On the effective elasticity tensor of cellular materials obtained by an energetic homogenization procedure

2000 ◽  
Vol 80 (S2) ◽  
pp. 381-382 ◽  
Author(s):  
J. Hohe ◽  
W. Becker
Keyword(s):  
Elasticity Tensor ◽  
Cellular Materials ◽  
Homogenization Procedure ◽  
Effective Elasticity

Determination of the effective elasticity tensor of a folded sandwich core

PAMM ◽  
2014 ◽  
Vol 14 (1) ◽  
pp. 535-536
Author(s):  
Holger Massow ◽  
Wilfried Becker
Keyword(s):  
Elasticity Tensor ◽  
Effective Elasticity ◽  
Sandwich Core

Which Elasticity Tensors are Realizable?

1995 ◽  
Vol 117 (4) ◽  
pp. 483-493 ◽  
Author(s):  
Graeme W. Milton ◽  
Andrej V. Cherkaev
Keyword(s):  
Building Blocks ◽  
Elasticity Tensor ◽  
Isotropic Phase ◽  
Order Tensor ◽  
Two Phase ◽  
Effective Elasticity ◽  
Elasticity Tensors ◽  
Two Phases ◽  
The Given ◽  
Fourth Order Tensor

It is shown that any given positive definite fourth order tensor satisfying the usual symmetries of elasticity tensors can be realized as the effective elasticity tensor of a two-phase composite comprised of a sufficiently compliant isotropic phase and a sufficiently rigid isotropic phase configured in an suitable microstructure. The building blocks for constructing this composite are what we call extremal materials. These are composites of the two phases which are extremely stiff to a set of arbitrary given stresses and, at the same time, are extremely compliant to any orthogonal stress. An appropriately chosen subset of the extremal materials are layered together to form the composite with elasticity tensor matching the given tensor.

scholarly journals Computational Engineering Homogenisation of Steel Reinforced Concrete Plates

1999 ◽  
Vol 8 (5) ◽  
pp. 096369359900800
Author(s):  
Marcin Kamiñski
Keyword(s):  
Reinforced Concrete ◽  
Composite Structures ◽  
Elasticity Tensor ◽  
Upper And Lower Bounds ◽  
Steel Reinforced Concrete ◽  
Concrete Plate ◽  
Computational Data ◽  
Effective Elasticity ◽  
Direct Approximation ◽  
Elastostatic Problem

The main subject of the paper is the application of the homogenisation method to the analysis of the elastostatic problem for the steel reinforced concrete plate. The upper and lower bounds on the effective elasticity tensor components as well as direct approximation of the tensor are used to build up the computational model of the structure. The results of numerical analysis are compared against the experimental data. The convergence of experimental and computational data for the real and homogenised composite plate confirms the effectiveness of the homogenisation method applied in the analysis of steel-reinforced uncracked concrete structures. To verify the generality of this approach, analogous tests should be carried out for other types of engineering composite structures, especially in continuous non-linear range.

Assessment of the effect of microstructural uncertainty on the macroscopic properties of random composite materials

2016 ◽  
Vol 51 (19) ◽  
pp. 2707-2725 ◽  
Author(s):  
Dimitrios Savvas ◽  
George Stefanou
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Composite Materials ◽  
Material Properties ◽  
Elasticity Tensor ◽  
Statistical Characteristics ◽  
Stochastic Finite Element ◽  
Homogenization Procedure ◽  
Random Composite ◽  
Geometrical Uncertainty

The linking of microstructural uncertainty with the random variation in the response of heterogeneous structures at the macroscale is particularly important in the framework of the stochastic finite element method. In this work, the effect of uncertainty in the constituent material properties and the geometry of the microstructure, on the macroscopic properties of composite materials is assessed through computational homogenization. Based on Hill–Mandel homogeneity condition, the homogenization procedure utilizes the excellent synergy of the extended finite element method and the Monte Carlo simulation. In this way, the computation of the statistical characteristics of the homogenized elasticity tensor of random composite materials reinforced with arbitrarily shaped inclusions is performed in a computationally efficient manner. The effect of stochastic variation in the elastic properties of the constituents as well as the effect of inclusion shape on the statistical characteristics of the homogenized elasticity tensor is assessed through probabilistic sensitivity analysis. A comparison is performed with regard to the relative influence of material and geometrical uncertainty which are considered separately. More realistic results are obtained by considering simultaneously material and geometrical uncertainty in the microstructural modeling of composite materials. The results can be further exploited in the stochastic finite element analysis of composite structures where material properties with random characteristics obtained by the presented multiscale homogenization procedure will be assigned to each finite element.

scholarly journals On the homogenization of a new class of locally periodic microstructures in linear elasticity with residual stress

2017 ◽  
Vol 23 (7) ◽  
pp. 1025-1039
Author(s):  
Brian Seguin
Keyword(s):  
Residual Stress ◽  
Residual Stresses ◽  
Broad Class ◽  
Elasticity Tensor ◽  
Unit Cell ◽  
Periodic Homogenization ◽  
New Class ◽  
Effective Elasticity ◽  
Periodic Microstructures ◽  
Macroscopic Equations

Many biological and engineering materials have nonperiodic microstructures for which classical periodic homogenization results do not apply. Certain nonperiodic microstructures may be approximated by locally periodic microstructures for which homogenization techniques are available. Motivated by the consideration that such materials are often anisotropic and can possess residual stresses, a broad class of locally periodic microstructures are considered and the resulting effective macroscopic equations are derived. The effective residual stress and effective elasticity tensor are determined by solving unit cell problems at each point in the domain. However, it is found that for a certain class of locally periodic microstructures, solving the unit cell problems at only one point in the domain completely determines the effective elasticity tensor.

scholarly journals Stochastic homogenization of plasticity equations

2018 ◽  
Vol 24 (1) ◽  
pp. 153-176 ◽  
Author(s):  
Martin Heida ◽  
Ben Schweizer
Keyword(s):  
Elasticity Tensor ◽  
Volume Element ◽  
Flow Rule ◽  
Stochastic Homogenization ◽  
Stress Evolution ◽  
Homogenization Procedure ◽  
Random Functions ◽  
Strain Plasticity ◽  
Hardening Parameter ◽  
Typical Length

In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flow-rule function are given through a dynamical system on a probability space. A parameter ε > 0 denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit ε → 0. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution \hbox{$[0,T]\ni t\mapsto \xi(t) \in \symM$} induces a stress evolution \hbox{$[0,T]\ni t\mapsto \Sigma(\xi)(t) \in \symM$}. Once the hysteretic evolution law Σ is justified for averages, we obtain that the macroscopic limit equation is given by −∇·Σ(∇su) = f.

Sign in / Sign up

Export Citation Format

Share Document