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Research Papers

Generalized Finite-Volume Theory for Elastic Stress Analysis in Solid Mechanics—Part II: Results

[+] Author and Article Information
Marcio A. A. Cavalcante

ASME Member Civil and Environmental Engineering Department,  University of Virginia, Charlottesville, VA 22904-4742

Marek-Jerzy Pindera1

ASME Member Civil and Environmental Engineering Department,  University of Virginia, Charlottesville, VA 22904-4742

1

Corresponding author.

J. Appl. Mech 79(5), 051007 (Jun 22, 2012) (13 pages) doi:10.1115/1.4006806 History: Received August 31, 2011; Revised February 22, 2012; Posted May 09, 2012; Published June 22, 2012; Online June 22, 2012

In Part I, a generalized finite-volume theory was constructed for two-dimensional elasticity problems on rectangular domains based on a higher-order displacement field representation within individual subvolumes of a discretized analysis domain. The higher-order displacement field was expressed in terms of elasticity-based surface-averaged kinematic variables that were subsequently related to corresponding static variables through a local stiffness matrix derived in closed form. The theory was constructed in a manner that enables systematic specialization through reductions to lower-order versions, including the original theory based on a quadratic displacement field representation, herein called the zeroth-order theory. Comparison of predictions generated by the generalized theory with its predecessor, analytical and finite-element results in Part II illustrates substantial improvement in the satisfaction of interfacial continuity conditions at adjacent subvolume faces, producing smoother stress distributions and good interfacial conformability. While in certain instances the first-order theory produces acceptably smooth stress distributions, concentrated loadings require the second-order (generalized) theory to reproduce stress and displacement fields with fidelity comparable to analytical and finite-element results.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 6

Shear stress distributions, σ23(x2,x3) (MPa), due to applied end load. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories and finite-element results.

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Figure 7

Normal (left) and shear (right) stress distributions, σ22(x20,x3), σ23(x20,x3) (MPa), at two cross sections due to applied end load, generated using 18 × 90 mesh. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories, finite-element method and a converged numerical result.

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Figure 5

Normal stress distributions, σ22(x2,x3) (MPa), due to applied end load. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories and finite-element results.

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Figure 4

Normal (left) and shear (right) stress distributions, σ22(x2,x3), σ23(x2,x3) (MPa), in the x20= 2.5625 m cross section (in the middle of the 21st column) due to applied end load, generated using 8 × 40 mesh. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories, analytical and finite-element results.

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Figure 3

Normal stress distributions, σ22(x2,x3) (MPa), due to applied end load. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories and finite-element results.

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Figure 2

Deformed cantilevered strip under applied end load. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories, analytical and finite-element results.

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Figure 12

Interfacial stress (a) and displacement (b) difference measures at the cross section x20=5.0mm as a function of mesh refinement. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories and finite-element results.

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Figure 13

Convergence of the global unbalanced average stresses Δσ¯22, Δσ¯33, (Δσ¯23+Δσ¯32)/2 (MPa) as a function of mesh refinement for: (a) cantilevered rectangular slab with square cutouts; (b) rectangular strip under concentrated normal tractions. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories and finite-element results.

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Figure 1

Homogeneous and heterogeneous rectangular strips subjected to different loadings analyzed using the generalized finite-volume theory for verification with analytical and finite-element results

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Figure 8

Deformed meshes in the vicinity of applied normal traction on the upper surface of rectangular strip magnified 400 times. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories and finite-element results.

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Figure 9

Normal and shear stress distributions, σ22(x2,x3), σ33(x2,x3), σ23(x2,x3) (MPa), in the vicinity of applied normal tractions on top and bottom surfaces of rectangular strip, generated using 287 × 28 mesh. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories, analytical and finite-element results.

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Figure 10

Global error measures relative to the analytical solution for the normal and shear stress fields σ22(x2,x3), σ33(x2,x3) and σ23(x2,x3) (MPa) as a function of mesh refinement. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories and finite-element results.

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Figure 11

Interfacial stress (a) and displacement (b) difference measures at the cross section x20=2.5mm as a function of mesh refinement. Comparison among predictions of zeroth-, first-, and second-order (generalized) finite-volume theories and finite-element results.

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