Tensile modulus calculation

Figure 1.6(a) The effect of fibre orientation on composite tensile modulus. Calculated for a 60 v/o, unidirectional laminate. The angle refers to that between the fibres and the load axis. 

The flexural modulus is represented by the slope of the initial straight-line portion of the stress-strain curve and is calculated by dividing the change in stress by the corresponding change in strain. The procedure to calculate flexural modulus is similar to the one described previously for tensile modulus calculations. 

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. 

Values of Mc calculated from the small-strain tensile modulus, i.e., using C- + C2 in equation 2 in place of Cj, were, of course, smaller than those obtained from Ci values. However, the general form of the dependence of threshold tear strength upon Mc and the relative values obtained for different polymers at the same Mc were not significantly affected by this alternate procedure for calculating the mean molecular weight of network strands. 

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 ... 

Most physical tests involve nondestructive evaluations. For our purposes, three types of mechanical stress measures (Figure 14.7) will be considered. The ratio of stress to strain is called Young s modulus. This ratio is also called the modulus of elasticity and tensile modulus. It is calculated by dividing the stress by the strain ... 

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 ... 

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 same laminate system was used by all laboratories, namely a laminar structure with five layers based on polypropylene (PP), an adhesive, an ethylene vinyl alcohol (EVOH) layer another adhesive and another polypropylene layer. This is designated PP/adh/EVOH/adh/PP and was used in both fixed arm peel tests and T-peel tests. Peel specimen were 15 mm in width and a notional 100 mm in length. The peel arms were PP/adh (with a thickness of 51pm) and EVOH/adh/PP (with a thickness of 75pm). Although there is no rigorous value for modulus for such multi-layered arms, we have obtained a tensile modulus value for each arm and assumed the materials of the arms to be uniform for the purposes of our calculations. 

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. 

From this point, the calculation of the complex tensile modulus, E, is straightforward. From Chapter 2, Section C, it will be remembered that E is the reciprocal of D, so we have... 

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 ... 

Calculations of tensile strength and tensile modulus of elasticity are described in the text above. 

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. 

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. 

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