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Research Papers

Stress Concentration in Low-Porosity Periodic Tessellations With Generic Patterns of Elliptical Holes Under Biaxial Strain

[+] Author and Article Information
Jiazhen Leng

Mem. ASME
Department of Mechanical Engineering,
McGill University,
817 Sherbrooke Street West,
Montreal, QC H3A 0C3, Canada
e-mail: jiazhen.leng@mcgill.ca

Gerard Reynolds

Department of Mechanical Engineering,
McGill University,
817 Sherbrooke Street West,
Montreal, QC H3A 0C3, Canada
e-mail: gerard.reynolds@mail.mcgill.ca

Megan Schaenzer

Siemens Power and Gas,
9545 Côte-de-Liesse,
Dorval, QC H9P 1A5, Canada
e-mail: megan.schaenzer@siemens.com

Minh Quan Pham

Siemens Power and Gas,
9545 Côte-de-Liesse,
Dorval, QC H9P 1A5, Canada
e-mail: minhquan.pham@siemens.com

Genevieve Bourgeois

Siemens Power and Gas,
9545 Côte-de-Liesse,
Dorval, QC H9P 1A5, Canada
e-mail: genevieve.bourgeois@siemens.com

Ali Shanian

Siemens Power and Gas,
9545 Côte-de-Liesse,
Dorval, QC H9P 1A5, Canada
e-mail: ali.shanian@siemens.com

Damiano Pasini

Mem. ASME
Department of Mechanical Engineering,
McGill University,
817 Sherbrooke Street West,
Montreal, QC H3A 0C3, Canada
e-mail: damiano.pasini@mcgill.ca

1Corresponding author.

Manuscript received March 24, 2018; final manuscript received May 28, 2018; published online July 6, 2018. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 85(10), 101010 (Jul 06, 2018) (13 pages) Paper No: JAM-18-1168; doi: 10.1115/1.4040539 History: Received March 24, 2018; Revised May 28, 2018

Stress concentration in porous materials is one of the most crucial culprits of mechanical failure. This paper focuses on planar porous materials with porosity less than 5%. We present a stress-prediction model of an arbitrarily rotated elliptical hole in a rhombus shaped representative volume element (RVE) that can represent a class of generic planar tessellations, including rectangular, triangular, hexagonal, Kagome, and other patterns. The theoretical model allows the determination of peak stress and distribution of stress generated near the edge of elliptical holes for any arbitrary tiling under displacement loading and periodic boundary conditions. The results show that the alignment of the void with the principal directions minimizes stress concentration. Numerical simulations support the theoretical findings and suggest the observations remain valid for porosity as large as 5%. This work provides a fundamental understanding of stress concentration in low-porosity planar materials with insight that not only complements classical theories on the subject but also provides a practical reference for material design in mechanical, aerospace, and other industry.

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Figures

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Fig. 1

Description of geometric variables and examples of tessellation: (a) RVE and geometric variables of one elliptical hole, (b) Rotation of two holes in the RVE, and (c) Void patterns with triangular, hexagonal, and Kagome tessellations

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Fig. 2

Displacement loading and deformation pattern of the RVE

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Fig. 3

Elliptical coordinate system and biaxial loading of the elliptical hole: (a) Illustration of the elliptical coordinate system, with constant coordinate curves of α and β and (b) Rotated elliptical hole under biaxial loading. σ1 is linearly proportional to σ2 by the factor r.

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Fig. 4

Curves of stress concentration factor (Kt) versus θ for given a/b, parameterized by σ1/σ2: (a) σ12=−1.0, (b) σ12 = −0.8, (c) σ12 = −0.4, (d) σ12 = −0.2, (e) σ12 = −0.1, (f) σ12 = 0, (g) σ12 = 0.1, (h) σ12 = 0.2, (i) σ12 = 0.4, (j) σ12 = 0.8, and (k) σ12 = 1.0

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Fig. 5

Optimal aspect ratio of elliptical holes versus principal stress ratio σ1/σ2

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Fig. 6

Contour plots of stress concentration factor (Kt) as a function of aspect ratio and rotation angle of elliptical hole in the RVE, for tessellation angle from 10 deg to 90 deg, under uniaxial tension, with optimal hole layout marked and plotted: (a) ϕ = 10 deg, (b) ϕ = 20 deg, (c) ϕ = 30 deg, (d) ϕ = 40 deg, (e) ϕ = 50 deg, (f) ϕ = 60 deg, (g) ϕ = 70 deg, (h) ϕ = 80 deg, and (i) ϕ = 90 deg

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Fig. 7

Stress distribution plots of selected holes and tessellations under uniaxial loading: (a) a/b = 1, θ = 0 deg, ϕ = 90 deg, (b) a/b = 2, θ = 0 deg, ϕ = 90 deg, (c) a/b = 5, θ = 0 deg, ϕ = 90 deg, (d) a/b = 5, θ = 0 deg, ϕ = 60 deg, (e) a/b = 5, θ = 10 deg, ϕ = 60 deg, and (f) a/b = 5, θ = 90 deg, ϕ = 60 deg

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Fig. 8

Stress concentration factor (Kt) contour plots of absolute rotation γ and relative rotation δ for three tessellation angles, RVE under uniaxial tension, with optimal hole layout marked and plotted: (a) a/b = 2, ϕ = 30 deg, (b) a/b = 5, ϕ = 60 deg, and (c) a/b = 20, ϕ = 90 deg

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Fig. 9

Simulation results of full size versus RVE model with a/b = 4, θ = −12 deg, ϕ = 60 deg and ψ = 2%: (a) Boundary conditions and loading of the full size model, (b) boundary conditions and loading of the RVE model, (c) von Mises stress distribution near the center hole of the full size model, and (d) von Mises stress distribution of the RVE model

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