Criteria of nonlinear fracture mechanics

Conclusions Difficulties are experienced in using some of the examined fracture criteria in the elastoplastic region. The apparent fracture toughness varies in a wide range. The effect of the thickness of the specimen on cracking resistance is relatively small.In a partial case examined in this work, i.e., in fracture of the low-strength aluminum alloy with slight hardening in the conditions of the plane stress state at the crack tip, the range of applicability of certain critria is very small so that it is necessary to use the fracture criterion which does not take into account the crack length. Consequently, in this case, the two-parametric fracture criterion is more suitable for the apparent fracture toughness K^*–a/W ratio curve. This is illustrated clearly by Fig. 9 where the fracture load, i.e., the fracture criterion belonging in the group of the fracture criteria of the intermediate state, was predicted by the Newman method.In fact, the criteria of nonlinear fracture mechanics can be used in the conditions restricted by two limiting states: fracture at linear elastic strains, and fracture as a result of the formation of a plastic hinge.The interpolated curve used in such a method is usually known as the two-parametric Newman method or the CEGB method which is the same in the present case. This is possible owing to the fact that the extent of hardening of the aluminum alloy examined in this work is only very small.Further investigations are required for predicting the critical crack growth with satisfactory accuracy.Metz Institute, France. Translated from Problemy Prochnosti, No. 11, pp. 14–23, November, 1988.     

On nonlinear effects in fracture mechanics

Linear elastic treatment of fracture is considered applicable for net section stress up to about 0.8 the uniaxial tensile yield stress. Crack front plastic yield is still small enough to be viewed and treated as a small perturbation to the local crack front elastic stress field.     

Weakly nonlinear fracture mechanics: experiments and theory

Material failure occurs at the small scales in the immediate vicinity of the tip of a crack. Due to its generally microscopic size and the typically high crack propagation velocity, direct observation of the dynamic behavior in this highly deformed region has been prohibitively difficult. Here we present direct measurements of the deformation surrounding the tip of dynamic mode I cracks propagating in brittle elastomers at velocities ranging from 0.2 to 0.8C _s . Both the detailed fracture dynamics and fractography of these materials are identical to that of standard brittle amorphous materials such as soda-lime glass. These measurements demonstrate how Linear Elastic Fracture Mechanics (LEFM) breaks down near the tip of a crack. This breakdown is quantitatively described by extending LEFM to the weakly nonlinear regime, by considering nonlinear elastic constitutive laws up to second order in the displacement-gradients. The theory predicts that, at scales within a dynamic lengthscale ? _nl from the tip of a single crack, significant logr displacements and 1/r displacement-gradient contributions arise, and provides excellent quantitative agreement with the measured near-tip deformation. As ? _nl is consistent with lengthscales that appear in crack tip instabilities, this “weakly nonlinear fracture mechanics” framework may serve as a springboard for the development of a comprehensive theory of fracture dynamics.     

An adaptive singular finite element in nonlinear fracture mechanics

In the case of nonlinear fracture mechanics the type of singularity induced by the crack tip is commonly not known. This results in a poor approximation of the near crack tip fields in a finite element setting and induces so called spurious—or residual—discrete material forces in the vicinity of the crack tip. Thus the numerical calculation of the crack driving material force in nonlinear fracture is often not that precise as in linear elasticity where we can use special crack tip elements and/or path independency. To overcome this problem we propose an adaptive singular element, which adapts automatically to the type of singularity. The adaption is based on an optimisation procedure using a variational principle.     

Coupling nonlinear beams and continua: Variational principles and finite element approximations

Two variational principles are proposed that describe equilibrium problems with connected nonlinear beams and solids. The principles extend the classical principle of minimum potential energy for beams and deformable bodies incorporating constraints that weakly enforce the beam kinematics at the common interface using either Lagrange multipliers or penalty terms. In contrast with existing alternatives, in the new approach, the surfaces of bodies connected to beams can deform in an energetically optimal way while globally behaving as beam cross sections. This allows, eg, warping and Poisson effects in beam/solid interfaces. Finite element implementations of the new principles are described in detail and application examples are provided that illustrate their use.     

Role of nonlinear fracture mechanics in assessing fracture and crack growth in welds

An experimental study was conducted to assess the structural performance of repair welds in an ex-service 1Cr-1Mo-0.25V steam turbine casing material. Material from two weld techniques, one involving a post-weld heat-treatment that produced undermatched welds and the other involving a temper bead welding technique that produced overmatched welds were tested. Both welding techniques were implemented in two base metal conditions giving rise to four different welds and two different base metal conditions. The tests conducted included tensile tests, creep tests, fracture toughness tests, fatigue crack growth tests, creep crack growth tests, and creep-fatigue crack growth tests on the base metal, weld metal and the weldment region.The yield strength of the weld metal in the undermatched condition was approximately 10% lower than the base metal, while the weld metal in the overmatched condition had a yield strength that was 30% higher than the base metal at 565 °C. The creep deformation rates in the undermatched welds were 60 times faster than the base metal at a stress of 207 MPa. In the overmatched welds, the creep rates at 207 MPa were about 2.8 times faster in one case and 2.8 times slower in the other.The crack path in fracture toughness specimens followed the interface between the transition layer and the weaker of the weld metal and the base metal. The J-resistance curves for the weldments at 565 °C showed significant variability among duplicate samples from the same welds. This scatter was caused by the variability in the location of the precrack with respect to the fusion line and the location of the low fracture toughness region in the weldment. This behavior was explained using a novel approach for characterizing the fracture of welds. The creep-fatigue crack growth rates at equivalent (C_t)_avg values in undermatched welds was higher than the crack growth rates in the overmatched weld samples. In all cases under creep-fatigue, the crack appeared to grow in the weaker of the base metal and the weld metal. Recommendations for future work are provided to enhance the theoretical underpinnings of the nonlinear fracture mechanics frame-work to rigorously address fracture and crack growth in welds.     

On the problem of fracture mechanics of physically nonlinear materials

For the construction of fracture mechanics of physically nonlinear materials, we join two different approaches. One of the approaches is the method of successive approximations in a small parameter characterizing a negligible deviation from the Hooke law. A solution of the proper problem of the linear theory of elasticity is considered as the zero approximation. By using the second method, the singularity of stresses in the vicinity of a cracklike defect is established in the process of solution of the elastic problem. According to the second method, the material is regarded as incompressible, deviations from a linear rheological law are considerable, and the material in the vicinity of a defect and in the whole plate obeys various power dependences of small elastic strains on the corresponding stresses. The interrelations between stress intensity factors are introduced for the first time, which are also valid for their critical values. By specific examples, we demonstrate the effects of the parameters of physical nonlinearity of materials, value of the applied load, and type of strain. In conclusion, we give a short survey of the scientific literature on the solution of problems concerning elastoplastic hardening of cracked bodies in the case of antiplane strain and their use for rupture cracks. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 3, pp. 28–38, May–June, 1999.     

Application of the principles of linear fracture mechanics to the composite materials

Application of the principles of linear fracture mechanics to the orthotropic composite materials is examined. It is shown that fracture toughness of unidirectional composites is independent of crack length but dependent on crack-fiber orientation. For experimental verification of the above principles, uni-directional glass epoxy material (Scotchply 1002) was used. The fracture toughness of Scotchply 1002 for different crack-fiber orientations is obtained by utilizing Solid Sap finite element program and compact tension specimens. An empirical formula relating the fracture toughness of the material for different crack-fiber orientation is found.

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Résumé On examine l'application du principe de la mécanique linéaire de rupture aux matériaux composites orthotropes. On montre que la ténacité à la rupture de composites unidirectionnels est indépendante de la longueur de la fissure mais dépend de l'orientation des fibres par rapport à cette fissure. Pour vérifier de manière expérimentale les principes ci-dessus, on a utilisé un matériau en verre epoxy unidirectionnel (Scotchply 1002). La ténacité à la rupture du Scotchply 1002 pour différentes orientations fibre/fissure a été obtenue en utilisant un programme à élément fini Solid Sap et des éprouvettes de traction compactes. On a trouvé une formule empirique mettant en relation la ténacité à la rupture du matériau pour différentes orientations fibre/fissure.

     

Combination of finite element and reliability methods in nonlinear fracture mechanics

This paper presents a probabilistic methodology for nonlinear fracture analysis in order to get decisive help for the reparation and functioning optimization of general cracked structures. It involves nonlinear finite element analysis. Two methods are studied for the coupling of finite element with reliability software: the direct method and the quadratic response surface method. To ensure the response surface efficiency, we introduce new quality measures in the convergence scheme. An example of a cracked pipe is presented to illustrate the proposed methodology. The results show that the methodology is able to give accurate probabilistic characterization of the J-integral in elastic–plastic fracture mechanics without obvious time consumption. By introducing an “analysis re-using” technique, we show how the response surface method becomes cost attractive in case of incremental finite element analysis.     

On theT- andS-criteria in fracture mechanics: new formulations and variational principles

Summary In this paper some new formulations of theT- and theS-criteria of fracture mechanics are obtained using the tools of inequality mechanics. These new formulations permit the derivation of some new variational principles for fractured bodies in inequality form. On the basis of the new formulations and variational expressions derived a comparison of the two fracture criteria is attempted showing the superiority of theT-criterion. Certain generalizations of the two criteria complete the presented theory.     

Variational principles in uncoupled thermoplasticity

Stress-rate and strain-rate variational principles are derived for a viscoplastic material characterized by additive elastic and inelastic strain rates. Elastic, viscous and plastic properties are assumed to be temperature dependent but all coupling effects are neglected.     

Variational principles and the patch test

The patch test is shown to be contained in the variational formulations of the finite element methods at the assembling level, all of which require the vanishing of the virtual work of interface connection loads. By a systematic introduction of stress generating functions, attention is drawn to the fact that any given finite element model can be assembled in two different ways: either by identification of a set of boundary displacements (leading to the direct stiffness method), or by identification of a set of local stress function values (leading to the dual direct flexibility method). Looking at any conjugate couple (generalized displacement-generalized surface traction) at an interface, one is strongly transmitted, the other weakly. Discretization of the zero virtual work condition at an interface of plate bending models, by means of Legendre polynomial expansions, allows a systematic construction of so-called ‘non-conforming’ elements that pass the patch test. They are in fact identified with weakly conforming, but strongly diffusive, hybrids, and the lowest degree element (quadratic) is in fact the Morley constant-moment element. Examples are given for higher degree displacement fields. The case of plate stretching elements can be handled by duality, the difficulties being here associated with the requirements for diffusivity. Non-diffusive elements that pass the zero interface virtual work test can be constructed systematically and are identifiable with weakly diffusive, but strongly conforming, hybrids of the type first proposed by T. H. H. Pian.     

Variational principles for BS dimension of subsets

Using the Carathéodory structure, we redefine a new BS-dimension with packing method and prove some properties of this dimension, especially Bowen's equation. We also consider the relation between the new BS-dimension and BS-density which extend the variational principle to the new BS-dimension.     

Generalized fracture mechanics

The application of generalized fracture mechanics to ductile materials is considered. The deformations leading to crack propagation in SEN specimens of a copper-beryllium alloy in two conditions were established experimentally and can be described in terms of the generalized theory. In particular, it was possible to define and measure a surface work for fracture propagation in specimens subject to general yielding. The values of surface work obtained were 7.5×10⁸ mJ m^–2 for the alloy in a low yield stress (235 MPa) condition and 1.2×10⁸ mJ m^–2 for the high yield-stress (725 MPa) condition.On secondment from RARDE (Ministry of Defence) Fort Halstead, Sevenoakes, Kent.     

Non-linear fracture mechanics

The fracture of solids is analyzed using the formalism of equilibrium mechanics. A partially cracked, mechanically loaded reversible system is considered. A stability criteria is formulated. The equilibrium theory is valid for non-linear load deflection relationships and is applied to linear elastic fracture mechanics, the Hertz test and J _IC tests.

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Résumé La rupture des solides est analysée en utilisant le formalisme de la mécanique de l'équilibre. On considère un système partiellement fissuré réversible et chargé mécaniquement. On formule un critère de stabilité. La théorie de l'équilibre est valide pour des relations non linéaires charge déflection et est appliquée à la mécanique de la rupture élastique, à l'essai de Hertz et aux essais J _IC.

     

Quantum fracture mechanics

Quantum fracture mechanics (QFM) is proposed by the author as a purely theoretical alternative of the generally accepted experimental-theoretical approach to fracture physics. The main concepts of theory and formulation of several main problems are examined. Using QFM to solve a problem of the initiation of cracks and dislocations makes it possible to predict the brittle-plastic nature of a specific crystal. Analytical and numerical results of calculations carried out using this theory are represented, including subcritical crack growth in a homogeneous lattice and intergranular layers: in loading and unloading, under stationary loading in creep conditions and in cyclic loading in fatigue conditions.Translated from Problemy Prochnosti, No. 2, pp. 3–9, February, 1990.The author is grateful to Jim Rice for reading the document and for useful comments.