Nonlinear composites: a linearization procedure, exact to second-order in contrast and for which the strain-energy and affine formulations coincide

2004 ◽  
Vol 332 (9) ◽  
pp. 693-700 ◽  
Author(s):  
Noël Lahellec ◽  
Pierre Suquet
Keyword(s):  
Strain Energy ◽  
Second Order ◽  
Nonlinear Composites

Local material symmetry group for first- and second-order strain gradient fluids

2021 ◽  
pp. 108128652110216
Author(s):  
Victor A. Eremeyev
Keyword(s):  
Symmetry Group ◽  
Strain Gradient ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Mass Density ◽  
Material Model ◽  
Second Order ◽  
Material Symmetry ◽  
Local Material ◽  
Current Mass

Using an unified approach based on the local material symmetry group introduced for general first- and second-order strain gradient elastic media, we analyze the constitutive equations of strain gradient fluids. For the strain gradient medium there exists a strain energy density dependent on first- and higher-order gradients of placement vector, whereas for fluids a strain energy depends on a current mass density and its gradients. Both models found applications to modeling of materials with complex inner structure such as beam-lattice metamaterials and fluids at small scales. The local material symmetry group is formed through such transformations of a reference placement which cannot be experimentally detected within the considered material model. We show that considering maximal symmetry group, i.e. material with strain energy that is independent of the choice of a reference placement, one comes to the constitutive equations of gradient fluids introduced independently on general strain gradient continua.

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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