Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials

2006 ◽  
Vol 74 (4) ◽  
pp. 741-753 ◽  
Author(s):  
D. Bigoni ◽  
W. J. Drugan
Keyword(s):  
Three Dimensional ◽  
Heterogeneous Material ◽  
Material Response ◽  
Spherical Inclusions ◽  
The Matrix ◽  
Cylindrical Inclusions ◽  
Dilute Suspension ◽  
Experimental Findings ◽  
Do So

Why do experiments detect Cosserat-elastic effects for porous, but not for stiff-particle-reinforced, materials? Does homogenization of a heterogeneous Cauchy-elastic material lead to micropolar (Cosserat) effects, and if so, is this true for every type of heterogeneity? Can homogenization determine micropolar elastic constants? If so, is the homogeneous (effective) Cosserat material determined in this way a more accurate representation of composite material response than the usual effective Cauchy material? Direct answers to these questions are provided in this paper for both two- (2D) and three-dimensional (3D) deformations, wherein we derive closed-form formulae for Cosserat moduli via homogenization of a dilute suspension of elastic spherical inclusions in 3D (and circular cylindrical inclusions in 2D) embedded in an isotropic elastic matrix. It is shown that the characteristic length for a homogeneous Cosserat material that best mimics the heterogeneous Cauchy material can be derived (resulting in surprisingly simple formulae) when the inclusions are less stiff than the matrix, but when these are equal to or stiffer than the matrix, Cosserat effects are shown to be excluded. These analytical results explain published experimental findings, correct, resolve and extend prior contradictory theoretical (mainly numerical and limited to two-dimensional deformations) investigations, and provide both a general methodology and specific results for determination of simple higher-order homogeneous effective materials that more accurately represent heterogeneous material response under general loading conditions. In particular, it is shown that no standard (Cauchy) homogenized material can accurately represent the response of a heterogeneous material subjected to a uniform plus linearly varying applied traction, while a homogenized Cosserat material can do so (when inclusions are less stiff than the matrix).

In-Plane Loading in an Elastic Matrix Containing Two Cylindrical Inclusions

1967 ◽  
Vol 1 (4) ◽  
pp. 404-412 ◽  
Author(s):  
James G. Goree
Keyword(s):  
Three Dimensional ◽  
Numerical Results ◽  
Elastic Matrix ◽  
Infinite Length ◽  
Dimensional Problem ◽  
Rigid Spheres ◽  
The Matrix ◽  
Cylindrical Inclusions

A solution is presented for the stresses and displacements in an infinite elastic matrix containing two perfectly bonded rigid circular cylindrical inclusions of different radii, and of infinite length normal to the x-y plane. The matrix is subjected to in-plane stresses Sx and Sy at infinity as well as loading due to radial expan sion of the inclusions. Numerical results are presented and a com parison is made with the associated three dimensional problem of unequal rigid spheres.

scholarly journals Flow Velocity and Effective Viscosity of a Fluid Containing Rigid Cylindrical Inclusions

2005 ◽  
Vol 60 (6) ◽  
pp. 401-407 ◽  
Author(s):  
Siegfried Hess
Keyword(s):  
Effective Viscosity ◽  
Volume Fraction ◽  
Dimensional Problem ◽  
Flow Properties ◽  
Cylindrical Inclusions ◽  
Dilute Suspension ◽  
Oriented Parallel ◽  
Relative Change ◽  
Special Case

The determination of the flow properties of a fluid containing a cylindrical inclusion with its long axis oriented parallel to the vorticity direction is a 2-dimensional problem which is treated as a special case in a calculation of the corresponding D-dimensional problem. The velocity and pressure are obtained from the solution of the equations of hydrodynamics where D-dimensional multipole potential tensors are used. The effective viscosity of a dilute suspension is evaluted via the entropy production, as suggested by Einstein, and via an effective stress tensor. The relative change of the viscosity is proportional to the volume fraction. For D = 2 the proportionality factor Z is found to be 2 and 3 when the inclusion rotates with an angular velocity equal to the vorticity and when the inclusion does not rotate, repectively. The corresponding results for D=3 are the well known number Z = 2.5 and Z = 4.

Determination of the Compressive Material Properties of the Supraspinatus Tendon

2000 ◽  
Vol 123 (1) ◽  
pp. 47-51 ◽  
Author(s):  
Mark E. Zobitz ◽  
Zong-Ping Luo ◽  
Kai-Nan An
Keyword(s):  
Material Properties ◽  
Three Dimensional ◽  
Supraspinatus Tendon ◽  
Hyperelastic Material ◽  
Compressive Properties ◽  
Energy Potential ◽  
The Matrix ◽  
Three Dimensional Models ◽  
Thickness Measurements

A methodology was developed for determining the compressive properties of the supraspinatus tendon, based on finite element principles. Simplified three-dimensional models were created based on anatomical thickness measurements of unloaded supraspinatus tendons over 15 points. The tendon material was characterized as a composite structure of longitudinally arranged collagen fibers within an extrafibrillar matrix. The matrix was formulated as a hyperelastic material described by the Ogden form of the strain energy potential. The hyperelastic material parameters were parametrically manipulated until the analytical load-displacement results were similar to the results obtained from indentation testing. In the geometrically averaged tendon, the average ratio of experimental to theoretical maximum indentation displacement was 1.00 (SD: 0.01). The average normalization of residuals was 2.1g (SD: 0.9g). Therefore, the compressive material properties of the supraspinatus tendon extrafibrillar matrix were adequately derived with a first-order hyperelastic formulation. The initial compressive elastic modulus ranged from 0.024 to 0.090 MPa over the tendon surface and increased nonlinearly with additional compression. Using these material properties, the stresses induced during acromional impingement can be analyzed.

scholarly journals DETERMINATION OF PIROCARBON MATRIX CHARACTERISTICS IN CARBON/CARBON COMPOSITES

2021 ◽  
Vol 64 (5) ◽  
pp. 44-49
Author(s):  
Maria V. Papkova ◽  
Sergei V. Tashchilov ◽  
Ilya V. Magnitsky ◽  
Alexander E. Dvoretsky
Keyword(s):  
Structural Characteristics ◽  
Three Dimensional ◽  
Carbon Composites ◽  
X Ray Diffraction ◽  
X Ray ◽  
Extinction Angle ◽  
Parallel Layer ◽  
The Matrix ◽  
Average Size

One of the methods of carbon/carbon composites (C/C composites) production is the deposition of a pyrocarbon (pyC) matrix in a porous preform. The investigation of the pyC matrix characteristics is based on the optical anisotropy with determination of the extinction angle Ae and X-ray diffraction determination of the interplanar spacing d002, crystallite size in the direction of stacking of graphite layers Lc and average size of graphite planes parallel layer in crystallites La. In this study, three previously produced by the thermal gradient method with different parameters specimens of C/C composites were investigated by optical microscopy and X-ray diffraction methods. The studied specimens have a different type of a texture and different structural characteristics of the pyC matrix. Extinction angle Ae for specimen 1, specimen 2 and 3 was 5°, 19° and 41°, respectively. The range of the extinction angle for the pyC matrix is wider than that presented in literature. And according to the classification of pyC the matrix of specimen 1, specimen 2 and 3 is dark laminar pyC, rough laminar pyC and highly textured pyC. For specimen 2 the largest d002 equal to 0.3476 nm was observed. The lowest degree of three-dimensional ordering relative other specimens was for the specimen 2 with rough laminar pyC matrix. The highest degree of three-dimensional ordering was for the specimen 3 with highly textured pyC matrix. However, there is no direct relationship between the textural and structural characteristics of the pyC matrix. Therefore, the study of the pyC matrix should be based on optical and X-ray diffraction methods.

Axisymmetric Torsional Stresses in a Solid Containing Two Partially Bonded Rigid Spherical Inclusions

1967 ◽  
Vol 34 (2) ◽  
pp. 313-320 ◽  
Author(s):  
J. G. Goree ◽  
H. B. Wilson
Keyword(s):  
Stress Field ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Torsional Stress ◽  
Mixed Boundary Value Problem ◽  
Singular Stress ◽  
Legendre Series ◽  
Spherical Inclusions ◽  
The Matrix ◽  
Singular Stress Field

An analysis of the three-dimensional mixed boundary-value problem of an infinite elastic body containing two identical partially bonded rigid spherical inclusions and subjected to an axisymmetric torsional stress field is presented. The order of the singularity in the stresses at the points of separation between the matrix and the inclusions is determined and the form of this singular stress field is utilized in determining a solution to the dual Legendre series generated from the boundary conditions. Numerical results are presented.

Hardness and flexural performance of 3D orthogonal carbon/glass fibers hybrid composites under thermal-oxidative aging

2021 ◽  
pp. 152808372198927
Author(s):  
Juanzi Li ◽  
Wei Fan ◽  
Tao Liu ◽  
Lili Xue ◽  
Linjia Yuan ◽  
...  
Keyword(s):  
Failure Modes ◽  
Hybrid Composites ◽  
Glass Fibers ◽  
Flexural Modulus ◽  
Three Dimensional ◽  
Accelerated Aging ◽  
Flexural Performance ◽  
Flexural Failure ◽  
The Matrix ◽  
Experimental Findings

This study reports the hardness and flexural performance of the three-dimensional (3 D) orthogonal carbon/glass hybrid fiber/bismaleimide composites subjected to the accelerated aging conditions for 10, 30, 90, 120, and 180 days at 250 °C in an air environment. The rate of reduction in the flexural performance and failure modes were observed, in general, to be related to the aging time. The experimental findings revealed that the significant decline in the flexural performance of the samples aged for less than 30 days was predominantly attributed to the matrix degradation, while for the longer aging durations, the cracks in the composites and decomposition of the residual matrix were responsible for the gradual reduction in the flexural performance. The unaged and 30 days aged samples suffered a brittle failure represented by the macro-cracks and fiber breakage, while the cracked fiber/matrix interface and loosened fiber bundles were the main failure modes for the samples aged for longer times. The changes in the flexural failure modes resulted due to the severe degradation of the matrix under an extreme thermo-oxidative environment. Subsequently, a nonlinear relationship relating the flexural modulus to hardness was proposed.

Development of early cell lineages in marsupial embryos: an overview

1994 ◽  
Vol 6 (4) ◽  
pp. 507 ◽  
Author(s):  
L Selwood
Keyword(s):  
Cell Fate ◽  
Three Dimensional ◽  
Embryonic Cell ◽  
Cleavage Pattern ◽  
Cell Lineages ◽  
Trophectoderm Cells ◽  
Maternal Determinants ◽  
Virginia Opossum ◽  
Do So

All major embryonic and extra-embryonic cell lineages are established before implantation in marsupials. In reptiles, birds, monotremes and most marsupials, the zygote is polarized, sometimes markedly so, and the cleavage pattern in relation to the polarized state provides the mechanism for the generation of positional signals. These ensure that the embryonic cell lineages develop in the centre of the developing blastoderm or blastocyst epithelium and the extra-embryonic lineages at the periphery. The evolution of the vertebrate yolky egg was accompanied by a decreasing dependence on maternal determinants and increasing dependence on positional signals to determine cell fate. It is proposed that when a less yolky egg evolved, the mechanisms for determination of cell fate in a developing epithelium were retained. It is proposed that in marsupials, positional signals are involved in the determination of cell fate of embryonic and trophectoderm cells but do so in a two-dimensional epithelium not a three-dimensional morula. The next lineage formed is the primary endoderm which separates off from the primitive ectoderm in the embryoblast and eventually lines the blastocyst cavity. Positional signals are responsible for the determination of primary endoderm in eutherian mammals, birds and probably also marsupials. Order of cell division during cleavage provides a mechanism whereby some cells in the embryoblast of marsupials have earlier and greater contact with their neighbouring cells. The mechanism for determination of primary endoderm cells in the blastocyst epithelium is examined in the Virginia opossum and the stripe-faced dunnart.

scholarly journals Targeted patterning of magnetic microparticles in a polymer composite

2020 ◽  
Vol 378 (2171) ◽  
pp. 20190256 ◽  
Author(s):  
Dmitry Borin
Keyword(s):  
Magnetic Field ◽  
Three Dimensional ◽  
Soft Magnetic Materials ◽  
Soft Magnetic ◽  
Parameter Variations ◽  
The Matrix ◽  
Viscous Media ◽  
Experimental Findings ◽  
External Stimuli ◽  
Further Development

Structured and polymerized in a uniform external magnetic field, polymer composites based on magnetic soft microparticles are considered. Variations of magnetic field parameters and material composition provide a possibility of targeted micro-structural patterning of these composites. The influences of parameter variations on the resulting internal micro-structure of the low concentrated specimens are evaluated and visualized using optical microscopy and microcomputed tomography. The experimental findings are discussed in order to provide advanced possibilities of controlled patterning of soft magnetic materials. It is experimentally demonstrated that the final three-dimensional morphology of composite structure is determined mainly by the concentration of magnetic powder. The intensity of the applied magnetic field influences the rate of structuring of particles in initially viscous media and, therefore, may provide a potential opportunity to obtain non-ergodic microstructures when the matrix is polymerized before the particles have completed the structuring process. The results obtained can serve as a basis for further development of the engineering method of targeted patterning. The method is intended to obtain a material with the desired microstructure by selecting specific parameters of external stimuli and components of the composite. This article is part of the theme issue ‘Patterns in soft and biological matters’.

Three-dimensional nanoprepreg and nanostitched aramid/phenolic multiwall carbon nanotubes composites: Experimental determination of in-plane shear

2019 ◽  
Vol 53 (28-30) ◽  
pp. 4077-4096 ◽  
Author(s):  
Kadir Bilisik ◽  
Gulhan Erdogan ◽  
Erdal Sapanci ◽  
Sila Gungor
Keyword(s):  
Carbon Nanotubes ◽  
Three Dimensional ◽  
Multiwall Carbon Nanotubes ◽  
Nano Composites ◽  
Plane Shear ◽  
Pull Out ◽  
Out Of Plane ◽  
The Matrix ◽  
Multiwall Carbon

In-plane shear of nanostitched three-dimensional para-aramid/phenolic composites were experimentally investigated. Adding the nanostitched fiber into nanoprepreg para-aramid fabric preform composites slightly improved their shear strengths. The carbon-stitched composite exhibited comparatively better performance compared to the para-aramid stitched composite probably due to well bonding between carbon fiber and phenolic resin. The stitched nano composites had mainly matrix breakages and micro shear hackles in the matrix; matrix debonding and filament pull-out in the composite interface; fibrillar peeling and stripping on the filaments due to angular deformation. This mechanism probably prohibited extensive interlaminar opening in the nanostitched composites. The result exhibited that the introducing of the nano stitched fiber where multiwall carbon nanotubes were transferred to the out-of-plane of the base structure enhanced its transverse fracture as a form of confined delamination area. Therefore, the damaged tolerance properties of the stitched nano composites were enhanced compared to the base.

Conformation of Ribosomes from Escherichia Coli

1974 ◽  
Vol 32 ◽  
pp. 210-211
Author(s):  
M. Boublik ◽  
W. Hellmann ◽  
F. Jenkins
Keyword(s):  
Binding Sites ◽  
Ribosomal Proteins ◽  
Fluorescent Dyes ◽  
Protein Biosynthesis ◽  
Three Dimensional ◽  
Spatial Arrangement ◽  
Dimensional Structure ◽  
Complete Understanding ◽  
And Function

The present knowledge of the three-dimensional structure of ribosomes is far too limited to enable a complete understanding of the various roles which ribosomes play in protein biosynthesis. The spatial arrangement of proteins and ribonuclec acids in ribosomes can be analysed in many ways. Determination of binding sites for individual proteins on ribonuclec acid and locations of the mutual positions of proteins on the ribosome using labeling with fluorescent dyes, cross-linking reagents, neutron-diffraction or antibodies against ribosomal proteins seem to be most successful approaches. Structure and function of ribosomes can be correlated be depleting the complete ribosomes of some proteins to the functionally inactive core and by subsequent partial reconstitution in order to regain active ribosomal particles.

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