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.

On inelasticity of damaged quasi rate independent anisotropic materials

2018 ◽  
Vol 24 (3) ◽  
pp. 778-795 ◽  
Author(s):  
Milan Mićunović ◽  
Ljudmila Kudrjavceva
Keyword(s):  
Effective Field ◽  
Short Fibers ◽  
Viscoplastic Strain ◽  
Order Tensor ◽  
Two Phase ◽  
Elastic Symmetry ◽  
Field Approach ◽  
Steel Aisi ◽  
Stainless Steel Aisi ◽  
Fourth Order Tensor

This paper deals with a body that has a random 3D-distribution of two phase inclusions: spheroidal mutually parallel voids, and differently oriented reinforcing parallel stiff spheroidal short fibers. By the effective field approach the effective stiffness fourth order tensor is formulated and found numerically. Simultaneous and sequential embeddings of inclusions are compared. Damage evolution is described by a modified Vakulenko approach to the endochronic thermodynamics. A brief account of the problem of effective elastic symmetry is considered. The results of the theory are applied to the damage-elasto-viscoplastic strain of a reactor stainless steel AISI 316H.

scholarly journals Predicting the Times of Retweeting in Microblogs

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Li Kuang ◽  
Xiang Tang ◽  
Kehua Guo
Keyword(s):  
Time Series ◽  
Time Distribution ◽  
Distribution Curve ◽  
Information Propagation ◽  
Web Portals ◽  
Traditional Media ◽  
Two Phase ◽  
Two Phases ◽  
The Times ◽  
The Given

Recently, microblog services accelerate the information propagation among peoples, leaving the traditional media like newspaper, TV, forum, blogs, and web portals far behind. Various messages are spread quickly and widely by retweeting in microblogs. In this paper, we take Sina microblog as an example, aiming to predict the possible number of retweets of an original tweet in one month according to the time series distribution of its topnretweets. In order to address the problem, we propose the concept of a tweet’s lifecycle, which is mainly decided by three factors, namely, the response time, the importance of content, and the interval time distribution, and then the given time series distribution curve of its topnretweets is fitted by a two-phase function, so as to predict the number of its retweets in one month. The phases in the function are divided by the lifecycle of the original tweet and different functions are used in the two phases. Experiment results show that our solution can address the problem of predicting the times of retweeting in microblogs with a satisfying precision.

scholarly journals Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions

2021 ◽  
pp. 3-22
Author(s):  
Magnus Herberthson ◽  
Evren Özarslan ◽  
Carl-Fredrik Westin
Keyword(s):  
Tensor Product ◽  
Fourth Order ◽  
Equivalence Problem ◽  
Elasticity Tensor ◽  
Two Dimensions ◽  
Second Order ◽  
Two Dimensional ◽  
Order Tensor ◽  
Symmetry Properties ◽  
Fourth Order Tensor

AbstractCalculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor $$R_{abcd}$$ R abcd . To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor $$R_{abcd}$$ R abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors $$R_{abcd}$$ R abcd and $$\widetilde{R}_{abcd}$$ R ~ abcd . In terms of components, such an equivalence means that components $$R_{ijkl}$$ R ijkl of the first tensor will transform into the components $$\widetilde{R}_{ijkl}$$ R ~ ijkl of the second tensor for some change of the coordinate system.

scholarly journals New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zhuanzhou Zhang ◽  
Jun He ◽  
Yanmin Liu ◽  
Zerong Ren
Keyword(s):  
Sufficient Conditions ◽  
Upper Bounds ◽  
Elasticity Tensor ◽  
Strong Ellipticity ◽  
Extreme Eigenvalues ◽  
Elasticity Tensors ◽  
The Given

In this paper, based on the extreme eigenvalues of the matrices arisen from the given elasticity tensor, S-type upper bounds for the M-eigenvalues of elasticity tensors are established. Finally, S-type sufficient conditions are introduced for the strong ellipticity of elasticity tensors based on the S-type M-eigenvalue inclusion sets.

scholarly journals A Two-Phase Approach for Semi-Supervised Feature Selection

Algorithms ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 215
Author(s):  
Amit Saxena ◽  
Shreya Pare ◽  
Mahendra Singh Meena ◽  
Deepak Gupta ◽  
Akshansh Gupta ◽  
...  
Keyword(s):  
Feature Selection ◽  
Classification Accuracy ◽  
Second Phase ◽  
Two Phase ◽  
Phase Approach ◽  
Whole Process ◽  
Novel Approach ◽  
Two Phases ◽  
Number Of Classes ◽  
The Given

This paper proposes a novel approach for selecting a subset of features in semi-supervised datasets where only some of the patterns are labeled. The whole process is completed in two phases. In the first phase, i.e., Phase-I, the whole dataset is divided into two parts: The first part, which contains labeled patterns, and the second part, which contains unlabeled patterns. In the first part, a small number of features are identified using well-known maximum relevance (from first part) and minimum redundancy (whole dataset) based feature selection approaches using the correlation coefficient. The subset of features from the identified set of features, which produces a high classification accuracy using any supervised classifier from labeled patterns, is selected for later processing. In the second phase, i.e., Phase-II, the patterns belonging to the first and second part are clustered separately into the available number of classes of the dataset. In the clusters of the first part, take the majority of patterns belonging to a cluster as the class for that cluster, which is given already. Form the pairs of cluster centroids made in the first and second part. The centroid of the second part nearest to a centroid of the first part will be paired. As the class of the first centroid is known, the same class can be assigned to the centroid of the cluster of the second part, which is unknown. The actual class of the patterns if known for the second part of the dataset can be used to test the classification accuracy of patterns in the second part. The proposed two-phase approach performs well in terms of classification accuracy and number of features selected on the given benchmarked datasets.

Estimating effective elasticity tensors from Christoffel equations

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB67-WB73 ◽  
Author(s):  
Mikhail Kochetov ◽  
Michael A. Slawinski
Keyword(s):  
Optimization Problem ◽  
Elasticity Tensor ◽  
Initial Guess ◽  
Statistical Information ◽  
Numerical Instability ◽  
First Finding ◽  
Effective Elasticity ◽  
Elasticity Tensors ◽  
Christoffel Equation ◽  
Anisotropic Tensor

We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular symmetry that corresponds to measurable traveltime and polarization quantities. These quantities — the wavefront-slowness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization problem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of particular symmetry without assuming its orientation is challenging. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we distinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.

scholarly journals Integrity basis for a second-order and a fourth-order tensor

1982 ◽  
Vol 5 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Josef Betten
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Tensor Function ◽  
Order Tensor ◽  
Isotropic Tensor ◽  
Anisotropic Properties ◽  
Isotropic Tensor Function ◽  
Fourth Order Tensor

In this paper a scalar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order greater than four, as shown in the paper, for instance, for a tensor of order six.

A Lightweight Formalism for Reference Lifetimes and Borrowing in Rust

2021 ◽  
Vol 43 (1) ◽  
pp. 1-73
Author(s):  
David J. Pearce
Keyword(s):  
Memory Management ◽  
Programming Model ◽  
Type System ◽  
Control Flow ◽  
Two Phase ◽  
Type Checking ◽  
Strong Focus ◽  
Reference Implementation ◽  
Two Phases ◽  
Complex Features

Rust is a relatively new programming language that has gained significant traction since its v1.0 release in 2015. Rust aims to be a systems language that competes with C/C++. A claimed advantage of Rust is a strong focus on memory safety without garbage collection. This is primarily achieved through two concepts, namely, reference lifetimes and borrowing . Both of these are well-known ideas stemming from the literature on region-based memory management and linearity / uniqueness . Rust brings both of these ideas together to form a coherent programming model. Furthermore, Rust has a strong focus on stack-allocated data and, like C/C++ but unlike Java, permits references to local variables. Type checking in Rust can be viewed as a two-phase process: First, a traditional type checker operates in a flow-insensitive fashion; second, a borrow checker enforces an ownership invariant using a flow-sensitive analysis. In this article, we present a lightweight formalism that captures these two phases using a flow-sensitive type system that enforces “ type and borrow safety .” In particular, programs that are type and borrow safe will not attempt to dereference dangling pointers. Our calculus core captures many aspects of Rust, including copy- and move-semantics, mutable borrowing, reborrowing, partial moves, and lifetimes. In particular, it remains sufficiently lightweight to be easily digested and understood and, we argue, still captures the salient aspects of reference lifetimes and borrowing. Furthermore, extensions to the core can easily add more complex features (e.g., control-flow, tuples, method invocation). We provide a soundness proof to verify our key claims of the calculus. We also provide a reference implementation in Java with which we have model checked our calculus using over 500B input programs. We have also fuzz tested the Rust compiler using our calculus against 2B programs and, to date, found one confirmed compiler bug and several other possible issues.

scholarly journals XCOVNet: Chest X-ray Image Classification for COVID-19 Early Detection Using Convolutional Neural Networks

2021 ◽  
Author(s):  
Vishu Madaan ◽  
Aditya Roy ◽  
Charu Gupta ◽  
Prateek Agrawal ◽  
Anand Sharma ◽  
...  
Keyword(s):  
Image Classification ◽  
Second Phase ◽  
Two Phase ◽  
X Ray ◽  
The Neural Network ◽  
Rapid Spread ◽  
Chest X Ray ◽  
Polymerase Chain ◽  
Two Phases ◽  
Active Patients

AbstractCOVID-19 (also known as SARS-COV-2) pandemic has spread in the entire world. It is a contagious disease that easily spreads from one person in direct contact to another, classified by experts in five categories: asymptomatic, mild, moderate, severe, and critical. Already more than 66 million people got infected worldwide with more than 22 million active patients as of 5 December 2020 and the rate is accelerating. More than 1.5 million patients (approximately 2.5% of total reported cases) across the world lost their life. In many places, the COVID-19 detection takes place through reverse transcription polymerase chain reaction (RT-PCR) tests which may take longer than 48 h. This is one major reason of its severity and rapid spread. We propose in this paper a two-phase X-ray image classification called XCOVNet for early COVID-19 detection using convolutional neural Networks model. XCOVNet detects COVID-19 infections in chest X-ray patient images in two phases. The first phase pre-processes a dataset of 392 chest X-ray images of which half are COVID-19 positive and half are negative. The second phase trains and tunes the neural network model to achieve a 98.44% accuracy in patient classification.

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