Computation of crack propagation in 3-dimensional anisotropic solids

This contribution presents ideas, how crack propagation in three-dimensional solids composed of anisotropic materials can be predicted using the Griffith energy principle. Since the work of Irwin the change of potential energy caused by a straight elongation of a crack in an isotropic two-dimensional homogeneous structure can be expressed in quadratic terms of the stress intensities at the crack tip. This result was generalized in the last decades using methods of asymptotic analysis by many authors [1] to more complicated geometries, to anisotropic and inhomogeneous materials. With the energy release rate at hand, quasi-static scenarios of crack propagation can be simulated for plane problems [2], but this is still a complicated task for three-dimensional problems [3]. We show an idea how the change of energy caused by propagation of a crack surface in a fully three-dimensional solid of nearly arbitrary shape can be computed in anisotropic materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermo-mechanical modeling of crack propagation in dynamically loaded elastomer specimens using a scaled boundary finite element approach

Recently, a scaled boundary finite element (SBFE) formulation for geometrically and physically nonlinear materials has been developed using the scaled boundary finite element method (SBFEM). The SBFE formulation has been employed to describe plane stress problems of notched and unnotched hyperelastic elastomer specimens. In this contribution, the derived SBFE formulation is extended to nonlinear time- and temperature-dependent material behavior. Subsequently, the SBFE formulation is incorporated into a crack propagation scheme to model crack propagation in cyclically loaded elastomer specimens of the so-called tear fatigue analyzer (TFA). (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Self-similar solution of a tensile crack problem in a coupled formulation

Using a self-similar variables, an asymptotic investigation is carried out into the stress fields and the rates of creep deformations and degree of damage close to the tip of a tensile crack under creep conditions in a coupled (creep - damage) plane formulation of the problem. It is shown that a domain of completely damaged material (DCDM) exists close to the crack tip. The geometry of this domain is determined for different values of the material parameters appearing in the constitutive relations of the Norton power law in the theory of steady-state creep and a kinetic equation which postulates a power law for the damage accumulation. It is shown that, if the boundary condition at the point at infinity is formulated as the condition of asymptotic approximation to the Hutchinson–Rice-Rosengren solution [Hutchinson JW. Singular behaviour at the end of a tensile crack in a hardening material. J Mech Phys Solids 1968;16(1):13–31; Rice JR, Rosengren GF. Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids. 1968;16(1):1–12], then the boundaries of the DCDM, which are defined by means of binomial and trinomial expansions of the continuity parameter, are substantially different with respect to their dimension and shape. A new asymptotic of the for stress field, which determines the geometry of the DCDM and leads to close configurations of the DCDM constructed using binomial and trinomial asymptotic expansions of the continuity parameter, are established by an asymptotic analysis and a numerical solution of the non-linear eigenvalue problem obtained.     

Dual boundary integral formulation for 2-D crack problems

An efficient integral equation formulation for two-dimensional crack problems is proposed with the displacement equation being used on the outer boundary and the traction equation being used on one of the crack faces. Discontinuous quarter point elements are used to correctly model the displacement in the vicinity of crack tips. Using this formulation a general crack problem can be solved in a single-region formulation, and only one of the crack faces needs to be discretised. Once the relative displacements of the cracks are solved numerically, physical quantities of interest, such as crack tip stress intensity factors can be easily obtained. Numerical examples are provided to demonstrate the accuracy and efficiency of the present formulation.     

Long-term failure of a layered viscoelastic composite material with a crack under a time-dependent load

The long-term failure of a layered viscoelastic composite caused by precritical propagation of a coin-shaped crack is studied. It is assumed that the crack is located inside a viscoelastic layer (the layer of binder) parallel to the layer orientation. The crack development due to stretching of the composite massive by uniformly distributed external forces increasing with time is described. It is assumed that these forces act perpendicularly to the plane of crack propagation. The investigation is carried out within the framework of Boltzmann-Volterra linear theory for resolving integral operators with difference kernels describing the deformation of a material with time-dependent rheological properties. An irrational function of the viscoelastic integral operator is presented in the form of a proper continued fraction and transformed using the method of operator continued fractions. Numerical solutions are obtained for resolving integral operators with the kernel in the form of Rabotnov exponential-fractional function. The kinetics of crack growth with a prefailure zone commensurable with the crack length is described. A comparison with the results obtained in terms of the concept of thin structure of the crak tip is given.Timoshenko Institute of Mechanics, Ukrainian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 545–558, July–August, 2000.     

Dynamic crack propagation in a 2D elastic body: The out-of-plane case

Already in 1920 Griffith has formulated an energy balance criterion for quasistatic crack propagation in brittle elastic materials. Nowadays, a generalized energy balance law is used in mechanics [F. Erdogan, Crack propagation theories, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York, 1968, pp. 498-586; L.B. Freund, Dynamic Fracture Mechanics, Cambridge Univ. Press, Cambridge, 1990; D. Gross, Bruchmechanik, Springer-Verlag, Berlin, 1996] in order to predict how a running crack will grow. We discuss this situation in a rigorous mathematical way for the out-of-plane state. This model is described by two coupled equations in the reference configuration: a two-dimensional scalar wave equation for the displacement fields in a cracked bounded domain and an ordinary differential equation for the crack position derived from the energy balance law. We handle both equations separately, assuming at first that the crack position is known. Then the weak and strong solvability of the wave equation will be studied and the crack tip singularities will be derived under the assumption that the crack is straight and moves tangentially. Using the energy balance law and the crack tip behavior of the displacement fields we finally arrive at an ordinary differential equation for the motion of the crack tip.     

Stress distribution in a transversely isotropic body with a thermally insulating parabolic crack subjected to a uniform heat flux

An explicit static thermoelastic solution is constructed for an infinite transversely isotropic body containing a thermally insulating parabolic crack in the plane of isotropy. The surface of the crack is free of stress. A uniform thermal flux is incident on the crack perpendicular to its surface. Formulas are obtained for the stress intensity factors near the tip of the crack. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; Catholic University, Portugal. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 54–66, 1999.     

功能梯度板中Griffith裂纹尖端应力场的三维解析研究

基于推广后的England-Spencer板理论,研究了横观各向同性功能梯度板中Griffith裂纹尖端的三维应力场.假定材料参数沿板厚方向可以任意连续变化,利用复变函数解法和保角变换技术分别给出了受无穷远处荷载作用和受均匀内压时裂纹尖端应力的三维解析解.当材料退化为各向同性均匀材料时,将该解答与经典二维解进行了比较,...     

Driving force acting on a running crack

The asymptotic analysis of the dynamic crack problem for the anti‐plane shear mode is provided. The field near the crack tip is studied in detail for a nonlinear elastic incompressible material whose stored energy behaves asymptotically as a power of the first invariant of the strain tensor at large strains. It is shown that the hardening parameter characterizes fully the singularity degree of the near‐crack‐tip field. Based on the latter knowledge the driving force acting on the crack tip is calculated. Possible scenarios of the crack propagation are discussed.     

Dynamic response of planar interface crack between viscoelastic bodies

IntroductionThe dynamic stress intensity factor (SIF) plays an important role in dynamic fracture underboth harmonic and transient loads. It predicts whether or not the fracture toughness of thematerial will be exceeded and catastrophic crack propagation will follow. The dynamic SIFof interfaCe cracks between two dissimilar elastic materials has been studied widely. Kundull]studied the dynamic SIF of interface crack under transient loading with the method based onBetti's reciprocal theore…     

Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

Simulation of short crack propagation using a hybrid boundary element technique

Service life of cyclically loaded components is often determined by the propagation of short fatigue cracks, which is highly influenced by microstructural features such as grain boundaries. A two-dimensional model to simulate the growth of such stage I-cracks is presented. The crack is discretised by dislocation discontinuity boundary elements and the direct boundary element method is used to mesh the grain boundaries. A superposition procedure couples these different boundary element methods to employ them in one model. Varying elastic properties of the grains are considered and their influence on short crack propagation is studied. A change in crack tip slide displacement determining short crack propagation is observed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implementation of a cohesive zone model for crack closure analysis of rotors

The presence of a crack in a rotor introduces a local flexibility which affects its dynamic response. Moreover, the crack may open and close during the vibration period. The crack status is a function of time and also depends on the rotational speed and the vibration amplitude of the rotor. This nonlinear case is still a challenging research topic especially in the field of closing crack in the rotating shaft. A cohesive zone model is developed in order to analyze the stiffness of a crack in a rotating shaft. The proposed expression will be compared to three different crack models, namely, a breathing crack model, a switching crack model and an open crack model. Moreover, a cohesive law to predict and to analyse the stress at the crack tip is presented. The numerical model is implemented using a finite element formulation. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals is considered. The Fourier transform technique is used to solve a moving crack problem under the action of antiplane shear and inplane electric field. Full elastic stresses of phonon and phason fields and electric fields are derived for a crack running with constant speed in the periodic plane. Obtained results show that the coupled elastic fields inside piezoelectric quasicrystals depend on the speed of crack propagation, and exhibit the usual square-root singularity at the moving crack tip. Electric field and phason stresses do not have singularity and electric displacement and phonon stresses have the inverse square-root singularity at the crack tip for a permeable crack. The field intensity factors and energy release rates are obtained in closed form. The crack velocity does not affect the field intensity factors, but alters the dynamic energy release rate. Bifurcation angle of a moving crack in a 1D hexagonal piezoelectric quasicrystal is evaluated from the viewpoint of energy balance. Obtained results are helpful to better understanding crack advance in piezoelectric quasicrystals.     

Simulation of short crack propagation under hydrogen influence using a hybrid boundary element technique for anisotropic elastic solids

A two-dimensional model for stage I short crack propagation on multiple slip planes under the influence of hydrogen is presented. It considers elastic-plastic material behaviour by allowing sliding on the active slip planes in the corresponding slip directions. A crack propagation law based on the crack tip sliding displacement is used to simulate crack growth. The activation of slip bands and the sliding on these active slip bands will be influenced by the local hydrogen concentration. The model is solved numerically using the boundary element method. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A BE based finite element for crack propagation processes

A boundary element based finite macro element for the simulation of 3D crack propagation in the framework of linear elastic fracture mechanics is presented. While the major part of the numerical model is discretized with finite elements, a small domain containing the crack is meshed with boundary elements. By means of the Symmetric Galerkin BEM a stiffness formulation for the cracked BE domain is obtained which enables a direct FEM/BEM coupling. All necessary operations for the crack propagation are carried out within this boundary element based finite macro element and exploit the potential of the boundary integral formulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Influence of Multiple Micro Cracks on the Damage Behavior of a Macro-Crack Tip北大核心CSCD

The solution of an infinite plane containing a macro crack and a cluster of micro cracks under uniaxial tensile load was presented based on Muskhelishvili’s complex function method and the stepwise recursive method. The stress field and stress intensity factor K were obtained. Combined with the damage mechanics, damage parameter D of the macro-crack tip and the micro-crack tip under uniaxial tension was redefined, and the influence of different damage zone forms on the damage of the crack tip was analyzed. The results show that, both the chain-distribution and the reverse-chain-distribution micro cracks have an amplifying effect on the macro crack growth, and the damage parameter increases with the decrease of the inclination angle of the micro crack and the reduction of the distance between the macro crack and the micro cracks. For a relatively small inclination angle of the micro crack, the damage parameters of the macro crack and the micro crack heightens, and the damage parameter of the macro crack increases with the micro-crack length. For evenly distributed micro cracks in the continuous damage zone, the micro cracks have an amplifying effect on the macro-crack growth, and the damage parameter of the macro crack increases with the micro-crack number. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

A generalized cohesive element technique for arbitrary crack motion

A computational method for arbitrary crack motion through a finite element mesh, termed as the generalized cohesive element technique, is presented. In this method, an element with an internal discontinuity is replaced by two superimposed elements with a combination of original and imaginary nodes. Conventional cohesive zone modeling, limited to crack propagation along the edges of the elements, is extended to incorporate the intra-element mixed-mode crack propagation. Proposed numerical technique has been shown to be quite accurate, robust and mesh insensitive provided the cohesive zone ahead of the crack tip is resolved adequately. A series of numerical examples is presented to demonstrate the validity and applicability of the proposed method.     

A numerical study of singular stress field of 3D cracks

We investigate the stress distribution and the variation of the mode I stress intensity factor along a straight three-dimensional (3D) crack by the finite element method. The results are checked against plane strain theory near the mid-crack and against the 3D theory of Zhu at the free surface. Although Zhu's formulation is not perfect and has some typographical errors. The surface stress distribution of his results are in line with the present study by the finite element method. The stress intensity factors at the free surface are found to be much lower than that at the mid-crack.     

Dynamic crack propagation in a 2D elastic body. The out-of plane case

We discuss the propagation of a running crack under shear waves in a rigorous mathematical way for a simplified model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u (y, t) and the one-dimensional crack tip trajectory h = h (t). We handle both equations separately, assuming at first that the crack position is known. Existence and uniqueness of strong solutions of the wave equation are studied and the crack-tip singularities are derived under the assumption that the crack is straight and moves tangentially. Using an energy balance law and the crack tip behaviour of the displacement fields we finally arrive at an ordinary differential equation for h (t), called equation of motion for the crack tip. We demonstrate the crack-tip motion with corresponding nonuniformly crack speed by numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)