Design sensitivity analysis for the homogenized elasticity tensor of a polymer filled with rubber particles

Kamiński, Marcin

doi:10.1016/j.ijsolstr.2013.10.025

Cited by 10 publications

(12 citation statements)

References 24 publications

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“…The homogenized medium characterized by

{C}_{ijkl}^{(eff)}(b)

is assumed to accumulate the same energy as original composite a medium with a series of effective elastic characteristics

{C}_{ijkl}(b)

, so that one uses the following equity applicable to the RVE:

U(b)=\frac{1}{2}{\int}_{\Omega}{C}_{ij kl}(b){\varepsilon}_{ij}(b){\varepsilon}_{kl}(b)d\Omega ={U}^{\mathrm{hom}}=\frac{1}{2}{\int}_{\Omega}{C}_{ij kl}^{(eff)}(b){\varepsilon}_{ij}^{\mathbf{x}}{\varepsilon}_{kl}^{\mathbf{x}}d\Omega,

where the so‐called macro strains in the effective medium are denoted by

{\varepsilon}_{ij}^{\mathbf{x}}

. The set of specific Dirichlet boundary conditions is imposed on the outer edges of the RVE, which corresponds to the uniform extension of this RVE along x 1 axis 53 :

\begin{matrix} lefttrue \\ ε_{italicij}^{x_{1}} : u_{1} (l_{1}, x_{2}, x_{3}) = {normalΔ}_{1}, u_{2} (x_{1} ...) \end{matrix}

…”

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

“…Elastic parameters of the particle have been taken as E p = 1.0 MPa, ν p = 0.4888, while for the polymer matrix as equal to E m = 4.0 GPa and ν m = 0.34. This case study has been discussed in Reference 53, where the deterministic sensitivity of the effective elasticity tensor with respect to all material parameters of this composite has been discussed. Both Young moduli of this composite have been adopted here in turn as Gaussian variables with given expectations and some variability interval for their coefficients of variation to see an influence of the input uncertainty (

\alpha \in \left[0.00,0.20\right]

).…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Numerical experiments are carried out for the RVE of particulate composite constating of a single spherical particle centrally embedded into a cubic matrix. Two different combinations of the constituents are analyzed namely ( i ) carbon‐black as well as ( ii ) rubber particles inserted into a polymeric matrix 52,53 . Uncertainty in these problems comes from material imperfections (Young modulus and Poisson ratio statistical scattering) or from micro‐geometry of the RVE (particle radius), and Gaussian distributions are taken each time due to the Maximum Entropy Principle.…”

Section: Introductionmentioning

confidence: 99%

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Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

Kamiński

2023

Numerical Meth Engineering

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The main idea of this work is an application of probabilistic entropy and also relative entropy in the numerical analysis of uncertainty propagation in the homogenization of some composite materials. The homogenization method is based on the determination of deformation energy for the representative volume elements computed with the use of some specific finite element method experiments. Uncertainty propagation concerns material and geometrical design parameters of particulate composites and is performed thanks to an application of polynomial responses; probabilistic moments of the effective tensor are computed via the iterative generalized stochastic perturbation technique and the semi‐analytical probabilistic method. Probabilistic entropy is determined according to Shannon theory, while relative entropies equations employed here follow mathematical models created by Kullback & Leibler, Jeffreys, Hellinger, and Bhattacharyya. Deterministic analyses have been performed with the use of the system ABAQUS, while all the remaining procedures have been programmed in the computer algebra system MAPLE.

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“…The homogenized medium characterized by

{C}_{ijkl}^{(eff)}(b)

is assumed to accumulate the same energy as original composite a medium with a series of effective elastic characteristics

{C}_{ijkl}(b)

, so that one uses the following equity applicable to the RVE:

U(b)=\frac{1}{2}{\int}_{\Omega}{C}_{ij kl}(b){\varepsilon}_{ij}(b){\varepsilon}_{kl}(b)d\Omega ={U}^{\mathrm{hom}}=\frac{1}{2}{\int}_{\Omega}{C}_{ij kl}^{(eff)}(b){\varepsilon}_{ij}^{\mathbf{x}}{\varepsilon}_{kl}^{\mathbf{x}}d\Omega,

where the so‐called macro strains in the effective medium are denoted by

{\varepsilon}_{ij}^{\mathbf{x}}

. The set of specific Dirichlet boundary conditions is imposed on the outer edges of the RVE, which corresponds to the uniform extension of this RVE along x 1 axis 53 :

\begin{matrix} lefttrue \\ ε_{italicij}^{x_{1}} : u_{1} (l_{1}, x_{2}, x_{3}) = {normalΔ}_{1}, u_{2} (x_{1} ...) \end{matrix}

…”

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

\alpha \in \left[0.00,0.20\right]

).…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

Kamiński

2023

Numerical Meth Engineering

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“…This strategy drastically reduces the computing time with respect to full online optimization procedures. Kamiński (2014) and Fachinotti et al (2015) have presented similar approaches to evaluate the sensitivity gradients of the computationally homogenized properties of random composites.…”

Section: The Parameterized Biomimetic Cellular Microstructurementioning

confidence: 99%

Multiscale design of elastic solids with biomimetic cancellous bone cellular microstructures

Colabella

Cisilino

Fachinotti

et al. 2019

Struct Multidisc Optim

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Natural (or biological) materials usually achieve outstanding mechanical performances. In particular, cancellous bone presents a high stiffness/strength to weight ratio, so its structure inspires the development of novel ultra-light cellular materials. A multiscale method for the design of elastic solids with a cancellous bone parameterized biomimetic microstructure is introduced in this work. The method combines a finite element model to evaluate the stiffness of the body at the macroscale with a gradient-based nonlinear constrained optimization solver to obtain the optimal values of the microparameters and microstructure orientation over the body domain. The most salient features of the implementation are an offline response surface methodology for the evaluation of the microstructure elastic tensor in terms of the microparameters, an adjoint method for the computation of the sensitivity of the macroscopic stiffness to the microparameters, a quasi-Newton approximation for the evaluation of the Hessian matrix of the nonlinear optimizer, and a distanceweighted filter of the microparameters to remediate checkerboard effects. The settings of the above features, the optimizer termination options, and the initial values of the microparameters are investigated for the best performance of the method. The effectiveness of the method is demonstrated for several examples, whose results are compared with the reference solutions calculated using a SIMP method. The method shows to be effective; it attains results coherent with SIMP approaches in terms of stiffness and spatial material distribution. The good performance of the multiscale method is attributed to the capability of the parameterized mimetic microstructure to attain bulk and shear moduli that are close to the Hashin-Shtrikman upper bounds over the complete solid volume fraction range.

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“…By accounting for the thermal coupling, this work constitutes a step further in sensitivity analysis of purely mechanical multiscale problems [9,10].…”

Section: Introductionmentioning

confidence: 99%

Sensitivity of the thermomechanical response of elastic structures to microstructural changes

Fachinotti

Toro

Sánchez

et al. 2015

International Journal of Solids and Structures

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The paper presents a useful technique for analyzing the sensitivity of the thermomechanical response of a body with variable microstructure. The technique can be applied to attain a desired macroscopic response by altering the body microstructure. This problem is a key issue in material design.The sensitivity analysis relies on an accurate determination of the effective properties of the heterogeneous material. These effective properties are determined by computational homogenization. And their sensitivities, with respect to the parameters defining the microstructure, are then computed.For an efficient evaluation of the thermomechanical response, we propose to build response surfaces for the effective material properties. The surfaces are generated in an offline stage, by solving a series of homogenization problems at the microscale. In such a way, the fully online multiscale response analysis reduces to a standard monoscale problem at the macroscale. Thus, an important reduction in computational time is achieved, which is a crucial * Corresponding author advantage for material design.The capability of the proposed methodology is shown in light of its application to the design of a thermally-loaded structure with variable microstructure. Perceptible improvements in the structural response are achieved.

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Design sensitivity analysis for the homogenized elasticity tensor of a polymer filled with rubber particles

Cited by 10 publications

References 24 publications

Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

Multiscale design of elastic solids with biomimetic cancellous bone cellular microstructures

Sensitivity of the thermomechanical response of elastic structures to microstructural changes

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