G.C. Sih

Stress Analysis of Cracks

and may b e t r a n s m i t t e d t o t h e E x e c u t i v e S e c r e t a r y , The p a p e r i s s u b j e c t t o m o d i f i c a t i o n and i s n o t t o b e p u b l i s h e d as a whole o r i n p a r t p e n d i n g i t s release by t h e S o c i e t y t h r o u g h t h e E x e c u t i v e S e c r e t a r y , T h i s advance copy T a b l e of C o n t e n t s. Ab s t r a c t I n t r o d u c t i o n Thermal Stresses S t r e s s-I n t e n s i t y-F a c t o r s f o r t h e Bending of P l a t e s and S h e l l s Couple S t r e s s Problems w i t h Cracks E s t i m a t i o n of S t r e s s I n t e n s i t y F a c t o r s f o r some Cases of P r a c t i c a l I n t e r e s t-i

Methods of analysis and solutions of crack problems

It is weH known that the traditional failure criteria cannot adequately explain failures which occur at a nominal stress level considerably lower than the ultimate strength of the material. The current procedure for predicting the safe loads or safe useful life of a structural member has been evolved around the discipline oflinear fracture mechanics. This approach introduces the concept of a crack extension force which can be used to rank materials in some order of fracture resistance. The idea is to determine the largest crack that a material will tolerate without failure. Laboratory methods for characterizing the fracture toughness of many engineering materials are now available. While these test data are useful for providing some rough guidance in the choice of materials, it is not clear how they could be used in the design of a structure. The understanding of the relationship between laboratory tests and fracture design of structures is, to say the least, deficient. Fracture mechanics is presently at astandstill until the basic problems of scaling from laboratory models to fuH size structures and mixed mode crack propagation are resolved. The answers to these questions require some basic understanding ofthe theory and will not be found by testing more specimens. The current theory of fracture is inadequate for many reasons. First of aH it can only treat idealized problems where the applied load must be directed normal to the crack plane.

SOME BASIC PROBLEMS IN FRACTURE MECHANICS AND NEW CONCEPTS

Abstract Although fracture mechanics has gained a sufficient amount of recognition in the past, it is not always clear how the laboratory measurements on fracture toughness or crack growth rate help engineers to design a safe structure. One is hopeful that the current theory can be generalized and extended to explain the various fracture phenomena. However, should it be discovered that the theory was developed on false premises and that the theory and experiment just happen to agree reasonably well, then, guided by the basic laws of physics, we must reconstruct our model. There is no doubt that new concepts are needed to explain many of the presently misunderstood fracture phenomena which are observed on the microscopic level as well as the macroscopic. Without fundamental new ideas, fracture mechanics will remain only a laboratory tool. The object of this paper is to re-examine some of the fundamental concepts of the classical theory of fracture and to discuss the numerous inconsistencies and misconceptions that have not received attention. The following topics have been selected for this discussion: 1. (1) Re-examination of the Griffith concept. 2. (2) Modified Griffith theory for ductile material. In addition, the concept of the strain-energy density field will be introduced and applied to the formulation of a new fracture theory.

Fracture mechanics applied to engineering problems-strain energy density fracture criterion

Abstract Traditional design criteria make no attempt to account for the failure mode which is characteristic of a flawed, frangible structure. A design rationale is outlined for materials which contain flaws caused, for example, by metallurgical inclusions, fabrication and erection overloads, and fatigue cracking. Linear elastic fracture mechanics can be employed successfully for high strength-low toughness materials design. Laboratory test results for critical values of K c , the stress intensity factor, are limited to specimens where the loads are applied symmetrically with respect to the crack plane. The evaluation of a structure due to the influence of a crack will in general involve a mixed or combined load situation where the crack will grow in a curved fashion. This necessitates the application of energy-density factor S , which has a geometrically controlled critical direction. θ and can also serve as a measure of the toughness of the material.

Mixed mode fatigue crack growth predictions

Abstract This work is aimed at developing a predictive capability for the quantitative assessment of crack growth under fatigue loadings. The crack growth rate relation, Δa ΔN , may involve all three stress intensity factors k1-k3 such that the direction of crack growth may not be known in advance and must be predicted from a preassumed criterion. In principle, both the stress amplitude and the mean stress level should be included in the original expression for Δa ΔN . The strain energy density factor range, ΔS, is found to be a convenient parameter for predicting fatigue crack growth and can be applied expediently to examine the combined influence of crack geometry, complex loadings and material properties. Assumed is the accumulation of energy, ΔW ΔV , stored in an element ahead of the crack which triggers subcritical crack growth upon reaching a number of loading cycle, say ΔN. The proposed δa ΔN relationship includes both the stress amplitude and mean stress effects.

Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation

Abstract While the method of solution for crack problems in anisotropic elasticity and magnetoelectroelasticity is nearly identical, the prediction of crack initiation and/or growth behavior based on the asymptotic stress fields can differ widely depending on the chosen failure criterion. The latter has no haven to hide. Avoidance of contraction and satisfaction of the first law of thermodynamics are the rules that need to be observed. A negative crack tip energy release rate would infer the creation of energy and a negative strain energy density would imply the non-uniqueness of solution in classical linear continuum mechanics theories. Under these conditions, the crack initiation and growth behavior in a magnetoelectroelastic material will be examined. For the sake of notation and continuity, this work gives a brief account of the well-known solution for a line crack in a magnetoelectroelastic medium. Not to be underestimated is the derivation of the asymptotic form of the strain energy density function that was first given for the piezoelectroelastic crack problem. The equivalent expression for the magnetoelectroelastic case is given here. This sets the stage for a physical interpretation of how a crack would behave in a composite possessing the mixed properties of piezoelectric and piezomagnetic materials. The individual terms in the strain energy density function shows how the applied electric and magnetic field would affect the critical applied mechanical normal and shear stress to trigger crack initiation, respectively. When both normal and shear action prevail, the direction of crack initiation is no longer obvious and it needs to be determined since the energy stored ahead of the crack depends on the direction of prospective crack initiation. Numerical results are given for a BaTiO 3 –CoFe 2 O 4 composite. The inclusion is the piezoelectric BaTiO 3 material and the matrix is the magnetostrictive CoFe 2 O 4 material. The proportion of the two phases can be varied by adjusting the volume fraction of the inclusions upon which the fluctuation of the energy density field near the crack depends and the material properties of the constituents. Directions of the applied electric and magnetic field can also be changed; their effects on crack initiation are discussed using the strain energy density criterion. Indeed, the additional magnetostrictive effect can have an influence on crack initiation as the applied field directions are altered. Possible behavior of crack initiation can be made for design-specific magnetoelectroelastic composites.

Magnetic and electric poling effects associated with crack growth in BaTiO3-CoFe2O4 composite

Abstract Magnetoelectroelastic composite possesses the dual feature that the application of magnetic field induces electric polarization and electric field induces magnetization. The poling directions introduced magnetically and electrically can be different in addition to those for the applied magnetic and electric field. Their choices can influence the character of crack growth which could be enhanced or retarded. The details of how the directions of poling and applied field would affect crack initiation and growth are discussed in relation to the volume fraction of inclusions for a BaTiO3–CoFe2O4 two phase composite. The multi-functional aspects of magnetoelectroelastic materials are involved since they entail multi-scaling features. Failure criteria that applies to isotropic elastic materials may not hold for composites exhibiting piezomagnetic and piezoelectric properties. For instance, a negative energy release rate has been obtained for cracks in piezoelectric materials. In view of what has been said with reference to the energy release rate approach, it is desirable to use the strain energy density function as a failure criterion, even if it is only for its positive definiteness character. Physically speaking, it is attractive to have a function that could rank the proportion of energy related to volume and shape change. They determine the proportion of the hard and soft phase of the composite and hence the volume fraction of the constituent. Strength and toughness parameters used for ranking isotropic and homogeneous materials will not apply for anisotropic and/or nonhomogeneous materials if these microstructure effects could not be suppressed to a lower scale and represented as an average at the macroscopic scale. Too much emphases cannot be placed on the need to clarify the multi-scaling aspects of piezoelectric and piezomagnetic materials. Their behavior as affected by the presence of crack-like defects should be understood prior to deciding whether the material characterization approach would be suitable. That is whether simplicity could justify at the expense of conceptual rigor. Much of this would depend on scaling the time and size related to loading and material structure interaction. The magnetoelectroelastic crack model selected in the work to follow perhaps will provide an insight into the complexicity of the state of affairs for treating the finer details of material behavior with rigor. The proposed test model shows that crack growth in the magnetoelectroelastic materials can be suppressed by increasing the magnitude of the piezomagnetic constants in relation to those for piezoelectricity. A more rational means of evaluating the resistance of materials against fracture is thus proposed, particularly when anisotropy and inhomogeneity might be present.

The strain energy density failure criterion applied to notched elastic solids

Abstract Application of the strain energy density failure criterion is made to plane notch problems, where the crack now becomes a special case of a more generalized approach to failure. The specific case considered is that of the plane elliptical cavity under remote tension and compression. Both failure loads and fracture trajectories are discussed. It is shown that an additional characteristic dimension provides satisfactory agreement of the theory with available data. Finally, known characteristics of fracture trajectories from a notch tip are shown to be predicted for unstable fracture conditions.

Impact response of a finite crack in plane extension

The transient stress and displacement fields around a finite crack opened out by normal and shear tractions applied to its surface are obtained using integral transforms coupled with the technique of Cagniard. The tractions are applied suddenly to the crack which simulates the case of impact loading. By subtracting a uniform state of stress, the present solution also applies to the problem of the sudden appearance of a crack in a pre-stressed body. The results show that significant differences exist between the dynamic stress-intensity factors obtained in this problem and those resulting from static loading. In particular, the energy released by the dynamically loaded crack which is associated with the stress-intensity factor varies with time reaching a maximum very quickly and then oscillates about the static value. The information gained is useful in determining the condition of crack propagation under impact.

Sharp notch fracture strength characterized by critical energy density

Abstract As the order of the stress singularity at a sharp notch tip varies with its vertex angle, the units for the coefficient that represents the amplitude of the local stress field changes accordingly. Only in the limit of a zero notch angle that the coefficient could be directly associated with the energy release concept infracture mechanics owing to the preservation of self-similar crack extension provided that the loads are applied symmetrically across the crack plane. For a finite notch angle, self-similarity would no longer be preserved before and after fracture even for symmetrical loading. The delineation of notch geometry from material behavior is a prerequisite for notch strength characterization and requires a more general consideration. By focusing attention on an element of material ahead of the notch, failure by yielding and/or fracture could be predicted from the stationary values of the volume energy density d W /d V regardless of the order of the notch tip stress singularity. Fracture initiation is associated with the critical value of d W /d V or (d W /d V ) c being characteristics of the material. As the stress singularity increases with decreasing notch angle, the critical applied stress to initiate failure also decreases and the initial ligament of fracture becomes more localized. This effect is much more pronounced for skew-symmetrical loading where the stress singularity becomes diminishingly weak as the half notch angle reaches 60°, a result that is not unexpected. The direction of fracture initiation for in-plane shear load was found to be away from the line that bisects the notch. It varied from ± 74.5° to ± 74.8° and ± 86.2° to ± 85.8° for Poissons ratios of 0.1 and 0.4, respectively as the half notch angle is increased from 0° (crack) to 60°. The Poissons ratio had a more appreciable influence on the crack initiation direction than the notch angle. This is in contrast to the maximum normal stress criterion that showed marked changes of the fracture angle with notch angle. The fracture angle decreases from ± 70.5° to ± 52.0° as the half notch angle is increased from 0° to 60°. Numerical solutions are obtained for different notch angles under symmetrical and skew-symmetrical loading. The rate change of volume to surface ratios for elements near the notch tip are compared with those obtained analytically and used as a guide for developing the finite element grid patterns. The distance of the nearest notch tip element in the numerical analysis tends to decrease with decreasing stress singularity. Such a knowledge can benefit the selection of finite element mesh sizes and distribution near sharp notches.