Viscoplastic BEM fracture analysis of creeping metallic cracked structures in plane stress using complex variable techniques

This paper extends a boundary element approach developed earlier by the present author to the solution of viscoplastic fracture problems arising in creeping cracked metallic structural components undergoing plane stress deformation conditions. By using a plane stress viscoplastic formulation this work adopts the procedure of Lyness and Moler to express the derivative of displacement rate as the ratio of the imaginary part of the displacement rate to the imaginary part of a complex variable representing the coordinates of a collocation point. In this manner the evaluation of creep stresses at internal points is greatly simplified. The results thus obtained are highly accurate in comparison with conventional BEM. The application of the present BEM approach is illustrated by obtaining creep stress and strain distribution for center cracked and standard CT fracture specimens subjected to remote tensile and shear stress loading under plane stress deformation conditions.     

Quasicontinuum simulation of crack propagation in bcc-Fe

We have investigated fracture in bcc-Fe through multiscale simulations. The quasicontinuum (QC) method with an embedded atom method (EAM) interatomic potential is applied. The analyses have been carried out assuming different crystallographic orientations and different T-stress under Mode I loading. Both anisotropic and isotropic formulations of the modified boundary layer (MBL) approach has here been investigated and compared. The results show that the mechanisms at the crack tip and the critical stress intensity factor K_Ic are sensitive to both the crystallographic orientation and whether or not the formulation of the boundary conditions are isotropic or anisotropic. Mechanisms such as cleavage crack propagation, twinning, and dislocation emission are observed in the analyses. A short literature review on atomistic and multiscale simulations of fracture in bcc-Fe has been performed and evaluated, and also compared with the current results.     

Monotonic vis-à-vis cyclic fracture behaviour of AISI 304LN stainless steel

This investigation is aimed to examine the monotonic and cyclic fracture behaviour of AISI 304LN stainless steel and its weldments, in order to assess their integrity under seismic loading conditions. The monotonic fracture resistance of the steel has been determined using standard J-integral technique; whereas the cyclic fracture resistance has been evaluated using periodic unloading to different extents fixed by pre-determined R-ratio. Comparison of the fracture toughness values of the steel estimated under monotonic and cyclic loading indicates that the latter could be as low as one-fifth of the former. The observed degradation in cyclic fracture resistance has been attributed to crack tip re-sharpening during cyclic loading.     

A general solution procedure for fracture mechanics weight function evalution based on the boundary element method

This paper reports on the development of an efficient and accurate means for the direct computation of crack surface weight functions for two dimensional fracture mechanics analysis. Weight functions are mathematical representations which can be used to efficiently calculate stress intensity factors for a variety of crack loading and boundary conditions. The method is inherently capable of handling mixed-mode problems. The weight function capability is especially important for problems of fatigue crack growth modeling where the efficient calculation of stress intensity factors is crucial.The basis of the new formulation and numerical solution method is the boundary element method (BEM), as implemented for two dimensional fracture mechanics analysis. The paper will review the analytical formulation of the new BEM, the numerical solution algorithm, and a limited number of validation examples.     

Quasi-static analysis of viscoplastic plates by the boundary element method

A direct boundary element method (BEM) is developed for the determination of the time-dependent inelastic deflection of plates of arbitrary planform and under arbitrary boundary conditions to general lateral loading history. The governing differential equation is the nonhomogeneous biharmonic equation for the rate of small transverse deflection. The boundary integral formulation is derived by using a combination of the BEM and finite element methodology. The plate material is modelled as elastic-viscoplastic. Numerical examples for sample problems are presented to illustrate the method and to demonstrate its merits.     

Creep-fatigue crack propagation in lead-free solder under cyclic loading with various waveforms

Crack propagation tests of lead-free solder were conducted using center-notched plate specimens under cyclic tension-compression of three load waveforms: pp waveform having fast loading and unloading, cp-h waveform having a hold time under tension, and cc-h waveform having a hold time under tension and compression. In the case of fatigue loading, i.e. pp waveform, the path of crack propagation was macroscopically straight and perpendicular to the maximum principal stress direction, showing tensile-mode crack propagation. The introduction of the creep components by hold time in cc-h and cp-h waveforms promoted shear-mode crack propagation. For fatigue loading of pp wave, the crack propagation rate was expressed as a power function of the fatigue J integral and the relation was identical for load-controlled and displacement-controlled conditions. The creep component due to the hold time greatly accelerates the crack propagation rate when compared at the same values of the fatigue J integral or the total J integral (the sum of fatigue J and creep J integrals). The creep crack propagation rate was expressed as a power function of the creep J integral for each case of cp-h and cc-h waveforms. The crack propagation rate for cp-h waveform is higher than that for cc-h waveform. The predominant feature of fracture surfaces was striations for pp waveform and grain boundary fracture for cp-h waveform. Grain fragmentation was abundantly observed on the fracture surface made under cc-h waveform.     

Numerical simulation with experimental verification of the fracture behavior in granite under confining pressures based on the tension-softening model

The use of the tension-softening model for analyzing fracture processes of rock is examined with special reference to the effect of confining pressure on the fracture extension. Tension-softening curves are measured by means of theJ-based technique from unconfined tests performed on compact tension (CT) specimens of granite. On the basis of the determined tension-softening relation, numerical analyses are executed using a boundary element method (BEM) to simulate fracture of the granite under confining pressures. Numerical results are compared to the experimental results of two series of tests for which CT specimens and thick-walled cylindrical specimens were loaded to failure under confining pressures ranging from 0 to 26.5 MPa. It is shown that the BEM analyses can predict the observed fracture behavior. Based on the results, it is demonstrated that the tension-softening relation provides a suitable model to analyze the fracture process in the rock. The source mechanism for the pressure sensitive fracture is discussed by examining the growth of the fracture process zone.     

Two-parameter characterization of elastic-plastic crack front fields: Surface cracked plates under tensile loading

In this paper the J-Q two-parameter characterization of elastic-plastic crack front fields is examined for surface cracked plates under uniaxial and biaxial tensile loadings. Extensive three-dimensional elastic-plastic finite element analyses were performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. Surface cracks with aspect ratios a/c = 0.2, 1.0 and relative depths a/t = 0.2, 0.6 were investigated. The loading levels cover from small-scale to large-scale yielding. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. In order to facilitate the determination of Q-factors, modified boundary layer analyses were also conducted. The J-Q two-parameter approach was then used in characterizing the elastic-plastic crack front stress fields along these 3D crack fronts. Complete distributions of the J-integral and Q-factors for a wide range of loading conditions were obtained. It is found that the J-Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis. These results are suitable for elastic-plastic fracture mechanics analysis of surface cracked plates.     

Boundary element analysis for viscoelastic solids containing interfaces/holes/cracks/inclusions

With the aid of the elastic–viscoelastic correspondence principle, the boundary element developed for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. Green's functions for the problems of two-dimensional linear anisotropic elastic solids containing holes, cracks, inclusions, or interfaces have been obtained analytically using Stroh's complex variable formalism. Through the use of these Green's functions and the correspondence principle, special boundary elements in the Laplace domain for viscoelastic solids containing holes, cracks, inclusions, or interfaces are developed in this paper. Subregion technique is employed when multiple holes, cracks, inclusions, and interfaces exist simultaneously. After obtaining the physical responses in Laplace domain, their associated values in time domain are calculated by the numerical inversion of Laplace transform. The main feature of this proposed boundary element is that no meshes are needed along the boundary of holes, cracks, inclusions and interfaces whose boundary conditions are satisfied exactly. To show this special feature by comparison with the other numerical methods, several examples are solved for the linear isotropic viscoelastic materials under plane strain condition. The results show that the present BEM is really more efficient and accurate for the problems of viscoelastic solids containing interfaces, holes, cracks, and/or inclusions.     

Quasi-static and dynamic fracture behavior of composite sandwich beams with a viscoelastic interface crack

Fracture analysis of sandwich beams with a viscoelastic interface crack under quasi-static and dynamic loading has been studied. Firstly, a three-parameter standard solid material model was employed to describe the viscoelasticity of the adhesive layer. And a novel interfacial fracture analysis model called three material media model was established, in which an interface crack was inserted in the viscoelastic layer. Secondly, a finite element procedure based on Rice J-integral and Kishimoto J-integral theories was used to analyze quasi-static and dynamic interface fracture behavior of the sandwich beam, respectively. Finally, the influence of viscoelastic adhesive layer on the quasi-static J-integral was discussed. In addition, comparison of quasi-static Rice J-integral with Kishimoto J-integral under various loading rates was carried out. The numerical results show that the oscillating characteristic of dynamic J-integral is more evident with shorter loading rise time.     

An analysis of the viscoelastic fracture toughness and crack growth in ice

In this paper, a numerical model is presented to predict the viscoelastic fracture toughness and crack growth in ice under different loading conditions. It is assumed that ice is a homogeneous, linear, and isotropic material. The model can predict the viscoelastic response of small ice samples under constant cross-head speed, constant strain rate, and constant stress rate loading conditions. The effect of different parameters such as test temperature, test sample size, and rate of loading is also investigated.     

A new M-integral parameter for mixed-mode crack growth in orthotropic viscoelastic material

This paper deals with a new independent path integral which provides the mixed-mode during a creep crack growth process in viscoelastic orthotropic media. The developments are based on an energetic approach using conservative laws. The mixed-mode fracture separation is introduced according to the generalization of the virtual work principle. The fracture algorithm is implemented in a finite element software and coupled with an incremental viscoelastic formulation and an automatic crack growth simulation. This M-integral provides the computation of stress intensity factors and energy release rate for each fracture mode. A numerical validation, in terms of energy release rate and stress intensity factors, is carried out on a CTS specimen under mixed-mode loading for different crack growth speeds.     

Experimental Evaluation of Mixed Mode Stress Intensity Factor for Prediction of Crack Growth by Phoelastic Method

Determination of stress intensity factors K _I, K _II, and constant stress term, σ _ox is investigated. A theory of determining the stress intensity factors using photo-elastic method is formulated taking three stress terms. Three-parameter method of fracture analysis for determining the mixed mode stress intensity factors under biaxial loading conditions from photo-elastic isochromatic fringe data is used. A special biaxial test rig is designed and fabricated for loading the specimen biaxially. A simplified and accurate method is proposed to collect the data from isochromatic fringes. Taking specimen geometry and boundary conditions into account, regression models are developed for estimation of fracture parameters.     

Time-domain BEM analysis of a rapidly growing crack in a bi-material

Rapid propagation of a matrix crack in a bi-material system is studied with emphasis on the dynamic interaction between the crack and the interface by combining the traditional time-domain displacement boundary element method (BEM) and the non-hypersingular traction BEM. The crack growth is controlled by the fracture criterion based on the maximum circumferential stress, and is modeled by adding new elements to the moving crack tip. Detailed computation is performed for an unbounded bi-material with a crack subjected to incident impact waves and a bounded rectangular bi-material plate under dynamic wedged loading. Numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions are presented for various material combinations and geometries. The effects of the interface on the crack growth are discussed.     

Solutions of the second elastic-plastic fracture mechanics parameter in test specimens

Extensive finite element analyses have been conducted to obtain solutions of the A-term, which is the second parameter in three-term elastic-plastic asymptotic expansion, for test specimens. Three mode I crack plane-strain test specimens, i.e. single edge cracked plate (SECP), center cracked plate (CCP) and double edge cracked plate (DECP) were studied. The crack geometries analyzed included shallow to deep cracks. Solutions of A-term were obtained for material following the Ramberg-Osgood power law with hardening exponent of n = 3, 4, 5, 7 and 10. Remote tension loading was applied which covers from small-scale to large-scale yielding. Based on the finite element results, empirical equations to predict the A-terms under small-scale yielding (SSY) to large-scale yielding conditions were developed. In addition, by using the relationships between A and other commonly used second fracture parameters such as Q factor and A₂-term, the present solutions can be used to calculate parameters A₂ and Q as well. The results presented in the paper are suitable to calculate the second elastic-plastic fracture parameters for test specimens for a wide range of crack geometries, material strain hardening behaviors and loading conditions.     

A hybrid micro–macro BEM formulation for micro-crack clusters in elastic components

Local analysis schemes capable of detailed representations of the micro-features of a problem are integrated with a macro-scale BEM technique capable of handling complex finite geometries and realistic boundary conditions. The micro-scale effects are introduced into the macro-scale BEM analysis through an augmented fundamental solution obtained from an integral equation representation of the micro-scale features. The proposed hybrid micro-macro BEM formulation allows decomposition of the complete problem into two sub-problems, one residing entirely at the micro-level and the other at the macro-level. This allows for investigations of the effects of the micro-structural attributes while retaining the macro-scale geometric features and actual boundary conditions for the component or structure under consideration. As a first attempt, elastic fracture mechanics problems with interacting cracks at close spacings are considered. The numerical results obtained from the hybrid BEM analysis establish the accuracy and effectiveness of the proposed micro–macro computational scheme for this class of problems. The proposed micro–macro BEM formulation can easily be extended to investigate the effects of other micro-features (e.g. interfaces, short or continuous fibre reinforcements, voids and inclusions, in the context of linear elasticity) on macroscopic failure modes observed in structural components.     

Optimum interpolation functions for boundary element method

In the Boundary Element Method (BEM) the density functions are approximated by interpolation functions which are chosen to satisfy appropriate continuity requirements. The error of approximation inside an element depends upon the location of the collocation points that are used in constructing the interpolation functions. The location of collocation points also affects the nodal values of the density function and, hence, the total error in the analysis if boundary conditions are satisfied in a collocation sense. In this paper, we minimize the error inside the element using the L₁ norm to obtain the optimum location of collocation points. Results show that irrespective of the continuity requirement at the element end, the location of collocation points computed by the algorithm presented in this paper results in an error that is less than the error corresponding to uniformly spaced collocation points. Results for optimum location of collocation points and the average error are presented for Lagrange polynomials up to order fifteen and for Hermite polynomials that ensure continuity up to the seventh order of derivative at the element end. The information of the optimum location of interpolation points for Lagrange and Hermite polynomials should be useful to other researchers in BEM who could incorporate it into their current programs without making significant changes that would be needed for incorporating the algorithm. The algorithm presented is independent of the BEM application in two-dimensions, provided that the density functions are approximated by polynomials and is applicable to direct and indirect formulations. Two numerical examples show the application of the algorithm to an elastostatic problem in which one boundary is represented by integrals of the Direct BEM while the other boundary by the Indirect BEM and a fracture mechanics problem by Direct method in which the crack is represented by displacement discontinuity density function.     

Electrical failure of piezoelectric ceramics with a conductive crack under electric fields

The failure behavior of piezoelectric ceramics with a conductive crack under purely electric loading is investigated. Electrical fracture tests are conducted to study the influence of the directions of poling and electric loading. Two failure modes of piezoelectric materials are observed: fracture that is accompanied with dielectric discharging and the formation of tubular channels without fracture. The critical J integrals at the onset of both fracture and breakdown are calculated numerically via finite element analysis. The effects of both the direction of the electric field and the poling direction on both fracture and breakdown resistance are discussed.     

Computation of thermoelastoplastic stresses in crack problems by the BEM

In the framework of deformation theory and assuming a power hardening material, a 2D elastoplastic crack problem is considered under loading conditions expressed in terms of a stationary temperature field. The HRR stress field is compared with the one obtained by a BEM analysis. It is shown that a three-term asymptotic expansion of stresses leads to a good approximation of the near field. The free coefficient in the asymptotic expansion has to be determined via the BEM results. The analysis is provided for a rectangular plate with a central crack under two kinds of boundary heat conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Numerical simulation with experimental verification of the fracture behavior in granite under confining pressures based on the tension-softening model

The use of the tension-softening model for analyzing fracture processes of rock is examined with special reference to the effect of confining pressure on the fracture extension. Tension-softening curves are measured by means of the J-based technique from unconfined tests performed on compact tension (CT) specimens of granite. On the basis of the determined tension-softening relation, numerical analyses are executed using a boundary element method (BEM) to simulate fracture of the granite under confining pressures. Numerical results are compared to the experimental results of two series of tests for which CT specimens and thick-walled cylindrical specimens were loaded to failure under confining pressures ranging from 0 to 26.5 MPa. It is shown that the BEM analyses can predict the observed fracture behavior. Based on the results, it is demonstrated that the tension-softening relation provides a suitable model to analyze the fracture process in the rock. The source mechanism for the pressure sensitive fracture is discussed by examining the growth of the fracture process zone.