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

2012 ◽  
Vol 79 (5) ◽  
Author(s):  
Marcio A. A. Cavalcante ◽  
Marek-Jerzy Pindera
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Displacement Field ◽  
Solid Mechanics ◽  
Order Theory ◽  
Substantial Improvement ◽  
Higher Order ◽  
Stress Distributions ◽  
Field Representation ◽  
First Order Theory

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.

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

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Marcio A. A. Cavalcante ◽  
Marek-Jerzy Pindera
Keyword(s):  
Displacement Field ◽  
Computer Code ◽  
Biological Tissues ◽  
Order Theory ◽  
Substantial Improvement ◽  
Higher Order ◽  
Finite Deformations ◽  
Stress Distributions ◽  
Field Representation ◽  
Periodic Materials

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.

Generalized Finite-Volume Theory for Elastic Stress Analysis in Solid Mechanics—Part I: Framework

2012 ◽  
Vol 79 (5) ◽  
Author(s):  
Marcio A. A. Cavalcante ◽  
Marek-Jerzy Pindera
Keyword(s):  
Finite Volume ◽  
Displacement Field ◽  
Stiffness Matrix ◽  
Solid Mechanics ◽  
Substantial Improvement ◽  
Higher Order ◽  
Standard Theory ◽  
Equilibrium Equations ◽  
Field Representation ◽  
Local Stiffness

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.

Analysis of three-layer composite plates with a new higher-order layerwise formulation

2014 ◽  
Vol 21 (3) ◽  
pp. 401-404
Author(s):  
Dalal A. Maturi ◽  
Antonio J.M. Ferreira ◽  
Ashraf M. Zenkour ◽  
Daoud S. Mashat
Keyword(s):  
Composite Plates ◽  
Order Theory ◽  
Higher Order ◽  
Basis Functions ◽  
Numerical Accuracy ◽  
First Order ◽  
The Core ◽  
First Order Theory ◽  
Layer Composite ◽  
Vibration Problems

AbstractIn this paper, we combine a new higher-order layerwise formulation and collocation with radial basis functions for predicting the static deformations and free vibration behavior of three-layer composite plates. The skins are modeled via a first-order theory, while the core is modeled by a cubic expansion with the thickness coordinate. Through numerical experiments, the numerical accuracy of this strong-form technique for static and vibration problems is discussed.

A Simple Higher-Order Theory for Laminated Composite Plates

1984 ◽  
Vol 51 (4) ◽  
pp. 745-752 ◽  
Author(s):  
J. N. Reddy
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Composite Plates ◽  
Order Theory ◽  
Higher Order ◽  
Laminated Composite Plates ◽  
First Order ◽  
First Order Theory

A higher-order shear deformation theory of laminated composite plates is developed. The theory contains the same dependent unknowns as in the first-order shear deformation theory of Whitney and Pagano [6], but accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact closed-form solutions of symmetric cross-ply laminates are obtained and the results are compared with three-dimensional elasticity solutions and first-order shear deformation theory solutions. The present theory predicts the deflections and stresses more accurately when compared to the first-order theory.

Application of Finite Volume Method for Solid Mechanics

2003 ◽  
Author(s):  
Bryce L. Fowler ◽  
Raymond K. Yee
Keyword(s):  
Finite Element ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Solid Mechanics ◽  
Preliminary Test ◽  
Test Case ◽  
Element Analysis ◽  
Vibration Isolators ◽  
Incompressible Materials ◽  
Volume Method

Polymers constitute a large class of nearly incompressible solid materials (i.e., Poisson’s Ratio near 0.5). These materials are often used as passive vibration isolators. Accurately modeling vibration isolators made of nearly incompressible materials has been extremely difficult with standard finite element analysis. This paper provides an alternative to the specialized finite element formulations currently used to model incompressible materials. The finite volume methodology of computational fluid dynamics is employed in this paper to solve the Hooke’s Law equations in solid mechanics. Test cases have been performed to evaluate the performance of finite volume method applied to solid mechanics problems. The formulation has been coded in Matlab for practical use. Based on the preliminary test case results, the finite volume formulation compares favorably to finite element method.

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