scholarly journals Calculation of Effective Elastic Moduli of Textured Materials

1995 ◽  
Vol 23 (1) ◽  
pp. 43-59 ◽  
Author(s):  
N. J. Park ◽  
H. J. Bunge ◽  
H. Kiewel ◽  
L. Fritsche
Keyword(s):  
Elastic Moduli ◽  
Expansion Method ◽  
Polycrystalline Materials ◽  
Effective Elastic Moduli ◽  
Actual Material ◽  
Finite Sum ◽  
Textured Materials ◽  
The Hill ◽  
Very High ◽  
Series Expansion Method

Effective elastic constants of polycrystalline materials were determined with a recently developed method. This method bases on the modelation of the actual material by a cluster of 100 to 500 single crystals. In the present version of the scheme parallelepipeds are used. The ODF was calculated with the series expansion method. The transformation of this ODF into a finite sum of single orientations permits to assign any grain an individual orientation.Reliable results for the effective elastic moduli of textured materials are reported. They lie always within the bounds of Voigt and Reuss. The very high anisotropic substances, e.g. shape-memory-alloys, show a significant deviation from the Hill values.

Effective Elastic Moduli of Ribbon-Reinforced Composites

1990 ◽  
Vol 57 (1) ◽  
pp. 158-167 ◽  
Author(s):  
Y. H. Zhao ◽  
G. J. Weng
Keyword(s):  
Aspect Ratio ◽  
Elastic Moduli ◽  
Volume Fraction ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Effective Elastic Moduli ◽  
Cross Sectional ◽  
Isotropic Composite ◽  
The Matrix ◽  
The Hill

Based on the Eshelby-Mori-Tanaka theory the nine effective elastic constants of an orthotropic composite reinforced with monotonically aligned elliptic cylinders, and the five elastic moduli of a transversely isotropic composite reinforced with two-dimensional randomly-oriented elliptic cylinders, are derived. These moduli are given in terms of the cross-sectional aspect ratio and the volume fraction of the elliptic cylinders. When the aspect ratio approaches zero, the elliptic cylinders exist as thin ribbons, and these moduli are given in very simple, explicit forms as a function of volume fraction. It turns out that, in the transversely isotropic case, the effective elastic moduli of the composite coincide with Hill’s and Hashin’s upper bounds if ribbons are harder than the matrix, and coincide with their lower bounds if ribbons are softer. These results are in direct contrast to those of circular fibers. Since the width of the Hill-Hashin bounds can be very wide when the constituents have high modular ratios, this analysis suggests that the ribbon reinforcement is far more effective than the traditional fiber reinforcement.

The iterative series-expansion method for quantitative texture analysis. II. Applications

1992 ◽  
Vol 25 (2) ◽  
pp. 259-267 ◽  
Author(s):  
M. Dahms
Keyword(s):  
Distribution Function ◽  
Series Expansion ◽  
Expansion Method ◽  
Orientation Distribution ◽  
Polycrystalline Materials ◽  
Pole Density ◽  
General Outline ◽  
Positivity Condition ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density functions (pole figures). This procedure, called pole-figure inversion, can be achieved by the series-expansion method (harmonic method). As a consequence of the (hkl)-({\bar h}{\bar k}{\bar l}) superposition, the solution is mathematically not unique. There is a range of possible solutions (the kernel) that is only limited by the positivity condition of the distribution function. The complete distribution function f(g) can be split into two parts, \tilde {f}(g) and \tildes {f}(q), expressed by even- and odd-order terms of the series expansions. For the calculation of the even part \tilde {f}(g), the positivity condition for all pole figures contributes essentially to an `economic' calculation of this part, whereas, for the odd part, the positivity condition of the ODF is the essential basis. Both of these positivity conditions can be easily incorporated in the series-expansion method by using several iterative cycles. This method proves to be particularly versatile since it makes use of the orthogonality and positivity at the same time. In the previous paper in this series [Dahms & Bunge (1989) J. Appl. Cryst. 22, 439–447] a general outline of the method was given. This, the second part, gives details of the system of programs used as well as typical examples showing the versatility of the method.

scholarly journals Improved Bounds on the Effective Elastic Moduli of Random Arrays of Cylinders

1992 ◽  
Vol 59 (1) ◽  
pp. 1-6 ◽  
Author(s):  
S. Torquato ◽  
F. Lado
Keyword(s):  
Elastic Moduli ◽  
Volume Fraction ◽  
Transversely Isotropic ◽  
Circular Cylinders ◽  
Effective Elastic Moduli ◽  
Random Arrays ◽  
The Matrix ◽  
Rigorous Bounds ◽  
Fiber Reinforced Material ◽  
The Hill

Improved rigorous bounds on the effective elastic moduli of a transversely isotropic fiber-reinforced material composed of aligned, infinitely long, equisized, circular cylinders distributed throughout a matrix are evaluated for cylinder volume fractions up to 70 percent. The bounds are generally shown to provide significant improvement over the Hill-Hashin bounds which incorporate only volume-fraction information. For cases in which the cylinders are stiffer than the matrix, the improved lower bounds provide relatively accurate estimates of the elastic moduli, even when the upper bound diverges from it (i.e., when the cylinders are substantially stiffer than the matrix). This last statement is supported by accurate, recently obtained Monte Carlo computer-simulation data of the true effective axial shear modulus.

scholarly journals Effect of the Grain Shape on the Elastic Constants of Polycrystalline Materials

1996 ◽  
Vol 28 (1-2) ◽  
pp. 17-33 ◽  
Author(s):  
H. Kiewel ◽  
H. J. Bunge ◽  
L. Fritsche
Keyword(s):  
Boundary Value Problem ◽  
Coordination Number ◽  
Elastic Constants ◽  
Elastic Moduli ◽  
Grain Shape ◽  
Polycrystalline Materials ◽  
Stress And Strain ◽  
Effective Elastic Moduli ◽  
Real Material ◽  
Effective Constants

We examine the influence of the grain shape on the effective elastic moduli of polycrystalline materials. For that purpose the real material is simulated by a cluster of Wigner-Seitz cells. For clarity each aggregate consists of grains with only one type of shape. Therefore we can classify each cluster by the coordination number of its grains. To determine the elastic moduli a homogeneous deformation is subjected to the surface of the cluster. The solution of this boundary value problem yields the average stress and strain governing inside the material whose interconnection by Hooke's law leads to the sought-for effective constants.The most important result is that with increasing coordination number the elastic moduli decrease.

HOLE–SPIN COUPLING AND SPIN DISTORTION IN HIGH Tc SUPERCONDUCTORS

1995 ◽  
Vol 09 (24) ◽  
pp. 1589-1594
Author(s):  
M. TIWARI ◽  
R. A. SINGH
Keyword(s):  
Correlation Function ◽  
Green Function ◽  
Low Temperature ◽  
Function Theory ◽  
Expansion Method ◽  
Temperature Series ◽  
Spin Coupling ◽  
High Tc Superconductors ◽  
High Tc ◽  
Series Expansion Method

The effect of hole–spin coupling together with spin distortion on the energy and hole correlation function have been studied in detail. Standard Green function theory and Low Temperature Series Expansion method have been utilised to get analytical results.

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