‫فرع السيراميك ومواد البناء‬/‫المرحلة الثالثة‬
Characteristic of Ceramic Materials/mechanical properties
1-7.2 ELASTIC CONSTANTS AND OTHER “CONSTANTS”
In this section we will define some of the parameters that describe the
mechanical behavior of materials. Some of these parameters are constants, like
Young’s modulus E. Some, like hardness, are not. Hardness depends on how the
material was tested. These are four constants that are most common.
1- E: Young’s modulus (also referred to as the elastic modulus) is a material constant
defined by Eq. 1 for a linear elastic material under uniaxial tensile or compressive
stress.
FIGURE A: Idealized stress–strain curves for
different materials classes.
FIGURE B: Factors affecting the mechanical
properties of ceramics.
It is therefore the slope of a (σ–ε) curve where only elastic deformation occurs.
2. ν: Poisson’s ratio is the negative ratio of the transverse strain (ε T) to longitudinal
strain (εL).
……………… (2)
For many ceramics and glasses it is in the range 0.18–0.30.
3. μ: Shear modulus is the ratio of shear stress to shear strain.
………… (3)
4. B: Bulk modulus is the ratio of stress to strain for hydrostatic compression.
…………… (4)
Although these constants are related directly to bonding forces between atoms, in real
ceramics they are affected by microstructure, e.g., porosity and the presence of
second phases. Because strain is dimensionless, elastic moduli have the same
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‫فرع السيراميك ومواد البناء‬/‫المرحلة الثالثة‬
Characteristic of Ceramic Materials/mechanical properties
dimensions as those of stress: force per unit area (N/m2) or in the SI classification Pa
and the relationship among them are:
E = 2μ (1+ν);
E = 3 B (1 -2ν)
Measurement of Young’s modulus
How is Young’s modulus measured? One way is to compress the material with
a known compressive force, and measure the strain. Young’s modulus is then given
by E = (σ/ε) ~ , each defined as described earlier. But this is not generally a good way
to measure the modulus. For one thing, if the modulus is large, the extension u may
be too small to measure with precision. And, for another, if anything else contributes
to the strain, like creep (which we will discuss in a later chapter), or deflection of the
testing machine itself, then it will lead to an incorrect value for E - and these spurious
strains can be serious.
A better way of measuring E is to measure the natural frequency of vibration of a
round rod of the material, simply supported at its ends (Fig. H) and heavily loaded by
a mass M at the middle (so that we may neglect the mass of the rod itself). The
frequency of oscillation of the rod, f cycles per second (or hertz), is given by
Fig. H: A vibrating bar with a central mass, M.
where I is the distance between the supports and d is the diameter of the rod. From
this,
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‫فرع السيراميك ومواد البناء‬/‫المرحلة الثالثة‬
Characteristic of Ceramic Materials/mechanical properties
Use of stroboscopic techniques and carefully designed apparatus can make this sort
of method very accurate. The best of all methods of measuring E is to measure the
velocity of sound in the material. The velocity of longitudinal waves, v l, depends on
Young’s modulus and the density, p:
vl is measured by ’striking’ one end of a bar of the material (by glueing a
piezoelectric crystal there and applying a charge-difference to the crystal surfaces)
and measuring the time sound takes to reach the other end (by attaching a second
piezoelectric crystal there). Most moduli are measured by one of these last two
methods.
1-7.3 Hardness
Hardness is an important property to quantify in ceramics. Measured hardness
indicates the ability of the ceramic to resist deformation by a hard object. Usually,
Knoop or Vickers diamond indenters are used in conjunction with a microindentation
hardness machine.8 Rarely are the popular Rockwell and Brinell indenters used for
ceramics research. Vickers indenters are used to characterize roughly 60% of the
ceramic hardness values that are published. The indentation force should always be
included with the hardness value.
Discrepancies can arise at different indentation forces. At higher forces, cracking
can complicate the measuring process or make measuring impossible. Measuring the
hardness from the indentation, especially at small forces, is also a significant source
of error in hardness testing. The hardness value can change based on the force value
applied to the test specimen at small forces. Volume 8 of the ASM Handbook8
recommends forces greater than or equal to 9.8 N for Vickers and Knoop
indentations. Errors in the measurement of the indentation diagonal length essentially
double the hardness error as the hardness value is proportional to the square of the
diagonal length. A Versailles Advanced Materials and Standards (VAMS) roundrobin test project conducted on alumina ceramic samples resulted in uncertainty in
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‫فرع السيراميك ومواد البناء‬/‫المرحلة الثالثة‬
Characteristic of Ceramic Materials/mechanical properties
the hardness values given by the laboratories involved. Although using numerous
indentations can reduce some of this uncertainty, engineers and scientists conducting
hardness tests should nevertheless keep this uncertainty in mind when considering
their data. Standards for measuring hardness for ceramics are listed below:
Vickers Hardness
ASTM, ‘‘Standard Test Method for Vickers Indentation Hardness of Advanced
Ceramics,’’ C1427-97
CEN, ‘‘Advanced Technical Ceramics—Monolithic Ceramics—Mechanical
Properties at Room Temperature—Part 4: Determination of Vickers, Knoop and
Rockwell Superficial Hardness Tests,’’ prEN834-4
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