Field statistics in nonlinear composites. II. Applications

Part I of this work provided a methodology for extracting the statistics of the local fields in nonlinear composites, from the effective potential of suitably perturbed composites. In particular, exact relations were given for the first and even moments of the fields in each constituent phase. In this part, use is made of these exact relations in the context of the ‘variational’, ‘tangent second-order’ and ‘second-order’ nonlinear homogenization methods to generate estimates for the phase averages and second moments of the fields for two-phase, power-law composites with isotropic and transversely isotropic microstructures. The accuracy of these estimates is assessed by confronting them against corresponding exact results for sequentially laminated composites. Among the nonlinear homogenization estimates considered in this work, the second-order estimates are found to be, in general, the most accurate, especially for large heterogeneity contrast and nonlinearity. Thus, these estimates are able to capture, for example, the strong anisotropy in the strain fluctuations that can develop inside nonlinear porous and rigidly reinforced composites.

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Field statistics in nonlinear composites. I. Theory

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2006.1756 ◽

2006 ◽

Vol 463 (2077) ◽

pp. 183-202 ◽

Cited By ~ 43

Author(s):

Martín I Idiart ◽

Pedro Ponte Castañeda

Keyword(s):

Effective Potential ◽

Second Order ◽

Local Fields ◽

Two Phase ◽

Effective Potentials ◽

Second Moments ◽

Local Potentials ◽

Homogenization Methods ◽

Derivatives Of ◽

Nonlinear Composites

This work presents a means for extracting the statistics of the local fields in nonlinear composites from the effective potential of suitably perturbed composites. The idea is to introduce a parameter in the local potentials, generally a tensor, such that differentiation of the corresponding effective potential with respect to the parameter yields the volume average of the desired quantity. In particular, this provides a generalization to the nonlinear case of well-known formulas in the context of linear composites, which express phase averages and second moments of the local fields in terms of derivatives of the effective potential. Such expressions are useful since they allow the generation of estimates for the field statistics in nonlinear composites, directly from homogenization estimates for appropriately defined effective potentials. Here, use is made of these expressions in the context of the ‘variational’, ‘tangent second-order’ and ‘second-order’ homogenization methods, to obtain rigorous estimates for the first and second moments of the fields in nonlinear composites. While the variational estimates for these quantities are found to be identical to those proposed in previous works, the tangent second-order and second-order estimates are found be different. In particular, the new estimates for the first moments given in this work are found to be entirely consistent with the corresponding estimates for the macroscopic behaviour. Sample results for two-phase, power-law composites are provided in part II of this work.

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Exact results in the micromechanics of fibrous piezoelectric composites exhibiting pyroelectricity

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1993.0048 ◽

1993 ◽

Vol 441 (1911) ◽

pp. 59-81 ◽

Cited By ~ 47

Keyword(s):

Transversely Isotropic ◽

Dielectric Constants ◽

Local Fields ◽

Piezoelectric Composite ◽

Uniform Temperature ◽

Piezoelectric Composites ◽

Three Phase ◽

Phase Systems ◽

Electromechanical Loading ◽

Exact Relations

Piezoelectric fibrous composites of two, three and four phases are considered. The phase boundaries are cylindrical but otherwise the microgeometry is totally arbitrary. The constituents are transversely isotropic, and exhibit pyroelectricity. Exact relations are derived between the local fields arising under a uniform electromechanical loading and a uniform temperature change in the piezoelectric composite. For given overall material symmetry, exact connections are obtained among the effective elastic, piezoelectric and dielectric constants of two- and three- phase systems. It is also shown that the effective thermal stress and pyroelectric coefficients can be expressed in terms of the effective elastic, piezoelectric, dielectric constants and constituent properties in two-, three- and four-phase composites.

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Local fields in two-phase fibrous piezo-composites with 622 constituents

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.3923 ◽

2016 ◽

Vol 40 (9) ◽

pp. 3221-3229 ◽

Cited By ~ 1

Author(s):

Ransés Alfonso-Rodríguez ◽

Julián Bravo-Castillero ◽

Leslie D. Pérez-Fernández

Keyword(s):

Local Fields ◽

Two Phase

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Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: II—applications

Journal of the Mechanics and Physics of Solids ◽

10.1016/s0022-5096(01)00098-9 ◽

2002 ◽

Vol 50 (4) ◽

pp. 759-782 ◽

Cited By ~ 65

Author(s):

Pedro Ponte Castañeda

Keyword(s):

Second Order ◽

Field Fluctuations ◽

Nonlinear Composites

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A second-order time-accurate implicit finite volume method with exact two-phase Riemann problems for compressible multi-phase fluid and fluid–structure problems

Journal of Computational Physics ◽

10.1016/j.jcp.2013.11.001 ◽

2014 ◽

Vol 258 ◽

pp. 613-633 ◽

Cited By ~ 11

Author(s):

Alex Main ◽

Charbel Farhat

Keyword(s):

Finite Volume Method ◽

Finite Volume ◽

Second Order ◽

Riemann Problems ◽

Two Phase ◽

Fluid Structure ◽

Multi Phase ◽

Volume Method ◽

Phase Fluid

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Exact results for weakly nonlinear composites and implications for homogenization methods

Comptes Rendus. Mécanique ◽

10.5802/crmeca.66 ◽

2021 ◽

Vol 348 (10-11) ◽

pp. 893-909

Author(s):

Joshua Furer ◽

Martin Idiart ◽

Pedro Ponte Castañeda

Keyword(s):

Exact Results ◽

Weakly Nonlinear ◽

Homogenization Methods ◽

Nonlinear Composites

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On transformation strains and uniform fields in multiphase elastic media

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1992.0062 ◽

1992 ◽

Vol 437 (1900) ◽

pp. 291-310 ◽

Cited By ~ 206

Keyword(s):

Concentration Factor ◽

Elastic Stiffness ◽

Local Fields ◽

Influence Functions ◽

Elastic Solids ◽

Two Phase ◽

General Properties ◽

Multiphase Systems ◽

Concentration Factors ◽

Mechanical Influence

The effect of local eigenstrain and eigenstress fields, or transformation fields, on the local strains and stresses is explored in multiphase elastic solids of arbitrary geometry and material symmetry. The residual local fields caused by such transformation fields are sought in terms of certain transformation influence functions and transformation concentration factor tensors. General properties of these functions and concentration factors, and their relation to the analogous mechanical influence functions and concentration factors, are established, in part, with the help of uniform strain fields in multiphase media. Specific estimates of the transformation concentration factor tensors are evaluated by the self-consistent and Mori-Tanaka methods. It is found here that although the two methods use different constraint tensors in solutions of the respective dilute problems, their estimates of the mechanical, thermal, and transformation concentration factor tensors, and of the overall stiffness of multiphase media have a similar structure. Proofs that guarantee that these methods comply with the general properties of the transformation influence functions, and provide diagonally symmetric estimates of the overall elastic stiffness, are given for two-phase and multiphase systems consisting of, or reinforced by, inclusions of similar shape and alignment. One of the possible applications of the results, in analysis of overall instantaneous properties and local fields in inelastic composite materials, is described in the following paper.

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Generalized second-order slip for unsteady convective flow of a nanofluid: a utilization of Buongiorno’s two-component nonhomogeneous equilibrium model

Nonlinear Engineering ◽

10.1515/nleng-2020-0005 ◽

2020 ◽

Vol 9 (1) ◽

pp. 156-168

Author(s):

Seyed Mahdi Mousavi ◽

Saeed Dinarvand ◽

Mohammad Eftekhari Yazdi

Keyword(s):

Boundary Layer ◽

Equilibrium Model ◽

Second Order ◽

Model Parameters ◽

Slip Parameter ◽

Two Phase ◽

Convective Boundary ◽

First Order ◽

Special Cases ◽

Two Component

AbstractThe unsteady convective boundary layer flow of a nanofluid along a permeable shrinking/stretching plate under suction and second-order slip effects has been developed. Buongiorno’s two-component nonhomogeneous equilibrium model is implemented to take the effects of Brownian motion and thermophoresis into consideration. It can be emphasized that, our two-phase nanofluid model along with slip concentration at the wall shows better physical aspects relative to taking the constant volume concentration at the wall. The similarity transformation method (STM), allows us to reducing nonlinear governing PDEs to nonlinear dimensionless ODEs, before being solved numerically by employing the Keller-box method (KBM). The graphical results portray the effects of model parameters on boundary layer behavior. Moreover, results validation has been demonstrated as the skin friction and the reduced Nusselt number. We understand shrinking plate case is a key factor affecting non-uniqueness of the solutions and the range of the shrinking parameter for which the solution exists, increases with the first order slip parameter, the absolute value of the second order slip parameter as well as the transpiration rate parameter. Besides, the second-order slip at the interface decreases the rate of heat transfer in a nanofluid. Finally, the analysis for no-slip and first-order slip boundary conditions can also be retrieved as special cases of the present model.

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