Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part I: Framework

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Marcio A. A. Cavalcante ◽  
Marek-Jerzy Pindera
Keyword(s):  
Finite Volume ◽  
Finite Deformation ◽  
Large Deformations ◽  
Finite Deformations ◽  
Stress Concentrations ◽  
Field Representation ◽  
Deformation Features ◽  
Elasticity Problems ◽  
Periodic Materials ◽  
Artificial Stress

The recently constructed generalized finite-volume theory for two-dimensional linear elasticity problems on rectangular domains is further extended to make possible simulation of periodic materials with complex microstructures undergoing finite deformations. This is accomplished by embedding the generalized finite-volume theory with newly incorporated finite-deformation features into the 0th order homogenization framework, and introducing parametric mapping to enable efficient mimicking of complex microstructural details without artificial stress concentrations by stepwise approximation of curved surfaces separating adjacent phases. The higher-order displacement field representation within subvolumes of the discretized unit cell microstructure, expressed in terms of elasticity-based surface-averaged kinematic variables, substantially improves interfacial conformability and pointwise traction and nontraction stress continuity between adjacent subvolumes. These features enable application of much larger deformations in comparison with the standard finite-volume direct averaging micromechanics (FVDAM) theory developed for finite-deformation applications by minimizing interfacial interpenetrations through additional kinematic constraints. The theory is constructed in a manner which facilitates systematic specialization through reductions to lower-order versions with the 0th order corresponding to the standard FVDAM theory. Part I presents the theoretical framework. Comparison of predictions by the generalized FVDAM theory with its predecessor, analytical and finite-element results in Part II illustrates the proposed theory's superiority in applications involving very large deformations.

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.

Tensorial Deformation Measures for Flexible Joints

2010 ◽  
Vol 6 (3) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Leihong Li ◽  
Pierangelo Masarati ◽  
Marco Morandini
Keyword(s):  
Finite Deformation ◽  
Nonlinear Behavior ◽  
Critical Issue ◽  
Finite Deformations ◽  
Flexible Joints ◽  
Elastic Bodies ◽  
Infinitesimal Deformations ◽  
Stiffness Characteristics ◽  
Definition Of

Flexible joints, sometimes called bushing elements or force elements, are found in all multibody dynamics codes. In their simplest form, flexible joints simply consist of sets of three linear and three torsional springs placed between two nodes of a multibody system. For infinitesimal deformations, the selection of the lumped spring constants is an easy task, which can be based on a numerical simulation of the joint or on experimental measurements. If the joint undergoes finite deformations, the identification of its stiffness characteristics is not so simple, especially if the joint itself is a complex system. When finite deformations occur, the definition of deformation measures becomes a critical issue. Indeed, for finite deformation, the observed nonlinear behavior of materials is partly due to material characteristics and partly due to kinematics. This paper focuses on the determination of the proper finite deformation measures for elastic bodies of finite dimension. In contrast, classical strain measures, such as the Green–Lagrange or Almansi strains, among many others, characterize finite deformations of infinitesimal elements of a body. It is argued that proper finite deformation measures must be of a tensorial nature, i.e., must present specific invariance characteristics. This requirement is satisfied if and only if the deformation measures are parallel to the eigenvector of the motion tensor.

Finite Deformations of a Rigid Perfectly Plastic Arch

1962 ◽  
Vol 29 (3) ◽  
pp. 549-553 ◽  
Author(s):  
E. T. Onat ◽  
L. S. Shu
Keyword(s):  
Closed Form ◽  
Yield Point ◽  
Finite Deformation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Finite Deformations ◽  
Deformation History ◽  
Rigid Plastic ◽  
Perfectly Plastic ◽  
History Of

The quasi-static postyield deformation of a rigid-plastic arch in the presence of geometry changes is considered. The problem is formulated in terms of a series of boundary-value problems concerned with rates of stress and velocities. In the present simple case, the consideration of the rate problem associated with the yield-point state of the structure enables one to construct a closed-form solution which describes the entire deformation history of the arch. However, the principal aim of the present study is to stress the central role played by the rate problem in the investigation of the finite deformation of structures.

Stress Analysis for a Nonlinear Viscoelastic Compressible Cylinder

1970 ◽  
Vol 37 (4) ◽  
pp. 1127-1133 ◽  
Author(s):  
E. C. Ting
Keyword(s):  
Large Deformations ◽  
Finite Deformations ◽  
Kernel Functions ◽  
Nonlinear Viscoelastic ◽  
Stress Strain Relation ◽  
Incompressible Materials ◽  
Small Deformations ◽  
Compressible Cylinder ◽  
Elastic Casing ◽  
Finite Difference Techniques

Real solids are not incompressible, although many viscoelastic materials which undergo large deformations show only small changes in volume under ordinary loading conditions. This paper is concerned with a pressurized isotropic viscoelastic hollow cylinder bonded to an elastic casing in which, during a finite deformation, the dilatational change in any element of the cylinder is a small quantity. The analysis is based in part upon the theory of small deformations superposed on finite deformations. Numerical calculations are evaluated by using finite-difference techniques and assuming particular forms of kernel functions in the stress-strain relation. The results for compressible and incompressible materials are compared.

On Boundary Condition Implementation Via Variational Principles in Elasticity-Based Homogenization

2016 ◽  
Vol 83 (10) ◽  
Author(s):  
Guannan Wang ◽  
Marek-Jerzy Pindera
Keyword(s):  
Variational Principle ◽  
Variational Principles ◽  
Transversely Isotropic ◽  
Homogenization Theory ◽  
Element Analysis ◽  
Field Representation ◽  
Unidirectional Composites ◽  
Exterior Problems ◽  
Periodic Microstructures ◽  
Periodic Materials

Convergence characteristics of the locally exact homogenization theory for periodic materials, first proposed by Drago and Pindera (2008, “A Locally-Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases,” ASME J. Appl. Mech., 75(5), p. 051010) and recently generalized by Wang and Pindera (“Locally-Exact Homogenization Theory for Transversely Isotropic Unidirectional Composites,” Mech. Res. Commun. (in press); 2016, “Locally-Exact Homogenization of Unidirectional Composites With Coated or Hollow Reinforcement,” Mater. Des., 93, pp. 514–528; and 2016, “Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers,” ASME J. Appl. Mech., 83(7), p. 071010), are examined vis-a-vis the manner of implementing periodic boundary conditions. The locally exact theory separates the unit cell problem into interior and exterior problems, with the separable interior problem solved exactly in cylindrical coordinates and the inseparable exterior problem tackled using a balanced variational principle. This variational principle leads to exceptionally fast and well-behaved convergence of the Fourier series coefficients in the displacement field representation of the unit cell's different phases. Herein, we compare the solution's convergence behavior based on the balanced variational principle with that based on the constrained energy-based principle originally proposed by Jirousek (1978, “Basis for Development of Large Finite Elements Locally Satisfying All Fields Equations,” Comput. Methods Appl. Mech. Eng., 14, pp. 65–92) in the context of locally exact finite-element analysis. The relevance of this comparison lies in the recently rediscovered implementation of Jirousek's constrained variational principle in the homogenization of periodic materials.

Finite Volume-Based Asymptotic Homogenization of Periodic Materials Under In-Plane Loading

2020 ◽  
Vol 87 (12) ◽  
Author(s):  
Zhelong He ◽  
Marek-Jerzy Pindera
Keyword(s):  
Finite Volume ◽  
Local Equilibrium ◽  
Shear Loading ◽  
Local Fields ◽  
Homogenization Theory ◽  
Asymptotic Homogenization ◽  
Plane Shear ◽  
Strain Gradients ◽  
Combination Approach ◽  
Periodic Materials

Abstract The previously developed finite volume-based asymptotic homogenization theory (FVBAHT) for anti-plane shear loading (He, Z., and Pindera, M.-J., “Finite-Volume Based Asymptotic Homogenization Theory for Periodic Materials Under Anti-Plane Shear,” Eur. J. Mech. A Solids (in revision)) is further extended to in-plane loading of unidirectional fiber reinforced periodic structures. Like the anti-plane FVBAHT, the present extension builds upon the previously developed finite volume direct averaging micromechanics theory applicable under uniform strain fields and further accounts for strain gradients and non-vanishing microstructural scale relative to structural dimensions, albeit with multidimensional in-plane loadings incorporated. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations and displacement and traction continuity in a surface-averaged sense which is unique among the existing asymptotic homogenization schemes, leading to microfluctuation functions that yield homogenized stiffness tensors at each order for use in macroscale problems. The newly extended multiscale theory is employed in the analysis of a structural boundary-value problem under in-plane loading, illustrating pronounced boundary effects. A combination approach proposed in the literature is subsequently employed to mitigate the boundary layer effects by explicitly accounting for the microstructural details in the boundary region. This combination approach produces accurate recovery of the local fields in both regions. The extension to in-plane problem marks FVBAHT as an alternative, self-contained asymptotic homogenization tool, with documented advantages relative to current numerical techniques, for the analysis of periodic materials in the presence of strain gradients produced by three-dimensional loading regardless of microstructural scale.

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.

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