Averaging Models for Heterogeneous Viscoplastic and Elastic Viscoplastic Materials

2001 ◽  
Vol 124 (1) ◽  
pp. 62-70 ◽  
Author(s):  
Alain Molinari
Keyword(s):  
Large Deformations ◽  
Approximate Solutions ◽  
Local Response ◽  
Nonlinear Properties ◽  
Average Strain Rate ◽  
The Matrix ◽  
Reference Medium ◽  
Homogeneous Matrix ◽  
Small Deformations ◽  
Eshelby Problem

Averaging models are proposed for viscoplastic and elastic-viscoplastic heterogeneous materials. The case of rigid viscoplastic materials is first discussed. Large deformations are considered. A first class of models is based on different linearizations of the nonlinear local response. A second class of models is obtained from approximate solutions of the nonlinear Eshelby problem. In this problem, an ellipsoid with uniform nonlinear properties is embedded in an infinite homogeneous matrix. An approximate solution is obtained by approaching the matrix behavior with an affine response. Using this solution of the nonlinear Eshelby problem, the average strain rate is calculated in each phase of the composite material, each phase being represented by an ellipsoid embedded in an infinite reference medium. By adequate choices of the reference medium, different averaging models are obtained (self-consistent scheme, nonlinear Mori Tanaka model…). Finally, elasticity is included in the modelling, but with a restriction to small deformations.

DMRG Approach to Fast Linear Algebra in the TT-Format

2011 ◽  
Vol 11 (3) ◽  
pp. 382-393 ◽  
Author(s):  
Ivan Oseledets
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
Direct Method ◽  
Fast Algorithms ◽  
Approximate Solutions ◽  
Vector Product ◽  
The Matrix ◽  
Minimization Scheme

AbstractIn this paper, the concept of the DMRG minimization scheme is extended to several important operations in the TT-format, like the matrix-by-vector product and the conversion from the canonical format to the TT-format. Fast algorithms are implemented and a stabilization scheme based on randomization is proposed. The comparison with the direct method is performed on a sequence of matrices and vectors coming as approximate solutions of linear systems in the TT-format. A generated example is provided to show that randomization is really needed in some cases. The matrices and vectors used are available from the author or at http://spring.inm.ras.ru/osel

scholarly journals Exotic garnet–clinopyroxene–K-feldspar granulites from the Chepelare shear zone, Central Rhodope massif, Bulgaria: implications for high-pressure granulite facies metamorphism

2019 ◽  
Vol 48 (3) ◽  
pp. 49-63
Author(s):  
Milena Georgirva ◽  
Tzvetomila Vladinova
Keyword(s):  
Shear Zone ◽  
Thick Layer ◽  
Granulite Facies ◽  
Mineral Association ◽  
Granulite Facies Metamorphism ◽  
Fine Grain ◽  
Fluid Infiltration ◽  
High Pressure Granulite ◽  
The Matrix ◽  
Homogeneous Matrix

Garnet–clinopyroxene–K-feldspar granulite occurs as a thick layer or boudin within the variegated rocks of the Chepelare shear zone in the Central Rhodope massif, Bulgaria. It consists of several domains: mesocratic homogeneous matrix (clinopyroxene–plagioclase–K-feldspar–quartz ± amphibole), porphyroblastic garnet, K-feldspar and clinopyroxene, and strongly foliated fine-grain bands (chloritized biotite–chlorite–prehnite–albite ± epidote). The origin and nature of the matrix mineral association is still unclear. The peak porphyroblast association forms at the expense of plagioclase from the matrix at higher pressure. The fine-grain deformation zones channel the lattermost fluid infiltration. The clinopyroxene-garnet and Zr-in-titanite thermometry give temperatures higher than 790–860 ºC at 2 GPa and, with thermodynamic modeling, suggests crystallization at ~1.8–2.1 GPa and temperature of ~850 ºC in HP granulite field for the porphyroblast granulite association.

scholarly journals Complexity in phase transforming pin-jointed auxetic lattices

2019 ◽  
Vol 475 (2224) ◽  
pp. 20180720 ◽  
Author(s):  
G. W. Hunt ◽  
T. J. Dodwell
Keyword(s):  
Large Deformations ◽  
Axial Stiffness ◽  
Local Response ◽  
Highly Nonlinear ◽  
Transitional Phase ◽  
Unstable States ◽  
Number Of Cells

We demonstrate the complexity that can exist in the modelling of auxetic lattices. By introducing pin-jointed members and large deformations to the analysis of a re-entrant structure, we create a material which has both auxetic and non-auxetic phases. Such lattices exhibit complex equilibrium behaviour during the highly nonlinear transition between these two states. The local response is seen to switch many times between stable and unstable states, exhibiting both positive and negative stiffnesses. However, there is shown to exist an underlying emergent modulus over the transitional phase, to describe the average axial stiffness of a system comprising a large number of cells.

Modeling Nanocomposite Piezoresistive Response With Electromechanical Cohesive Zone Material Point Method

2016 ◽  
Author(s):  
Adarsh K. Chaurasia ◽  
Gary D. Seidel
Keyword(s):  
Cohesive Zone ◽  
Large Deformations ◽  
Material Point Method ◽  
Material Point ◽  
Volume Element ◽  
Point Method ◽  
Current Work ◽  
Secondary Field ◽  
Small Strains ◽  
The Matrix

In the current work, the Material Point Method (MPM) is extended to allow for interfacial discontinuities in problems with composite materials using cohesive zone (CZ) techniques. The proposed CZMPM is observed to result in smaller errors in the primary and secondary field variables, especially near the interface, for a given boundary value problem in comparison to the traditional MPM solution. The proposed CZMPM is used to solve an electromechanical test problem with a single fiber in the matrix medium. It is observed that the proposed CZMPM results in smaller local and volume averaged errors. The CZMPM is further used to evaluate the effective piezoresistive response of the nanoscale carbon nanotube (CNT)-polymer composite with electron hopping in between the nanotubes. The observed effective piezoresistive response exhibits features similar to those reported in the literature using finite element techniques for small strains. However, CZMPM allows for large deformations of the nanoscale representative volume element as presented in the current work.

On Multiple Inhomogeneities in Plane Elasticity

2019 ◽  
Author(s):  
Elie Honein ◽  
Tony Honein ◽  
Michel Najjar ◽  
Habib Rai
Keyword(s):  
Homogeneous Problem ◽  
Analytical Techniques ◽  
Plane Elasticity ◽  
Plane Case ◽  
Schwarz Alternating Method ◽  
Innovative Methods ◽  
Arbitrary Loading ◽  
The Matrix ◽  
Homogeneous Matrix ◽  
Made In

Abstract In this paper we present some new analytical techniques which have been recently developed to solve for problems of circular elastic inhomogeneities in anti-plane and plane elasticity. The inhomogeneities may be composed of different materials and have different radii. The matrix may be subjected to arbitrary loadings or singularities. The solution to this heterogeneous problem is sought as a transformation performed on the solution of the corresponding homogeneous problem, i.e., the problem when all the inhomogeneities are removed and the homogeneous matrix is subjected to the same loading/singularities, a procedure which has been dubbed ‘heterogenization’. In previous works, a single inhomogeneity or hole has been considered and the transformation has been shown to be purely algebraic in the antiplane case and involves differentiation of the Kolosov-Mushkelishvili complex potentials in the plane case. Universal formulas, i.e., formulas which are independent of the loading/singularities, that express the stresses at the inter-face of the inhomogeneity in terms of the stresses that would have existed at the same interface had the inhomogeneity been absent, have been be derived. The solution for a single inhomogeneity bonded to a matrix which is subjected to arbitrary loading/singularities can then in principle be used systematically in a Schwarz alternating method to obtain the solution for multiple inhomogeneities to any degree of accuracy. However alternative and innovative methods have been sought which lead to a much faster convergence and in some cases to exact expressions in terms of infinite series. The aim of this paper is to present some of the progress that has been made in this direction.

Legendre Collocation Method to Solve the Riccati Equations with Functional Arguments

2020 ◽  
Vol 17 (10) ◽  
pp. 2050011
Author(s):  
Şuayip Yüzbaşı ◽  
Gamze Yıldırım
Keyword(s):  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix ◽  
Legendre Collocation Method

In this study, a method for numerically solving Riccatti type differential equations with functional arguments under the mixed condition is presented. For the method, Legendre polynomials, the solution forms and the required expressions are written in the matrix form and the collocation points are defined. Then, by using the obtained matrix relations and the collocation points, the Riccati problem is reduced to a system of nonlinear algebraic equations. The condition in the problem is written in the matrix form and a new system of the nonlinear algebraic equations is found with the aid of the obtained matrix relation. This system is solved and thus the coefficient matrix is detected. This coefficient matrix is written in the solution form and hence approximate solution is obtained. In addition, by defining the residual function, an error problem is established and approximate solutions which give better numerical results are obtained. To demonstrate that the method is trustworthy and convenient, the presented method and error estimation technique are explicated by numerical examples. Consequently, the numerical results are shown more clearly with the aid of the tables and graphs and also the results are compared with the results of other methods.

scholarly journals Run-Out Simulation of a Landslide Triggered by an Increase in the Groundwater Level Using the Material Point Method

Water ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 2817
Author(s):  
Antonello Troncone ◽  
Luigi Pugliese ◽  
Enrico Conte
Keyword(s):  
Groundwater Level ◽  
Large Deformations ◽  
Material Point Method ◽  
Material Point ◽  
Point Method ◽  
Deformation Mechanisms ◽  
The Finite Element Method ◽  
Numerical Techniques ◽  
Rainy Period ◽  
Small Deformations

Deformation mechanisms of the slopes are commonly schematized in four different stages: pre-failure, failure, post-failure and eventual reactivation. Traditional numerical methods, such as the finite element method and the finite difference method, are commonly employed to analyse the slope response in the pre-failure and failure stages under the assumption of small deformations. On the other hand, these methods are generally unsuitable for simulating the post-failure behaviour due to the occurrence of large deformations that often characterize this stage. The material point method (MPM) is one of the available numerical techniques capable of overcoming this limitation. In this paper, MPM is employed to analyse the post-failure stage of a landslide that occurred at Cook Lake (WY, USA) in 1997, after a long rainy period. Accuracy of the method is assessed by comparing the final geometry of the displaced material detected just after the event, to that provided by the numerical simulation. A satisfactory agreement is obtained between prediction and observation when an increase in the groundwater level due to rainfall is accounted for in the analysis.

A Simplified Method to Account for Plastic Rate Sensitivity With Large Deformations

1979 ◽  
Vol 46 (4) ◽  
pp. 811-816 ◽  
Author(s):  
N. Perrone ◽  
P. Bhadra
Keyword(s):  
Strain Rate ◽  
Large Deformations ◽  
Three Dimensional ◽  
Rate Sensitivity ◽  
Approximate Solutions ◽  
Dimensional Structure ◽  
Initial Kinetic Energy ◽  
Maximum Strain ◽  
Membrane Action ◽  
Deformation State

A string supported impulsively loaded mass is used to study large deformation rate sensitivity effects where membrane action is dominant. It is found that an overall correction factor can be devised using physical properties associated with the average strain rate. Maximum strain rate occurs with a velocity field corresponding to the deformation state wherein half the initial kinetic energy has been dissipated. (If V0 is initial velocity, V0/2 is associated with maximum strain rate.) Exact and approximate solutions for a broad range of parameters serve to illustrate and verify the procedure. A discussion is presented to show how the same methodology could also be applied via a modal approach to an arbitrary three-dimensional structure undergoing large deformations, if the primary mechanism for energy absorption is from membrane action.

Stress Analysis for a Nonlinear Viscoelastic Compressible Cylinder

1970 ◽  
Vol 37 (4) ◽  
pp. 1127-1133 ◽  
Author(s):  
E. C. Ting
Keyword(s):  
Large Deformations ◽  
Finite Deformations ◽  
Kernel Functions ◽  
Nonlinear Viscoelastic ◽  
Stress Strain Relation ◽  
Incompressible Materials ◽  
Small Deformations ◽  
Compressible Cylinder ◽  
Elastic Casing ◽  
Finite Difference Techniques

Real solids are not incompressible, although many viscoelastic materials which undergo large deformations show only small changes in volume under ordinary loading conditions. This paper is concerned with a pressurized isotropic viscoelastic hollow cylinder bonded to an elastic casing in which, during a finite deformation, the dilatational change in any element of the cylinder is a small quantity. The analysis is based in part upon the theory of small deformations superposed on finite deformations. Numerical calculations are evaluated by using finite-difference techniques and assuming particular forms of kernel functions in the stress-strain relation. The results for compressible and incompressible materials are compared.

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