A new fracture mechanics theory for orthotropic materials like wood

A new fracture mechanics theory is derived based on a new orthotropic-isotropic transformation of the Airy stress function, making the derivation of the Wu-“mixed mode I-II” fracture criterion possible, based on the failure criterion of a flat elliptic crack. As a result of this derivation, the right fracture energy and theoretical relation between mode I and II stress intensities and energy release rates are obtained.     

Griffith theory and development of fracture mechanics criteria

A. A. Blagonravov Institute of Engineering, Russian Academy of Sciences, Moscow. Translated from Fiziko-Khimicheskya Mekhnika, Vol. 29, No. 3, pp. 140–145, May–June, 1993.     

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.     

From Griffith's theory to the fractal fracture mechanics

Conclusion The above analysis shows the necessity for further development of fractal fracture mechanics at micro-, meso-, and macrolevels, using fractal theory and the general principle of synergetics.A. A. Baikov Institute of Metallurgy, Russian Academy of Sciences, Moscow. Published in Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 29, No. 3, pp. 101–106, May–June, 1993.     

A new fracture mechanics

Published in Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 6, pp. 15–22, November–December, 1991.     

Energy theory of linear fracture mechanics of shells with crack-cuts

A variational approach to calculating shells with crack-cuts based on a nonclassical Timoshenko-type theory is proposed and realized. Compliance and virtual crack growth methods are proposed and developed for calculating the stress intensity factor, which is the main characteristic of the local stress field at the crack tip. These methods are realized by the finite-element method in the displacement and finite-difference variant. Analytical and mathematical models of the crack-cut are proposed. A spherical shell with one and two through cracks rectilinear in plan and a square cylindrical panel with a crack located in the center of the diagonal were calculated.Translated from Problemy Prochnosti, No. 7, pp. 59–67, July, 1995.     

A thermodynamic framework of fracture mechanics

A rigorous theory of non-linear fracture mechanics consistent with the basic laws of thermodynamics is proposed. It is shown that the local balance equation(s) can be derived from the postulated global balance equation using a generalized Reynolds transport equation appropriate to a body containing a growing crack. The necessary conditions to be fulfilled on the newly created fracture surfaces are derived. Within the framework of local theory the governing equations are applicable to crackproblems in materials with internal variables or fading memory and to geometrically and/or physically non-linear problems. As a special case, the governing equations for infinitesimal thermoviscoelastic fracture mechanics are given.     

Study of fatigue crack initiation from flaws using fracture mechanics theory

This report presents theoretical and experimental results on fatigue crack initiation from flaws in cyclic loaded structures. The results indicated that the fracture mechanics stress-intensity-factor range. ΔK, is the governing parameter for inducing fatigue crack initiation from a flaw. Theoretical crack growth analysis indicated that when initiating an ‘engineering size’ fatigue crack from a flaw, most of the cyclic behavior was crack growth and only a small part was nucleation.     

Towards a catastrophe theory for the mechanics of plasticity and fracture

The catastrophe theory has established that the conservative systems have only a few ways to lose their stability. We sketch a similar classification for a class of dissipative systems. Although our scope is limited to the case of two dissipative variables, we find six elementary catastrophes. We present examples from discrete plasticity, crack and friction mechanics.     

A variational formulation in fracture mechanics

A variational approach to linear elasticity problems is considered. The family of variational principles is proposed based on the linear theory of elasticity and the method of integrodifferential relations. The idea of this approach is that the constitutive relation is specified by an integral equality instead of the local Hooke’s law and the modified boundary value problem is reduced to the minimization of a nonnegative functional over all admissible displacements and equilibrium stresses. The conditions of decomposition on two separated problems with respect to displacements and stresses are found for the variational problems formulated and the relation between the approach under consideration and the minimum principles for potential and complementary energies is shown. The effective local and integral criteria of solution quality are proposed. A numerical algorithm based on the piecewise polynomial approximations of displacement and stress fields over an arbitrary domain triangulation are worked out to obtained numerical solutions and estimate their convergence rates. Numerical results for 2D linear elasticity problems with cracks are presented and discussed.     

A fracture mechanics analysis of ultrasonic fatigue

The method of ultrasonic fatigue finds increasing interest in materials science. Especially, fatigue crack growth rates near the threshold stress intensity range, $\text{ΔK}{}_0$, can be determined with this method in reasonable times providing no frequency and corrosion effects exist. But for an accurate application of this technique it is necessary to improve the testing systems and also the determination of the dynamic cyclic stress intensity range, $\text{ΔK}$. In this paper, fatigue crack growth experiments at ultrasonic frequencies with different mean stresses and also the calculation of the dynamic stress intensity range with finite elements are treated. On this basis fatigue crack growth curves at room temperature of the alloys Hastelloy X and IN 800 were measured and compared with results obtained at low frequencies. No significant influence of frequency could be found in these materials.     

Fractional generalized Hamiltonian mechanics

In this paper, we present a new fractional theory of dynamics, i.e., the dynamics of generalized Hamiltonian system with fractional derivatives (fractional generalized Hamiltonian mechanics). Based on the definition of Riemann–Liouville fractional derivatives, the fractional generalized Hamiltonian equations are obtained, the gradient representation and second-order gradient representation of the fractional generalized Hamiltonian system are studied, and then the conditions on which the system can be considered as a gradient system and a second-order gradient system are given, respectively. By using the method and results of this paper, the conditions under which a fractional generalized Hamiltonian equation can be reduced to a generalized Hamiltonian equation, a fractional Hamiltonian equation and a Hamiltonian equation are given, respectively, and then the existing conditions and their form of gradient equation and second-order gradient equation are investigated. Finally, an example of a fractional dynamical system is given to illustrate the method and results of the application.     

Adhesive fracture mechanics

Recently Williams, Malyshev and Salganik and others have applied the concepts of fracture mechanics to predict adhesive fracture. The specific adhesive fracture energy, γ_a, is defined as the energy released per unit of new surface created on separation of dissimilar materials. Williams has elucidated the similarity between adhesive and cohesive fracture from the standpoint of a Griffith energy balance analysis. One finds that for either cohesive or adhesive fracture, crack instability (where the crack has a characteristic dimension,a) is predicted by a general equation of the form \(\sigma _{cr} = k\sqrt {E\gamma /a}\) wherek=f [geometry and loading] and includes all loading and geometric factors,E and γ are Young's modulus and specific fracture energy, respectively, and σ_cr is the applied load at incipient failure. Within the fracture mechanics interpretation the adhesive fracture energy γ_a is viewed as a fundamental property of the adhesive system. It is important to note, however, that it may depend on surface preparation, curing conditions, absorbed monolayers, etc. It is, therefore, essential that if γ_a is used to predict adhesive fracture for different geometries, then the surface preparation must be identical with that for the test specimen. If γ_a is a system parameter, then it would be possible to predict fracture by conducting an energy balance analysis of the configuration, utilizing values of γ_a, Young's modulus, and Poisson's ratio as determined from separate simple test specimens. There is, however, a need to establish that γ_a is a system parameter which is independent of geometry. One can in principle perform a number of tests on several specimen configurations or, more effectively, several tests on a single specimen which changes configuration between successive tests. In this latter case the surface and dissimilar materials remain constant. A specimen which is suitable for our purpose was developed by Williams and Jones by incorporating aspects of tests first suggested by Dannenburg and Salganik and Malyshev. The test method considers a disk or plate which has been bonded to a substrate material except for a central portion of radiusa. When pressure,p, is injected into the unbonded region, the plate lifts off the substrate and forms a blister whose radius stays fixed until a critical pressure,p _cr, is reached. At this critical value the radius of the blister increases in size, signifying an adhesive failure along the interface. An energy balance analysis is available for the circular blister specimen in the two limiting cases of a thick plate (Williams) or a very thick medium (Mossakovskii), each with an infinite outer radius. Having established the utility of this general test method, we have considered it necessary to extend the analysis and test calibration capability to other thicknesses for more general engineering applications, as for example, very thin membranes which are used in paint coatings. An axisymmetric finite element numerical analysis was, therefore, conducted for specimens of different thickness and debond radii to establish the energy balance for the various arbitrary thicknesses. A continuous curve for arbitrary specimen thickness was then produced on a dimensionless plot ofp ² a/Eγ versush/a whereh is the specimen thickness. The region ofh/a over which the limiting case equations are valid was also established within the accuracy of the numerical analysis. Since many, if not most, bond geometries are not readily analyzed in closed form, the numerical procedures for energy balance analysis is included and may be used for analyzing geometries other than the blister test specimens. Experiments were conducted over a broad range ofh/a and found to agree with the analytical elasticity solutions which assumed a constant (for given surface preparation, temperature, loading conditions, strain rate independence, etc.) value for γ_a. These experiments confirm that for this system at least γ_a is a constant independent of geometry.     

Generalized fracture mechanics

According to Andrews' generalized fracture mechanics theory [1], the fracture energy of a solid is given by = ₀ where ₀ is a surface energy and a loss function whose form is explicit. The loss function has been evaluated experimentally for four highly extensible materials, styrenebutadiene rubber, ethylene-propylene rubber, plasticized PVC and polyethylene and at various rates of crack propagation. The quantity ₀ has also been calculated from existing theory and a prediction thus obtained for fracture energy. The results indicate good agreement between experiment and theory and thus appear to corroborate the generalized formulation of fracture mechanics in its application to non-linear inelastic materials.     

Short-pulse fracture mechanics

This paper summarizes the results of a multiyear research effort at SRI International aimed at extending classical static fracture-mechanics concepts to apply to cases where a load is applied as a short-duration pulse. A criterion for crack instability under short-pulse loads was deduced from considerations of Sih-Achenbach stress-intensity solutions. Impact experiments were performed on polymers and metallic alloys to observe crack-instability behavior and verify the short-pulse criteria. This new understanding of short-pulse fracture mechanics is being used to help develop procedures for high-rate fracture testing and to guide investigations of fracture incubation time.     

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.