RELAXATION OF A MODEL IN FINITE PLASTICITY WITH TWO SLIP SYSTEMS

2013 ◽  
Vol 23 (11) ◽  
pp. 2111-2128 ◽  
Author(s):  
SERGIO CONTI ◽  
GEORG DOLZMANN ◽  
CAROLIN KREISBECK
Keyword(s):  
Three Dimensional ◽  
Material Behavior ◽  
Variational Model ◽  
Geometrically Nonlinear ◽  
Slip Systems ◽  
Two Dimensional ◽  
Finite Plasticity ◽  
Elasto Plasticity ◽  
Moduli Of Elasticity ◽  
Active Slip

The macroscopic material response of a variational model in geometrically nonlinear elasto-plasticity with two active slip systems, rigid elasticity, and hardening is determined. In particular, an explicit formula for the relaxation of the underlying energy density is given, both in the two-dimensional and a related three-dimensional setting. Finally, it is shown that the assumption of elastically rigid material behavior is justified since models with rigid elasticity can be obtained as Γ-limits of models with finite elastic energy for diverging moduli of elasticity.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

scholarly journals Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170612 ◽  
Author(s):  
Malena I. Español ◽  
Dmitry Golovaty ◽  
J. Patrick Wilber
Keyword(s):  
Continuum Model ◽  
Domain Walls ◽  
Three Dimensional ◽  
Variational Model ◽  
Dimensional System ◽  
Two Dimensional ◽  
Starting Point ◽  
Continuum Modelling ◽  
Three Dimensional System ◽  
The Continuum

In this paper, we derive a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The starting point is a discrete atomistic model for the two lattices which are assumed to have slightly different lattice parameters and, possibly, a small relative rotation. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. We use a discrete-to-continuum procedure to obtain the continuum model which recovers both qualitatively and quantitatively the behaviour observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching and bending deformation that accommodates the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters, which can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the details of the deformation associated with the domain walls.

scholarly journals Infinite-order laminates in a model in crystal plasticity

2009 ◽  
Vol 139 (4) ◽  
pp. 685-708 ◽  
Author(s):  
Nathan Albin ◽  
Sergio Conti ◽  
Georg Dolzmann
Keyword(s):  
Energy Density ◽  
Lower Bounds ◽  
Crystal Plasticity ◽  
Infinite Order ◽  
Two Dimensions ◽  
Upper And Lower Bounds ◽  
Geometrically Nonlinear ◽  
Slip Systems ◽  
Quasiconvex Envelope ◽  
Active Slip

We consider a geometrically nonlinear model for crystal plasticity in two dimensions, with two active slip systems and rigid elasticity. We prove that the rank-1 convex envelope of the condensed energy density is obtained by infinite-order laminates, and express it explicitly via the 2F1 hypergeometric function. We also determine the polyconvex envelope, leading to upper and lower bounds on the quasiconvex envelope. The two bounds differ by less than 2%.

High Resolution Microscopy of Multi-Layered Biological Crystals

1982 ◽  
Vol 40 ◽  
pp. 80-83
Author(s):  
H.A. Cohen ◽  
T.W. Jeng ◽  
W. Chiu
Keyword(s):  
High Resolution ◽  
Low Dose ◽  
Three Dimensional ◽  
Data Interpretation ◽  
Two Dimensional ◽  
Biological Objects ◽  
Density Maps ◽  
Density Map ◽  
Complex Crystal ◽  
High Resolution Microscopy

This tutorial will discuss the methodology of low dose electron diffraction and imaging of crystalline biological objects, the problems of data interpretation for two-dimensional projected density maps of glucose embedded protein crystals, the factors to be considered in combining tilt data from three-dimensional crystals, and finally, the prospects of achieving a high resolution three-dimensional density map of a biological crystal. This methodology will be illustrated using two proteins under investigation in our laboratory, the T4 DNA helix destabilizing protein gp32*I and the crotoxin complex crystal.

Instrumentation For Quantitative Microscopy

1977 ◽  
Vol 35 ◽  
pp. 206-209
Author(s):  
B. Ralph ◽  
A.R. Jones
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Automatic Data ◽  
Phase Volume Fraction ◽  
Statistical Confidence ◽  
Resultant Data ◽  
Automatic Data Collection ◽  
Third Dimension ◽  
Evolution Of Microstructure

In all fields of microscopy there is an increasing interest in the quantification of microstructure. This interest may stem from a desire to establish quality control parameters or may have a more fundamental requirement involving the derivation of parameters which partially or completely define the three dimensional nature of the microstructure. This latter categorey of study may arise from an interest in the evolution of microstructure or from a desire to generate detailed property/microstructure relationships. In the more fundamental studies some convolution of two-dimensional data into the third dimension (stereological analysis) will be necessary.In some cases the two-dimensional data may be acquired relatively easily without recourse to automatic data collection and further, it may prove possible to perform the data reduction and analysis relatively easily. In such cases the only recourse to machines may well be in establishing the statistical confidence of the resultant data. Such relatively straightforward studies tend to result from acquiring data on the whole assemblage of features making up the microstructure. In this field data mode, when parameters such as phase volume fraction, mean size etc. are sought, the main case for resorting to automation is in order to perform repetitive analyses since each analysis is relatively easily performed.

A linearity test for thick specimens imaged by HVEM over a 180° angular range

1991 ◽  
Vol 49 ◽  
pp. 552-553
Author(s):  
Yu Liu
Keyword(s):  
Phase Contrast ◽  
Three Dimensional ◽  
Angular Range ◽  
Two Dimensional ◽  
Back Projection ◽  
Line Integral ◽  
Exponential Law ◽  
Transmission Electron ◽  
Absorption Law ◽  
Linearity Test

The image obtained in a transmission electron microscope is the two-dimensional projection of a three-dimensional (3D) object. The 3D reconstruction of the object can be calculated from a series of projections by back-projection, but this algorithm assumes that the image is linearly related to a line integral of the object function. However, there are two kinds of contrast in electron microscopy, scattering and phase contrast, of which only the latter is linear with the optical density (OD) in the micrograph. Therefore the OD can be used as a measure of the projection only for thin specimens where phase contrast dominates the image. For thick specimens, where scattering contrast predominates, an exponential absorption law holds, and a logarithm of OD must be used. However, for large thicknesses, the simple exponential law might break down due to multiple and inelastic scattering.

An Image Reconstruction System Applied to High Voltage Microscopy

1975 ◽  
Vol 33 ◽  
pp. 292-293
Author(s):  
D. E. Johnson
Keyword(s):  
High Voltage ◽  
Prior Information ◽  
Block Diagram ◽  
Three Dimensional ◽  
Real Space ◽  
Two Dimensional ◽  
Efficient Manner ◽  
Fourier Methods ◽  
Tilt Series ◽  
Methods Of Reconstruction

Increased specimen penetration; the principle advantage of high voltage microscopy, is accompanied by an increased need to utilize information on three dimensional specimen structure available in the form of two dimensional projections (i.e. micrographs). We are engaged in a program to develop methods which allow the maximum use of information contained in a through tilt series of micrographs to determine three dimensional speciman structure.In general, we are dealing with structures lacking in symmetry and with projections available from only a limited span of angles (±60°). For these reasons, we must make maximum use of any prior information available about the specimen. To do this in the most efficient manner, we have concentrated on iterative, real space methods rather than Fourier methods of reconstruction. The particular iterative algorithm we have developed is given in detail in ref. 3. A block diagram of the complete reconstruction system is shown in fig. 1.

Computer Assisted Image Reconstruction of Oligomer Structure

1976 ◽  
Vol 34 ◽  
pp. 472-473
Author(s):  
A.M. Jones ◽  
A. Max Fiskin
Keyword(s):  
Three Dimensional ◽  
Depth Of Field ◽  
Computer Assisted ◽  
Projection Data ◽  
Two Dimensional ◽  
Single Particles ◽  
Dimensional Reconstruction ◽  
Computer Assisted Image ◽  
The Fourier Transform ◽  
Space Approach

If the tilt of a specimen can be varied either by the strategy of observing identical particles orientated randomly or by use of a eucentric goniometer stage, three dimensional reconstruction procedures are available (l). If the specimens, such as small protein aggregates, lack periodicity, direct space methods compete favorably in ease of implementation with reconstruction by the Fourier (transform) space approach (2). Regardless of method, reconstruction is possible because useful specimen thicknesses are always much less than the depth of field in an electron microscope. Thus electron images record the amount of stain in columns of the object normal to the recording plates. For single particles, practical considerations dictate that the specimen be tilted precisely about a single axis. In so doing a reconstructed image is achieved serially from two-dimensional sections which in turn are generated by a series of back-to-front lines of projection data.

Polymorphic phase transition of a membrane protein - analysis of continuous diffuse electron diffraction intensity

1986 ◽  
Vol 44 ◽  
pp. 166-167
Author(s):  
Douglas L. Dorset ◽  
Andrew K. Massalski
Keyword(s):  
Molecular Sieve ◽  
Three Dimensional ◽  
Pore Geometry ◽  
Two Dimensional ◽  
Diffraction Intensity ◽  
Polymorphic Phase Transition ◽  
E Coli ◽  
Crystallographic Study ◽  
Hexagonal Form ◽  
Polymorphic Forms

Matrix porin, the ompF gene product of E. coli, has been the object of a electron crystallographic study of its pore geometry in an attempt to understand its function as a membrane molecular sieve. Three polymorphic forms have been found for two-dimensional crystals reconstituted in phospholipid, two hexagonal forms with different lipid content and an orthorhombic form coexisting with and similar to the hexagonal form found after lipid loss. In projection these have been shown to retain the same three-fold pore triplet geometry and analyses of three-dimensional data reveal that the small hexagonal and orthorhombic polymorphs have similar structure as well as unit cell spacings.

Two-Dimensional Crystallization of Tetanus Toxin

1985 ◽  
Vol 43 ◽  
pp. 528-529
Author(s):  
Jeffry A. Reidler ◽  
John P. Robinson
Keyword(s):  
Electron Microscopy ◽  
Tetanus Toxin ◽  
Structural Information ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Membrane Interface ◽  
3D Reconstructions ◽  
Cellular Receptors ◽  
Computer Reconstruction ◽  
2D Crystals

We have prepared two-dimensional (2D) crystals of tetanus toxin using procedures developed by Uzgiris and Kornberg for the directed production of 2D crystals of monoclonal antibodies at an antigen-phospholipid monolayer interface. The tetanus toxin crystals were formed using a small mole fraction of the natural receptor, GT1, incorporated into phosphatidyl choline monolayers. The crystals formed at low concentration overnight. Two dimensional crystals of this type are particularly useful for structure determination using electron microscopy and computer image refinement. Three dimensional (3D) structural information can be derived from these crystals by computer reconstruction of photographs of toxin crystals taken at different tilt angles. Such 3D reconstructions may help elucidate the mechanism of entry of the enzymatic subunit of toxins into cells, particularly since these crystals form directly on a membrane interface at similar concentrations of ganglioside GT1 to the natural cellular receptors.

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