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.

scholarly journals Non-orthogonal elastoplastic constitutive model for sand with dilatancy

2021 ◽  
Author(s):  
Jingyu Liang ◽  
Dechun Lu ◽  
Xiuli Du ◽  
Wei Wu ◽  
Chao Ma
Keyword(s):  
Constitutive Model ◽  
Plastic Flow ◽  
Flow Direction ◽  
Model Performance ◽  
Yield Function ◽  
Flow Rule ◽  
Shear Process ◽  
Hardening Parameter ◽  
Elastoplastic Constitutive Model ◽  
Plastic Flow Rule

A non-orthogonal elastoplastic constitutive model for sand with dilatancy is presented in the characteristic stress space. Dilatancy of sand is represented by the direction of plastic flow, which can be directly determined by applying the non-orthogonal plastic flow rule to an improved elliptic yield function. A new hardening parameter is developed to describe the contractive and dilative volume change during the shear process, which is co-ordinated with the non-orthogonal plastic flow direction. The combination of the non-orthogonal plastic flow rule and the proposed hardening parameter renders the proposed model with the ability to reasonably describe the stress-strain relationship of sand with dilatancy. The model performance is evaluated by comparing with the experimental data of sand under triaxial stress conditions.

scholarly journals Homogenization of Random Porous Materials With Low-Order Virtual Elements

2019 ◽  
Vol 5 (3) ◽  
Author(s):  
Marco Pingaro ◽  
Emanuele Reccia ◽  
Patrizia Trovalusci
Keyword(s):  
Elastic Moduli ◽  
Volume Element ◽  
Homogenization Procedure ◽  
Virtual Element Method ◽  
Low Contrast ◽  
Statistical Homogenization ◽  
Random Composites ◽  
Material Contrast ◽  
Virtual Element ◽  
Definition Of

A fast statistical homogenization procedure (FSHP) based on virtual element method (VEM)—previously developed by the authors has been successfully adopted for the homogenization of particulate random composites, via the definition of the representative volume element (RVE), and of the related equivalent elastic moduli. In particular, the adoption of virtual elements of degree one for modeling the inclusions provided reliable results for materials with low contrast, defined as the ratio between mechanical properties of inclusions and matrix. Porous media are then here described as bimaterial systems in which soft circular inclusions, with a very low value of material contrast, are randomly distributed in a continuous stiffer matrix. Several simulations have been performed by varying the level of porosity, highlighting the effectiveness of FSHP in conjunction with virtual elements of degree one.

Second-Order Computational Homogenization Scheme Preserving Microlevel C1 Continuity

2014 ◽  
Vol 627 ◽  
pp. 381-384 ◽  
Author(s):  
Tomislav Lesičar ◽  
Zdenko Tonković ◽  
Jurica Sorić
Keyword(s):  
Gradient Elasticity ◽  
Multiscale Model ◽  
Computational Homogenization ◽  
Second Order ◽  
Volume Element ◽  
Homogenization Procedure ◽  
Triangular Finite Element ◽  
Small Strains ◽  
Mesh Density ◽  
Homogenization Scheme

The paper deals with a new second-order computational homogenization procedure for modeling of heterogeneous materials at small strains, whereC1continuity is preserved at the microlevel. The multiscale model is based on the Aifantis theory of gradient elasticity. TheC1two dimensional triangular finite element used for the discretization of macro-and microlevel is described. Contrary to theC1-C0transition, here besides the displacements, the displacement gradients are included into the boundary conditions on the representative volume element (RVE). According to the second order continuum at microlevel, the relevant homogenization relations are derived. Finally, the performance of the algorithms derived is investigated. Dependency of homogenized stresses on mesh density and microstructural parameterlare examined in simple loading cases.

Yield surfaces of heterogeneous media with debonded inclusions

2015 ◽  
Vol 32 (6) ◽  
pp. 1802-1813 ◽  
Author(s):  
Deniz D. Somer ◽  
D Peric ◽  
Eduardo Alberto de Souza Neto ◽  
Wulf G Dettmer
Keyword(s):  
Design Methodology ◽  
Heterogeneous Media ◽  
Present Knowledge ◽  
Volume Element ◽  
Homogenization Procedure ◽  
Content Type ◽  
Compressive Loads ◽  
Coulomb Type ◽  
Plastic Matrix ◽  
Yield Surfaces

Purpose – The purpose of this paper is to present knowledge in estimating yield surfaces of heterogeneous media by use of homogenization, especially where the macroscopic behaviour is driven by weak interfaces between phase constituents. Design/methodology/approach – A computational homogenization procedure is used to determine the yield surface of a Representative Volume Element (RVE) that contains a fully debonded inclusion embedded within ideally plastic matrix, whereby the interface is modelled by a Coulomb type friction law. Findings – The macroscopic behaviour of the RVE is shown to coincide an RVE with a hole for expanding loads, whereas for compressive loads, it was shown to approach an RVE with a fully bonded inclusion. Originality/value – The present paper builds on Gurson’s work in estimating macroscopic yield surfaces of heterogeneous materials. The work is novel in the sense that there had been no previous publications discussing influence of weak interfaces on yield surfaces.

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.

The Theoretical Contrast in Scanning and Conventional High Resolution Microscopes

1969 ◽  
Vol 27 ◽  
pp. 170-171
Author(s):  
E. Zeitler ◽  
M. G. R. Thomson
Keyword(s):  
Small Volume ◽  
Volume Element ◽  
Optical Processing ◽  
Image Construction ◽  
Object Interaction ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Optical Treatment ◽  
Electron Microscopes ◽  
Intensity Contrast

In the formation of an image each small volume element of the object is correlated to an areal element in the image. The structure or detail of the object is represented by changes in intensity from element to element, and this variation of intensity (contrast) is determined by the interaction of the electrons with the specimen, and by the optical processing of the information-carrying electrons. Both conventional and scanning transmission electron microscopes form images which may be considered in this way, but the mechanism of image construction is very different in the two cases. Although the electron-object interaction is the same, the optical treatment differs.

Trace nanoanalysis

1994 ◽  
Vol 52 ◽  
pp. 940-941
Author(s):  
D. E. Newbury ◽  
R. D. Leapman
Keyword(s):  
High Performance ◽  
Secondary Ion Mass Spectrometry ◽  
Detection Efficiency ◽  
Volume Element ◽  
And Performance ◽  
Trace Constituents ◽  
Secondary Ion ◽  
Concentration Levels ◽  
Structure Properties

Trace constituents, which can be very loosely defined as those present at concentration levels below 1 percent, often exert influence on structure, properties, and performance far greater than what might be estimated from their proportion alone. Defining the role of trace constituents in the microstructure, or indeed even determining their location, makes great demands on the available array of microanalytical tools. These demands become increasingly more challenging as the dimensions of the volume element to be probed become smaller. For example, a cubic volume element of silicon with an edge dimension of 1 micrometer contains approximately 5×1010 atoms. High performance secondary ion mass spectrometry (SIMS) can be used to measure trace constituents to levels of hundreds of parts per billion from such a volume element (e. g., detection of at least 100 atoms to give 10% reproducibility with an overall detection efficiency of 1%, considering ionization, transmission, and counting).

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