Abstract
A crack initiates frequently at a vertex in three-dimensional joints under an external load and a thermal load. In the present paper, the stress distributions near a very small crack occurring at the vertex of the interface in a three-dimensional joint are analyzed under a tensile load using a boundary element method, and the stress intensity factor of mode II is investigated along the crack front. The joint model is composed of silicon and resin, which is modeled on a material combination in electronic devices. Three kinds of crack shape, triangular, quarter circular, and concave shapes, are supposed as a crack shape. First, the stress distributions near the vertex in the model without a crack are obtained and are used for normalizing the singular stress at the front of the crack. Dimensionless stress intensity factor for an interface crack is defined and determined from the distribution of the normalized stress. Next, the stress distribution near the intersection point of the crack front and the side surface is precisely investigated. An eigenanalysis at the intersection point is conducted, and eigenvalues yielding the stress singularity are obtained. Then, it is found that there are two values yielding the stress singularity. The stress distributions near the intersection point are expressed using the angular functions for each value yielding the singularity. Finally, it is shown that the stress intensity factor for mode II along the crack front varies following the summation of functions composed of the distance from the intersection point with the power indices of (\({0.5 - {\lambda _1}}\)) and (\({0.5 - {\lambda _2}}\)), where \({\lambda _1}\) and \({\lambda _2}\) are the orders of stress singularity at the intersection point.
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References
Bogy D.B.: Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. J. Appl. Mech. 38, 377–386 (1971)
Bogy D.B., Wang K.C.: Stress singularities at interface corners in bonded dissimilar isotropic elastic materials. Int. J. Solids Struct. 7, 993–1005 (1971)
Dundurs J.: Edge-bonded dissimilar orthogonal elastic wedges under normal and shear loading. J. Appl. Mech. 36, 650–652 (1969)
Dempsey J.P., Sinclair G.B.: On the stress singularities in the plane elasticity of the composite wedge. J. Elast. 9, 373–391 (1979)
Dempsey J.P., Sinclair G.B.: On the singular behavior at the vertex of a bi-material wedge. J. Elast. 11, 317–327 (1981)
Munz D., Yang Y.Y.: Stress near the edge of bonded dissimilar materials described by two stress intensity factors. Int. J. Fract. 60, 169–177 (1993)
Banks-Sills L., Sherer A.: A conservative integral for determining stress intensity factors of a bimaterial notch. Int. J. Fract. 115, 1–26 (2002)
Luangarpa C., Koguchi H.: Intensity of singularity in three-material joints under shear loading: two-real singularities and power-logarithmic singularities. Euro. J. Mech. A Solids 40, 60–68 (2013)
Glushkov E.V., Glushkova N.V., Munz D., Yang Y.Y.: Analytical solution for bonded wedges under thermal stresses. Int. J. Fract. 106, 321–339 (2000)
Qian Z.Q., Akisanya A.R., Thompson D.S.: Stress behavior at the interface junction of an elastic inclusion. J. Appl. Mech. 69, 844–852 (2002)
Goglio L., Rossetto M.: Stress intensity factor in bonded joints: influence of the geometry. Int. J. Adhes. 30, 313–321 (2010)
Qian Z.Q., Akisanya A.R.: Stress distribution at the interface corner of a Tri-material structure. J. Eng. Mech. 127, 747–753 (2001)
Koguchi H., Luangarpa C.: Two-dimensional joint analysis under shear loading using enriched finite element. J. Solid Mech. Mater. Eng. 2, 319–332 (2008)
Kuo T-L., Hwu C.: Multi-order stress intensity factors along three-dimensional interface corners. J. Appl. Mech. 77, 031020-1–031020-12 (2010)
Hwu C., Huang H.Y.: Investigation of the stress intensity factors for interface corners. Eng. Fract. Mech. 93, 204–224 (2012)
Pageau S.S., Joseph P.F., Biggers S.B.: The order of stress singularities for bonded and disbanded three-material junction. Int. J. Solids Struct. 31, 2979–2997 (1994)
Pageau S.S., Bigger S.B. Jr.: Finite element evaluation of free-edge singular stress fields in anisotropic materials. Int. J. Numer. Methods Eng. 38, 2225–2239 (1995)
Koguchi H., Muramoto T.: The order of stress singularity near the vertex in three-dimensional joints. Int. J. Solids Struct. 37, 4737–4762 (2000)
Mittelstedt C., Becker W.: Efficient computation of order and mode of three-dimensional stress singularities in linear elasticity by the boundary finite element method. Int. J. Solids Struct. 43, 2868–2903 (2006)
Koguchi H., Meo N.: An evaluation of interface strength at a vertex in a three-dimensional joint considering residual thermal stresses using three dimensional boundary element method. Trans. Jpn. Soc. Mech. Eng. 72-723, 1598–1606 (2006)
Monchai P., Koguchi H.: Boundary element analysis of the stress field at the singularity lines in three-dimensional bonded joints under thermal loading. J. Mech. Mater. Struct. 2-1, 149–166 (2007)
Koguchi H., Taniguchi T.: Characteristics of stress singularity field of residual thermal stresses at vertex in three-dimensional bonded joints. Trans. Jpn. Soc. Mech. Eng. Ser. A 74-724, 864–872 (2008)
Koguchi H., Konno N.: Intensity of residual thermal stresses at the vertex in three-dimensional joints with three layers. Trans. Jpn. Soc. Mech. Eng. Ser. A 75, 1148–1155 (2009)
Koguchi H., Nakajima M.: Influence of interlayer thickness on the intensity of singular field in 3D three-layered joints under an external load. Trans. Jpn. Soc. Mech. Eng. Ser. A 76, 1110–1118 (2010)
Koguchi H., da Costa J.A.: Analysis of the stress singularity field at a vertex in 3D-bonded structures having a slanted side surface. Int. J. Solids Struct. 47, 3131–3140 (2010)
Chiu Sou-Hsiung J., Chaudhuri A.: A three-dimensional eigenfunction expansion approach for singular stress field near an adhesively-bonded scarf joint interface in a rigidly-encased plate. Eng. Fract. mech. 78, 2220–2234 (2011)
Koguchi H., Hoshi K.: Evaluation of joining strength of silicon–resin interface at a vertex in a three-dimensional joint structure. J. Electron. Packag. 134, 020902-1–020902-7 (2012)
Koguchi H., Kimura N.: Stress analysis near a small crack within singular stress field in a three-dimensional bonded joint under a tensile load. Trans. Jpn. Soc. Mech. Eng. 80, 1148–1155 (2014)
Koguchi, H., Yokoyama, K., Luangarpa, C.: Variation of stress intensity factor along a small interface crack front in singular stress fields. Int. J. Solids Struct. 71, 156–168 (2015)
Nagai M., Ikeda T., Miyazaki N.: Stress intensity factor analysis of a three-dimensional interface crack between dissimilar anisotropic materials. Eng. Fract. Mech. 74, 2481–2497 (2007)
Zhu B., Shi Y., Qin T., Sukop M., Yu S., Li Y.: Mixed-mode stress intensity factors of 3D interface crack in fully coupled electromagnetothermoelastic multiphase composites. Int. J. Solids Struct. 46, 2669–2679 (2009)
Chiu T.-C., Lin H.-C.: Analysis of stress intensity factors for three-dimensional interface crack problems in electronic packages using the virtual crack closure technique. Int. J. Fract. 156, 75–96 (2009)
Veluri B., Jensen H.M.: Steady-state propagation of interface corner crack. Int. J. Solids Struct. 50, 1613–1620 (2013)
Luangarpa C., Koguchi H.: Analysis of a three-dimensional dissimilar material joint with one real singularity using a conservative integral. Int. J. Solids Struct. 51, 2908–2919 (2014)
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Koguchi, H., Yokoyama, K. Stress analysis in three-dimensional joints with a crack at the vertex of the interface. Acta Mech 228, 2759–2773 (2017). https://doi.org/10.1007/s00707-015-1525-x
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DOI: https://doi.org/10.1007/s00707-015-1525-x