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

2002 ◽  
Vol 50 (4) ◽  
pp. 759-782 ◽  
Author(s):  
Pedro Ponte Castañeda
Keyword(s):  
Second Order ◽  
Field Fluctuations ◽  
Nonlinear Composites

Improving the Self-Consistent Predictions of Texture Development of Polycrystals Incorporating Intragranular Field Fluctuations

2005 ◽  
Vol 495-497 ◽  
pp. 955-964 ◽  
Author(s):  
Ricardo A. Lebensohn ◽  
Carlos Tomé ◽  
Pedro Ponte Castañeda
Keyword(s):  
Basal Slip ◽  
Texture Evolution ◽  
Rate Sensitivity ◽  
Second Order ◽  
Homogenization Procedure ◽  
Compression Direction ◽  
Self Consistent ◽  
Field Fluctuations ◽  
Standard Linear ◽  
Derivatives Of

In this contribution we present how to implement the calculation of average field fluctuations inside the grains of a thermoelastic aggregate in terms of the derivatives of the stress potential given by the standard linear self-consistent (SC) model, and how this statistical information can be used to generate second-order estimates for the mechanical behavior of nonlinear viscoplastic polycrystals, by means of a rigorous non-linear homogenization procedure. To illustrate the differences between this second-order (SO) self-consistent approach and the classical first-order SC approximations, we compare them in terms of their predictions of the effective behavior of random fcc polycrystals as a function of their rate-sensitivity, and of the texture evolution in hcp ice polycrystals under uniaxial compression. In the latter case, the SO approximation is the only one able to predict a substantial accommodation of deformation by basal slip, even when the basal poles become strongly aligned with the compression direction and the basal slip systems became unfavorably oriented.

scholarly journals Field statistics in nonlinear composites. I. Theory

2006 ◽  
Vol 463 (2077) ◽  
pp. 183-202 ◽  
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.

scholarly journals Second-order theory for nonlinear composites and application to isotropic constituents

2006 ◽  
Vol 334 (10) ◽  
pp. 575-581 ◽  
Author(s):  
Martín I. Idiart ◽  
Kostas Danas ◽  
Pedro Ponte Castañeda
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Nonlinear Composites

scholarly journals Field statistics in nonlinear composites. II. Applications

2006 ◽  
Vol 463 (2077) ◽  
pp. 203-222 ◽  
Author(s):  
Martín I Idiart ◽  
Pedro Ponte Castañeda
Keyword(s):  
Transversely Isotropic ◽  
Second Order ◽  
Local Fields ◽  
Two Phase ◽  
Second Moments ◽  
Constituent Phase ◽  
Homogenization Methods ◽  
Exact Relations ◽  
Nonlinear Homogenization ◽  
Nonlinear Composites

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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