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


Corresponding author.

Appl. Mech. Rev 61(2), 020801 (Mar 18, 2008) (22 pages) doi:10.1115/1.2885134 History: Published March 18, 2008

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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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

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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.

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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.

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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).

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Figure 5

Scheme of a multimaterial junction

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Figure 6

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

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Figure 7

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

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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

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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.

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Figure 10

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

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Figure 11

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

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Figure 12

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

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Figure 13

A scheme of a corner negotiated by three immiscible viscous fluids

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Figure 14

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

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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.

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Figure 16

Scheme of a composite material subjected to antiplane shear deformation

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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).

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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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