Research Papers

Localized Structures in Indented Shells: A Numerical Investigation

[+] Author and Article Information
Alice Nasto

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: anasto@mit.edu

Pedro M. Reis

Department of Mechanical Engineering
and Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: preis@mit.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 16, 2014; final manuscript received October 9, 2014; accepted manuscript posted October 15, 2014; published online October 30, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(12), 121008 (Oct 30, 2014) (8 pages) Paper No: JAM-14-1427; doi: 10.1115/1.4028804 History: Received September 16, 2014; Revised October 09, 2014

We present results from a numerical investigation of the localization of deformation in thin elastomeric spherical shells loaded by differently shaped indenters. Beyond a critical indentation, the deformation of the shell ceases to be axisymmetric and sharp structures of localized curvature form, referred to as “s-cones,” for “shell-cones.” We perform a series of numerical experiments to systematically explore the parameter space. We find that the localization process is independent of the radius of the shell. The ratio of the radius of the shell to its thickness, however, is an important parameter in the localization process. Throughout, we find that the maximum principal strains remain below 6%, even at the s-cones. As a result, using either a linear elastic (LE) or hyperelastic constitutive description yields nearly indistinguishable results. Friction between the indenter and the shell is also shown to play an important role in localization. Tuning this frictional contact can suppress localization and increase the load-bearing capacity of the shell under indentation.

Copyright © 2014 by ASME
Topics: Shells , Friction
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Grahic Jump Location
Fig. 4

The effect of shell radius on mechanical response. Nondimensionalized force–indentation curves for shells with various radii, ranging from R1 = 0.01 m to R1 = 1000 m with the same thickness ratio of t/R1 = 0.01. Force is nondimensionalized with the Young's modulus E and thickness t. The solid (dashed) vertical lines correspond to the average (standard deviation) of the critical indentation for the formation of 3 and 4 s-cones (across the six shell radii).

Grahic Jump Location
Fig. 2

Changing the shape of the indenter. (a) Schematic of the indentation of a shell by an indenter of radii R1 and R2, respectively. The indenter to shell ratio is Γ = R2/R1. (b) Sharp indenters (Γ < 1) remain in contact with the pole of the shell during indentation. (c) Blunt indenters (Γ > 1) delaminate from the pole of the shell during indentation.

Grahic Jump Location
Fig. 1

Point indentation. (a) Experimental snapshots (axonometric view) of the evolution of the pattern of localization for an elastomeric shell under point indentation at its pole. (b) Experimental snapshots captured from underneath the shell. The white reflection corresponds to the location at which the shell inverts. (c) Snapshots from FEM simulations of the same scenario corresponding to (b). Color map represents the strain energy density. Further details on the material and geometric properties are given in the text.

Grahic Jump Location
Fig. 3

The effect of self-weight on the mechanical response. The shell thickness to radius ratio is t/R1 = 0.012. (a) Load–indentation curves for a shell under point indentation when weight is included (dashed blue) or excluded (solid red). (b) Relative difference of the two load curves, ϕ (defined in Eq. (2)). The vertical dashed lines correspond to indentation values at which three s-cones and four s-cones form, ε = 0.21 and ε = 0.69, respectively.

Grahic Jump Location
Fig. 5

The effect of shell thickness on the formation and evolution of s-cones. Representative snapshots of shells for t/R1 = (0.004, 0.02, 0.05), under point indentation. The color map corresponds to strain energy density.

Grahic Jump Location
Fig. 6

The effect of shell thickness. (a) Snapshots of shells with increasing t/R1 ratios at ε = 0.75, under point indentation. The color map indicates strain energy density. The black dashed lines trace a path along ridges between adjacent s-cones. (b) Strain energy density, W, along the paths in (a), for shells with thickness to radius ration in the range 0.002 < t/R1 < 0.03, indented to ε = 0.75 under point indentation. (c) Wmax (at the s-cones) and Wmin (at the ridges), as a function of t/R1. The solid red line indicates a power-law fit to the data, with exponent 2.1 ± 0.1.

Grahic Jump Location
Fig. 7

Deformed configurations of a shell under indentation using (a) a LE or (b) a NH material model. The shell thickness to radius ratio is t/R1 = 0.012. In both cases the ratio between the radii of the shell and the indenter, is varied from point load (Γ = 0) to plate load (Γ = ). The color map represents to strain energy density. Red circles indicate the location of s-cones. Dashed lines are drawn over ridges that connect s-cones, along which the curvature of the shell is inverted. Solid red lines are drawn over gullies that connect s-cones to the pole of the shell (only present for point load). No localization occurs with indenters with Γ ∼ 1 (shaded region).

Grahic Jump Location
Fig. 8

Quantitative comparison of mechanical response between LE and NH material models. (a) Load-indentation curves for a variety of indenters, Γ = (0, 1, 3, 10), using the LE (solid lines) and NH (dashed lines) material model. (b) Critical indentation at the onset of localization, comparing the LE and NH models. No localization occurs for Γ ∼ 1 (vertical dashed line). The shell thickness to radius is t/R1 = 0.012.

Grahic Jump Location
Fig. 9

Maximum principal strain, εmax, during indentation a shell (t/R1 = 0.01) for (a) sharp indenters, Γ ≤ 1, and (b) blunt indenters, Γ > 1. The black diamond symbol on each curve indicates the critical indentation at which s-cones form.

Grahic Jump Location
Fig. 13

The effect of friction on the mechanical response. Load-indentation curves for an indenter with Γ = 10 for friction coefficients μ = 0, 0.25, 0.5, 1.0, 1.3, 1.5, 1.7.

Grahic Jump Location
Fig. 12

(a) The critical indentation for the onset of localization for an indenter with Γ = 10 versus μ. (b) Load, P, at the critical indentation, versus μ. The shell thickness to radius ratio is t/R1 = 0.01.

Grahic Jump Location
Fig. 10

The effect of friction on localization. (a) FEM snapshots for indentation of a shell (t/R1 = 0.01) with Γ = 10, for various friction coefficients, μ, at ε = 0.4 indentation. The color map indicates strain energy density. (b) Strain energy density plotted along paths over ridges between adjacent s-cones, black dashed lines in (a), for shells different values of μ. (c) Wmax (at the s-cones) and Wmin (at the ridges midpoint) as a function of μ.

Grahic Jump Location
Fig. 11

(a) Isometric view of the global buckling mode for a shell (t/R1 = 0.01) under large indentations (configuration shown at ε = 0.51) for an indenter with Γ = 10 and μ = 1.7. (b) Side view. Color map indicates strain energy density.



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