Path-independent integral for the sharp V-notch in longitudinal shear problem. (English) Zbl 1236.74262
Summary: By applying Noether’s theorem to the elastic energy density in longitudinal shear problem, it is shown that its symmetry-transformations of material space can be expressed by the real and imaginary parts of an analytic function. This kind of the symmetry-transformations leads to the existence of a conservation law in material space, which does not belong to trivial conservation laws and whose divergence-free expression gives a path-independent integral. It is found that by adjusting the analytic function, a finite value can be obtained from this path-independent integral calculated around the material point with any order singularity. For a sharp V-notch placed on the edge of homogeneous materials and/or the interface of bi-materials, application shows that the finite value obtained from this path-independent integral is directly related to the notch stress intensity factor (NSIF) and does not depend on the location of integral endpoints chosen respectively along two traction-free surfaces of which form a notch opening angle. Usability is presented in an example to estimate the NSIF of a bi-material plate.
Keywords:
sharp V-notch; notch stress intensity factor; conserved quantity; path-independent integral; Noether’s theoremReferences:
| [1] | Berto, F.; Lazzarin, P.: Relationships between J-integral and the strain energy evaluated in a finite volume surrounding the tip of sharp and blunt V-notches, Int. J. Solids struct. 44, 4621-4645 (2007) · Zbl 1166.74412 · doi:10.1016/j.ijsolstr.2006.11.041 |
| [2] | Cherepanov, G. P.: Crack propagation in continuous media, J. appl. Math. mech. 31, 503-512 (1967) · Zbl 0288.73078 · doi:10.1016/0021-8928(67)90034-2 |
| [3] | Cherepanov, G. P.: Mechanics of brittle fracture, (1979) · Zbl 0442.73100 |
| [4] | Caviglia, G.; Morro, A.: Conservation laws in anisotropic elastostatics, Int. J. Eng. sci. 26, 393-400 (1988) · Zbl 0634.73007 · doi:10.1016/0020-7225(88)90118-8 |
| [5] | Chen, D.; Nisitani, H.: Singular stress field near a corner of jointed dissimilar materials under antiplane loads, Bull. Japan soc. Mech. eng., 399-403 (1992) |
| [6] | Chen, Y. H.; Lu, T. J.: On the path dependence of the J-integral in notch problems, Int. J. Solids struct. 41, 607-618 (2004) · Zbl 1075.74618 · doi:10.1016/j.ijsolstr.2003.10.007 |
| [7] | Dunn, M. L.; Suwito, W.; Cunningham, S.: Stress intensities at notch singularities, Eng. fract. Mech. 57, 417-430 (1997) · Zbl 0942.74632 |
| [8] | Eshelby, J. D.: The force on an elastic singularity, Philos. trans. Roy. soc. London A 244, 87-112 (1951) · Zbl 0043.44102 · doi:10.1098/rsta.1951.0016 |
| [9] | Eshelby, J. D.: Energy relations and the energy-momentum tensor in continuum mechanics, Inelastic behavior of solids, 77-114 (1970) |
| [10] | Fletcher, D. C.: Conservation laws in linear elastodynamics, Arch. rat. Mech. anal. 60, 329-353 (1976) · Zbl 0353.73024 · doi:10.1007/BF00248884 |
| [11] | Gross, R.; Mendelson, A.: Plane elastostatic analysis of V-notched plates, Int. J. Fract. mech. 8, 267-276 (1972) |
| [12] | Knowles, J. K.; Sternberg, E.: On a class of conservation laws in linearized and finite elastostatics, Arch. rat. Mech. anal. 44, 187-211 (1972) · Zbl 0232.73017 · doi:10.1007/BF00250778 |
| [13] | Herrmann, A. G.: On conservation laws of continuum mechanics, Int. J. Solids struct. 17, 1-9 (1981) · Zbl 0445.73002 · doi:10.1016/0020-7683(81)90042-1 |
| [14] | Honein, T.; Herrmann, G.: Conservation laws in nonhomogeneous plane elastostatics, J. mech. Phys. solids 45, 789-805 (1997) · Zbl 0974.74507 · doi:10.1016/S0022-5096(96)00087-7 |
| [15] | Irwin, G. R.: Fracture mechanics, Structural mechanics, 557-591 (1960) |
| [16] | Lazzarin, P.; Livieri, P.; Zambardi, R.: A J-integral-based approach to predict the fatigue strength of components weakened by sharp V-shaped notches, Int. J. Comput. appl. Technol. 15, 202-210 (2002) |
| [17] | Lazzarin, P.; Zappalorto, M.: Plastic notch stress intensity factors for pointed V-notches under antiplane shear loading, Int. J. Fract. 152, 1-25 (2008) · Zbl 1264.74246 |
| [18] | Livieri, P.: A new path independent integral applied to notched components under mode I loadings, Int. J. Fract. 123, 107-125 (2003) |
| [19] | Livieri, P.: Use of J-integral to predict static failures in sharp V-notches and rounded U-notches, Eng. fract. Mech. 75, 1779-1793 (2008) |
| [20] | Matvienko, Y. G.; Morozov, E. M.: Calculation of the energy J-integral for bodies with notches and cracks, Int. J. Fract. 125, 249-261 (2004) · Zbl 1187.74193 · doi:10.1023/B:FRAC.0000022241.23377.91 |
| [21] | Noether, E., 1918. Invariant variation problems. Math-Phys. Klasse (German), pp. 235 – 257. (English Translation: Transport Theory and Statistical Physics, vol. 1, 1971, pp. 183 – 207). · JFM 46.0770.01 |
| [22] | Noda, N.; Takase, Y.: Generalized stress intensity factors of V-shaped notch in a round bar under torsion, tension and bending, Eng. fract. Mech. 70, 1447-1466 (2003) |
| [23] | Olver, P. J.: Conservation laws in elasticity- I: General results, Arch. rat. Mech. anal. 85, 119-129 (1984) · Zbl 0559.73019 · doi:10.1007/BF00281447 |
| [24] | Olver, P. J.: Conservation laws in elasticity- II: Linear homogeneous isotropic elastostatics, Arch. rat. Mech. anal. 85, 131-160 (1984) · Zbl 0582.73024 · doi:10.1007/BF00281448 |
| [25] | Olver, P. J.: Applications of Lie groups to differential equations, (1993) · Zbl 0785.58003 |
| [26] | Qian, J.; Hasebe, N.: Property of eigenvalues and eigenfunctions for an interface V-notch in antiplane elasticity, Eng. fract. Mech. 56, 729-734 (1997) |
| [27] | Rice, J. R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. appl. Mech. 35, 379-386 (1968) |
| [28] | Rice, J. R.: Mathematical analysis in the mechanics of fracture, Fracture: an advanced treatise II, 191-310 (1968) · Zbl 0214.51802 |
| [29] | Seweryn, A.; Molski, K.: Elastic stress singularities and corresponding generalized stress intensity factors for angular corners under various boundary conditions, Eng. fract. Mech. 55, 529-556 (1996) |
| [30] | Shi, Weichen: On the dynamic energy release rate in functionally graded materials, Int. J. Fract. 131, L31-L35 (2005) · Zbl 1196.74224 · doi:10.1007/s10704-005-2597-8 |
| [31] | Shi, Weichen; Gao, Qinghai; Li, Huanhuan: On conservation laws in geometrically nonlinear elasto-dynamic field of non-homogenous materials, Int. J. Eng. sci. 44, 1007-1022 (2006) · Zbl 1213.74153 · doi:10.1016/j.ijengsci.2006.05.014 |
| [32] | Williams, M. L.: Stress singularities resulting from various boundary conditions in angular corners of plates in extension, ASME J. Appl. mech. 19, 526-528 (1952) |
| [33] | Williams, M. L.: On the stress distribution at the base of a stationary crack, ASME J. Appl. mech. 24, 109-114 (1957) · Zbl 0077.37902 |
| [34] | Zappalorto, M.; Lazzarin, P.; Yates, J. R.: Elastic stress distributions resulting from hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings, Int. J. Solids struct. 45, 4879-4901 (2008) · Zbl 1169.74339 · doi:10.1016/j.ijsolstr.2008.04.020 |
| [35] | Zappalorto, M.; Lazzarin, P.; Berto, F.: Elastic notch stress intensity factors for sharply V-notched rounded bars under torsion, Eng. fract. Mech. 76, 439-453 (2009) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
