The Linear Modulus

In the ordered state, lamellar block copolymers frequently show departures from linear viscoelastic behavior at low strain amplitudes of around 1% (Rosedale and Bates 1990 Winey et al. 1993a). Homogeneous polymers, on the other hand, typically show departures 

Both Figs. 13-14 and 13-15 indicate that there is a critical reduced frequency coc above which the storage moduli are similar in the ordered and disordered states (see also Fig. 13-13). Hence, for co cOc, the modulus is controlled by molecular relaxations, while 

The fluctuation contribution to the modulus is smaller or not even visible in other block copolymers (Rosedale et al. 1995 Han et al. 1995). Bates and coworkers (Rosedale et al. 1995) have shown that in polyolefin block copolymers, the fluctuation contributions to the rheology occur within the range of /iV values for which the structure factor S(k) deviates from mean-field theory, namely the range 10.5 (xN)od- This range of 

Within the fluctuation-dominated region of the disordered phase of a diblock near Todt shearing can apparently induce a transition to the ordered state (Koppi et al. 1993). Cates and Milner (1989) predicted such a phenomenon, based on a shear-induced suppression of 

For higher-molecular-weight, strongly segregated materials, such as PS-PI with M  

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Data in Table 30.1 clearly show that, whatever the position of the test sample along the compounding line, there is a substantial difference between run 1 and run 2 data, particularly in what the linear modulus data are concerned. However, G is an extrapolated value and quite unrealistic values are obtained on certain samples, e.g., TR and AA, and it might be safer to consider modulus variations along the compounding line by using the (recalculated) complex modulus at 10% strain (Figure 30.13). 

Poisson s61 ratio a is the change in length per unit length, usually the contraction in one direction due to the dilatation in the perpendicular direction (for isotropic elastic bodies, —1.0 < a < + 0.5). Young s62 modulus Y, also called the linear modulus of elasticity, is the 3x3 tensor of the stress P divided by the strain s ... 

Thus, the linear modulus is controlled by the curvature of the particle-particle potential W at its minimum. Of course, this local curvature is extremely sensitive to the details of the particle-particle interactions at close separations (Goodwin et al.l986), and thus the modulus will also depend strongly on these details. Nevertheless, if we... 

Nevertheless, some of the predictions of simple regular foam models are relevant to real foams. One such property is the linear modulus Go, which is the slope of the stress-strain curve at zero strain. From Eq. (9-57a), we obtain... 

Let us first examine the sign of this third coefficient. If it is negative, the viscosity decreases as time goes on. If it is positive, the viscosity may deviate upwards from the baseline = git fixed by the linear modulus. In other words, it shows an upturn at a certain time from the linear baseline. Such a stress growth beyond the linear baseline caused by a large deformation is called strain hardening. 

Tensile Strength and Elongation. The tensile strength of latex mbber foam has been shown to depend on the density of the foam (149,177) and on the tensile strength of the parent mbber (177,178). At low densities the tensile modulus approximates a linear relation with density but kicreases with a higher power of density at higher densities. Similar relations hold for polyurethane and other flexible foams (156,179,180). 

Fig. 19. Generalized modulus—temperature curves for polymeric materials showing the high modulus glassy state, glass-transition regions for cured and uncured polymers, plateau regions for cross-linked polymers, and the dropoff in modulus for a linear polymer.

Here c[-], which will be called the elastic modulus tensor, is a fourth-order linear mapping of its second-order tensor argument, while b[-], which will be called the inelastic modulus tensor, is a linear mapping of k whose form will depend on the specific properties assigned to k. They depend, in general, on and k. For example, if k consists of a single second-order tensor, then in component form... 

While c in (5.112) is a linear function of d, it may be an arbitrary function of s. Truesdell considered cases where c is a polynomial in s, terming (5.112) a hypoelastic equation of grade n, where n is the power of the highest-order term in the polynomial. For a hypoelastic equation of grade zero, the elastic modulus c is independent of s and linear in dand therefore has the representation (A.89). It is convenient to nondimensionalize the stress by defining s = sjljx. Since the stress rate must vanish when d is zero, Cq = 0 and the result is... 

Like the modulus, the tensile and compressive strengths depend mainly on the density (Fig. 26.6). The strength parallel to the grain varies linearly with density, for the same reason that the axial modulus does it measures the strength of the cell wall, scaled by the fraction of the section it occupies, giving... 

Via an ad hoc extension of the viscoelastic Hertzian contact problem, Falsafi et al. [38] incorporated linear viscoelastic effects into the JKR formalism by replacing the elastic modulus with a viscoelastic memory function accounting for time and deformation, K t) ... 

Tests by Roe et al. [63] with unidirectional jute fiber-reinforced UP resins show a linear relationship (analogous to the linear mixing rule) between the volume content of fiber and Young s modulus and tensile strength of the composite over a range of fiber content of 0-60%. Similar results are attained for the work of fracture and for the interlaminate shear strength (Fig. 20). Chawla et al. [64] found similar results for the flexural properties of jute fiber-UP composites. 

In the region where the relationship between stress and strain is linear, the material is said to be elastic, and the constant of proportionality is E, Young s modulus, or the elastic modulus. 

The secant modulus measurement is used during the designing of a product in place of a modulus of elasticity for materials where the stress-strain diagram does not demonstrate a linear proportionality of stress to strain or E is difficult to locate. 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles, follows ... 

It is interesting to point out the small variation of the average value of the variable Ef(r)-modulus of the mesophase, which reflects the uniformity of the adhesion quality of these series of composites, which is also indicated by the almost linear variation of the composite modulus, versus the volume content. 

The second version of the unfolding model uses, instead of three, two terms, on of which is a negative-power function of the polar distance, acting upon the inclusion modulus, and the other one, acting upon the matrix modulus, is expressed as a linear function of r. 

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