scholarly journals Model reduction by mean-field homogenization in viscoelastic composites. I. Primal theory

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200407 ◽  
Author(s):  
Martín I. Idiart ◽  
Noel Lahellec ◽  
Pierre Suquet
Keyword(s):  
Strain Field ◽  
Mean Field ◽  
Inelastic Strain ◽  
Mathematical Structure ◽  
Schwarz Inequality ◽  
Internal Variables ◽  
Legendre Transform ◽  
Time Step ◽  
Legendre Transforms ◽  
Viscoelastic Composites

A homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44 , 507–529 ( doi:10.1016/j.ijsolstr.2006.04.038 )) is revisited. The scheme relies upon an incremental variational formulation providing the inelastic strain field at a given time step in terms of the inelastic strain field from the previous time step, along with a judicious use of Legendre transforms to approximate the relevant functional by an alternative functional depending on the inelastic strain fields only through their first and second moments over each constituent phase. As a result, the approximation generates a reduced description of the microscopic state of the composite in terms of a finite set of internal variables that incorporates information on the intraphase fluctuations of the inelastic strain and that can be evaluated by mean-field homogenization techniques. In this work we provide an alternative derivation of the scheme, relying on the Cauchy–Schwarz inequality rather than the Legendre transform, and in so doing we expose the mathematical structure of the resulting approximation and generalize the exposition to fully anisotropic material systems.

scholarly journals Legendre transforms and Apéry's sequences

1995 ◽  
Vol 58 (3) ◽  
pp. 358-375 ◽  
Author(s):  
Asmus L. Schmidt
Keyword(s):  
Legendre Transform ◽  
Image Position ◽  
Legendre Transforms

AbstractThis article studies particular sequences satisfying polynomial recurrences, among those Apéry's sequence which is shown to be the Legendre transform of the sequence. This results in the construction of simultaneous approximations of π 2/8 and ζ(3).

On the Modeling of Evolving Elastic and Plastic Anisotropy in the Finite Strain Regime – Application to Deep Drawing Processes

2012 ◽  
Vol 504-506 ◽  
pp. 679-684 ◽  
Author(s):  
Ivaylo N. Vladimirov ◽  
Michael P. Pietryga ◽  
Vivian Tini ◽  
Stefanie Reese
Keyword(s):  
Deep Drawing ◽  
Evolution Equations ◽  
Finite Strain ◽  
Kinematic Hardening ◽  
Plastic Anisotropy ◽  
Material Model ◽  
Internal Variables ◽  
Time Step ◽  
Finite Element Software ◽  
Structure Tensors

In this work, we discuss a finite strain material model for evolving elastic and plastic anisotropy combining nonlinear isotropic and kinematic hardening. The evolution of elastic anisotropy is described by representing the Helmholtz free energy as a function of a family of evolving structure tensors. In addition, plastic anisotropy is modelled via the dependence of the yield surface on the same family of structure tensors. Exploiting the dissipation inequality leads to the interesting result that all tensor-valued internal variables are symmetric. Thus, the integration of the evolution equations can be efficiently performed by means of an algorithm that automatically retains the symmetry of the internal variables in every time step. The material model has been implemented as a user material subroutine UMAT into the commercial finite element software ABAQUS/Standard and has been used for the simulation of the phenomenon of earing during cylindrical deep drawing.

scholarly journals Model reduction by mean-field homogenization in viscoelastic composites. II. Application to rigidly reinforced solids

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200408
Author(s):  
Martín I. Idiart ◽  
Noel Lahellec ◽  
Pierre Suquet
Keyword(s):  
Elastic Moduli ◽  
Solid Phase ◽  
Mean Field ◽  
Free Energy Density ◽  
Companion Paper ◽  
Viscoelastic Solid ◽  
Complex Deformation ◽  
Link Type ◽  
The Mean ◽  
Viscoelastic Composites

The mean-field homogenization scheme proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44 , 507–529 ( doi:10.1016/j.ijsolstr.2006.04.038 )) and revisited in a companion paper (Idiart et al . 2020 Proc. R. Soc. A 20200407 ( doi:10.1098/rspa.2020.0407 )) is applied to random mixtures of a viscoelastic solid phase and a rigid phase. Two classes of mixtures with different microstructural arrangements are considered. In the first class the rigid phase is dispersed within the continuous viscoelastic phase in such a way that the elastic moduli of the mixture are given exactly by the Hashin–Shtrikman formalism. In the second class, both phases are intertwined in such a way that the elastic moduli of the mixture are given exactly by the Self-Consistent formalism. Results are reported for specimens subject to various complex deformation programmes. The scheme is found to improve on earlier approximations of common use and even recover exact results under several circumstances. However, it can also generate highly inaccurate predictions as a result of the loss of convexity of the free-energy density. An auspicious procedure to partially circumvent this issue is advanced.

Conjugates and Legendre Transforms of Convex Functions

1967 ◽  
Vol 19 ◽  
pp. 200-205 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Strict Convexity ◽  
Legendre Transformation ◽  
Legendre Transform ◽  
Implicit Functions ◽  
The One ◽  
Legendre Transforms ◽  
Effective Domain ◽  
The Relationship

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.

scholarly journals An entropy minimization approach to second-order variational mean-field games

2019 ◽  
Vol 29 (08) ◽  
pp. 1553-1583 ◽  
Author(s):  
Jean-David Benamou ◽  
Guillaume Carlier ◽  
Simone Di Marino ◽  
Luca Nenna
Keyword(s):  
Efficient Algorithm ◽  
Mean Field ◽  
Time Discretization ◽  
Second Order ◽  
Mean Field Games ◽  
Time Step ◽  
Entropy Minimization ◽  
Convergence Results ◽  
Quadratic Hamiltonian ◽  
Minimization Approach

We propose an entropy minimization viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We carefully analyze the time discretization of such problems, establish [Formula: see text]-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.

scholarly journals ON THE INVERSE MULTIFRACTAL FORMALISM

2009 ◽  
Vol 52 (1) ◽  
pp. 179-194 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Multifractal Analysis ◽  
Regularity Condition ◽  
Upper Bounds ◽  
Legendre Transform ◽  
Self Similarity ◽  
Multifractal Formalism ◽  
Multifractal Spectra ◽  
Physics Literature ◽  
Image Position ◽  
Legendre Transforms

AbstractTwo of the main objects of study in multifractal analysis of measures are the coarse multifractal spectra and the Rényi dimensions. In the 1980s it was conjectured in the physics literature that for ‘good’ measures the following result, relating the coarse multifractal spectra to the Legendre transform of the Rényi dimensions, holds, namely This result is known as the multifractal formalism and has now been verified for many classes of measures exhibiting some degree of self-similarity. However, it is also well known that there is an abundance of measures not satisfying the multifractal formalism and that, in general, the Legendre transforms of the Rényi dimensions provide only upper bounds for the coarse multifractal spectra. The purpose of this paper is to prove that even though the multifractal formalism fails in general, it is nevertheless true that all measures (satisfying a mild regularity condition) satisfy the inverse of the multifractal formalism, namely

A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials

2010 ◽  
Vol 467 (2128) ◽  
pp. 1186-1196 ◽  
Author(s):  
Klaus Hackl ◽  
Franz Dieter Fischer ◽  
Jiri Svoboda
Keyword(s):  
Heat Flux ◽  
Evolution Equations ◽  
Mathematical Structure ◽  
Dissipation Function ◽  
Internal Variables ◽  
Efficient Tool ◽  
Equilibrium Processes ◽  
Rigorous Treatment ◽  
Isothermal Processes ◽  
Principle Of Maximum

Onsager’s principle of maximum dissipation (PMD) has proven to be an efficient tool to derive evolution equations for the internal variables describing non-equilibrium processes. However, a rigorous treatment of PMD for several simultaneously acting dissipative processes is still open and presented in this paper. The coupling or uncoupling of the processes is demonstrated via the mathematical structure of the dissipation function. Examples are worked out for plastic deformation and heat flux.

Two-time-scales and time-averaging approaches for the analysis of cyclic creep based on Armstrong–Frederick type constitutive model

2018 ◽  
Vol 233 (5) ◽  
pp. 1690-1700 ◽  
Author(s):  
Holm Altenbach ◽  
Dmitry Breslavsky ◽  
Konstantin Naumenko ◽  
Oksana Tatarinova
Keyword(s):  
Cyclic Loading ◽  
Constitutive Model ◽  
Inelastic Strain ◽  
Cyclic Creep ◽  
Analytical Form ◽  
Type Model ◽  
Time Averaging ◽  
Inelastic Behavior ◽  
Time Step ◽  
The Mean

The aim of this paper is the analysis of inelastic behavior under periodic cyclic loading regimes for materials showing recovery effects. Starting with the Armstrong–Frederick type model and applying the two-time-scale asymptotic technique, the constitutive equation for the mean inelastic strain rate and the evolution equation for the mean backstress variable are derived. The advantage of the presented technique is the closed analytical form of the solutions such that the influence of the loading profiles on the resulting slow cycle-by-cycle strain accumulation can be analyzed explicitly. To validate the derived equations, the original constitutive model is integrated numerically by applying the time-step procedure for various cyclic loading profiles. Numerical examples are presented to illustrate cyclic creep behavior of X20CrMoV12-1 heat resistant steel.

Initial Strain Field and Fatigue Crack Initiation Mechanics

1983 ◽  
Vol 50 (2) ◽  
pp. 367-372 ◽  
Author(s):  
S. R. Lin ◽  
T. H. Lin
Keyword(s):  
Shear Stress ◽  
Stress Field ◽  
Initial Stress ◽  
Strain Field ◽  
Fatigue Crack Initiation ◽  
Inelastic Strain ◽  
Slip Lines ◽  
Thin Slice ◽  
Initial Stress Field ◽  
Reversed Loading

On single aluminum crystals under cyclic loadings, fresh slip lines appeared during the reversed loading, lying very close to, but not coincident with the slip lines formed in the forward loading. These slip lines indicate the start of extrusion or intrusion as commonly observed in fatigue specimens. An initial stress field is present in all metals. The initial stress field favorable to the aforementioned sequence of slip is one having a positive shear stress in one thin slice P and a negative one in a closely located thin slice Q. A forward loading causes a positive shear stress, which is of the same sign as the initial shear stress in P, but of opposite sign to that in Q. Hence the shear stress in P will reach the critical value first to cause slip. Due to the continuity of the stress field, slip in P relieves not only the positive shear stress in P but also in Q. This has the same effect as increasing negative shear stress in Q. During the reversed loading, Q has the highest negative shear stress and hence slides. Similarly, this slip causes P to be more ready to slide in the next forward loading. This process is repeated to cause a monotonic alternate sliding in P and Q. In this way, an extrusion or intrusion is nucleated. A crack can be started from an intrusion. The thin slices P and Q are considered to be in a most favorably oriented crystal located at a free surface. An initial stress field giving positive shear stress in P and negative in Q is calculated from an assumed initial inelastic strain field which, in turn, can be caused by distribution of dislocations. The buildup of plastic shear strain in P and Q causing the start of extrusion or intrusion is shown.

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