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Research Papers

Analytical Solutions and Stress Concentration Factors for Annuli With Inhomogeneous Boundary Conditions

[+] Author and Article Information
S. Shahzad

Department of Civil Engineering,
Aalto University,
Rakentajanaukio 4,
Espoo 02150, Finland

J. Niiranen

Department of Civil Engineering,
Aalto University,
Rakentajanaukio 4,
Espoo 02150, Finland

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 20, 2017; final manuscript received April 18, 2018; published online May 10, 2018. Assoc. Editor: Thomas Siegmund.

J. Appl. Mech 85(7), 071008 (May 10, 2018) (14 pages) Paper No: JAM-17-1686; doi: 10.1115/1.4040079 History: Received December 20, 2017; Revised April 18, 2018

Analytical displacement and stress fields with stress concentration factors (SCFs) are derived for linearly elastic annular regions subject to inhomogeneous boundary conditions: an infinite class of the mth order polynomial antiplane tractions or displacements. The solution of the Laplace equation governing the out-of-plane problem covers both rigid and void circular inclusions forming the core of the annulus. The results show first that the SCF and the loading order are inversely proportional. In particular, the SCF approaches value 2 when either the outer boundary of the annulus tends to infinity or the order of the polynomial loading increases. Second, the number of peculiar points on the inner contour having null stress increases with the increasing loading order. The analytical solution is confirmed and extended to noncircular enclosures via finite element analysis by exploiting the heat-stress analogy. The results show that the closed-form solution for a circular annulus can be used as an accurate approximation for noncircular enclosures. Altogether, the results shown can be exploited for analyzing complex loading conditions and/or multiple rigid or void inclusions for enhancing the design of hollow and reinforced composites materials.

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Figures

Grahic Jump Location
Fig. 1

Annular cross sections composed of an inclusion having a circular contour of radius a embedded in a matrix with elliptical contour (defined by the semimajor axis b and the semiminor axis R̂) and of regular polygons circumscribed by circle of radius R̂. In particular, the extreme cases of circular inclusions defined by the Greek letter χ are taken into consideration, see Eq. (5).

Grahic Jump Location
Fig. 2

Contour plots under 3D shaded surface plots for a circular annulus undergoing nonuniform polynomial antiplane displacement fields w(r, θ) corresponding to uniform, linear, and quadratic (m = 0, 1, 2) antiplane shearing loads (b0(m)=0)

Grahic Jump Location
Fig. 3

Cases of dimensionless radial shear stress τrz(m)(r,0)/(amF(m)(θ)) (flat curves between the stress values of 2 and 1) and circumferential shear stress component τθz(m)(r,0)/(am∂F(m)(θ)/∂θ) (exploding curves between the stress values of 2 and ) as a function of the volume fractions f undergoing different loading orders (m = 0, 1, 2, 5, 10)

Grahic Jump Location
Fig. 4

Dimensionless shear stress τ(m)(r,0)/τ∞(m)(a,0) along the horizontal line bisecting the circular annulus with different values of the volume fractions f as a function of dimensionless radial distance r/a is reported. In particular, cases of uniform (m = 0) and linear (m = 1) antiplane loading conditions activating stress concentrations for void (b0(m)=0) and for rigid inclusion (c0(m)=0) are considered. The results show that for thin annulus, the void becomes very dangerous from the failure point of view due to the high stress concentration with respect to the rigid inclusion. If inclusion becomes very small, then the SCF value 2 is reached for both the boundary value problems but the void approaches the value from above and the rigid inclusion from below.

Grahic Jump Location
Fig. 5

Loading order m = 0 remains the same, while aspect ratio f varies. Contour plots of the dimensionless shear stress modulus τ(0)(r,θ)/τ∞(0)(r,θ) (top row) and the values of SCF τ(0)(a,θ)/τ∞(0)(a,θ) (bottom row) along the inner contour subject to uniform antiplane shear (b0(0)=0) are reported. Results show that, when f → 1, then SCF tends to infinity, while for f → 0 the SCF value tends to 2 (as shown in Refs. [19], [21], and [32]). Notice that the analytical solution is confirmed through the comparison of SCF analytical curves (solid line) and numerical curves (dashed line) obtained via finite element analysis discussed comprehensively in Sec. 4.

Grahic Jump Location
Fig. 6

The aspect ratio of the annulus remains constant, while the loading order m varies. Contour plots of the dimensionless shear stress modulus τ(m)(r,θ)/τ∞(m)(a,θ) (top row) and the values of SCFs τ(m)(a,θ)/τ∞(m)(a,θ) (bottom row) along the inner contour of the circular hole subject to uniform (m = 0), linear (m = 1) and quadratic (m = 2) antiplane loading (b0(m)=0) are reported. Results show that the value of SCF along the boundary of circular hole is inversely proportional to the order of the polynomial loading. Whereas the number of annihilation points (with null stress) occurring at the inner contour is directly proportional to the loading order. The number of annihilation points is given by 2(m + 1).

Grahic Jump Location
Fig. 7

A rectangular (I) and an elliptical enclosure (II) modeling hollow core slab and symmetrical hollow airfoil, respectively, are meshed at two levels with free triangular elements

Grahic Jump Location
Fig. 8

Contour lines of the dimensionless shear stress modulus τ(m)(r,θ)/τ∞(m)(r,θ) for the value of volume fraction f = 0.04 (i.e., a/R = 1/5, top row) and f = 0.25 (i.e., a/R = 1/2, bottom row) and the values of SCF (middle column) along the inner contour of a circular annulus subject to uniform (b0(0)=0) show that, even a small amount of relative eccentricity d¯=d/a leads to the higher SCF

Grahic Jump Location
Fig. 9

Values of the SCF are shown for noncircular annulus (circular void and square enclosure) subject to uniform (top row) and linear (bottom row) antiplane loading condition (b0(m)=0). Different aspect ratios are considered for the noncircular annulus to evaluate the influence on the SCF value. Results show that, for uniform loading, the error is greater than 1%, while already for the linear loading, the analytical solution is very close to the numerical one.

Grahic Jump Location
Fig. 10

Cracking of mortar specimens containing cylindrical inclusion. Source: Courtesy by Han and co-workers [15].

Grahic Jump Location
Fig. 11

Simulation of a hollow core slab subject to uniform (m = 0) antiplane shear stress (b0(0)=0) is addressed showing that the interaction between the inclusions does not have significant influence on the stress concentration values. While the horizontal vicinity of the external boundary increases significantly the SCF value.

Grahic Jump Location
Fig. 12

An elliptical enclosure approximating a symmetrical airfoil subject to uniform (m = 0) antiplane shear stress (b0(0)=0) is reported and the interaction between the inclusions does not have significant influence on the stress concentration values

Grahic Jump Location
Fig. 13

A quarter of a square prism with collinear circular void subject to uniform antiplane shear (m = 0): dimensionless shear stress field (top left), cut plane (bottom left), and dimensionless stress curves on the perimeter p of the circular arc of the cut plane

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