Path-independent H integrals for three-dimensional fracture mechanics

In fracture mechanics, a number of real applications have intrinsically three-dimensional crack geometries, thereby requiring a means of extracting stress intensity factors under such circumstances. Two approaches to this end are examined here: one, a three-dimensional J-integral; the other, three-dimensional H integrals for each mode. The first integral is well accepted by the fracture mechanics community; the second integrals are newly developed herein. The two are compared on three-dimensional test problems with closed-form solutions that are constructed for this purpose. Analysis is via quarter-point elements on two successively refined grids for each test problem. The results demonstrate that both types of path-independent integral can furnish estimates of stress intensity factors which converge to good levels of accuracy in return for reasonable levels of computational effort. This revised version was published online in July 2006 with corrections to the Cover Date.     

Path-independent integrals in anisotropic media

The concept proposed by the author for the construction of path-independent integrals (PI Is) in isotropic media is generalized for the construction of PIIs in anisotropic plane bodies weakened by a number of collinear cracks. The PIIs are classified into three classes satisfying well-defined restrictions. Using this classification about fifty general forms of PIIs are given, from which an infinity of new PI Is can be constructed. In many cases the form of the proposed integrals is simpler than the already known ones. This will facilitate their use in conjunction with finite element or experimental methods.A number of illustrative examples is also given.

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Résum'e L'auteur a proposé un concept pour construire des intégrals indépendantes du parcours dans les milieux isotropes. Il généralise à présent son concept pour construire ces intégrales dans le cas des corps plans anisotropes affaiblis par la présence d'un grand nombre de fissures colinéaires. Ces intégrales sont classées suivant trois catégories correspondant à des restrictions bien définies. En utilisant cette classification, près de cinquante formes générales d'intégrales indépendantes du parcours sont fournies, et, à partir d'elles, une infinité d'intégrales peuvent être construites. Dans de nombreux cas, la forme des intégrales proposées est plus simple que celles déjà connues. Ceci va faciliter leur utilisation, en association avec la méthode des éléments finis ou des methodes experimentales.On donne également plusieurs exemples d'application.

     

Path-independent line integrals for steady-state,two-dimensional thermoelasticity

Three path-independent line integrals J' _k, M', and L' ₃ are derived for steady-state, two-dimensional thermoelasticity. These integrals are similar to the J _k, M, and L ₃ presented by Knowles and Sternberg [1], but include additional terms of either free expansion displacement vector u _k ^* or temperature and its conjugate harmonic function in their formulation. These new line integrals enable us to avoid the undesirable area integration [3–8] when calculating the strain energy release rate for crack problems. Application of J/ _k, M', and L' ₃ is demonstrated through a sample problem of a constant heat flux disturbed by a finite crack in an infinite plate.

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Résumé On établit les intégrales linéaires J _k, M' et L' ₃ indépendantes du parcours applicables aux problèmes de thermo-élasticité en état stable et suivant deux dimensions. Ces intégrales sont similaires aux intégrales de conservation J _k, M et L ₃ introduites par Knowles et Sternberg, mais comportent des termes supplémentaires prenant en compte dans la formulation le vecteur de déplacement u _k ^* en dilatation libre ou la température et sa fonction complexe .Ces nouvelles intégrales linéaires permettent aux auteurs d'éviter une intégration sur des zones inutiles lors du calcul, dans des problèmes de fissuration, de la vitesse de relaxation de l'énergie de déformation.On démontre l'applicabilité de J' _k, M' et L' ₃ dans un exemple de problème de flux de chaleur constant soumis à une perturbation par une fissure finie dans une plaque infinie.

     

Path-independent integrals and crack extension force for functionally graded materials

Path-independent integrals for functionally graded materials are constructed and calculated around a tip of crack. It is found that the stress intensity factor is not equivalent to the crack extention force in estimating crack growth. The effect of functional gradient on the crack extension force is presented.     

Path-independent integrals around two circular holes in an infinite plate under biaxial loading conditions

An analytic solution is presented for stresses induced in an infinite plate with two unequal circular holes by remote uniform loadings and arbitrary internal pressures in the holes. The solution is obtained by using the general expression for a biharmonic function in bipolar coordinates. The Airy stress function is decomposed in the sum of a fundamental stress function for an infinite plate remotely loaded, which gives non vanishing tractions on the circular boundaries, and an auxiliary stress function required to satisfy the boundary conditions on the pressures at the edges of the holes, which produces vanishing stresses at infinity. Correspondingly, the variations of the stress concentration factor are determined in terms of the holes geometry and loading conditions. The path independent J_k- (k = 1, 2), M- and L-integrals are analytically calculated on a closed contour encircling the two holes, under remote loading, in order to evaluate the energy release rates accompanying unit translation, self similar expansion and rotation of the holes, respectively. Results are then presented for varying loading orientation angle, biaxial loading ratio and holes geometry.     

Inverse extraction of cohesive zone laws by field projection method using numerical auxiliary fields

The cohesive zone law relates the cohesive tractions with the cohesive separations within the fracture process zone of a material and is used to quantify the strength and toughness of the material. Determining the material's cohesive zone law, however, is a nontrivial inverse problem of finding unknown tractions and separations from measurement data. Previously, a field projection method was established to extract the cohesive zone laws from far‐field data using interaction J‐integrals between the physical field of interest and auxiliary analytical probing fields. Here, we extend the universality of the field projection method and its ease of numerical implementation by using numerical auxiliary fields. These numerical fields are generated by systematically imposing uniform surface tractions element‐by‐element along the crack faces in finite element models. Then, interaction J‐ and M‐integrals between these auxiliary probing fields and the measurement field are used to reconstruct the traction and separation relationship along the crack faces. The effectiveness of this method in extracting the cohesive zone law from measured displacements in the far‐field region is demonstrated through numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Investigation of viscoelastic fracture fields in asphalt mixtures using digital image correlation

In this work we have studied the fracture behavior of asphalt mixtures, a heterogeneous mix of hard aggregates (usually in the form of crushed quarried rock) with a petroleum based asphalt binder, used in paving applications. Specifically, we studied the dependence of asphalt mixes’ fracture response on loading rate, temperature, and recycled content—the latter used primarily to replace virgin materials like aggregates and binder. Fracture tests were conducted on semi-circular bend edge cracked specimens obtained from mixes with different compositions, and the fracture event was recorded with a camera to allow for digital image correlation (DIC) measurements. DIC, with a spatial resolution of about 40 \(\upmu \)m/pixel, measured the far-field strain and displacement fields developing around a preexisting notch tip. Our focus here is on characterizing the material behavior by quantifying its viscoelastic response and fracture properties. The elastic–viscoelastic correspondence principle was used to extract viscous and elastic components from the full-field DIC-measured strain and displacement fields. Various energy dissipation mechanisms other than the fracture itself were evaluated. Stress–strain response and energy dissipated in the far-field regions were quantified. The pseudo-elastic stress intensity factor was then used to study the fracture properties, and quantify the effects on fracture properties of loading rate, temperature, and recycled content in the binder. It was seen that the viscoelastic characteristics of the material were a dominant factor in the material behavior obtained at room temperature. In general, the elastic component of the displacement was only up to about 30% of the total displacement, indicating a strong influence of viscoelasticity in this state. Loading rate, temperature and recycled asphalt shingles (RAS) content all affected the viscous response by introducing more elastic response when loading rate or recycled content increased or when temperature decreased. It became clear from these macroscopic measurements that the increase of RAS content considerably embrittles the material producing less viscous effects and less energy dissipated in the far-field, almost comparable to reductions associated with the loading rate increase (from 6.25 to 50 mm/min) or the temperature change (\(-12\) to \(25\,{^{\circ }}\)C).     

Higher moments of weighted integrals of non-Gaussian fields

In general, the exact probability distribution of a definite integral of a given non-Gaussian random field is not known. Some information about this unknown distribution can be obtained from the 3rd and 4th moment of the integral. Approximations to these moments can be calculated by discretizing the integral and replacing the integrand by third-degree polynomials of correlated Gaussian variables which reproduce the first four moments and the correlation function of the field correctly. The method described (see Ditlevsen O, Mohr G, Hoffmeyer P. Integration of non-Gaussian fields. Probabilistic engineering mechanics, 1996) based on these ideas is discussed and further developed and used in a computer program which produces fairly accurate approximations to the mentioned moments with no restrictions put on the weight function applied to the field and the correlation function of the field. A pathological example demonstrating the limitations of the method is given.     

Finite element methodology for path integrals in fracture mechanics

Based on the energy foundation of the path-independent integral in non-linear fracture mechanics, I* integral as the dual form of Rice's J is presented, it is also path-independent and is equivalent to J in value but it relates to the complementary energy. It is proved that, in numerical implementation, the path independence of J and I* can be ensured by using the assumed displacement finite elements and the assumed stress finite elements, respectively. Regarding the bounds of crack parameters, it is demonstrated that the lower bound of J can be estimated by the displacement compatible elements, and the upper bound of I* can be estimated by the stress equilibrium elements. In view of the difficulties in formulating stress equilibrium model, instead of it, a quasi-equilibrium model is proposed, which makes hybrid stress elements be able to estimate the bound of I*, and do not lose the characteristics of stiffness formulation. Two four-node plane elements are suggested; of them, the incompatible one can be used in incompressible/fully plastic fracture analysis, and the penalty-equilibrium one can be implemented to estimate the bound of I*. Furthermore, an incremental formulation is developed for I*, and can be extended into the calculations of ductile fracture under monotonic loading. For attestation, quite a number of numerical experiments is carried out, and some significant results are offered. © 1998 John Wiley & Sons, Ltd.     

Incremental path-independent integrals in inelastic and dynamic fracture mechanics

Certain incremental path-independent integrals, of relevance in the mechanics of fracture of elastic-plastic materials described by a classical flow theory of plasticity, are presented. Both quasi-static as well as dynamic fracture situations are considered. The topics discussed include: (i) incremental path-independent integrals that characterize the crack-tip fields in elastic-plastic materials; (ii) incremental integrals related to the incremental total potential energy difference; and (iii) the complementary or dual representations of these integrals. The use of these integrals is illustrated through some numerical examples. Comments are made on the utility of these integrals in postulating rational fracture criteria.     

New path independent integrals in linear elastic fracture mechanics

In this paper the properties of eigenfunction expansion form (abbreviated as EEF) in the crack problems of plane elasticity and antiplane elasticity are discussed in details. After using the Betti's reciprocal theorem to the cracked body, several new path independent integrals are obtained. All the coefficients in the EEF at the crack-tip, including the K₁, K₂ and k₃ values, can be related to corresponding path-independent integrals.     

Closed-form solutions for finite-part integrals occurring in shear-mode fracture mechanics

Summary This contribution is a continuation of a recent paper by K. Mayrhofer and F. D. Fischer [1], deriving an analytical solution for a general two-dimensional finite-part integral appearing in tension-mode fracture mechanics. This paper corresponds to shear-mode fracture mechanics where three further types of finite-part integrals occur. Their closed-form solutions will be introduced on the basis of an inclined, elliptically shaped crack. In general they can be expressed in terms of powered input values and by means of the Gauss hypergeometric function₂ F ₁. Formulae for different special cases are listed in the Appendices. Their correctness has been checked by means of two different numerical methods.     

Interaction integrals for thermal fracture of functionally graded materials

This paper addresses finite element evaluation of the non-singular T-stress and mixed-mode stress intensity factors in functionally graded materials (FGMs) under steady-state thermal loads by means of interaction integral. Interaction integral provides an accurate and efficient numerical framework in evaluating these fracture parameters in FGMs under thermal as well as mechanical loads. We use a non-equilibrium formulation and the corresponding auxiliary (secondary) fields tailored for FGMs. Graded finite elements have been developed to account for the spatial gradation of thermomechanical properties. This paper presents various numerical examples in which the accuracy of the present method is verified.     

Path-independent integral for the dilatation symmetry group in thermoplasticity

The aim of this study is to establish a fatigue criteria based on an energy threshold corresponding to crack initiation in the case of dilatational symmetry of a structure under thermomechanical loading. An extension of the M integral to thermoplasticity relating to the scaling symmetry is presented. Based on a discrete Lagrangian describing the thermo-inelastic system and Noether's theorem defining the conservation law, the null Lagrangian theorem is applied in order to introduce thermoplasticity. By the use of the divergence theorem, the modified M integral is obtained. Path-domain independence is discussed illustrated by a numerical example.     

Mixed-mode separation in dynamic fracture mechanics: New path independent integrals

This paper presents a theoretical and numerical analysis of mixed-mode separation in fracture dynamics, based on new path independent integrals. The M-integral method proposed by Chen and Shield is generalized for dynamic fracture applications. Strain energy density is expressed as a function of the actual displacement field and of an auxiliary kinematically admissible field. On the other hand, the concept of the G-integral developed by Destuynder using the Rice J-integral is extended to dynamic problems. Introducing the same concept in the M-integral formulation leads to the new path-independent integral M in elastodynamics. Numerical tests give us accurate results of separated mode.     

Interaction integrals for fracture analysis of functionally graded magnetoelectroelastic materials

This paper presents the domain form of interaction integrals based on three independent formulations for computation of stress intensity factors, electric displacement intensity factors and magnetic induction intensity factors for cracks in functionally graded magnetoelectroelastic materials. Conservation integrals of J-type are derived based on the governing equations for magnetoelectroelastic media and the crack tip asymptotic fields of homogeneous magnetoelectroelastic medium as auxiliary fields. Each of the formulations differs in the way auxiliary fields are imposed in the evaluation of interaction integrals and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded magnetoelectroelastic medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.