Calculating the Tensile Modulus

Rem the longitudinal and transverse (tensile) moduli Eji and E22 are identical when particles are spherical 

---

C being a factor that depends on the approach followed to obtain the shear correction. The values of d/l are 0.129, 0.153, 0.167, respectively. These results suggest that if the accuracy of the experiment device to calculate the tensile modulus is better than 5%, corrections due to shear effects are unnecessary for values of d/l lower than 0.1. 

The standard test methods, calculate the tensile modulus by drawing a tangent to the initial linear part of the stress-strain curve and calculating the slope of the line. In cases where no clearly defined linear portion exists, the secant modulus should be determined. 

When specific stress is used in calculating the tensile modulus, the modulus unit becomes gram per denier or gram per tex. 

According to the composite theory, tensile modulus of fiber reinforced composites can be calculated by knowing the mechanical constants of the components, their volume fraction, the fiber aspect ratio, and orientation. But in the case of in situ composites injection molded, the TLCP fibrils are developed during the processing and are still embedded in the matrix. Their modulus cannot be directly measured. To overcome this problem, a calculation procedure was developed to estimate the tensile modulus of the dispersed fibers and droplets as following. 

The tensile modulus can be determined from the slope of the linear portion of this stress-strain curve. If the relationship between stress and strain is linear to the yield point, where deformation continues without an increased load, the modulus of elasticity can be calculated by dividing the yield strength (pascals) by the elongation to yield ... 

The stress-strain curves for cortical bones at various strain rates are shown in Figure 5.130. The mechanical behavior is as expected from a composite of linear elastic ceramic reinforcement (HA) and a compliant, ductile polymer matrix (collagen). In fact, the tensile modulus values for bone can be modeled to within a factor of two by a rule-of-mixtures calculation on the basis of a 0.5 volume fraction HA-reinforced... 

Let us return to the proposed problem of calculating the displacement of a viscoelastic thin hollow sphere after a sudden internal pressurization. According to Eq. (16.66a), the determination of the displacement u requires to obtain an expression for (1 — v)/E in the viscoelastic system. From the differential operators of a standard solid and the equations for the tensile modulus and the Poisson ratio developed, respectively, in Problems 16.2 and 16.4 at the end of this chapter, the following expression for (1 —v)/E is obtained ... 

Over the six-week period of permeation testing, sample containers from each type and temperature set were randomly selected and used for tensile testing. The tensile specimens were prepared and tested following the procedures of ASTM Test Method D1708-79 with the following modifications (i) the bottles were drained, and five specimens were cut parallel to the long axis of the cylinder (machine direction), (ii) the specimens were tested immediately after blotting to remove any surface solvent, (iii) a test speed of 5.08 cm/min. (2 in./min.) was used, and (iv) the tensile modulus of elasticity was calculated from the initial slope of the load-extension curve. 

Note that the choice of the tensile modulus is arbitrary, the shear modulus could have been calculated. In this case the factor of 3 would be absent. Remember that the variable N represents the number of chains per unit volume. Alternatively, equation (3-77) can be written ... 

The ratio of tensile stress to corresponding strain below the proportional limit. Many polymers/blends do not obey Hooke s law through out the elastic range but deviate therefrom even at stresses well below the yield stress. However, stress-strain curves almost always show a linear region at low stresses, and a straight line drawn tangent to this portion of the curve permits calculation of tensile modulus. 

The following values of the tensile modulus E and rigidity modulus G, expressed in gigapascals, were found for a random co-polyester fibre at two different temperatures E = 125, G = 1.1 and E = 62, G = 0.28. Assuming that equation (12.19) applies, calculate the value of Ej ax and the order parameter S for the fibre. 

The test data are shown in Table III. The mechanical tests were performed at room and liquid nitrogen temperatures, with no apparent differences for the results at the lower temperature. The tensile strength on the composite was within 10% of the predicted value and the tensile modulus was within 20%, as calculated by the rule of mixtures, i.e., 96 vs. 117 GPa. The density of the composite was measured at 5.34 g/cm. ... 

A composite material consists of 60% (by volume) continuous, uniaxialfy aligned, glass fibres in a matrix of epo. A tensile stress of ISO MPa is applied in a direction inclined at 30 to the fibres. Calculate the tensile strain which results parallel to the fibres. Take the tensile modulus and Poisson s ratio of glass to be 76 GPa and 0.22, and of epo to be 2.4 GPa and 0.34, respectively. 

Apart from the filler, the stiffening effect can also arise from the rubber network itself such as the crosslink concentration, as shown in Figure 3.9. The results shown here are based on compound formulations given in Table 3.4. The tensile modulus MlOO increased linearly with increasing crosslink concentration of the rubber network. The crosslink concentration was calculated by the Flory-Rehner equilibrium swelling equation ... 

Piezoresistivity [66] was observed in cement matrix composites with 2.6-7.4 vol% unidirectional continuous carbon fibers. The dc electrical resistance in the fiber direction increased upon tensile loading in the same direction, such that the effect was mostly reversible when the stress was below that required for the tensile modulus to decrease. The gage factor was up to 60. The resistance increase was due to the degradation of the interface of the fiber and matrix, which was mostly reversible. Above the stress at which the modulus started to decrease, the resistance abruptly increased with stress/strain, due to fiber breakage. The tensile strength and modulus of the composites were 88% and 84%, respectively, of the calculated values based on the rule of mixtures. 

The authors calculated the strain hardening intensity values as a measure of melt strength using the ratio of riE.noniinear/ HE,linear ( )- The slope of log versus Hencky strain defines the strain hardening intensity. As seen in Fig. 7.42 (a), effective improvements in melt strength required over 50 wt% PBSA which decreases the tensile modulus. 

An interpolation procedure applied by Halpin and Tsai [17,18] has led to general expressions for the moduli of composites, as given by Eqs. (2.18) and (2.19). Note that for = 0, Eq. (2.18) reduces to that for the lower hmit, Eq. (2.8), and for = infinity, it becomes equal to the upper limit for continuous composites, Eq. (2.7). By empirical curve fitting, the value of = 2(l/d) has been shown to predict the tensile modulus of aligned short-fiber composites in the direction of the fibers, and the value of = 0.5 can be used for the transverse modulus. Other mathematical relationships for modulus calculations of composites with discontinuous fillers include the Takaya-nagi and the Mori-Tanaka equations [20]. 

It is apparent that on the basis of the ultimate tensile strength any of the materials would perform at reasonable thicknesses. Applying the criterion of restricting the deformation to 5% we can calculate the apparent modulus at 5000 hours (the initial stress divided into the... 

Hydrostatic pressure is supplied in the hole, either with compressed air or with a fluid (oil, mercury, etc.). The pressure is the primary measure of the progress of the debonding process. The diameter and height of the blister will determine the work of adhesion. From these data together with the tensile modulus of the film and its thickness a critical pressure value can be calculated ... 

---