A generalized finite-volume theory is proposed for two-dimensional elasticity problems on rectangular domains. The generalization is based on a higher-order displacement field representation within individual subvolumes of a discretized analysis domain, in contrast with the second-order expansion employed in our standard theory. The higher-order displacement field is expressed in terms of elasticity-based surface-averaged kinematic variables, which are subsequently related to corresponding static variables through a local stiffness matrix derived in closed form. The novel manner of defining the surface-averaged kinematic and static variables is a key feature of the generalized finite-volume theory, which provides opportunities for further exploration. Satisfaction of subvolume equilibrium equations in an integral sense, a defining feature of finite-volume theories, provides the required additional equations for the local stiffness matrix construction. The theory is constructed in a manner which enables systematic specialization through reductions to lower-order versions. Part I presents the theoretical framework. Comparison of predictions 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.

References

References
1.
Versteeg
,
H. K.
, and
Malalasekera
,
W.
, 1995,
An Introduction to Computational Fluid Dynamics: The Finite Volume Method
,
Prentice Hall
,
New York
.
2.
Demirdzic
,
I.
,
Martinovic
,
D.
, and
Ivankovic
,
A.
, 1988, “
Numerical Simulation of Thermomechanical Deformation Processes in a Welded Work-Piece
,”
Zavarivanje
,
31
, pp.
209
219
(in Serbo-Croat).
3.
Fryer
,
Y. D. C.
,
Bailey
,
C.
,
Cross
,
M.
, and
Lai
,
C.-H.
, 1991, “
A Control Volume Procedure for Solving the Elastic Stress-Strain Equations on an Unstructured Mesh
,”
Appl. Math. Model.
,
15
, pp.
639
645
.
4.
Demirdzic
,
I.
, and
Martinovic
,
D.
, 1993, “
Finite Volume Method for Thermo-Elastic-Plastic Stress Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
109
, pp.
331
349
.
5.
Demirdzic
,
I.
, and
Muzaferija
,
S.
, 1994, “
Finite Volume Method for Stress Analysis in Complex Domains
,”
Int. J. Numer. Methods Eng.
,
37
, pp.
3751
3766
.
6.
Bailey
,
C.
, and
Cross
,
M.
, 1995, “
A Finite Volume Procedure to Solve Elastic Solid Mechanics Problems in Three Dimensions on an Unstructured Mesh
,”
Int. J. Numer. Methods Eng.
,
38
, pp.
1757
1776
.
7.
Taylor
,
G. A.
,
Bailey
,
C.
, and
Cross
,
M.
, 1995, “
Solutions of the Elastic/Visco-Plastic Constitutive Equations: A Finite Volume Approach
,”
Appl. Math. Model.
,
19
, pp.
746
760
.
8.
Wheel
,
M. A.
, 1996, “
A Finite-Volume Approach to the Stress Analysis of Pressurized Axisymmetric Structures
,”
Int. J. Pressure Vessels Piping
,
68
, pp.
311
317
.
9.
Taylor
,
G. A.
,
Bailey
,
C.
, and
Cross
,
M.
, 2003, “
A Vertex-Based Finite Volume Method Applied to Non-Linear Material Problems in Computational Solid Mechanics
,”
Int. J. Numer. Methods Eng.
,
56
, pp.
507
529
.
10.
Wenke
,
P.
, and
Wheel
,
M. A.
, 2003, “
A Finite Volume Method for Solid Mechanics Incorporating Rotational Degrees of Freedom
,”
Comput. Struct.
,
81
, pp.
321
329
.
11.
Fallah
,
N.
, 2004, “
A Cell Vertex and Cell Centred Finite Volume Method for Plate Bending Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
193
, pp.
3457
3470
.
12.
Fallah
,
N.
, 2005, “
A New Approach in Cell Centred Finite Volume Formulation for Plate Bending Analysis
,”
Int. Conf. Comput. Methods Sci. and Eng. (ICCMSE 2005), Lect. Ser. Comput. Comput. Sci.
,
4
, pp.
187
190
.
13.
Fallah
,
N.
, 2005, “
Using Shape Function in Cell Centred Finite Volume Formulation for Two Dimensional Stress Analysis
,”
Int. Conf. Comput. Methods Sci. and Eng. (ICCMSE 2005), Lect. Ser. Comput. Comput. Sci.
,
4
, pp.
183
186
.
14.
Fallah
,
N.
, 2006, “
On the Use of Shape Functions in the Cell Centered Finite Volume Formulation for Plate Bending Analysis Based on Mindlin-Reissner Plate Theory
,”
Comput. Struct.
,
84
, pp.
1664
1672
.
15.
Basic
,
H.
,
Demirdzic
,
I.
, and
Muzaferija
,
S.
, 2005, “
Finite Volume Method for Simulation of Extrusion Processes
,”
Int. J. Numer. Methods Eng.
,
62
, pp.
475
494
.
16.
Bijelonja
,
I.
,
Demirdzic
,
I.
, and
Muzaferija
,
S.
, 2006, “
A Finite Volume Method for Incompressible Linear Elasticity
,”
Comput. Methods Appl. Mech. Eng.
,
195
, pp.
6378
6390
.
17.
Wheel
,
M. A.
, 2008, “
A Control Volume-Based Finite Element Method for Plane Micropolar Elasticity
,”
Int. J. Numer. Methods Eng.
,
75
, pp.
992
1006
.
18.
Paulino
,
G. H.
,
Pindera
,
M.-J.
,
Dodds
,
R. H.
,
Rochinha
,
F. E.
,
Dave
,
E. V.
, and
Chen
,
L.
, 2008, “
Multiscale and Functionally Graded Materials
,”
AIP Conf. Proc.
,
973
,
Melville
,
New York
.
19.
Achenbach
,
J. D.
, 1975,
A Theory of Elasticity With Microstructure for Directionally Reinforced Composites
,
Springer-Verlag
,
New York
.
20.
Cavalcante
,
M. A. A.
, 2006, “
Modelling of the Transient Thermo-Mechanical Behavior of Composite Material Structures by the Finite-Volume Theory
,” M.S. thesis, Federal University of Alagoas, Maceio, Alagoas, Brazil.
21.
Cavalcante
,
M. A. A.
,
Marques
,
S. P. C.
, and
Pindera
,
M.-J.
, 2007, “
Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials Part I: Analysis
,”
J. Appl. Mech.
,
74
(
5
), pp.
935
945
.
22.
Cavalcante
,
M. A. A.
,
Marques
,
S. P. C.
, and
Pindera
,
M.-J.
, 2007, “
Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials Part II: Numerical Results
,”
J. Appl. Mech.
,
74
(
5
), pp.
946
957
.
23.
Bansal
,
Y.
, and
Pindera
,
M.-J.
, 2003, “
Efficient Reformulation of the Thermoelastic Higher-Order Theory for FGMs
,”
J. Therm. Stresses
,
26
(
11–12
), pp.
1055
1092
.
24.
Zhong
,
Y.
,
Bansal
,
Y.
, and
Pindera
,
M.-J.
, 2004, “
Efficient Reformulation of the Thermal Higher-Order Theory for FGM’s With Variable Thermal Conductivity
,”
Int. J. Comput. Eng. Sci.
,
5
(
4
), pp.
795
831
.
25.
Gattu
,
M.
,
Khatam
,
H.
,
Drago
,
A. S.
, and
Pindera
,
M.-J.
, 2008, “
Parametric Finite-Volume Micromechanics of Uniaxial, Continuously-Reinforced Periodic Materials With Elastic Phases
,”
J. Eng. Mater. Technol.
,
130
(
3
), p.
31015
.
26.
Khatam
,
H.
, and
Pindera
,
M.-J.
, 2009, “
Parametric Finite-Volume Micromechanics of Periodic Materials With Elastoplastic Phases
,”
Int. J. Plast.
,
25
(
7
), pp.
1386
1411
.
27.
Khatam
,
H.
, and
Pindera
,
M.-J.
, 2010, “
Plasticity-Triggered Architectural Effects in Periodic Multilayers With Wavy Microstructures
,”
Int. J. Plast.
,
26
(
2
), pp.
273
287
.
28.
Bansal
,
Y.
, and
Pindera
,
M.-J.
, 2005, “
A Second Look at the Higher-Order Theory for Periodic Multiphase Materials
,”
J. Appl. Mech.
,
72
, pp.
177
195
.
29.
Bansal
,
Y.
, and
Pindera
,
M.-J.
, 2006, “
Finite-Volume Direct Averaging Micromechanics of Heterogeneous Materials With Elastic-Plastic Phases
,”
Int. J. Plast.
,
22
(
5
), pp.
775
825
.
30.
Aboudi
,
J.
,
Pindera
,
M.-J.
, and
Arnold
,
S. M.
, 1999, “
Higher-Order Theory for Functionally Graded Materials
,”
Composites, Part B
,
30
(
8
), pp.
777
832
.
31.
Aboudi
,
J.
,
Pindera
,
M.-J.
, and
Arnold
,
S. M.
, 2001, “
Linear Thermoelastic Higher-Order Theory for Periodic Multiphase Materials
,”
J. Appl. Mech.
,
68
(
5
), pp.
697
707
.
32.
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
, pp.
422
438
.
33.
Pan
,
W.
,
Wheel
,
M. A.
, and
Qin
,
Y.
, 2010, “
Six-Node Triangle Finite Volume Method for Solids With a Rotational Degree of Freedom for Incompressible Material
,”
Comput. Struct.
,
88
, pp.
1506
1511
.
34.
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
.
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