Closure to “Discussion of ‘On the Parametrization of the Elastic Moduli of Two-Phase Materials’” (1965, ASME J. Appl. Mech., 32, pp. 953–954)

1965 ◽  
Vol 32 (4) ◽  
pp. 954-954
Author(s):  
Tai Te Wu
Keyword(s):  
Elastic Moduli ◽  
Two Phase

scholarly journals Model Calculations of Elastic Moduli in Highly Deformed Composites

1989 ◽  
Vol 10 (2) ◽  
pp. 153-164 ◽  
Author(s):  
H. J. Bunge
Keyword(s):  
Plastic Deformation ◽  
Simple Model ◽  
Elastic Moduli ◽  
Opposite Sign ◽  
Internal Stresses ◽  
Minor Importance ◽  
High Increase ◽  
Two Phase ◽  
Linear Elasticity Theory ◽  
Model Calculations

Young's modulus of heavily deformed two-phase composites shows an unusually high increase after plastic deformation. It is assumed that this is due to two reasons, i.e. texture changes and changes of the moduli of the constitutive phases on the basis of non-linear elasticity theory and internal stresses of opposite sign in the phases. Expressions of the two contributions are given on the basis of simple model assumptions. It is estimated that the changes of shape and arrangement of the phases and shape and arrangement of the crystallites in the phases are only of minor importance.

Mechanical Properties of Diverse High-Temperature Compounds–Thermal Variation of Microhardness and Crack Formation

1988 ◽  
Vol 133 ◽  
Author(s):  
Robert L. Fleischer
Keyword(s):  
Elastic Moduli ◽  
Crack Formation ◽  
The Other ◽  
Terminal Phase ◽  
Thermal Variation ◽  
Two Phase ◽  
Thermally Activated ◽  
High Melting Temperature ◽  
Binary Compounds ◽  
Single Crystal Plasticity

ABSTRACTMicrohardness vs temperature and elastic moduli have been measured for a suite of intermetallic compounds that melt above 1400°C. Binary intermetallics were selected to represent a variety of crystal structures and yet have optimal combinations of high melting temperature (Tm) and low specific gravity. Some deliberately two-phase alloys were prepared in which one phase is a terminal-phase metal and the other an intermetallic compound.Binary compounds can be described by two patterns. In those where plasticity is difficult, hardness decreased slowly with temperature up to Tm/2, the decrease being no more than that normally shown by the elastic moduli. In those compounds where single crystal plasticity is known (or at least plausible), microhardness decreases more rapidly than do elastic moduli, presumably due to thermally activated slip.

On a Spring-Network Model and Effective Elastic Moduli of Granular Materials

1999 ◽  
Vol 66 (1) ◽  
pp. 172-180 ◽  
Author(s):  
K. Alzebdeh ◽  
M. Ostoja-Starzewaski
Keyword(s):  
Granular Materials ◽  
Disordered Systems ◽  
Granular Media ◽  
Elastic Moduli ◽  
Volume Fraction ◽  
Effective Medium ◽  
Solid State Physics ◽  
Two Phase ◽  
Effective Medium Theories ◽  
The Individual

Two challenges in mechanics of granular media are taken up in this paper: (i) development of adequate numerical discrete element models of topologically disordered granular assemblies, and (ii) calculation of macroscopic elastic moduli of such materials using effective medium theories. Consideration of the first one leads to an adaptation of a spring-network (Kirkwood) model of solid-state physics to disordered systems, which is developed in the context of planar Delaunay networks. The model employs two linear springs: a normal one along an edge connecting two neighboring vertices (grain centers) which accounts for normal interactions between the grains, as well as an angular one which accounts for angle changes between two edges incident onto the same vertex; edges remain straight and grain rotations do not appear. This model is then used to predict elastic moduli of two-phase granular materials—random mixtures of soft and stiff grains —for high coordination numbers. It is found here that an effective Poisson’s ratio, νeff, of such a mixture is a convex function of the volume fraction, so that νeff may become negative when the individual Poisson’s ratios of both phases are both positive. Additionally, the usefulness of three effective medium theories—perfect disks, symmetric ellipses, and asymmetric ellipses—is tested.

Elastic Moduli of Thickly Coated Particle and Fiber-Reinforced Composites

1991 ◽  
Vol 58 (2) ◽  
pp. 388-398 ◽  
Author(s):  
Y. P. Qiu ◽  
G. J. Weng
Keyword(s):  
Shear Modulus ◽  
Bulk Modulus ◽  
Elastic Moduli ◽  
Poisson Ratio ◽  
Fiber Reinforced Composites ◽  
Reinforced Composites ◽  
Fiber Reinforced ◽  
Two Phase ◽  
Coated Particle ◽  
Replacement Method

Based on the models of Hashin (1962) and Hashin and Rosen (1964), the effective elastic moduli of thickly coated particle and fiber-reinforced composites are derived. The microgeometry of the composite is that of a progressively filled composite sphere or cylinder element model. The exact solutions of the effective bulk modulus κ of the particle-reinforced composite and those of the plain-strain bulk modulus κ23, axial shear modulus μ12, longitudinal Young’s modulus E11, major Poisson ratio ν12, of the fiber-reinforced one are derived by the replacement method. The bounds for the effective shear modulus μ and the effective transverse shear modulus μ23 of these two kinds of composite, respectively, are solved with the aid of Christensen and Lo’s (1979) formulations. By considering the six possible geometrical arrangements of the three constituent phases, the values of κ, and of κ23, μ12, E11, and ν12 are found to always lie within the Hashin-Shtrikman (1963) bounds, and the Hashin (1965), Hill (1964), and Walpole (1969) bounds, respectively, but unlike the two-phase composites, none coincides with their bounds. The bounds of μ and μ23 derived here are consistently tighter than their bounds but, as for the two-phase composites, one of the bounds sometimes may fall slightly below or above theirs and therefore it is suggested that these two sets of bounds be used in combination, always choosing the higher for the lower bound and the lower for the upper one.

Elastic Moduli of Composites Reinforced by Multiphase Particles

1995 ◽  
Vol 62 (4) ◽  
pp. 1023-1028 ◽  
Author(s):  
M. L. Dunn ◽  
H. Ledbetter
Keyword(s):  
Elastic Moduli ◽  
Aluminum Matrix ◽  
Recent Analysis ◽  
Simple Extension ◽  
Two Phase ◽  
Elastic Fields ◽  
Three Phase ◽  
Theoretical Predictions ◽  
Concentration Factors ◽  
Pulse Echo

A theoretical approach is proposed to estimate the elastic moduli of three-phase composites consisting of a matrix phase reinforced by two-phase particles. The theoretical predictions are based on a simple extension to nondilute concentrations of the mechanical concentration factors obtained from the recent analysis of the average elastic fields in a double inclusion by Hori and Nemat-Nasser (1993). The proposed micromechanics theory can account for the effects of shapes and concentrations of both the particles and the dispersed phase in the particles. Theoretical estimates of the concentration factors and the effective elastic moduli are obtained in closed form and are diagonally symmetric and fall within the Hashin-Shtrikman-Walpole bounds for all cases considered. The theoretical predictions are in excellent agreement with experimental results obtained from pulse-echo and rod-resonance measurements of the elastic moduli of a three-phase composite consisting of an aluminum matrix reinforced by mullite/alumina particles.

Determination of closed form expressions of the second-gradient elastic moduli of multi-layer composites using the periodic unfolding method

2018 ◽  
Vol 24 (5) ◽  
pp. 1475-1502 ◽  
Author(s):  
Jean-François Ganghoffer ◽  
Gérard Maurice ◽  
Yosra Rahali
Keyword(s):  
Composite Materials ◽  
Closed Form ◽  
Elastic Moduli ◽  
Gradient Elasticity ◽  
Constitutive Law ◽  
Two Phase ◽  
Linear Elastic ◽  
Second Gradient ◽  
Two Phases ◽  
Unfolding Method

The present paper aims at introducing a homogenization scheme for the identification of strain–gradient elastic moduli of composite materials, based on the unfolding mathematical method. We expose in the first part of this paper the necessary mathematical apparatus in view of the derivation of the effective first- and second-gradient mechanical properties of two-phase composite materials, focusing on a one-dimensional situation. Each of the two phases is supposed to obey a second-gradient linear elastic constitutive law. Application of the unfolding method to the homogenization of multi-layer materials provides closed form expressions of all effective first- and second-gradient elastic moduli as well as coupling moduli between first- and second-gradient elasticity. A comparison between the unfolding method and the method of oscillating functions shows that both methods, despite their differences, deliver the same effective second-gradient elastic constitutive law for stratified materials.

VELOCITY AND ATTENUATION OF SEISMIC WAVES IN TWO‐PHASE MEDIA: PART I. THEORETICAL FORMULATIONS

Geophysics ◽  
1974 ◽  
Vol 39 (5) ◽  
pp. 587-606 ◽  
Author(s):  
Guy T. Kuster ◽  
M. Nafi Toksöz
Keyword(s):  
Elastic Moduli ◽  
Seismic Waves ◽  
Matrix Material ◽  
Composite Medium ◽  
Solid Matrix ◽  
Seismic Velocities ◽  
Displacement Fields ◽  
Two Phase ◽  
The Matrix ◽  
In Series

The propagation of seismic waves in two‐phase media is treated theoretically to determine the elastic moduli of the composite medium given the properties, concentrations, and shapes of the inclusions and the matrix material. For long wavelengths the problem is formulated in terms of scattering phenomena in an approach similar to that of Ament (1959). The displacement fields, expanded in series, for waves scattered by an “effective” composite medium and individual inclusions are equated. The coefficients of the series expansions of the displacement fields provide a relationship between the elastic moduli of the effective medium and those of the matrix and inclusions. The expressions are derived for both solid and liquid inclusions in a solid matrix as well as for solid suspensions in a fluid matrix. Both spherical and oblate spheroidal inclusions are considered. Some numerical calculations are carried out to demonstrate the effects of fluid inclusions of various shapes on the seismic velocities in rocks. It is found that the concentration, shapes, and properties of the inclusions are important parameters. A concentration of a fraction of one percent of thin (small aspect ratio) inclusions could affect the compressional and shear velocities by more than ten percent. For both sedimentary and igneous rock models, the calculations for “dry” (i.e.,air‐saturated) and water‐saturated states indicate that the compressional velocities change significantly while the shear velocities change much less upon saturation with water.

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