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Research Papers

# Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part II: Results

[+] Author and Article Information
Marek-Jerzy Pindera

Mem. ASME
Civil and Environmental Engineering Department,
University of Virginia,
Charlottesville, VA 22904

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 24, 2012; final manuscript received February 16, 2013; accepted manuscript posted May 7, 2013; published online September 16, 2013. Assoc. Editor: Krishna Garikipati.

J. Appl. Mech 81(2), 021006 (Sep 16, 2013) (12 pages) Paper No: JAM-12-1493; doi: 10.1115/1.4024407 History: Received October 24, 2012; Revised February 16, 2013; Accepted May 07, 2013

## Abstract

In Part I, a generalized finite-volume direct averaging micromechanics (FVDAM) theory was constructed for periodic materials with complex microstructures undergoing finite deformations. The generalization involves the use of a higher-order displacement field representation within individual subvolumes of a discretized analysis domain whose coefficients were expressed in terms of surface-averaged kinematic variables required to be continuous across adjacent subvolume faces. In Part II of this contribution we demonstrate that the higher-order displacement representation leads to a substantial improvement in subvolume interfacial conformability and smoother stress distributions relative to the original theory based on a quadratic displacement field representation, herein called the $0th$ order theory. This improvement is particularly important in the finite-deformation domain wherein large differences in adjacent subvolume face rotations may lead to the loss of mesh integrity. The advantages of the generalized theory are illustrated through examples based on a known analytical solution and finite-element results generated with a computer code that mimics the generalized theory's framework. An application of the generalized FVDAM theory involving the response of wavy multilayers confirms previously generated results with the $0th$ order theory that revealed microstructural effects in this class of materials which are important in bio-inspired material architectures that mimic certain biological tissues.

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## References

Cavalcante, M. A. A. and Pindera, M.-J., 2012, “Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part I: Analytical Framework,” ASME J. Appl. Mech., xx(x), p. xxx.
Chen, D. J., Shah, D. K., and Chan, W. S., 1996, “Interfacial Stress Estimation Using Least Square Extrapolation and Local Stress Smoothing in Laminated Composites,” Comput. Struct., 58, pp. 765–774.
Cavalcante, M. A. A., Khatam, H., and Pindera, M.-J., 2011, “Homogenization of Elastic-Plastic Periodic Materials by FVDAM and FEM Approaches—An Assessment,” Composites, Part B, 42(6), pp. 1713–1730.
Khatam, H., and Pindera, M.-J., 2012, “Microstructural Scale Effects in the Nonlinear Elastic Response of Bio-Inspired Wavy Multilayers Undergoing Finite Deformation,” Composites, Part B, 43(3), pp. 869–884.
Timoshenko, S. P., and Goodier, N. J., 1970, Theory of Elasticity, 3rd ed., McGraw-Hill, New York.
Bansal, Y., and Pindera, M.-J., 2009, “Finite-Volume Micromechanics of Heterogeneous Periodic Materials,” VDM Verlag Dr. Muller, Saarbrucken, Germany.
Sussman, T., and Bathe, K. J., 1987, “A Finite Element Deformation of Nonlinear Incompressible Elastic and Inelastic Analysis,” Comput. Struct., 26, pp. 357–409.
Cavalcante, M. A. A., and Pindera, M.-J., 2012, “Generalized Finite-Volume Theory for Elastic Stress Analysis in Solid Mechanics—Part II: Results,” ASME J. Appl. Mech., 79(5), p. 051007.
Cavalcante, M. A. A., Marques, S. P. C., and Pindera, M.-J., 2008, “Computational Aspects of the Parametric Finite-Volume Theory for Functionally Graded Materials,” J. Comput. Mater. Sci., 44(2), pp. 422–438.

## Figures

Fig. 3

Convergence of the global unbalanced average first Piola–Kirchhoff stresses as a function of mesh refinement

Fig. 2

Interfacial first Piola–Kirchhoff traction and displacement difference (normalized by the porosity radius r) measures as a function of mesh refinement

Fig. 1

(a) Square array of circular porosities, and (b) the coarsest mesh used in the analysis with 16×8 circumferential and radial subvolumes/elements

Fig. 4

Comparison of the local Cauchy stress fields σ22(y2,y3), σ33(y2,y3), and σ23(y2,y3) obtained from the analytical and FVDAM solutions for the dilute cylindrical porosity array under uniaxial far-field transverse stress σ22∞ = 1 MPa and unit cell discretization of 64×32 subvolumes

Fig. 5

Deformed meshes obtained from the analytical and FVDAM solutions using coarse and fine mesh discretizations, generated by amplifying the displacements 50,000 times

Fig. 6

(a) Hexagonal array of circular porosities, and (b) one of the coarsest unit cell meshes used in the analysis with 18×3 subvolumes/elements

Fig. 15

Homogenized T¯22-λ2 response due to the transverse load T¯22≠0 obtained from the FVDAM and finite-element methods for different unit cell discretizations

Fig. 16

Comparison of the Cauchy stress fields σ22(y2,y3) due to the transverse load T¯22≠0 at λ2 = 1.05 obtained from the FVDAM and finite-element methods for two different mesh discretizations

Fig. 17

Comparison of the Cauchy stress fields σ22(y2,y3) due to the transverse load T¯22≠0 at λ2 = 1.25 obtained from the FVDAM and finite-element methods for two different mesh discretizations

Fig. 7

Homogenized T¯22-λ2 response obtained from the numerical solutions using different mesh discretizations for the uniaxial macroscopic transverse loading T¯22≠0

Fig. 8

Comparison of the local Cauchy stress fields σ22(y2,y3) obtained from the numerical solutions using different mesh discretizations for the uniaxial macroscopic transverse loading T¯22≠0 at the macroscopic stretch λ2 = 2.0

Fig. 9

Deformed meshes with 66×11 subvolumes (the finest mesh used in the analysis) due to the uniaxial macroscopic transverse loading T¯22≠0 at the macroscopic stretch of (a) λ2 = 1.01 (generated by amplifying the displacements 100 times), and (b) λ2 = 2.0

Fig. 10

Interfacial first Piola–Kirchhoff traction difference measure as a function of mesh refinement at (a) λ2 = 1.01, and (b) λ2 = 2.0

Fig. 11

Interfacial displacement difference measure (normalized by the porosity radius r) as a function of mesh refinement at (a) λ2 = 1.01, and (b) λ2 = 2.0

Fig. 12

Convergence of the global unbalanced average first Piola–Kirchhoff stress component ΔT¯22 as a function of mesh refinement at (a) λ2 = 1.01, and (b) λ2 = 2.0

Fig. 13

Convergence of the global unbalanced average first Piola–Kirchhoff stress components (ΔT¯23+ΔT¯32)/2 as a function of mesh refinement at (a) λ2 = 1.01, and (b) λ2 = 2.0

Fig. 14

(a) A wavy lamellar material with a highlighted unit cell of the periodic microstructure, and (b) the coarsest unit cell mesh used in the analysis with 48×4 subvolumes/elements

Fig. 18

Homogenized T¯22-λ2 response due to the transverse load T¯22≠0 obtained from the FVDAM and finite-element methods for unit cells with progressively finer microstructures (thinner layers)

Fig. 19

Comparison of the Cauchy stress fields σ22(y2,y3) at three transverse stretch values generated by the 2nd order FVDAM theory for two unit cells with one and two stiff layers subjected to the transverse load T¯22≠0

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