Young’s modulus defined

The constant is called the modulus of elasticity (E) or Young s modulus (defined by Thomas Young in 1807 although the concept was used by others that included the Roman Empire and Chinese-BC), the elastic modulus, or just the modulus. This modulus is the straight line slope of the initial portion of the stress-strain curve, normally expressed in terms such as MPa or GPa (106 psi or Msi). A... 

Young s modulus defines the response of a body, the strain, to a linear stress tending to stretch or compress it (Figure S4.1). The relationship between these quantities is ... 

A solid, therefore, is characterized by its modulus of longitudinal elasticity, known as Young s modulus, defined by ... 

Young s modulus defined from the line between origin and a certain point of the stress-strain curve (secant modulus) e.g. E g is the Young s modulus at 50% of the failure stress ... 

The factor 3 appears because the viscosity is defined for shear deformation - as is the shear modulus G. For tensile deformation we want the viscous equivalent of Young s modulus . The answer is 3ri, for much the same reason that = (8/3)G 3G - see Chapter 3.) Data giving C and Q for polymers are available from suppliers. Then... 

When we consider the mechanical properties of polymeric materials, and in particular when we design methods of testing them, the parameters most generally considered are stress, strain, and Young s modulus. Stress is defined as the force applied per unit cross sectional area, and has the basic dimensions of N m in SI units. These units are alternatively combined into the derived unit of Pascals (abbreviated Pa). In practice they are extremely small, so that real materials need to be tested with a very large number of Pa... 

NR with standard recipe with 10 phr CB (NR 10) was prepared as the sample. The compound recipe is shown in Table 21.2. The sectioned surface by cryo-microtome was observed by AFM. The cantilever used in this smdy was made of Si3N4. The adhesion between probe tip and sample makes the situation complicated and it becomes impossible to apply mathematical analysis with the assumption of Hertzian contact in order to estimate Young s modulus from force-distance curve. Thus, aU the experiments were performed in distilled water. The selection of cantilever is another important factor to discuss the quantitative value of Young s modulus. The spring constant of 0.12 N m (nominal) was used, which was appropriate to deform at rubbery regions. The FV technique was employed as explained in Section 21.3.3. The maximum load was defined as the load corresponding to the set-point deflection. 

Dynamic mechanical results are generally given in terms of complex moduli or compliances (3,4), The notation will be illustrated in terms Of shear modulus G, but exactly analogous notation holds for Young s modulus F. The complex moduli are defined by... 

As the temperature is raised, the vibrational energy increases, because it is kBT in each direction. If we have a simple cubic crystal in which the intermolecular spacing is r then the molar volume is Nar3. The Young s modulus for the crystal is Y and we assume a Hooke s law spring. We can define the local stress as the applied force per molecule, Fm, divided by r2, giving a local strain of x/r where x is the extension caused by the oscillation. Hence ... 

Define Young s modulus. Should it be high or low for a fiber ... 

The ratio between these relative deformations is Ifn and can be used to define the deformation profile or length scale. Due to the presence of a softer back pad, more deformation is expected for the stacked pad but the shape, which is the main concern, will be approximately similar [45,46], The deformation is relatively small compared to the region of apphcation of the force. Using approximate material properties for the ICIOOO pad (Young s modulus of 2.9 x 10 Pa [41] and approximate Poisson ratio of 1/3) with force applied in a circular region of radius 2 mm, and a local pressure of 7 psi, the maximum deflection is about 6 fira. This deformation is referenced to the origin as illustrated in Fig. 13. It is also important to note that the transition shape is very gradual and this sets the polish limit for the down areas. 

Consider a sphere of diameter D of material 1 in contact with a planar surface of material 2. The reaction of the load P is determined by an effective Young s modulus E, which is defined as... 

We have written Eq. (5.4) with variables grouped as they are in order to define two very important quantities. The first quantity in parentheses is called the modulus—or in this case, the tensile modulus, E, since a tensile force is being applied. The tensile modulus is sometimes called Young s modulus, elastic modulus, or modulus of elasticity, since it describes the elastic, or recoverable, response to the applied force, as represented by the springs. The second set of parentheses in Eq. (5.4) represents the tensile strain, which is indicated by the Greek lowercase epsilon, e. The strain is defined as the displacement, r — rp, relative to the initial position, rp, so that it is an indication of relative displacement and not absolute displacement. This allows comparisons to be made between tensile test performed at a variety of length scales. Equation (5.4) thus becomes... 

Figure 3.3 shows representative stress-strain curves for a variety of polymeric materials. At normal use temperatures, such as room temperature, rigid polymers such as polystyrene (PS) exhibit a rapid increase in stress with increasing strain until sample failure. This behavior is typical of brittle polymers with weak interchain secondary bonding. As shown in the top curve in Figure 3.3, the initial stress-strain relation in such polymers is approximately linear and can be described in terms of Hooke s law, i.e., S = Ee, where E is Young s modulus, typically defined as the slope of the stress-strain plot. At higher stresses, the plot becomes nonlinear. The point at which this occurs is called the proportional limit. 

The coefficients Cn are called elasticity constants and the coefficients Su elastic compliance constants (Azaroff, 1960). Generally, they are described jointly as elasticity constants and constitute a set of strictly defined, in the physical sense, quantities relating to crystal structure. Their experimental determination is impossible in principle, since Cu = (doildefei, where / i, and hence it would be necessary to keep all e constant, except et. It is easier to satisfy the necessary conditions for determining Young s modulus E, when all but one normal stresses are constant, since... 

Finding a correlation between Young s modulus and material hardness required the use of hardness standards on the Mohs scale (quartz, topaz, corundum) whose hardness calculated using various methods was shown in Table 9.3. These data helped us to define the relation between Young s modulus and hardness of standard materials (Fig. 9.12), and then to read the degree of hardness of various tested materials from a chart (Table 9.4, Fig. 9.13). 

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