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Applied Mechanics Reviews | Volume 61 | Issue 2 | Review Article
Review Articles

On the Stress Singularities at Multimaterial Interfaces and Related Analogies With Fluid Dynamics and Diffusion

[+] Author and Article Information
Marco Paggi1

Department of Structural and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italymarco.paggi@polito.it

Alberto Carpinteri

Department of Structural and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

1

Corresponding author.

Appl. Mech. Rev 61(2), 020801 (Mar 18, 2008) (22 pages) doi:10.1115/1.2885134 History: Published March 18, 2008
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Joining of different materials is a situation frequently observed in mechanical engineering and in materials science. Due to the difference in the elastic properties of the constituent materials, the junction points can be the origin of stress singularities and a possible source of damage. Hence, a full appreciation of these critical situations is of fundamental importance both from the mathematical and the engineering standpoints. In this paper, an overview of interface mechanical problems leading to stress singularities is proposed to show their relevance in engineering. The mathematical methods for the asymptotic analysis of stress singularities in multimaterial junctions and wedges composed of isotropic linear-elastic materials are reviewed and compared, with special attention to in-plane and out-of-plane loadings. This analysis mathematically demonstrates in a historical retrospective the equivalence of the eigenfunction expansion method, of the complex function representation, and of the Mellin transform technique for the determination of the order of the stress singularity in such problems. The analogies between linear elasticity and the Stokes flow of dissimilar immiscible fluids, the steady-state heat transfer across different materials, and the St. Venant torsion of composite bars are also discussed. Finally, advanced issues for the stress singularities due to joining of angularly nonhomogeneous elastic wedges are presented. This review article contains 147 references.
Copyright © 2008 by American Society of Mechanical Engineers
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Sinclair, G. B., 2005, “Asymptotic Fields for Reentrant Corners in Antiplane Shear With Cohesive Laws in Boundary Conditions,” Department of Mechanical Engineering, Louisiana State University, Technical Report No. ME-MA2-05.
130
Sinclair, G. B., 1980, “On the Singular Eigenfunctions for Plane Harmonic Problems in Composite Regions,” ASME J. Appl. Mech., 47 , pp. 87–92.
131
Gadre, H. R., and Sinclair, G. B., 2005, “Finite Element Analysis of Sharp Corners With and Without Stress Singularities, Antiplane Shear States,” Department of Mechanical Engineering, Louisiana State University, Technical Report No. ME-MA3-05.
132
Higgins, T. J., 1942, “A Comprehensive Review of Saint-Venant Torsion Problem,” Am. J. Phys.  [CrossRef], 10 , pp. 248–259.
133
Higgins, T. J., 1945, “Analogic Experimental Methods in Stress Analysis as Exemplified by Saint-Venant’s Torsion Problem,” Proc. Soc. Exp. Stress Anal., 2 , pp. 17–27.
134
Todhunter, I., and Pearson, K., 1886, "A History of the Theory of Elasticity and of the Strength of Materials from Galilei to Lord Kelvin. Vol. II. Saint-Venant to Kelvin", Cambridge University Press, Cambridge.
135
de Saint-Venant, B., 1843, “Sur la torsion des prismes a base rectangle et a base losange, et sur une petite correction numerique a faire subir,” Mem. Acad. Sci. Inst. Fr., 17 , pp. 1180–1190.
136
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137
Beltrami, E., 1889, “Note Fisico-Matematiche,” "Opere Matematiche", Hoepli, Milano, Vol. IV , pp. 320–329.
138
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139
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141
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142
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146
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Figures

Grahic Jump Location
+Subproblems for the evaluation of the eigenequation for a bimaterial junction with (a) E1→∞ and (b) E2→∞
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Subproblems for the evaluation of the eigenequation for a bimaterial junction with (a) E1→∞ and (b) E2→∞
Figure 10

Subproblems for the evaluation of the eigenequation for a bimaterial junction with (a) E1→∞ and (b) E2→∞

Grahic Jump Location
+Subproblems for the evaluation of the eigenequation for a trimaterial junction with (a) E3→∞ and (b) E3→0
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Subproblems for the evaluation of the eigenequation for a trimaterial junction with (a) E3→∞ and (b) E3→0
Figure 11

Subproblems for the evaluation of the eigenequation for a trimaterial junction with (a) E3→∞ and (b) E3→0

Grahic Jump Location
+Indentation of an elastic half-plane (a) by a frictionless rigid flat punch and (b) by an elastic punch with corner angles other than π∕2
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Indentation of an elastic half-plane (a) by a frictionless rigid flat punch and (b) by an elastic punch with corner angles other than π∕2
Figure 1

Indentation of an elastic half-plane (a) by a frictionless rigid flat punch and (b) by an elastic punch with corner angles other than π∕2

Grahic Jump Location
+Intersection of a solid or liquid island with a solid substrate. θ is the wetting angle and γsv, γis, and γvi are, respectively, the interfacial tensions associated with substrate-vapor, island-substrate, and vapor-island interfaces.
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Intersection of a solid or liquid island with a solid substrate. θ is the wetting angle and γsv, γis, and γvi are, respectively, the interfacial tensions associated with substrate-vapor, island-substrate, and vapor-island interfaces.
Figure 2

Intersection of a solid or liquid island with a solid substrate. θ is the wetting angle and γsv, γis, and γvi are, respectively, the interfacial tensions associated with substrate-vapor, island-substrate, and vapor-island interfaces.

Grahic Jump Location
+Scheme of the geometry of a grain triple junction. The grains are composed of the same material, and grain boundaries are assumed to be freely sliding.
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Scheme of the geometry of a grain triple junction. The grains are composed of the same material, and grain boundaries are assumed to be freely sliding.
Figure 3

Scheme of the geometry of a grain triple junction. The grains are composed of the same material, and grain boundaries are assumed to be freely sliding.

Grahic Jump Location
+Atomic structure of the multiple-twin junction obtained after MD simulation (a). Shaded rings indicate the grain boundaries. Contour plot of the stress field in the vertical direction around the triple junction (b). The right scale stress are in units of kbar (reprinted from Ref. 34).
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Atomic structure of the multiple-twin junction obtained after MD simulation (a). Shaded rings indicate the grain boundaries. Contour plot of the stress field in the vertical direction around the triple junction (b). The right scale stress are in units of kbar (reprinted from Ref. 34).
Figure 4

Atomic structure of the multiple-twin junction obtained after MD simulation (a). Shaded rings indicate the grain boundaries. Contour plot of the stress field in the vertical direction around the triple junction (b). The right scale stress are in units of kbar (reprinted from Ref. 34).

Grahic Jump Location
+Scheme of a multimaterial junction
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Scheme of a multimaterial junction
Figure 5

Scheme of a multimaterial junction

Grahic Jump Location
+Some interface problems addressed by Rao (reprinted from Ref. 64)
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Some interface problems addressed by Rao (reprinted from Ref. 64)
Figure 6

Some interface problems addressed by Rao (reprinted from Ref. 64)

Grahic Jump Location
+Schemes of a two-dimensional single-material wedge (a) and of a two-dimensional composite wedge (b)
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Schemes of a two-dimensional single-material wedge (a) and of a two-dimensional composite wedge (b)
Figure 7

Schemes of a two-dimensional single-material wedge (a) and of a two-dimensional composite wedge (b)

Grahic Jump Location
+Scheme of a multimaterial junction with a geometric symmetry and subjected to symmetric (Mode I) (a) and to skew-symmetric (Mode II) (b) deformation
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Scheme of a multimaterial junction with a geometric symmetry and subjected to symmetric (Mode I) (a) and to skew-symmetric (Mode II) (b) deformation
Figure 8

Scheme of a multimaterial junction with a geometric symmetry and subjected to symmetric (Mode I) (a) and to skew-symmetric (Mode II) (b) deformation

Grahic Jump Location
+Scheme of a multimaterial junction with a reentrant corner (a) and with a crack (b). Case (a) is usually referred to as a multimaterial wedge.
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Scheme of a multimaterial junction with a reentrant corner (a) and with a crack (b). Case (a) is usually referred to as a multimaterial wedge.
Figure 9

Scheme of a multimaterial junction with a reentrant corner (a) and with a crack (b). Case (a) is usually referred to as a multimaterial wedge.

Grahic Jump Location
+Three examples of trimaterial junctions whose limit eigenvalues are reported in Table 3
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Three examples of trimaterial junctions whose limit eigenvalues are reported in Table 3
Figure 12

Three examples of trimaterial junctions whose limit eigenvalues are reported in Table 3

Grahic Jump Location
+A scheme of a corner negotiated by three immiscible viscous fluids
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A scheme of a corner negotiated by three immiscible viscous fluids
Figure 13

A scheme of a corner negotiated by three immiscible viscous fluids

Grahic Jump Location
+Sketch of the stream lines for symmetric (a) and (b) and skew-symmetric (c) and (d) flows around a corner
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Sketch of the stream lines for symmetric (a) and (b) and skew-symmetric (c) and (d) flows around a corner
Figure 14

Sketch of the stream lines for symmetric (a) and (b) and skew-symmetric (c) and (d) flows around a corner

Grahic Jump Location
+Sketch of the stream lines in corner eddies for γ=60deg, reprinted from Ref. 123. The relative dimensions of the eddies are approximately correct and the relative intensities are indicated.
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Sketch of the stream lines in corner eddies for γ=60deg, reprinted from Ref. 123. The relative dimensions of the eddies are approximately correct and the relative intensities are indicated.
Figure 15

Sketch of the stream lines in corner eddies for γ=60deg, reprinted from Ref. 123. The relative dimensions of the eddies are approximately correct and the relative intensities are indicated.

Grahic Jump Location
+Scheme of a composite material subjected to antiplane shear deformation
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Scheme of a composite material subjected to antiplane shear deformation
Figure 16

Scheme of a composite material subjected to antiplane shear deformation

Grahic Jump Location
+Scheme of the St. Venant torsion problem with a composite cylinder (a). Relationship between the displacements in polar coordinates computed in the reference systems with origin O (singular point) and C (center of twist) (b).
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Scheme of the St. Venant torsion problem with a composite cylinder (a). Relationship between the displacements in polar coordinates computed in the reference systems with origin O (singular point) and C (center of twist) (b).
Figure 17

Scheme of the St. Venant torsion problem with a composite cylinder (a). Relationship between the displacements in polar coordinates computed in the reference systems with origin O (singular point) and C (center of twist) (b).

Grahic Jump Location
+Scheme of a FGM bimaterial junction (a) and variation of the elastic moduli in the two-material regions (b). This variation can be completely general, without any kind of symmetry.
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Scheme of a FGM bimaterial junction (a) and variation of the elastic moduli in the two-material regions (b). This variation can be completely general, without any kind of symmetry.
Figure 18

Scheme of a FGM bimaterial junction (a) and variation of the elastic moduli in the two-material regions (b). This variation can be completely general, without any kind of symmetry.

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