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Research Papers

Incompatible Graded Finite Elements for Analysis of Nonhomogeneous Materials

[+] Author and Article Information
Asmita Rokaya, Jeongho Kim

Department of Civil and Environmental
Engineering,
University of Connecticut,
261 Glenbrook Road, U-3037,
Storrs, CT 06269

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 2, 2018; final manuscript received November 26, 2018; published online December 17, 2018. Assoc. Editor: N. R. Aluru.

J. Appl. Mech 86(2), 021009 (Dec 17, 2018) (9 pages) Paper No: JAM-18-1456; doi: 10.1115/1.4042132 History: Received August 02, 2018; Revised November 26, 2018

A six-node incompatible graded finite element is developed and studied. Such element is recommended for use since it is more accurate than four-node compatible element and more efficient than eight-node compatible element in two-dimensional plane elasticity. This paper presents comparison between six-node incompatible (QM6) and four-node compatible (Q4) graded elements. Numerical solution is obtained from abaqus using UMAT capability of the software and exact solution is provided as reference for comparison. A graded plate with exponential and linear gradation subjected to traction and bending load is considered. Additionally, three-node triangular (T3) and six-node triangular (T6) graded elements are compared to QM6 element. Incompatible graded element is shown to give better performance in terms of accuracy and computation time over other element formulations for functionally graded materials (FGMs).

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Figures

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Fig. 1

Q4 element in physical space and QM6 element in physical space with curved lines in the boundary showing the added displacement functions

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Fig. 2

Geometry of plate (left) loaded in tension load perpendicular to the gradation, and geometry of plate (right) loaded in bending load perpendicular to the gradation. Linear and exponential variation of modulus along the width E = E(x) where E1 = E(x = 0) and E2 = E(x = L).

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Fig. 5

(a) Strain distribution for loading applied parallel to exponential gradation and (b) strain distribution for loading applied parallel to linear gradation

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Fig. 6

Stress distribution for Q4 and QM6 elements with bending load applied perpendicular to the exponential material gradation

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Fig. 3

Nonaveraged nodal stress results for tension load applied perpendicular to exponential material gradation

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Fig. 4

(a) Stress distribution for tension load applied parallel to exponential material gradation and (b) stress distribution for tension load applied parallel to linear gradation

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Fig. 10

(a) Stress distribution for tension load applied parallel to exponential material gradation for T3 and QM6 elements and (b) stress distribution for tension load applied parallel to exponential material gradation for T6 and QM6 elements

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Fig. 11

(a) Stress distribution for tension load applied parallel to exponential material gradation for T3 and QM6 elements and (b) stress distribution for tension load applied parallel to exponential material gradation for T6 and QM6 elements (diagonal of mesh swapped)

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Fig. 12

(a) Stress distribution comparison for tension load applied parallel to linear material gradation for T3 and QM6 elements and (b) stress distribution for tension load applied parallel to exponential material gradation for T6 and QM6 elements

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Fig. 7

(a) Stress distribution for tension load applied perpendicular to exponential material gradation for T3 and QM6 elements and (b) stress distribution for tension load applied perpendicular to exponential material gradation for T6 and QM6 elements

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Fig. 8

Triangular elements (T3 and T6): regular setup (left) and diagonals swapped (right)

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Fig. 9

Stress distribution for tension load applied perpendicular to exponential material gradation for (a) T3 and QM6 elements (regular mesh in Figs. 8(a)) and 8(b) T6 and QM6 elements (mesh swapped in Fig. 8(b))

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Fig. 13

(a) Stress distribution for tension load applied parallel to linear material gradation for T3 and QM6 elements and (b) stress distribution for tension load applied parallel to exponential material gradation for T6 and QM6 elements (diagonal of triangular mesh swapped)

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Fig. 14

(a) Stress distribution for bending load applied perpendicular to exponential material gradation for T3 and QM6 elements and (b) stress distribution for bending load applied perpendicular to exponential material gradation for T3 and QM6 elements

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Fig. 15

(a) Stress distribution for bending load applied perpendicular to exponential material gradation for T3 and QM6 elements and (b) stress distribution for bending load applied perpendicular to exponential material gradation for T3 and QM6 elements (diagonal of mesh swapped)

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