Статті в журналах: "Crack-tip asymptotic expansion"

1

Gerasimova, T. E., P. N. Lomakov, and L. V. Stepanova. "NUMERICAL PHOTOMECHANICS: NUMERICAL PROCESSING OF PHOTOELASTICITY EXPERIMENTS AND ITS APPLICATION TO THE PROBLEMS OF FRACTURE MECHANICS PROBLEMS." Vestnik of Samara University. Natural Science Series 19, no. 9.2 (June 6, 2017): 63–73. http://dx.doi.org/10.18287/2541-7525-2013-19-9.2-63-73.

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On the basis of photoelasticity method the experimental study of near crack tip stressed strain state in specimens under mixed loading conditions is performed. Carried out experimental investigation allows to obtain coefficients of full asymptotic expansions of stress and displacement fields in the vicinity of the crack tip and alos to find coefficients of highest approach in Williams full asymptotic expansion.

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2

Dai, Yao, Xiu Fa Yan, Chang Qing Sun, and Wei Tan. "The Crack-Tip Higher Order Asymptotic Fields for a Mode III Crack in a Functionally Gradient Material." Advanced Materials Research 33-37 (March 2008): 713–18. http://dx.doi.org/10.4028/www.scientific.net/amr.33-37.713.

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Crack-tip higher order stress and displacement fields for a mode III crack along the direction of property variation in a functionally gradient material (FGM), which has a power variation of shear modulus along the gradient direction, are obtained through the asymptotic analysis. The asymptotic expansions of crack tip stress fields are derived to explicitly bring out the influence of non-homogeneity on the structure of the stress field. The analysis reveals that only the higher order terms in the expansion are influenced by the material non-homogeneity. Moreover, it can be seen from expressions of higher order stress fields that at least three terms must be considered in the case of FGMs in order to explicitly and theoretically account for non-homogeneity effects on crack tip stress fields.

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3

Stepanova, L. V., and P. S. Roslyakov. "MULTIPARAMETRIC ANALYSIS OF THE STRESS FIELD NEAR THE CRACK TIP." Vestnik of Samara University. Natural Science Series 21, no. 10 (May 15, 2017): 52–76. http://dx.doi.org/10.18287/2541-7525-2015-21-10-52-76.

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Анотація:

The paper is devoted to analytical determination of coefficients of the Williams asymptotic expansion of the stress field in the neighborhood of two collinear crack tips in an infinite plate under mixed mode loading. On the basis of the Kolosof-Muskhelishvili approach the complete asymptotic expansion of the stress field in the vicinity of the crack tips of two collinear cracks of equal lengths under mixed mode loading is derived. The analysis of the higher order terms in the asymptotic expansion series is performed. It is clear that it is necessary to take into account the higher order terms.

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4

Dai, Yao, and Xiao Chong. "The Higher Order Crack Tip Fields for Physical Weak-- Discontinuous Crack of FGM Plate with Reissner’s Effect." Advanced Materials Research 664 (February 2013): 841–45. http://dx.doi.org/10.4028/www.scientific.net/amr.664.841.

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The physical weak-discontinuous problem of an interfacial crack between homogeneous material and functionally graded materials (FGMs) is studied based on Reissner’s plates considering transverse shear deformation effect. The crack-tip higher order asymptotic fields of homogeneous materials and FGMs regions are obtained by the asymptotic expansion method, respectively. Finally, the whole crack tip high order fields are assembled and given. The results provide a theoretical basis for solving interfacial crack problems of FGMs plates and their engineering application.

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5

Stepanova, L. V., and V. S. Dolgikh. "EXPERIMENTAL DETERMINATION OF COEFFICIENTS OF A MULTIPARAMETER DECOMPOSITION OF FIELD OF CRACK TIP STRESSES: PHOTOELASTICITY METHOD." Vestnik of Samara University. Natural Science Series 23, no. 1 (September 20, 2017): 59–68. http://dx.doi.org/10.18287/2541-7525-2017-23-1-59-68.

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The purpose of this study is multiparameter asymptotic analysis of the stress field in the immediate vicinity of the crack tip in a linearly elastic material and construction of complete asymptotic expansion of M. Williams stress field in the vicinity of the crack tip. Multiparametric analysis of the stress field is based on the polarization-optical methods of mechanics of a deformable solid (the method of photoelasticity). Digital processing of the results of optoelectronic measurements performed on a series of samples with cracks and notches is carried out. Different classes of samples from optically sensitive materials, in particular a sample with two collinear cracks under conditions of normal detachment, were considered. A set of programs has been prepared that makes it possible to determine the scale (amplitude) multipliers of complete asymptotic expansion of M.Villiams for the stress field at the crack tip. Using the basic law of photoelasticity, first five coefficients of complete asymptotic expansion of M. Williams are calculated. The results of the experiments are compared with the available analytical solution. It is shown that the results of processing optoelectronic measurements are in good agreement with the analytical solution obtained for an infinite plate with two collinear cracks.

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6

Dai, Yao, Jun Feng Liu, Lei Zhang, Xiao Chong, and Hong Qian Chen. "Higher Order Crack-Tip Fields of Reissner’s Linear Functionally Graded Materials Plates." Advanced Materials Research 549 (July 2012): 914–17. http://dx.doi.org/10.4028/www.scientific.net/amr.549.914.

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Reissner’s plate bending fracture theory with consideration of transverse shear deformation effects is adopted for the crack problem of functionally graded materials (FGMs) plates. Assume that the crack is perpendicular to the material property gradient. By applying the asymptotic expansion method, the crack-tip higher order asymptotic fields which are similar to Williams’ solutions of homogeneous materials are obtained.

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7

Dai, Yao, Shi Min Li, Peng Zhang, and Xiao Chong. "The Higher Order Asymptotic Fields for Anti-Plane Oblique Crack with Physical Weak-Discontinuity." Advanced Materials Research 217-218 (March 2011): 1309–13. http://dx.doi.org/10.4028/www.scientific.net/amr.217-218.1309.

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An arbitrarily oriented anti-plane crack with its tip at the physical weak-discontinuous line of the structure which is made up of homogeneous material and functionally graded materials (FGMs) is studied. The analytic solution of the higher order crack tip fields (similar to the Williams’ solution of homogenous material) is obtained by applying the asymptotic series expansion. When non-homogeneous material parameters are degenerated, the solutions become the same as the asymptotic crack tip fields of the homogeneous material. Therefore, the solutions are the basic results of non-homogeneous fracture mechanics, and provide a theoretical basis for solving the fracture problems of one common structure with physical weak-discontinuity.

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8

Dai, Yao, Jun Feng Liu, Xiao Chong, and Lei Zhang. "The Crack-Tip Higher Order Field of Power-Law FGMs Plates with Reissner’s Effect for Crack Parallel to Material Property Gradient." Advanced Materials Research 664 (February 2013): 821–24. http://dx.doi.org/10.4028/www.scientific.net/amr.664.821.

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Reissner’s plate bending fracture theory with consideration of transverse shear deformation effects is adopted for the crack problem of power-law functionally graded materials (FGMs) plates. Assume that the crack is parallel to the material property gradient. By applying the asymptotic expansion method, the crack-tip higher order asymptotic fields which are similar to Williams’ solutions of homogeneous materials are obtained.

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9

Dai, Yao, Lei Zhang, Xiao Chong, and Chun Fang Xue. "The Higher Order Crack-Tip Field of Circumferential Crack for Reissners FGMs Cylindrical Shell." Advanced Materials Research 791-793 (September 2013): 754–57. http://dx.doi.org/10.4028/www.scientific.net/amr.791-793.754.

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Reissners shell theory is employed to analyze the circumferential crack problem for FGMs cylindrical shell by using the asymptotic expansion method. The higher order crack tip fields for circumferential FGMs cylindrical shell which is similar to Williams solution are obtained.

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10

Dai, Yao, Lei Zhang, Jun Feng Liu, Xiao Chong, and Hong Qian Chen. "The Higher Order Asymptotic Crack-Tip Field for Reissner’s Linear Functionally Graded Shell." Advanced Materials Research 549 (July 2012): 826–29. http://dx.doi.org/10.4028/www.scientific.net/amr.549.826.

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The eigen-problem of a crack in functionally graded Reissner’s spherical shell is analyzed. By adopting the asymptotic expansion method, the higher order crack tip asymptotic fields which are similar to the Williams’ solutions of plane crack problems in homogenous materials are obtained. The grade direction is assumed to be parallel to the crack. The results can be widely adopted in numerical analysis, experimental investigation and the engineering application of FGM shell structure.

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11

Dai, Yao, Peng Zhang, Zhao Quan Zheng, and Wei Tan. "The Higher Order Asymptotic Fields for a Anti-Plane Crack in a Linear Functionally Gradient Material." Advanced Materials Research 97-101 (March 2010): 1782–85. http://dx.doi.org/10.4028/www.scientific.net/amr.97-101.1782.

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The exponential and power material functions are often applied to functionally gradient materials (FGM). Obviously, it is of fundamental significance to study FGM with arbitrary material function. Because an arbitrary function can be treated as finite linear segments approximately, it is essential to research FGM with a linear material function. Crack-tip higher order stress and displacement fields for an anti-plane crack perpendicular to the direction of property variation in a FGM with a linear shear modulus along the gradient direction are obtained through the asymptotic analysis. The asymptotic expansions of crack tip stress fields bring out explicitly the influence of non-homogeneity on the structure of the stress field. The analysis reveals that only the higher order terms in the expansion are influenced by the material non-homogeneity. Moreover, it can be seen from expressions of higher order stress fields that at least three terms must be considered in the case of FGM in order to explicitly and theoretically account for non-homogeneity effects on crack tip stress fields.

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12

Dai, Yao, and Xiao Chong. "The Higher Order Crack Tip Fields of Exponential Gradient FGMs Plates with Reissner’s Effect." Applied Mechanics and Materials 278-280 (January 2013): 491–94. http://dx.doi.org/10.4028/www.scientific.net/amm.278-280.491.

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Анотація:

The Reissner’s plate bending theory with consideration of transverse shear deformation effects is adopted to study the fundamental fracture problem in functionally graded materials (FGMs) plates for a crack perpendicular to material gradient. The crack-tip higher order asymptotic fields of FGMs plates are obtained by the asymptotic expansion method. This study has fundamental significance as Williams’ solution.

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13

Dai, Yao, Lei Zhang, and Xiao Chong. "The Higher Order Crack-Tip Field for FGMs Shells with Reissner's Effect." Applied Mechanics and Materials 427-429 (September 2013): 129–32. http://dx.doi.org/10.4028/www.scientific.net/amm.427-429.129.

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Анотація:

The theory for plates and shells with Reissners effect is adopted to analyze the crack problem for FGMs cylindrical and spherical shells. The higher order crack-tip fields for power function FGMs shells obtained by asymptotic expansion method are used, the eigen-solutions of the crack tip fields for arbitrary material functions of FGMs cylindrical and spherical shells which are similar to Williams solution are given by superposition principle.

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14

Dai, Yao, Chang Qing Sun, Sun Qi, and Wei Tan. "Analytical Solutions for Crack-Tip Higher Order Stress Fields in Thermal Barrier Coating." Advanced Materials Research 47-50 (June 2008): 1023–26. http://dx.doi.org/10.4028/www.scientific.net/amr.47-50.1023.

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Анотація:

Analytical expressions for crack-tip higher order stress functions for a plane crack in a special functionally graded material (FGM), which has an variation of elastic modulus in 1 2 power form along the gradient direction, are obtained through an asymptotic analysis. The Poisson’s ratio of the FGM is assumed to be constant in the analysis. The higher order fields in the asymptotic expansion display the influence of non-homogeneity on the structure of crack-tip fields obviously. Furthermore, it can be seen from expressions of higher order stress fields that at least three terms must be considered in the case of FGMs in order to explicitly account for non-homogeneity effects on the crack- tip stress fields. These results provide the basis for fracture analysis and engineering applications of this FGM.

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15

Dai, Yao, Jun Feng Liu, Lei Zhang, and Xiao Chong. "The Crack-Tip Higher Order Fields of Linear FGM Plates with Reissner’s Effect." Advanced Materials Research 476-478 (February 2012): 1421–24. http://dx.doi.org/10.4028/www.scientific.net/amr.476-478.1421.

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Анотація:

The Reissner’s plate bending theory with consideration of transverse shear deformation effects is adopted to study the fundamental fracture problem in functionally graded materials(FGMs) plates for a crack parallel to material gradient. By means of the asymptotic expansion method, the crack-tip higher order asymptotic fields which are similar to the famous Williams’ solutions to homogeneous materials are obtained.

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16

Dai, Yao, Xiao Chong, and Shi Min Li. "The Higher Order Crack Tip Fields for Anti-Plane Crack in Functionally Graded Piezoelectric Materials." Applied Mechanics and Materials 472 (January 2014): 617–20. http://dx.doi.org/10.4028/www.scientific.net/amm.472.617.

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The anti-plane crack problem is studied in functionally graded piezoelectric materials (FGPMs). The material properties of the FGPMs are assumed to be the exponential function of y. The crack is electrically impermeable and loaded by anti-plane shear tractions and in-plane electric displacements. Similar to the Williams solution of homogeneous material, the high order asymptotic fields are obtained by the method of asymptotic expansion. This investigation possesses fundamental significance as Williams solution.

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17

Dai, Yao, Lei Zhang, Xiao Chong, and Ying Chen. "The Higher Order Crack-Tip Field for Crack along x-Axis in Reissner's Bidirectional FGMs Cylindrical Shell." Advanced Materials Research 791-793 (September 2013): 758–61. http://dx.doi.org/10.4028/www.scientific.net/amr.791-793.758.

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Анотація:

The crack located along x-axis in bi-directional functionally graded cylindrical Reissners shell is studied. The asymptotic expansion method is used to obtain the eigen-solution of the higher order crack tip fields. The results are similar to Williams solutions to plane problems for homogeneous materials.

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18

Stepanova, Larisa, Pavel Roslyakov, and Tatjana Gerasimova. "Complete Williams Asymptotic Expansion near the Crack Tips of Collinear Cracks of Equal Lengths in an Infinite Plane." Solid State Phenomena 258 (December 2016): 209–12. http://dx.doi.org/10.4028/www.scientific.net/ssp.258.209.

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The present study is aimed at analytical determination of coefficients in crack tip expansion for two collinear finite cracks of equal lengths in an infinite plane medium. The study is based on the solutions of the complex variable theory in plane elasticity theory. The analytical dependence of the coefficients on the geometrical parameters and the applied loads for two finite cracks in an infinite plane medium is given. It is shown that the effect of the higher order terms of the Williams series expansion becomes more considerable at large distances from the crack tips. The knowledge of more terms of the stress asymptotic expansions allows us to approximate the stress field near the crack tips with high accuracy.

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19

Dai, Yao, Lei Zhang, and Xiao Chong. "The Higher Order Circumferential Crack-Tip Field for Power FGMs Cylindrical Shell with Reissner's Effect." Applied Mechanics and Materials 419 (October 2013): 107–10. http://dx.doi.org/10.4028/www.scientific.net/amm.419.107.

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Анотація:

The theory for plate and shell with Reissners effect is adopted to analyze the circumferential crack problem for FGMs cylindrical shell by using the asymptotic expansion method. The eigen-solution of the crack tip field for circumferential power type FGMs cylindrical shell which is similar to Williams solution is given.

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20

Zhabbarov, R. M., and L. V. Stepanova. "COMPARATIVE ANALYSIS OF STRESS STATE IN THE VICINITY OF THE CRACK-TIPS AND NOTCHES USING TRUNCATED WILLIAMS SERIES EXPANSION." Vestnik of Samara University. Natural Science Series 27, no. 4 (October 14, 2022): 30–67. http://dx.doi.org/10.18287/2541-7525-2021-27-4-30-67.

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In the present study the asymptotic analysis of the multi-parameter Williams series expansion of the stress-strain state in the vicinity of seven different configurations is performed. The aim of the asymptotic analysis is to educate the sensitivity of different cracked and notched configurations to the number of kept terms in Williams series expansion. It is shown that all configurations with the concentrated forces possess the more sensitivity to the number of terms in the truncated Williams series expansions and, therefore, require the more terms in the series expansion. The cracked configurations with distributed loadings are less sensitive to the number of terms in Williams series expansion. As a whole, it is demonstrated that high order terms have principal role in the description of the stress-strain in the vicinity of the crack tip.

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21

Dai, Yao, Zhang Lei, and Xiao Chong. "The Higher Order Crack-Tip Field for Reissners Power FGMs Spherical Shell." Advanced Materials Research 748 (August 2013): 354–57. http://dx.doi.org/10.4028/www.scientific.net/amr.748.354.

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The crack problem of power functionally graded spherical shell with Reissners effect is studied. Based on the Reissners theory, the governing equation of power functionally graded spherical shell is given. The eigen-solution of the crack tip field is obtained by using the asymptotic expansion method. The result is similar to Williams solution for homogeneous material.

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22

Dai, Yao, Lei Zhang, Xiao Chong, and Chun Fang Xue. "The Higher Order Axis-Directional Crack-Tip Field for Reissner's FGMs Cylindrical Shell." Advanced Materials Research 791-793 (September 2013): 746–49. http://dx.doi.org/10.4028/www.scientific.net/amr.791-793.746.

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Reissners theory for cylindrical shell is adopted to analyze the axis-directional crack problem for FGMs cylindrical shell by using the asymptotic expansion method. The eigen-solution of the crack-tip fields for the cylindrical shell is obtained. The results are similar to Williams solution for the plane problems in homogeneous materials, and will be applied widely to engineering structures.

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23

Dai, Yao, Lei Zhang, and Xiao Chong. "The Higher Order Axis-Directional Crack-Tip Field for Power FGMs Cylindrical Shell with Reissner's Effect." Applied Mechanics and Materials 419 (October 2013): 103–6. http://dx.doi.org/10.4028/www.scientific.net/amm.419.103.

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Анотація:

The theory for cylindrical shell with Reissners is adopted to analyze the axis-directional crack problem for FGMs cylindrical shell by using the asymptotic expansion method. The eigen-solution of the crack-tip fields for the cylindrical shell are obtained. The results are similar to Williams solution for the plane problems in homogeneous materials, and will be of wider application to engineering structures.

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24

Kubair, Dhirendra V., Philippe H. Geubelle, and John Lambros. "Asymptotic Analysis of a Mode III Stationary Crack in a Ductile Functionally Graded Material." Journal of Applied Mechanics 72, no. 4 (October 20, 2004): 461–67. http://dx.doi.org/10.1115/1.1876434.

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Анотація:

The dominant and higher-order asymptotic stress and displacement fields surrounding a stationary crack embedded in a ductile functionally graded material subjected to antiplane shear loading are derived. The plastic material gradient is assumed to be in the radial direction only and elastic effects are neglected. As in the elastic case, the leading (most singular) term in the asymptotic expansion is the same in the graded material as in the homogeneous one with the properties evaluated at the crack tip location. Assuming a power law for the plastic strains and another power law for the material spatial gradient, we derive the next term in the asymptotic expansion for the near-tip fields. The second term in the series may or may not differ from that of the homogeneous case depending on the particular material property variation. This result is a consequence of the interaction between the plasticity effects associated with a loading dependent length scale (the plastic zone size) and the inhomogeneity effects, which are also characterized by a separate length scale (the property gradient variation).

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25

Profant, Tomáš, Jan Klusák, Oldřich Ševeček, Michal Kotoul, Miroslav Hrstka, and Petr Marcián. "An Effect of the First Non-Singular Term of the Williams Asymptotic Expansion to the Stability of the Bi-Material Orthotropic Notch." Key Engineering Materials 592-593 (November 2013): 745–48. http://dx.doi.org/10.4028/www.scientific.net/kem.592-593.745.

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The domain of the generalized stress intensity factors dominance ahead of the notch tip can be rather small with respect to the length of the perturbing cracks initiated from the tip of the notch. Thus the non-singular terms of the stress asymptotic expansion at the notch tip would play an important role in the notch tip stability. Following the procedures dealing with complex potential theory and path-independent two-state integrals developed for the singular stress analysis of the stress concentrators one can evaluate their magnitude and include them to the energy release rate of the preexisting crack initiated from the notch tip applying the matched asymptotic procedure. The presented analysis should lead to better understanding of the notch stability process and precising of the notch stability criteria.

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26

Hamdi, A., N. Benseddiq, and F. Mejni. "Rectangular Strain-Rosette Method for Measuring the Mode I Stress-Intensity Factor KI and T-stress." Engineering, Technology & Applied Science Research 7, no. 5 (October 19, 2017): 1922–29. http://dx.doi.org/10.48084/etasr.1396.

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In this paper, a new experimental technique for measuring Stress Intensity Factor (SIF) and T-stress under mode I loading is developed. The expressions of the normal and tangential strains close to the crack tip are given using the first five terms of the generalized Westergaard formulation. In order to accurately determine the SIF and T-stress, the method exploits the optimal positioning of a rectangular strain gage rosette near a crack tip in mode I. Thus, errors due to the higher order terms of the asymptotic expansion are eliminated. Finally, a comparison of the analytical results with a finite element calculations, for different specimen dimensions, is carried out.

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27

Kotoul, Michal, Oldřich Ševeček, and Tomáš Profant. "Modelling of Crack Bifurcation in Laminar Ceramics with Large Compressive Stress." Key Engineering Materials 488-489 (September 2011): 130–33. http://dx.doi.org/10.4028/www.scientific.net/kem.488-489.130.

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Ceramic laminates designed with strong interfaces have shown crack growth resistance (R-curve) behaviour through microstructural design (e.g. grain size, layer composition) and/or due to the presence of compressive residual stresses, acting as a barrier to crack propagation. The goal of the contribution is to model the mechanism of crack bifurcation in laminar ceramics with large compressive stress which still have not been satisfactory explained. Experimental observations of the crack path in the multilayered ceramics tested under several kinds of loading showed crack penetration (i.e. crack propagating normal to the layers followed by crack bifurcation when the crack propagated from the tensile to the compressive layer. Numerical results [1] show that the initiation of crack bifurcation can be explained by the near-tip J-integral, provided that micro-cracks exist near the crack tip. We revisit the problem using the concept of Finite fracture mechanics and the matched asymptotic expansion method in order to evaluate the energy release rate criteria describing the competition of the crack bifurcation and straight crack propagation near behind the bimaterial interface.

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28

Xu, Feng Lin, Jun Yu Liu, Bao Kuan Ning, and He Fan. "Evaluation of the Higher Order Terms of the Wedge-Splitting Specimen Based on the SBFEM." Applied Mechanics and Materials 477-478 (December 2013): 25–29. http://dx.doi.org/10.4028/www.scientific.net/amm.477-478.25.

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The scaled boundary finite element method (abbr. SBFEM) is a semi-analytical method developed by Wolf and Song. The analytical advantage of the solution in the radial direction allows SBFEM converge to the Williams expansion. The coefficients of the Williams expansion, including the stress intensity factor, the T-stress, and higher order terms can be calculated directly without further processing. In the paper the coefficients of higher order terms of the crack tip asymptotic field of typical wedge splitting specimens with two different loading arrangements are evaluated using SBFEM. Numerical results show the method has high accuracy and effectiveness. The results have certain significance on determining crack stability of the wedge-splitting specimen.

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29

Hwang, K. C., S. W. Yu, and W. Yang. "Theoretical Study of Crack-Tip Singularity Fields in China." Applied Mechanics Reviews 43, no. 2 (February 1, 1990): 19–33. http://dx.doi.org/10.1115/1.3119159.

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Анотація:

Crack-tip singularity studies play a dominant role in various aspects concerning fracture and fracture mechanics. Significant advances have been made by Chinese scholars engaged in this field. A systematic, but by no means conclusive, review is attempted here to outline their major progress and to guide the international reader to obtain an access to a portion of fracture research data contributed from Chinese literature. Theoretical framework on crack-tip singularity expansion, classification of the governing differential equations, and the contiguity conditions crucial to the asymptotic assembly are discussed briefly at the beginning part of the review. Attention is then focused on various particular cases in stationary cracks, quasistatically growing cracks, and dynamically propagating cracks. A great number of solutions concerning the above-mentioned crack status with respect to different nonlinear material prescriptions are reviewed. Preliminary studies of the three-dimensional effect on crack-tip singularity fields (most have been pursued for elastic material) as well as the interplay of damage during the process of crack separation have also been carried out actively in China. Some typical results are cited here to give insights into research in these two aspects. The applications of crack-tip singularity study are highlighted by the establishment of theoretical resistance curves based on singularity field calculations. Experimental verifications of those theoretically derived curves as well as their employment in structure integrity assessment are described.

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30

Hundzina, M. A., and O. V. Yuhnovskaya. "Application of Flow Theory Relations for Solving Problems of Steady-State Crack Growth." Science & Technique 21, no. 3 (June 2, 2022): 229–35. http://dx.doi.org/10.21122/2227-1031-2022-21-3-229-235.

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Анотація:

To represent local displacement fields in the problem of the steady-state growth of a crack, which contains a plate of incompressible material, the strain intensity formula is used in the form of a polynomial of the second degree. The case of plane deformation for an elastoplastic material is considered. The solution is obtained by the method of asymptotic expansions. Numerical analysis is carried out for the first term of the expansion. The aim of the work is the process of obtaining analytical solutions to applied problems of the theory of plasticity: finding the components of stress and strain tensors. The paper considers a variant of the method of asymptotic expansions and its application for the problem of the distribution of the stress-strain state in an elastoplastic specimen with a crack. The method of asymptotic expansions has some advantages over the numerical approach in studying the stress-strain state in the vicinity of a crack. It allows to establish exact quantitative relationships between the radial component, the angle, and the components of the stress and strain tensor. Another advantage of this method is the possibility of compiling the mechanical characteristics of an object at the design stage. A system of differential equations has been developed that contains V0 and its derivatives up to the third order. An example of stress distribution in the vicinity of a crack tip in a steel sample, obtained in a computer system by a numerical method, is given. The deformation diagram has been constructed for the material steel 40. The research results can be used to construct stress and strain fields in the vicinity of a crack, as well as to predict the further direction of crack development.

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31

Li, Chunyu, and G. J. Weng. "Antiplane Crack Problem in Functionally Graded Piezoelectric Materials." Journal of Applied Mechanics 69, no. 4 (June 20, 2002): 481–88. http://dx.doi.org/10.1115/1.1467091.

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In this paper the problem of a finite crack in a strip of functionally graded piezoelectric material (FGPM) is studied. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric permitivity of the FGPM vary continuously along the thickness of the strip, and that the strip is under an antiplane mechanical loading and in-plane electric loading. By using the Fourier transform, the problem is first reduced to two pairs of dual integral equations and then into Fredholm integral equations of the second kind. The near-tip singular stress and electric fields are obtained from the asymptotic expansion of the stresses and electric fields around the crack tip. It is found that the singular stresses and electric displacements at the tip of the crack in the functionally graded piezoelectric material carry the same forms as those in a homogeneous piezoelectric material but that the magnitudes of the intensity factors are dependent upon the gradient of the FGPM properties. The investigation on the influences of the FGPM graded properties shows that an increase in the gradient of the material properties can reduce the magnitude of the stress intensity factor.

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32

Belova, O. N., and L. V. Stepanova. "DETERMINATION OF THE COEFFICIENTS OF ASYMPTOTIC CRACK—TIP STRESS EXPANSION. MIXED MODE LOADING OF THE PLATE." Vestnik of Samara University. Natural Science Series 26, no. 3 (May 6, 2020): 40–62. http://dx.doi.org/10.18287/2541-7525-2020-26-3-40-62.

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The aim of the study is to calculate the coefficients of M. Williams asymptotic expansion of stress anddisplacement fields using the data of finite element modeling of a plate with an inclined central crack in a uniaxial tension field. In this work, we also simulated the loading of a half-disk with a vertical and oblique notch under conditions of three-point bending. The simulation was carried out in the multifunctional software SIMULIA Abaqus. The paper proposes an algorithm for calculating the coefficients. The program, written in the MAPLE computer algebra system, allows calculating any predetermined number of M. Williams expansioncoefficients (amplitude or scale factors) and uses the values of the stress tensor components at points in the vicinity of the crack and their coordinates as input. The analysis of the influence of the number of calculated coefficients on the accuracy of their determination is carried out. Recommendations on the choice of points for calculating the coefficients are given.

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33

Stepanova, L. V. "Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I." PNRPU Mechanics Bulletin, no. 4 (December 15, 2020): 237–49. http://dx.doi.org/10.15593/perm.mech/2020.4.20.

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This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

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34

Zhabbarov, R. M., and L. V. Stepanova. "Numerical determination of coefficients of multi-parameter asymptotic expansion of the crack-tip stress field using FEM." Procedia Structural Integrity 28 (2020): 1768–73. http://dx.doi.org/10.1016/j.prostr.2020.10.153.

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35

Belova, O. N., and L. V. Stepanova. "Photoelastic evaluation of stress fields and coefficients of multi-parameter asymptotic expansion of the crack-tip stress field." Procedia Structural Integrity 32 (2021): 32–41. http://dx.doi.org/10.1016/j.prostr.2021.09.006.

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36

Zhabbarov, R. M., and L. V. Stepanova. "Experimental evaluation of coefficients of multi-parameter asymptotic expansion of the crack-tip stress field using digital photoelasticity." Procedia Structural Integrity 28 (2020): 1774–80. http://dx.doi.org/10.1016/j.prostr.2020.10.154.

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37

Kotoul, Michal, Tomáš Profant, Oldřich Ševeček, and Martin Krejcir. "Solution Methods for General Stress Concentrators in Anisotropic Heterogeneous Media." Key Engineering Materials 348-349 (September 2007): 677–80. http://dx.doi.org/10.4028/www.scientific.net/kem.348-349.677.

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The increasing use of fibre-reinforced composites in high performance structures has brought a renewed interest in the analysis of cracks, wedges, and multi-material wedges in anisotropic materials. This paper will address three crucial stages of the general stress concentrator analysis: i) numerical procedures for the determination of eigenvalues and eigenvectors in Williams-like asymptotic expansion for multi-material wedge; ii) approaches to an accurate calculation of the near crack tip fields – the application of so-called two-state (or mutual) conservation integrals; iii) application of fracture criteria for the assessment of fracture inception at the general stress concentrators - concept of the so called finite fracture mechanics.

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38

Hundzina, M. A. "ENERGY INVARIANTS IN THEORY OF ELASTOPLASTIC CRACKS." Science & Technique 16, no. 4 (July 6, 2017): 355–62. http://dx.doi.org/10.21122/2227-1031-2017-16-4-355-362.

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The paper considers a problem on a rectilinear crack in hardening elastoplastic material with load which is applied at infinity under plane-strain deformation conditions. While distributing J-integral in this case it is necessary to take into account specific characteristics associated with strain potential for environments with nonholonomic state equations. While considering a problem on a crack in elastoplastic material a principal term of asymptotic expansion in crack tip vicinity has an unknown singularity index in addition to an indefinite multiplier. It has been shown for steel 12X18H9T that while having invariance of energy integral it is possible to trace a singularity index for a principal term of stresses. The paper presents dependences of crack length compared to permissible Griffith’s length in accordance with the applied load which is associated with yield strength. Conceptions of J-integrals have been described for solution of a quasi-static problem. The developed approach can be used to formulate a criterion for destruction of elastoplastic material containing a rectilinear crack. The obtained theoretical dependences pertaining to determination of structure limit state characteristics have permitted to make a motivated selection of geometric parameters with due account of material strength properties. Results of the investigations can be used while preparing recommendations for development of structures with prescribed properties. The given approach makes most sense to be applied for determination of critical forces and critical value of crack length for elastoplastic material.

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39

Stepanova, L. V. "ASYMPTOTIC ANALYSIS OF THE STRESS FIELD AT A CRACK TIP IN A LINEARLY ELASTIC MATERIAL: EXPERIMENTAL DETERMINATION OF WILLIAMS EXPANSION COEFFICIENTS." Diagnostics, Resource and Mechanics of materials and structures, no. 2 (April 2018): 29–41. http://dx.doi.org/10.17804/2410-9908.2018.2.029-041.

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40

Nikishkov, G. P. "An algorithm and a computer program for the three-term asymptotic expansion of elastic-plastic crack tip stress and displacement fields." Engineering Fracture Mechanics 50, no. 1 (January 1995): 65–83. http://dx.doi.org/10.1016/0013-7944(94)00139-9.

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41

Stepanova, L. V. "Asymptotic stress fields in the vicinity of the crack in perfectly plastic solids under mixed mode loading." PNRPU Mechanics Bulletin, no. 3 (December 15, 2020): 73–89. http://dx.doi.org/10.15593/perm.mech/2020.3.08.

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In the paper presents the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for a full range of the mode mixities. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities both for plane strain and plane stress conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions, all values of the mixity parameter within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained using the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the solution whereas six and five sectors define the solution for the mixity parameter higher 0.33 and less than 0.89 and higher 0.89 respectively for plane strain conditions and seven sectors determine the asymptotic solution for the mixity parameter less than 0.39, while five sectors determine the solution for other values of the mixity parameter for plane stress conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. It is interesting to note that the characteristic feature of the asymptotic solution obtained is the presence of a segment of values of the mixity parameter for which the solution does not depend on the mixity parameter (the solution does not depend on the mixity parameter for the mixity parameter from 0.89 to 1 and the solution coincides with the solution for mode I crack in perfect plastic materials for plane strain conditions). Thus, the salient point of the study is that the asymptotic solution is described by the same formulae for all values of the mixity parameter from 0.89 to 1 for plane strain. For plane stress conditions this segment can’t be observed. The solution in each sector corresponds to the certain value of the mixity parameter. The obtained solutions for plane strain and plane stress conditions can be considered as the limit solution for power law hardening materials and creeping power law materials.

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42

Repka, M., J. Sládek, V. Sládek, and M. Wünsche. "Evaluation of Fracture Parameters for Cracks in Coupled Thermoelasticity for Functionally Graded Materials." Strojnícky casopis – Journal of Mechanical Engineering 65, no. 1 (November 1, 2015): 57–76. http://dx.doi.org/10.1515/scjme-2016-0004.

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Abstract The finite element method (FEM) is developed for coupled thermoelastic crack problems if material properties are continuously varying. The weak form is utilized to derive the FEM equations. In conventional fracture theories the state of stress and strain at the crack tip vicinity is characterized by a single fracture parameter, namely the stress intensity factor or its equivalent, J-integral. In the present paper it is considered also the second fracture parameter called as the T-stress. For evaluation of both fracture parameters the quarter-point crack tip element is developed. Simple formulas for both fracture parameters are derived comparing the variation of displacements in the quarter-point element with asymptotic expression of displacement at the crack tip vicinity. The leading terms of the asymptotic expansions of fields in the crack-tip vicinity in a functionally graded material (FGM) are the same as in a homogeneous one with material coefficients taken at the crack tip.

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43

Deng, X. "General Crack-Tip Fields for Stationary and Steadily Growing Interface Cracks in Anisotropic Bimaterials." Journal of Applied Mechanics 60, no. 1 (March 1, 1993): 183–89. http://dx.doi.org/10.1115/1.2900743.

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This study builds upon some recent results in the literature regarding the asymptotic behavior of bimaterial interface cracks, and gives the general form, both oscillatory and nonoscillatory, of the crack-tip stress and displacement fields for stationary and steadily growing interface cracks in anisotropic bimaterials, which are equivalent to complete Williams-type series expansions. Special cases, such as cracks in homogeneous anisotropic materials and interface cracks with decoupled antiplane shear and in-plane deformations, are discussed briefly. Explicit series expansions of the stress and displacement fields in crack-tip polar coordinates are derived for both stationary and steadily propagating interface cracks in isotropic bimaterials.

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44

Drugan, W. J. "Radial Dependence of Near-Tip Continuum Fields for Plane Strain Tensile Crack Growth in Elastic-Ideally Plastic Solids." Journal of Applied Mechanics 53, no. 1 (March 1, 1986): 83–88. http://dx.doi.org/10.1115/1.3171743.

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This paper is an extension of work by Drugan et al. (1982) who derive the stress and deformation fields at the tip of a plane strain tensile crack that grows quasi-statically, under general nonsteady conditions, in an elastic-ideally plastic solid. Here I perform a higher-order analysis of the near-tip fields for this growing crack problem. My principal objectives are to determine the radial variation of the near-tip stress field and elucidate the structure of the deformation fields in the 90-deg sector ahead of the growing crack; this information was not provided by the lowest-order solution of Drugan et al. (1982). I also derive a crucial asymptotic expression for the normal radial component of the deformation rate tensor in a moving “centered fan” plastic sector, which was given without complete proof by Rice (1982). The analysis presented herein differs from typical perturbation analyses in that I am able to derive the higher-order structure of the continuum fields rather than having to assume expansions for them. Among the results, normal polar components of deviatoric stress are shown to vary as (ln r)−1, while the in-plane polar shear component varies as (ln r)−2, for small r > 0 in moving “centered fan” plastic sectors, r denoting distance from the (moving) crack tip. Further, in-plane strains proportional to ln|ln r| as r → 0 appear not to be precluded in the 90-deg sector ahead of the growing crack.

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45

"The asymptotic structure of transient elastodynamic fields at the tip of a stationary crack." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 446, no. 1926 (July 8, 1994): 1–13. http://dx.doi.org/10.1098/rspa.1994.0088.

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The asymptotic structure of the transient elastodynamic near-tip fields around a stationary crack is investigated for all three fracture modes. The transient fields are obtained as the sum of their quasi-static counterparts and corresponding transient correction terms, in terms of variable-separable expansions. By allowing the coefficients of terms in the quasi-static expansion to deviate from their quasi-static restrictions, the correction terms are shown to be the particular solutions of a set of first order (for mixed mode I and II) or second order (for mode III) ordinary differential equations with constant coefficients and non-homogeneous terms involving only sine and cosine functions of the independent variable. It is found that the transient effects of dynamic loading on the near-tip fields are to alter the universal angular variations of the quasi-static field quantities for the fifth and higher order terms in their variable-separable expansions; thus the first four terms in the expansions have the same angular variations under both quasi-static and dynamic loading conditions. This seems to suggest that transient effects on the crack-tip fields are in general less severe for a stationary crack than for a propagating crack where only the first two terms in the expansions hold the same angular variations under both steady-state and transient crack growth conditions. Furthermore, the transient higher order terms for a stationary crack do not depend on the time-rate of the stress intensity factors; in fact, they only relate to the even order time-derivatives of the instantaneous values of the coefficients of the terms in the quasi-static expansions. This is also in contrast with the case of transient crack propagation where the time rates of the dynamic stress intensity factors play important roles in the higher order transient terms. Explicit expressions for the transient near-tip stress and displacement fields are provided.

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46

Lee, Kwang Ho, Vijaya Bhaskar Chalivendra, and Arun Shukla. "Dynamic Crack-Tip Stress and Displacement Fields Under Thermomechanical Loading in Functionally Graded Materials." Journal of Applied Mechanics 75, no. 5 (July 10, 2008). http://dx.doi.org/10.1115/1.2932093.

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Thermomechanical stress and displacement fields for a propagating crack in functionally graded materials (FGMs) are developed using displacement potentials and asymptotic analysis. The shear modulus, mass density, and coefficient of thermal expansion of the FGMs are assumed to vary exponentially along the gradation direction. Temperature and heat flux distribution fields are also derived for an exponential variation of thermal conductivity. The mode mixity due to mixed-mode loading conditions around the crack tip is accommodated in the analysis through the superposition of opening and shear modes. Using the asymptotic stress fields, the contours of isochromatics (contours of constant maximum shear stress) are developed and the results are discussed for various crack-tip thermomechanical loading conditions.

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47

Qi, Baoqiang, Zhihui Wang, Hong Chen, Yuqing Jian, and Shiqian He. "Computer Art Design Model Based on Nonlinear Fractional Differential Equations." Applied Mathematics and Nonlinear Sciences, July 15, 2022. http://dx.doi.org/10.2478/amns.2022.2.0180.

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Abstract The influence of the environment will deform materials in computer art design. Based on nonlinear fractional differential equations, the paper constructs the change of material mechanical properties in computer art design. This paper uses the asymptotic expansion method to transform the higher-order partial differential equations into nonlinear fractional-order differential equations. In this paper, the equations are solved to obtain the stress function. Then the analytical formula of the high-order asymptotic field of the stress at the crack tip in the functionally graded material is obtained. In this paper, the separation method of variables is used to obtain the solution of the equation expressed in rectangular coordinates, and the expressions of displacement and stress are obtained. The study found that the order of the model can quantitatively describe the evolution of the mechanical properties of plastic metals in computer art.

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48

"Plane strain fracture in poroelastic media." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 434, no. 1892 (September 9, 1991): 605–33. http://dx.doi.org/10.1098/rspa.1991.0116.

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The effect of fully coupled poroelasticity on an impulsively loaded crack in plane strain is investigated. A formally exact solution for a semi-infinite crack in a linear, isotropic, poroelastic medium with a prescribed internal stress is considered; the solution is obtained using Laplace and Fourier transforms in time and space respectively and then using the Wiener-Hopf technique to solve the resulting functional equations. The stress intensity factor is found as a function of the Laplace variable s and is evaluated explicitly for small times and numerically for all times. The problem of a finite length crack embedded in a poroelastic medium under uniform impulsively applied tension at infinity is solved using the method of matched asymptotic expansions for small times. The formal solution for a steadily propagating semi-infinite crack under tension is outlined, the crack-tip fields are examined and the crack-tip stress intensity factors are found as functions of the crack velocity. Analytical solutions for the pore pressure and stress ahead of the crack are obtained and their relevance to the retardation of fracture discussed. The results extend the range of possible solutions of the fully coupled poroelastic equations to mixed boundary-value problems in fracture mechanics. These are fundamental to the study of the interaction between a diffusing pore fluid and the solid elastic skeleton. In particular, time dependent solutions to the symmetric problems of impulsive loadings and explicit solutions to the steady problems are considered.

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49

"Crack problems in a poroelastic medium: an asymptotic approach." Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 346, no. 1680 (February 15, 1994): 387–428. http://dx.doi.org/10.1098/rsta.1994.0026.

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We consider problems involving semi-infinite cracks in a porous elastic material. The cracks are loaded with a time dependent internal stress, or pore pressure. Either mixed or unmixed pore pressure boundary conditions on the fracture plane are considered. An asymptotic procedure that partly uncouples the elastic and fluid responses is used, allowing an asymptotic expression for the stress intensity factors as time progresses to be obtained. The method allows the physical processes involved at the crack tip and their interactions to be studied. This is an advance on previous methods where results were obtained in Laplace transform space and inverted numerically to obtain real-time solutions. The crack problems are formulated using distributions of dislocations (and pore pressure gradient discontinuities when necessary) to generate integral equations of the Wiener—Hopf type. The resulting functional equations are, of course, identical to those considered by C. Atkinson and R. V. Craster, but with the alternative formulation we develop an asymptotic procedure which should be applicable to other problems (e.g. finite length cracks). This asymptotic procedure can be used to derive asymptotic expansions for more complicated loadings when the numerical effort involved in evaluating results would be excessive. A large-time asymptotic method is also briefly described which complements the small-time method. The operators for poroelastic crack problems are inverted for a particular loading; the reciprocal theorem for poroelasticity is used together with eigensolutions of the fundamental problems to deduce the stress (or where necessary the pore pressure gradient) intensity factors for any loading. These formulae extend previous results allowing a wide range of different loadings to be considered. As an example, the stress intensity factor for a point loaded crack is derived and the asymptotic method is applied to this problem to derive a simple asymptotic formula. Finally, an invariant integral, which is a generalization of the Eshelby energy-momentum tensor, is used to derive integral identities which serve as a check on the intensity factors in some situations.

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