An analytical stress field for bi-material V-notches with end hole: New solution and effects of higher order terms

Abstract

Current manuscript develops an analytical stress solution for the bi-material V-notches with an end hole (VO). To do so, based on the Kolosov-Muskhelishvili’s approach, while appropriate potential functions are employed, the boundary conditions of the problem are imposed to the solution to reduce the number of unknown parameters. Subsequently, the analytical stress field is derived as an asymptotic series solution, where each term possessing a constant coefficient and an order of singularity. The order of singularity for each term is obtained from the characteristic equations of the problem which is dependent on the notch geometry and material combinations. The so-called least square method (LSM) is then used to compute the constant coefficients of the asymptotic series for several case studies. Special attention is given to the bi-material notch stress intensity factors (BNSIFs) and the coefficients of the higher order terms (HOTs) in the stress series expansion. The accuracy of the presented stress solution is verified by benchmarking the results with numerical values obtained from finite element (FE) method. In this process, several notch opening angles and notch radii are simulated using the three-point bend (3PB) specimen. The developed asymptotic stress solution is demonstrated to be capable of accurately evaluating the stress field around bi-material VO-notched structures.

Keywords

Analytical solution asymptotic stress expansion bi-material components complex eigenvalues VO-notches geometrical and material singularities

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References

Labossiere

Dunn

Cunningham

. Application of bimaterial interface corner failure mechanics to silicon/glass anodic bonds. J Mech Phys Solids 2002; 50(3): 405–433.

Greco

Leonetti

Lonetti

. A two-scale failure analysis of composite materials in presence of fiber/matrix crack initiation and propagation. Comp Struct 2013; 95: 582–597.

Chen

Beyerlein

Brinson

. Bridged crack models for the toughness of composites reinforced with curved nanotubes. J Mech Phys Solids 2011; 59(9): 1938–1952.

Chaudhuri

Chiu

S-HJ

. Three-dimensional asymptotic stress field at the front of an unsymmetric bimaterial wedge associated with matrix cracking or fiber break. Comp Struct 2007; 78(2): 254–263.

Ling

Hirvi

Suvanto

, et al. A computational study of adhesion between rubber and metal sulfides at rubber–brass interface. Chem Phys 2015; 453: 7–12.

Gong

Wang

, et al. Effect of metal surface state on injection joining strength of aluminum-rubber composite part. J Manuf Process 2020; 49: 365–372.

Yang

Dudek

Kumpfert

. Interfacial reaction and stability of SCS-6 SiC/Ti–25Al–10NB–3V–1MO composites. Mater Sci Eng: A 1998; 246(1–2): 213–220.

Zhang

Wang

, et al. Crack path selection at the interface of wrought and wire+ arc additive manufactured Ti–6Al–4V. Mater Design 2016; 104: 365–375.

Kishen

Singh

. Stress intensity factors based fracture criteria for kinking and branching of interface crack: application to dams. Eng Fract Mech 2001; 68(2): 201–219.

10.

Rice

Sih

. Plane problems of cracks in dissimilar media. J Appl Mech 1965; 32(2): 418–423.

11.

Hutchinson

Mear

Rice

. Crack paralleling an interface between dissimilar materials. J Appl Mech 1987; 54(4): 828–832.

12.

Clements

. A crack between dissimilar anisotropic media. Int J Eng Sci 1971; 9(2): 257–265.

13.

Erdogan

. Stress distribution in bonded dissimilar materials with cracks. J Appl Mech 1965; 32(2): 403–410.

14.

England

. A crack between dissimilar media. J Appl Mech 1965; 32(2): 400–402.

15.

Williams

. The stresses around a fault or crack in dissimilar media. Bullet Seismol Soc Am 1959; 49(2): 199–204.

16.

Bogy

. Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. J Appl Mech 1968; 35(3): 460–466.

17.

Rao

. Stress concentrations and singularities at interface corners. ZAMM: J Appl Math Mech/Zeitschrift Angew Math Mech 1971; 51(5): 395–406.

18.

Hein

Erdogan

. Stress singularities in a two-material wedge. Int J Fract Mech 1971; 7(3): 317–330.

19.

Groth

. Stress singularities and fracture at interface corners in bonded joints. Int J Adhes Adhesiv 1988; 8(2): 107–113.

20.

Lazzarin

Quaresimin

Ferro

. A two-term stress function approach to evaluate stress distributions in bonded joints of different geometries. J Strain Anal Eng Design 2002; 37(5): 385–398.

21.

Yao

Wang

. An analytical singular element for interface cracks in bi-material Kirchhoff plate bending. Eng Fract Mech 2012; 91: 103–116.

22.

Arabi

Mirsayar

Samaei

, et al. Study of characteristic equation of the elastic stress field near bimaterial notches. Streng Mater 2013; 45(5): 598–606.

23.

Mirsayar

. Calculation of stress intensity factors for an interfacial notch of a bi-material joint using photoelasticity. Eng Solid Mech 2013; 1(4): 149–153.

24.

Cheng

Niu

Recho

. Analysis of the stress singularity for a bi-material V-notch by the boundary element method. Appl Math Model 2013; 37(22): 9398–9408.

25.

Treifi

Oyadiji

. Bi-material V-notch stress intensity factors by the fractal-like finite element method. Eng Fract Mech 2013; 105: 221–237.

26.

Mieczkowski

. Stress fields at the tip of a sharp inclusion on the interface of a biomaterial. Mech Comp Mater 2016; 52(5): 601–610.

27.

Bui

, et al. Bi-material V-notched SIFs analysis by XFEM and conservative integral approach. Comp Struct 2018; 196: 217–232.

28.

Kubair

. Alternate expressions for the near-tip in-plane mixed-mode K-field stresses and displacements. Eng Fract Mech 2010; 77(14): 2927–2934.

29.

Zappalorto

Lazzarin

. In-plane and out-of-plane stress field solutions for V-notches with end holes. Int J Fract 2011; 168(2): 167–180.

30.

Mirzaei

Ayatollahi

Bahrami

. Asymptotic stress field and the coefficients of singular and higher order terms for V-notches with end holes under mixed-mode loading. Int J Solids Struct 2019; 172: 51–69.

31.

Mirzaei

Ayatollahi

Bahrami

, et al. A new unified asymptotic stress field solution for blunt and sharp notches subjected to mixed mode loading. Int J Mech Sci 2021; 193: 106176.

32.

Qian

Akisanya

. Analysis of free-edge stress and displacement fields in scarf joints subjected to a uniform change in temperature. Fatig Fract Eng Mater Struct 1998; 21(6): 687–703.

33.

Ayatollahi

Mirsayar

Nejati

. Evaluation of first non-singular stress term in bi-material notches. Comp Mater Sci 2010; 50(2): 752–760.

34.

Qian

Akisanya

. An investigation of the stress singularity near the free edge of scarf joints. Eur J Mech: A/Solids 1999; 18(3): 443–463.

35.

Bahrami

Ayatollahi

Mirzaei

, et al. Improved stress and displacement fields around V-notches with end holes. Eng Fract Mech 2019; 217: 106539.

36.

Bahrami

Ayatollahi

Mirzaei

. Analysis of stresses and displacements in the vicinity of blunt V-notches by considering higher order terms. Fatig Fract Eng Mater Struct 2019; 42(8): 1760–1774.

37.

Nejati

Ghouli

Ayatollahi

. Crack tip asymptotic fields in anisotropic planes: Importance of higher order terms. Appl Math Model 2021; 91: 837–862.