitudinal shear load. By fully comparing the current numerical results with known analytical and boundary element solutions, the high precision of the proposed solution is verified. Furthermore, the effects of periodic parameters and orthotropic parameter ratio on the stress intensity factor, crack tearing displacement, and effective shear modulus are studied, and an analytically polynomial estimation for the equivalent shear modulus is proposed in a certain range. The interaction distances among the vertical and horizontal periodic cracks are quite different, and their effects vary with the orthotropic parameter ratio. In addition, the dynamic problem is discussed briefly in the case where the material is subjected to harmonic longitudinal shear stress waves. Further work will continue the in-depth study of the dynamics problem of the doubly periodic arrays of cracks.     

The Method of Smooth Domains in the Equilibrium Problem for a Plate with a Crack

A free boundary problem is considered of the equilibrium of an elastic plate with a crack. We suppose that some boundary mutual nonpenetration conditions are given on the crack faces in the form of simultaneous equalities and inequalities. We suggest a new approach to posing the problem in a smooth domain although it was stated in a domain with cuts originally. We treat the constraints on the components of the displacement vector and stress tensor on the crack faces as interior constraints, i.e., constraints given on subsets of the smooth domain of a solution.     

Analysis of Calving Events in Antarctic Ice Shelves Using Configurational Forces

Previous studies on the sensitivity of cracks in ice shelves with different boundary conditions, stress states and density profiles revealed the need for further analyses. As the transfer of boundary conditions from dynamic ice flow simulations to the linear elastic fracture analyses proved to be a critical point in previous studies, a new approach to relate viscous and elastic material behaviour is proposed. The numerical simulations are conducted using Finite Elements utilizing the concept of configurational forces. To show the applicability of the approach, a 2-dimensional plane stress geometry with volume loads due to the ice shelf flow is analyzed. The resulting crack path is compared to available crack paths from satellite images. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The free-discontinuity problem of inverse crack identification from boundary measurements

The inverse crack identification of planar cracks from elastostatics boundary measurements is regarded as free-discontinuity problem in respect to the unknown displacement field and the discontinuity region of the cracked body. The proposed solution strategy is based on the variational approximation of the sharp interface problem by elliptic functionals developed by Ambrosio and Tortorelli. The numerical calculation is realized by the finite element method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asymptotic approach in problems of crack identification

An asymptotic approach to solving problems of the identification of a rectilinear crack of small relative size is presented. The solution of the direct problem is reduced to solving a boundary integral equation. Using the proposed approach, its kernel is investigated, and the main part of the asymptotic form is singled out. The inverse problem of determining the crack parameters from prescribed information on the amplitudes of the displacement on the boundary of a layer is solved. Transcendental equations are obtained, from which the characteristics of a crack are determined in stages. Numerical results of the solution of the inverse problem are presented.     

Optimal Control of Inclusion and Crack Shapes in Elastic Bodies

The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provide a mutual nonpenetration between crack faces. Inclusion and crack shapes are considered as control functions and control objectives, respectively. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry. We prove an existence of optimal solutions.     

A phase field model for fracture

The variational formulation of brittle fracture as formulated for example by Francfort and Marigo in [1], where the total energy is minimized with respect to any admissible crack set and displacement field, allows the identification of crack paths, branching of preexisting cracks and even crack initiation without additional criteria. For its numerical treatment a continuous approximation of the model in the sense of Γ-convergence has been presented by Bourdin in [2]. In the regularized Francfort–Marigo model cracks are represented by an additional field variable (secondary variable) s∈[0,1] which is 0 if the material is cracked and 1 if it is undamaged. In this work, we reinterpret the crack variable as a phase field order parameter and address cracking as a phase transition problem. The crack growth is governed by the evolution equation of the order parameter which resembles the Ginzburg–Landau equation. The numerical treatment is done by finite elements combined with an implicit Euler scheme for the time integration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals

Based on the Stroh-type formalism for anti-plane deformation, the fracture mechanics of four cracks originating from an elliptical hole in a one-dimensional hexagonal quasicrystal are investigated under remotely uniform anti-plane shear loadings. The boundary value problem is reduced to Cauchy integral equations by a new mapping function, which is further solved analytically. The exact solutions in closed-form of the stress intensity factors for mode III crack problem are obtained. In the limiting cases, the well known results can be obtained from the present solutions. Moreover, new exact solutions for some complicated defects including three edge cracks originating from an elliptical hole, a half-plane with an edge crack originating from a half-elliptical hole, a half-plane with an edge crack originating from a half-circular hole are derived. In the absence of the phason field, the obtainable results in this paper match with the classical ones.     

Distributionally robust parameter identification of a time-delay dynamical system with stochastic measurements

In this paper, we consider a parameter identification problem involving a time-delay dynamical system, in which the measured data are stochastic variable. However, the probability distribution of this stochastic variable is not available and the only information we have is its first moment. This problem is formulated as a distributionally robust parameter identification problem governed by a time-delay dynamical system. Using duality theory of linear optimization in a probability space, the distributionally robust parameter identification problem, which is a bi-level optimization problem, is transformed into a single-level optimization problem with a semi-infinite constraint. By applying problem transformation and smoothing techniques, the semi-infinite constraint is approximated by a smooth constraint and the convergence of the smooth approximation method is established. Then, the gradients of the cost and constraint functions with respect to time-delay and parameters are derived. On this basis, a gradient-based optimization method for solving the transformed problem is developed. Finally, we present an example, arising in practical fermentation process, to illustrate the applicability of the proposed method.     

Steady-state response of a geared rotor system with slant cracked shaft and time-varying mesh stiffness

The dynamic behavior of geared rotor system with defects is helpful for the failure diagnosis and state detecting of the system. Extensive efforts have been devoted to study the dynamic behaviors of geared systems with tooth root cracks. When surface cracks (especially for slant cracks) appear on the transmission shaft, the dynamic characteristics of the system have not gained sufficient attentions. Due to the parametric excitations induced by slant crack breathing and time-varying mesh stiffness, the steady-state response of the cracked geared rotor system differs distinctly from that of the uncracked system. Thus, utilizing the direct spectral method (DSM), the forced response spectra of a geared rotor system with slant cracked shaft and time-varying mesh stiffness under transmission error, unbalance force and torsional excitations are, respectively, obtained and discussed in detail. The effects of crack types (straight or slant crack) and crack depth on the forced response spectra of the system without and with torsional excitation are considered in the analysis. In addition, how the frequency response characteristics change after considering the crack is also investigated. It is shown that the torsional excitations have significant influence on the forced response spectra of slant cracked system. Sub-critical resonances are also found in the frequency response curves. The results could be used for shaft crack detection in geared rotor system.     

Approximate methods for analyzing the growth of large cracks in fatigue damaged aircraft sheet material and lap joints

Several levels of approximation are investigated to account for the effect of small fatigue cracks on the residual strength of aircraft sheet materials and fuselage lap joints containing major cracks. A version of the Dugdale model is proposed which accounts for strain hardening of the sheet in an approximate way and which incorporates a criterion for crack advance leading to crack growth resistance. This model builds upon the model proposed by Nilsson and Hutchinson [1] and accounts for the detailed interaction between the major crack and the small damage cracks. A simpler version of the model uses the damage-reduced local strength of the sheet or joint in assessing the effect of the major crack on residual strength. The simpler approach thus bypasses the necessity of a direct determination of the highly complicated details of the interaction of the small cracks in a lap joint with a major crack.     

Thermo-mechanical modeling of turbine blades taking into account structural defects

In turbine blades of aero-engines typical defects are cracks due to high mechanical and thermal loads. The extended finite element method (XFEM) is used for simulations of fracture mechanics problems with cracks. Discontinuities in the displacement and temperature field are allowed and the crack opening displacement and crack tip stress field are reproduced accurately. Since crack closure and non-physical penetration of the crack surfaces may occur under certain load conditions, it becomes necessary to enforce the non-penetration condition for crack surfaces. This contact formulation is assumed to be frictionless. The node-to-segment approach proposed in [3] is extended to ten-node tetrahedral elements with quadratic shape functions. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)