Calculation of elastic modulus

Bending beam theory calculation of elastic modulus, 361-362 calculation of glass temperature, 362 calculation of thermal expansion coefficient, 362 layer stress determination, 361 Benzophenone-3,3, 4,4 -tetracarboxydi-anhydride-oxydianiline-m-phenylenediamine (BTDA-ODA-MPDA) polyimide, properties, 115-116 Bilayer beam analysis schematic representation of apparatus, 346,348/ thermal stress, 346 Binary mixtures of polyamic acids curing, 116-124 exchange reactions, 115 Bis(benzocyclobutenes) heat evolved during polymerization vs. 

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

A and D indicate the two parameters most commonly extracted from a creep curve. A represents the instantaneous elastic compliance and can be used to calculate an elastic modulus. D represents the limiting viscosity, which is related to the reciprocal of the slope. In some cases, parameters from creep testing have been related to molecular mechanisms (Shama and Sherman, 1970 Davis, 1973 deMan et al., 1985). The parameters have also been correlated with hardness and spreadability (Scott-Blair, 1938). 

Fig.5 shows the calculated curvature and temperature evolution for an FGM deposit with thickness of about 180 im, which is consistent with the experimental results shown in Fig.4 except for the transient oscillations. Fig.6 (a) shows the calculated stress distributions in 2-layer and FGM deposits. The gradual stress variation in the FGM can be observed. In Fig.6 (b) effects of model parameters such as the substrate temperature and elastic modulus of YPSZ on the stress distribution in 2-layer deposits are demonstrated. As the substrate temperature is raised from 600 to 825K, the tensile stress in the NiCrAlY layer is significantly reduced. If a value of elastic modulus of 190GPa of a dense bulk material was used, the compressive residual stress in the YPSZ is excessively overestimated. This example clearly demonstrates the importance of using realistic values for modeling thermal and mechanical behavior of sprayed deposits. 

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 problem of calculating the moduli of semicrystalline polymers is even more difficult. It involves in principle four steps (i) the calculation of the modulus of the amorphous material, (ii) the calculation of the elastic constants of the anisotropic crystalline material, (iii) the averaging of the elastic constants of the crystalline material to give an effective isotropic modulus and (iv) the averaging of the isotropic amorphous and crystalline moduli to give the overall modulus. The second of these steps can now be done fairly accurately, but the other three present serious difficulties. 

High fibre volume is essential for good aircraft stmcture performance. It is also important that distribution of both fibre and resin is uniform throughout the component. To illustrate, the simple mle of mixtures (ROM) approach for calculation of longitudinal modulus falls into the mechanics of materials category the modulus of elasticity in the longitudinal direction ( l), which is the direction parallel fibres, is given as ... 

Table 2 and Table 3 respectively. The equation (Eqn. 1) provided by Barton (1991) was used for calculation of dynamic modulus of elasticity. 

The technique of calculation of elastic constants consists of the following. A value of bulk modulus is found as has been described in Section 10.3. 

The three resins above were tested by thermomechanical analysis (TMA) on a Met-tler 40 apparatus. Triplicate samples of beech wood alone, and of two beech wood plys each 0.6 mm thick bonded with each resin system were tested. Sample dimensions were 21 mm x 6 mm x 1.2 mm. The samples were tested in non-isothermal mode from 40°C to 220°C at heating rates of 10°C/min, 20°C/min and 40°C/min with a Mettler 40 TMA apparatus in three-point bending on a span of 18 mm. A continuous force cycling between 0.1 N and 0.5 N and back to 0.1 N was applied on the specimens with each force cycle duration being 12 s. The classical mechanics relation between force and deflection E = [L /(4bh )][AF/(Af)] (where L is the sample length, AF the force variation applied and A/ the resulting deflection, b the width and h the thickness of the sample) allows calculation of the modulus of elasticity E for each case tested and to follow its rise as functions of both temperature and time. The deflections A/ obtained and the values of E obtained from them proved to be constant and reproducible. 

Nanoindentation is a technique gaining increasing popularity [74-76]. Actually, the technique is sometimes abused by attempts to calculate the elastic modulus E on the basis of a model valid for fully elastic materials only [74]. While such attempts fail, a connection has been found by Fujisawa and Swain between E and the unloading strain rate [75]. As shown by Tweedie and Van Vliet [76], spherical indentation provides lower contact strains and more reliable results than conical indentation. A modification providing repetitive indenter hits perpendicular to the specimen surface at the same spot and thus nanoindentation fatigue testing (NIFT) exists also [77]. 

The comparison of experimental i and calculated according to the equation (12) elasticity modulus values for the studied HDPE has been adduced in Fig. 9.1. As one can see, the good correspondence between theoiy and experiment is obtained (the average discrepancy between E and E7 does not exceed 6%, that is, comparable with an error of elasticity modulus experimental determination). 

FIGURE 14.1 The dependences of elasticity modulus E on testing temperature T for HDPE, obtained in impact (1,3) and quasistatic (2,4) tests. 1,2 - the experimental data 3, 4 - calculation according to the equation (12). 

FIGURE 3.3 The experimental (1 4) and calculated according to the Eq. (3.10) (5 8) dependences of elasticity modulus E on sharp notch length a for HOPE at testing temperatures... 

The only system for which detailed investigations of the longitudinal Young modulus are available is polyethylene (PE). The most detailed study of this problem was performed by Suhai. His first step in calculating the elastic modulus was to optimize the geometrical structure of PE at both the HF and correlated levels. In stretching the polymer he... 

The interlayer model was developed by Maurer et al. The model of the particulate-filled system is taken in which a representative volume element is assumed which contains a single particle with the interlayer surrounded by a shell of matrix material, which is itself surrounded by material with composite properties (almost the same as Kemer s model). The radii of the shell are chosen in accordance with the volume fraction of the fQler, interlayer, and matrix. Depending on the external field applied to the representative volmne element, the physical properties can be calculated on the basis of different boundary conditions. The equations for displacements and stresses in the system are derived for filler, interlayer, matrix, and composite, assuming the specific elastic constants for every phase. This theory enables one to calculate the elastic modulus of composite, depending on the properties of the matrix, interlayer, and filler. In... 

In Figure 10.2 the dependenee El (WJ, calculated according to the Eq. (10.21) in supposition =eons1 21.3 GPa, is also adduced. As one can see, in this case the theoretieal values of elasticity modulus El exceed essentially experimentally received ones E. Henee, the good correspondence of experiment and calculation according to the Eq. (10.21) is due to real values usage only. 

---