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Research Papers

Elastic Soft-Core Sandwich Plates: Critical Loads and Energy Errors in Commercial Codes Due to Choice of Objective Stress Rate

[+] Author and Article Information
Jan Vorel

Assistant Professor
Czech Technical University in Prague,
Prague 16629, Czech Republic
e-mail: jan.vorel@fsv.cvut.cz

Zdeněk P. Bažant

ASME Fellow
McCormick Institute Professor
W.P. Murphy Professor of Civil Eng. and Materials Science;
Northwestern University,
Evanston, IL 60208
e-mail: z-bazant@northwestern.edu

Mahendra Gattu

Graduate Research Assistant
Northwestern University,
Evanston, IL 60208

1Visiting Scholar, Northwestern University.

2Corresponding author.

Manuscript received September 17, 2012; final manuscript received October 27, 2012; accepted manuscript posted November 19, 2012; published online May 31, 2013. Editor: Yonggang Huang.

J. Appl. Mech 80(4), 041034 (May 31, 2013) (10 pages) Paper No: JAM-12-1450; doi: 10.1115/1.4023024 History: Received September 17, 2012; Revised October 27, 2012; Accepted November 19, 2012

Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress rate the Jaumann rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since it is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the Truesdell objective stress rate, which is work-conjugate to the Green–Lagrangian finite strain tensor. The commercial codes should switch to this rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell rate and Green–Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell rate. Comparisons of the errors of various approaches are tabulated.

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References

Figures

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Fig. 1

Element of sandwich panel: (a) dimensions, (b) rotation angles and shear forces

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Fig. 2

Plate analyzed: both edges perpendicular to the axis x1 are clamped (C) and the longer edges are simply supported (SS)

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Fig. 3

Cylindrical buckling: homogenized plate simulation by OOFEM: (a) buckling mode, (b) sharp-break plot

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Fig. 4

Discretized simulation of rectangular plate by OOFEM (last line in Table 5): (a) buckling mode, (b) sharp-break plot

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Fig. 5

Cylindrical buckling: (a) evolution of normalized critical buckling load of rectangular plates of different span ratios a/b, for fixed b=2,540 mm (Engesser, Hencky, Haringx formulas given by Eq. (B3), cases of m=0,±2 given by Eq. (B1)), (b) deflection curve for the first buckling mode for m=2, a=3,380 mm

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Fig. 6

Analytical solution of homogenized plate: (a) normalized critical buckling load versus span ratio a/b, b=2,540 mm, (b) error when m=0 is used, err=(Pcr(0)/Pcr(2)-1)×100 %

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