A finite fracture mechanics approach to structures with sharp V-notches

Criteria assuming that failure of quasi-brittle materials is affected by the stresses acting over a finite distance from the crack tip are widely used inside the scientific community. For instance, they have been applied to predict the failure load of specimens containing sharp V-notches, assuming as a critical parameter the average stress ahead the notch tip. However, this kind of approaches disregards energy balance considerations, which, as well known, are the basis of linear elastic fracture mechanics (LEFM). In order to overcome these drawbacks, the present paper uses a recently introduced finite fracture mechanics (FFM) criterion, i.e. a fracture criterion assuming that crack grows by finite steps. The length of this finite extension is determined by a condition of consistency of both energy and stress requirements; as a consequence, the crack advancement is not a material constant but a structural parameter. The criterion is applied to structures with sharp V-notches. The expression of the generalized fracture toughness, which is a function of material tensile strength, fracture toughness and notch opening angle, is given analytically. Finally, we provide comparisons with: (i) the experimental data we obtained from testing Polystyrene specimens under three point bending; (ii) some experimental data available in the literature. The agreement between theoretical predictions and experimental results is generally satisfactory and, for most of the cases analyzed, the FFM predictions are better than the ones provided by the simple average stress approach.     

Brittle failures at rounded V-notches: a finite fracture mechanics approach

Finite Fracture Mechanics (FFM) is applied to investigate the brittle failure behavior of rounded V-notched elements subjected to mode I loading. According to the criterion, fracture does not propagate continuously, but by finite crack extensions, whose value is determined by the contemporaneous fulfilment of a stress requirement and an energy balance. Consequently, the crack advance becomes a structural parameter. By assuming the generalized apparent stress intensity factor as the governing failure parameter, as expected for a brittle structural behavior, the expression of the apparent generalized fracture toughness as a function of the material properties as well as of the notch opening angle and root radius is achieved. FFM predictions are then successfully compared to experimental data available in the Literature and to results provided by other theoretical approaches based on a critical distance.     

Use of J-integral to predict static failures in sharp V-notches and rounded U-notches

In the present paper, the physical meaning of J_V (namely, the classic J-integral applied to either sharp V-notch) is discussed. Consider a Cartesian reference frame having the x-axis parallel to the notch bisector, each mode of J_V, for a given circular path, is proportional to the correspondent mode of the classic J-integral of a virtual crack having length equal to the path radius and emanating from the tip of the V-notch. Analytical and numerical results have been performed for linear elastic materials. Additionally, in order to verify the formulations of J_V, experimental result of embedded cracks of sharp V-notch was considered.Then, by introducing a characteristic path radius ρ^∗, assumed to be dependent only on the material properties, the J_V parameter was used for the estimation of the static failure load of sharp V-notches specimens under mode I loading.Furthermore, the J_Vρ parameter (namely, the classic J-integral applied to U-rounded notches) was used to analyze the static failure of two new series of specimens with double U-notches made of brittle material (PMMA and PVC glass) subjected to tensile loading. This method allowed us to prove that when the ratio between the notch tip radius and ρ^∗ is small the approach agrees with the classic J-integral, whereas when ρ^∗ becomes small with respect to the notch tip radius, the J_Vρ method agrees with the classic peak stress approach.     

The use of linear elastic fracture mechanics for particulate solids

The fracture of model porous particulate solids consisting of sand and low volume fractions of a polymeric binder has been investigated. In order to apply a linear elastic fracture mechanics analysis it was necessary to use an effective crack length comprising the sum of the notch depth and a quantity a. The behaviour of the model solids was intermediate between that exhibited by ceramics and polymers. A possible interpretation of a is that it represents a process zone size where the energy dissipation mechanism could involve either microcracking or yielding of the interparticle junctions in the zone. The former would correspond to ceramic-like behaviour while the latter is characteristic of polymers.     

Isovector fields for problems in the mechanics of solids and fluids

Practical aspects of the application of isovector methods to equations of balance are summarized. The dimension reduction argument given in the appendix together with programs in the REDUCE-2 algebraic manipulation language reduce the computation time from man-years to a few mornings work at a computer terminal. Specific applications to commonly encountered problems are given. All isovector fields for the following systems are exhibited: steady incompressible viscous flows in 2 spatial dimensions; steady thermal boundary layer equations in 2 spatial dimensions; unsteady incompressible viscous flows in 3 spatial dimensions; and dynamic thermoelasticity in 3 spatial dimensions. These results give immediate information concerning the connectivity of the solution sets and similarity variables. In particular, it is shown that only solutions of the unsteady incompressible viscous flow equations with the same local continuity classes can be connected.     

An approximate, analytical approach to the `HRR'-solution for sharp V-notches

The well-known so-called `HRR-solution' (Hutchinson, 1968 and Rice and Rosengren, 1968) considers the elasto-plastic stress field in a power-law strain hardening material near a sharp crack. It provides a closed form explicit expression for the stress singularity as a function of the power-law exponent `n' of the material, but the stress angular variation functions are not found in closed form. More recently, similar formulations have appeared in the literature for sharp V-notches under mode I and II loading conditions. In such cases not only is the angular variation of the stress fields obtained numerically, but so is the singularity exponent of the stress field. In the present paper, approximate but accurate closed form solutions are first reported for sharp V-notches with an included angle greater than /6 radians. Such solutions, limited here to Mode I loading conditions, allow a very satisfactory estimate of the angular stress components in the neighbourhood of the notch tip, in the entire range of notch angles and for the most significant values of n (i.e. from 1 to 15). When the notch opening angle tends towards zero, and the notch approaches the crack case, the solution becomes much more complex and a precise evaluation of the parameters involved requires a best-fitting procedure which, however, can be carried out in an automatic way. This solution is also reported in the paper and its degree of accuracy is discussed in detail.     

A criterion for low-stress brittle fracture of materials with crack or sharp notch based on combined micro-and-macro fracture mechanics

For fracture of solids with crack, the following two requisites should be satisfied:

1. (1) The local stress at or near the tip of the crack should exceed the ideal strength, that is, atomic cohesion, and

2. (2) The free energy should decrease with propagation of the crack, that is, the energy unstable condition should be satisfied.

In low-stress brittle fracture of materials with crack or sharp notch, it has been usually considered that the local stress at or near the tip of the crack could not reach the atomic cohesion level. Then many investigators have been led to the conclusions that critical displacement should be assumed instead of critical stress or energy unstable condition. This appears to be the reason why the COD concept has been widely assumed. Thus it is to be noted that the explanation of the phenomenon by the COD concept concerns essentially the rupture or slipping off mechanism rather than fracture.

On the other hand, when plastic flow occurs at or near the tip of the crack, dislocation will move and thus, if there are any obstacles near, then pile-up will occur. In this article, the stress concentration caused by such microscopic mechanism is considered, and thus the local stress near the tip of the crack will reach the ideal strength, say, atomic cohesion, and, that is, the first requisite mentioned above is satisfied.

Then, the energy balance condition as the second requisite is studied in the article.

Comparing the two critical stresses as satisfying the first and the second requisites respectively, the criterion for low-stress brittle fracture is discussed.

Furthermore, the experimental study was made on polymethyl-methacrylate, and the result is in good agreement with the energy unstable condition.     

Some problems in experimental fracture mechanics

A standard procedure for the determination of fracture toughness K_IC is discussed. The insufficiency of the existing K_ic determination confidence criteria is stressed and the following criteria are proposed instead: φ_max ? 1.5%; $\text{σ}{}_{\mathrm{fr}}{}^{\mathrm{net}}\text{σ}{}_{\mathrm{0.2\; ?\; 0.8}}$, in conjunction with the old criterion $\text{P}{}_{\max }\text{P}{}_{\mathrm{Q}}\text{? 1.1}$. Determination of K_IC from P_max should be used instead of from P_Q.A method for the determination of a point on the “force-displacement” diagram corresponding to crack growth initiation is set forth. The method is based on specimen compliance tests under repeated load-relief cycles. The crack growth initiation point is used to determine both the critical crack opening and plane strain fracture toughness. The indefinite effect of the growing crack (in the ease of crack opening or Cherepanov-Rice integral calculations) is thereby eliminated. Necessity is emphasized to determine the share of the J-integral which contributes to fracture process. A method for plotting the elastic displacement diagram is proposed which allows on the basis of preliminary estimates to determine fracture toughness of small-sized specimens without using special setups. The area ratio between the plastic and elastic strain diagrams is proposed to be adopted as fracture type criterion. Certain experiments to determine crack resistance of material specimens are described and discussed.     

Two-dimensional dynamical problems for incompressible isotropic linear elastic solids with time dependent moduli and variable density

Summary This paper describes a new general method of solution of two-dimensional problems involving the dynamical behaviour of incompressible linear elastic solids. As formulated, the solution takes account of density variation with both position and time and a shear modulus which is purely time dependent. Several simple applications are made to problems involving thick cylinders and an infinite plate containing a circular hole.

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Zusammenfassung Diese Arbeit beschreibt eine neue, allgemeine Methode zur Lösung von zweidimensionalen, dynamischen Problemen von inkompressiblen, linear-elastischen Materialien. Die Lösung berücksichtigt die Abhängigkeit der Dichte von Ort und Zeit und die Abhängigkeit des Schubmoduls von der Zeit allein. In einigen einfachen Anwendungen werden Probleme gelöst, die den dicken Zylinder und die unendliche Platte mit einem kreisrunden Loch enthalten.

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With 2 Figures     

The micro-statistical fracture mechanics approach to dynamic fracture problems

Continuum fracture mechanics concepts should be applied to solve dynamic fracture problems wherever a continuum approach can provide sufficient answers. Many dynamic fracture problems, however, involve multiple cracks or voids and are not well-suited to the relatively simple continuum approach. This paper describes a statistical fracture mechanics concept on a microscopi scale and illustrates its use for the case of shock-wave-induced ductile spall fracture. The paper further shows how micro-statistical fracture mechanics (MSFM) merges with continuum fracture mechanics by treating a macrocrack propagating in a DCB specimen using MSFM and the MSFM parameters deduced from the spall work.Thus, the two approaches are consistent. Although significantly more complicated, the MSFM approach promises to be helpful in improving our understanding of the influences of microstructure on toughness and in extending continuum approaches to more ductile materials and to smaller specimens.

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Résumé Les concepts de fracture mécanique d'un continuum devraient être appliqués à la solution des problèmes de fracture de rupture dynamique dès lors qu'une approche par continuum peut fournir des réponses suffisantes. Cependant, plusieurs problèmes de rupture dynamique comportent des fissures ou des cavités multiples et ne sont pas bien adaptés à une approche par continuum relativement simple. Le mémoire décrit un concept de mécanique de la rupture statistique à une échelle microscopique et illustre son utilisation au cas d'une rupture par décollement ductile sous l'effet d'une onde de choc. On montre en outre qu'une mécanique de rupture micro-statistique peut être envisagée dans le mécanisme de rupture d'un continuum en traitant la propagation d'une macro-fissure dans une éprouvette Cantilever et en utilisant ce concept et les paramètres qui y sont liés, déduits du travail relatif à l'écaillage.Ainsi, les deux approches se révèlent applicables. Bien que considérablement compliquées, l'approche de mécanique de rupture micro-statistique se révèle prometteuse pour améliorer notre compréhension des influences de la microstructure sur la ténacité et pour une extension de l'approche du continuum de milieu continu à des matériaux plus ductiles ou à des éprouvettes plus petites.

     

Relationship between the stress intensity and stress concentration factors for sharp and rounded notches

The method of singular integral equations is used to find the solution of the plane problem of the theory of elasticity for a plane containing an infinite V-shaped rounded notch. This enables us to establish the relationship between the stress intensity factor at the tip of the sharp V-shaped notch, the stress concentration factor for the corresponding rounded notch, and the radius of curvature at the notch tip. It is shown that the indicated relationship is not unique. Indeed, for the same curvature at the notch tip, we get different dependences for different shapes of its vicinity. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 6, pp. 17–26, November–December, 2006.     

Deformation fracture criterion for bodies with V-notches under the conditions of longitudinal shear

The force fracture criterion is especially extensively used for the analysis of the limiting equilibrium of bodies with V-notches. In this case, the critical stress intensity factor depends on the opening angle of the notch. We propose a deformation fracture criterion containing only standard characteristics of the material, i.e., the yield strength and crack resistance. The criterion is developed on the basis of the closed solution of the elastoplastic problem for an infinite wedge under the conditions of longitudinal shear. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 43, No. 1, pp. 49–53, January–February, 2007.     

Application of infrared spectroscopy in fracture mechanics problems

Presented at First All-Union Conference on Fracture Mechanics of Materials, Lvov, October 20–22, 1987.