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Research Papers

Simplified Approach to Solution of Mixed Boundary Value Problems on Homogeneous Circular Domain in Elasticity

[+] Author and Article Information
Gaurav Singh

Department of Mechanical Engineering,
Indian Institute of Technology Bombay,
Powai,
Mumbai 400076, India
e-mail: singh_gaurav@iitb.ac.in

Tanmay K. Bhandakkar

Department of Mechanical Engineering,
Indian Institute of Technology Bombay,
Powai,
Mumbai 400076, India
e-mail: tbhanda2@iitb.ac.in

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 11, 2018; final manuscript received November 6, 2018; published online December 12, 2018. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 86(2), 021007 (Dec 12, 2018) (9 pages) Paper No: JAM-18-1525; doi: 10.1115/1.4041965 History: Received September 11, 2018; Revised November 06, 2018

This work proposes a novel strategy to render mixed boundary conditions on circular linear elastic homogeneous domain to displacement-based condition all along the surface. With Michell solution as the starting point, the boundary conditions and extent of the domain are used to associate the appropriate type and number of terms in the Airy stress function. Using the orthogonality of sine and cosine functions, the modified boundary conditions lead to a system of linear equations for the unknown coefficients in the Airy stress function. Solution of the system of linear equations provides the Airy stress function and subsequently stresses and displacement. The effectiveness of the present approach in terms of ease of implementation, accuracy, and versatility to model variants of circular domain is demonstrated through excellent comparison of the solution of following problems: (i) annulus with mixed boundary conditions on outer radius and prescribed traction on the inner radius, (ii) cavity surface with mixed boundary conditions in an infinite plane subjected to far-field uniaxial loading, and (iii) circular disc constrained on part of the surface and subjected to uniform pressure on rest of the surface.

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Figures

Grahic Jump Location
Fig. 1

A linear elastic, homogeneous cylinder with inner radius r1 and outer radius r2 is subjected to prescribed radial (p(θ)) and tangential (q(θ)) traction on the inner surface (r = r1) and the outer surface (r = r2) is subjected to mixed bcs, i.e., part of the surface (0 ≤ θα and 2π − αθ ≤ 2π) is constrained while rest of the outer surface is traction free

Grahic Jump Location
Fig. 2

(a) A circular cavity of radius a in a homogenous, isotropic, infinite plate subjected to mixed bcs at the surface of the cavity, i.e., part of the cavity surface (0 ≤ θα and 2π − αθ ≤ 2π) is constrained while rest of the surface is traction free. In the far field, the plate is subjected to uni-axial tension (σ) (b) A homogenous, isotropic, linear elastic solid disk of radius a is subjected to mixed bcs at the outer surface (r = a), i.e., part of the surface (−αθα) is constrained while rest of the surface is under constant pressure p.

Grahic Jump Location
Fig. 3

Comparison of (a) radial stress σr, (b) shear stress σ, (c) tangential stress σθ, (d) displacement ur, uθ, along the outer surface (r = r2) generated using the present approach without any filtering (solid lines) and comsol-based FE solution (dash lines) for the cylinder shown in Fig. 1 with inner radius r1 = 0.5, outer radius r2 = 1, α = π/4, radial traction p(θ)=cos(2θ)+sin(3θ) and tangential traction q(θ)=cos(3θ)+sin(2θ) applied along the inner radius r = r1 and elastic properties μ = 3/8 and ν = 1/3. The result using the present approach is calculated for N = 150, while the FE solution is based on 6398 quadrilateral elements with smallest element size ∼ 4 × 10−5r2. The insets in Fig. 3(c) show the magnified view of oscillations in the stress component σθ(r2, θ) and the effect of filtering in the constrained region 0 ≤ θπ/4.

Grahic Jump Location
Fig. 4

(a) Comparison of displacement components ur, uθ along the inner radius r = r1 of cylinder (Fig. 1) generated using the present approach and without using any filter for N = 30, 50, 150 shown using diamond marker, dot marker, and solid line, respectively. (b) Variation of the strain energy density per unit length U¯ for the cylinder is shown in Fig. 1 as a function of N, the number of terms in the series solution of the present approach. The solution is based on r1=0.5, r2=1, α=π/4, p(θ)=cos(2θ)+sin(3θ), q(θ)=cos(3θ)+sin(2θ), μ=3/8 and ν = 1/3. For N > 50, U¯ converges to 2.07 shown through dash line. The corresponding value of U¯ for the comsol-based FE calculation, which uses 6398 quadrilateral elements and smallest element size ∼ 4 × 10−5r2, is 2.06.

Grahic Jump Location
Fig. 5

Comparison of (a) radial stress σr, (b) shear stress σ, (c) tangential stress σθ, (d) displacement ur, uθ, along the surface (r = r1) of the cavity, generated using the present approach with filtering (solid lines) and complex variable solution (solid circle) (details in Supplemental Material, which is available under the “Supplemental Data” tab for this paper on the ASME Digital Collection) for the circular cavity shown in Fig. 2(a). The result using the present approach is calculated under plain strain condition for N = 100, a = 1, α = π/4, κ = 1.8, μ = 1. Variation of σθ in the region 7π/4 ≤ θ ≤ 2π not shown in (c) is same as in 0 ≤ θπ/4.

Grahic Jump Location
Fig. 6

Comparison of (a) radial stress σr, (b) shear stress σ, (c) tangential stress σθ, (d) displacement ur, uθ, along the surface (r = a) of the solid disk, generated using the present approach with filtering (solid lines) and complex variable solution (solid circle) (details in Supplemental Material, which is available under the “Supplemental Data” tab for this paper on the ASME Digital Collection) for the solid disk shown in Fig. 2(b). The result using the present approach is calculated under plain strain condition for N = 300, a = 1, α = π/4, κ = 2, μ = 1. Variation of σθ in the region 7π/4 ≤ θ ≤ 2π not shown in (c) is same as in 0 ≤ θπ/4.

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