Stress-strain functions

Eqn(7) is the Gaussian stress-strain function, with a the force per unit area of the undeformed network, G the shear modulus and A the deformation ratio. In eqn(8), A is the so-called front factor, p the density of the... 

Linear fiscoelasticity stress/strain = function of time t... 

Our goal was to measure the viscoelastic properties of the human brain under practical conditions. Therefore, we used the tactile resonant sensor with the stress-strain function that simulated manual palpation. In this study, the stiffness was 2.837 0.709 (N), Young s elastic modulus was E = 5.08 1.31, and the shear modulus was G = 1.94 0.49 for a depth of 3.0 mm. Poisson s ratio (u) was calculated as 0.31-0.62 using the equation E = 2G (1h- u). These values were approximately equal to those previously reported for the viscoelasticity properties of the brain in vivo [1-7]. The results of indentation fitted the Maxwell model as expressed by the equation G = Ge - Gi exp (-t/x), where Ge is the instantaneous modulus in shear, Gi is the relaxation in the shear modulus, t is time, and x is the relaxation time. Thus, G = 1.94-1- 3.3 exp(-t/0.5) under the assumption that Ge = 1.94, Gi = 3.3, t = h/1.5, and x = 0.5. The results obtained in this study by an indentation method, reflected those of a previous model [9-12]. However, this measurement method evaluated brain viscoelasticity via multiple structural layers including the skin, subcutaneous tissues, muscle fascia, and dura. Moreover, some assumptions had to be made to approximate the expression for elasticity. 

Clearly equation (3.4) contains the Mullins type hysteresis discussed earlier. Data from tests like the one illustrated above to increasing values of e can be used to determine the distribution function N(x) independent of the local stress-strain function f(x ). Differentiating equation (3.8a) and 3.8b) with respect to strain, substracting one from the other, and evaluating each at = e, produces... 

Farris, R. J., "The Character of the Stress-Strain Function for Solid Propellants," Trans. Soc. Rheol., 12, 281-301 (1968). 

Bueche attempted to calculate the distribution of local strain (2) he used the inverse Langevin function for the local stress-strain behavior (3) he assumed a critical strain failure criterion and (4) his representation was one-dimensional. In my model, I proposed a simple representation wherein the local stress-strain function and the distribution of local strains were arbitrary and for the case of a critical strain failure criteria, I have described how these two functions can be assessed from experimental tests. This portion of my work can be looked upon as an extension of Bueche s research. The main difference between my work and Bueche s is that I also handle the case when the local... 

The electromagnetic spectrum measures the absorption of radiation energy as a function of the frequency of the radiation. The loss spectrum measures the absorption of mechanical energy as a function of the frequency of the stress-strain oscillation. 

Another aspect of plasticity is the time dependent progressive deformation under constant load, known as creep. This process occurs when a fiber is loaded above the yield value and continues over several logarithmic decades of time. The extension under fixed load, or creep, is analogous to the relaxation of stress under fixed extension. Stress relaxation is the process whereby the stress that is generated as a result of a deformation is dissipated as a function of time. Both of these time dependent processes are reflections of plastic flow resulting from various molecular motions in the fiber. As a direct consequence of creep and stress relaxation, the shape of a stress—strain curve is in many cases strongly dependent on the rate of deformation, as is illustrated in Figure 6. 

The mechanical piopeities of stmctuial foams and thek variation with polymer composition and density has been reviewed (103). The variation of stmctural foam mechanical properties with density as a function of polymer properties is extracted from stress—strain curves and, owkig to possible anisotropy of the foam, must be considered apparent data. These relations can provide valuable guidance toward arriving at an optimum stmctural foam, however. 

Stress—Strain Curve. Other than the necessity for adequate tensile strength to allow processibiUty and adequate finished fabric strength, the performance characteristics of many textile items are governed by properties of fibers measured at relatively low strains (up to 5% extension) and by the change ia these properties as a function of varyiag environmental conditions (48). Thus, the whole stress—strain behavior of fibers from 2ero to ultimate extension should be studied, and various parameters should be selected to identify characteristics that can be related to performance. 

Figure 6.3. Stress-strain response of shock-loaded 6061-T6 A1 as a function of peak shock pressure showing minimal shock strengthening.

Figure 6.14 shows the reload compressive stress-strain response of shock-loaded copper as a function of pulse duration [40]. For copper shock loaded to 10 GPa the yield strength is observed to increase with increasing pulse... 

Figure 6.14. Stress-strain response of copper shock loaded to 10 GPa as a function of duration.

Perhaps the most significant complication in the interpretation of nanoscale adhesion and mechanical properties measurements is the fact that the contact sizes are below the optical limit ( 1 t,im). Macroscopic adhesion studies and mechanical property measurements often rely on optical observations of the contact, and many of the contact mechanics models are formulated around direct measurement of the contact area or radius as a function of experimentally controlled parameters, such as load or displacement. In studies of colloids, scanning electron microscopy (SEM) has been used to view particle/surface contact sizes from the side to measure contact radius [3]. However, such a configuration is not easily employed in AFM and nanoindentation studies, and undesirable surface interactions from charging or contamination may arise. For adhesion studies (e.g. Johnson-Kendall-Roberts (JKR) [4] and probe-tack tests [5,6]), the probe/sample contact area is monitored as a function of load or displacement. This allows evaluation of load/area or even stress/strain response [7] as well as comparison to and development of contact mechanics theories. Area measurements are also important in traditional indentation experiments, where hardness is determined by measuring the residual contact area of the deformation optically [8J. For micro- and nanoscale studies, the dimensions of both the contact and residual deformation (if any) are below the optical limit. 

Substitute the deflection function, Equation (3.95), the shear strain expression, Equation (3.130), and the stress-strain relation. Equation (3.131), in Equation (3.132) to get... 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

MFI of the composition to that of the matrix, as a function of the filler concentration. It can be seen that, as the concentration of a particular filler increases, the index increases too for one matrix but decreases for another, and varies by a curve with an extremum for a third one. Even for one and the same polymerfiller system and a fixed concentration of filler, the stress-strain characteristics, such as ultimate stress, may, depending on the testing conditions (temperature, rate of deformation, etc.) be either higher or lower than in the reference polymer sample [36],... 

As an example, for room-temperature applications most metals can be considered to be truly elastic. When stresses beyond the yield point are permitted in the design, permanent deformation is considered to be a function only of applied load and can be determined directly from the stress-strain diagram. The behavior of most plastics is much more dependent on the time of application of the load, the past history of loading, the current and past temperature cycles, and the environmental conditions. Ignorance of these conditions has resulted in the appearance on the market of plastic products that were improperly designed. Fortunately, product performance has been greatly improved as the amount of technical information on the mechanical properties of plastics has increased in the past half century. More importantly, designers have become more familiar with the behavior of plastics rather than... 

Stress relaxation. In a stress-relaxation test a plastic is deformed by a fixed amount and the stress required to maintain this deformation is measured over a period of time (Fig. 2-33) where (a) recovery after creep, (b) strain increment caused by a stress step function, and (c) strain with stress applied (1) continuously and (2) intermittently. The maximum stress occurs as soon as the deformation takes place and decreases gradually with time from this value. From a practical standpoint, creep measurements are generally considered more important than stress-relaxation tests and are also easier to conduct. 

The process of design for static loads involves a great deal more than the mechanical operation of the stress-strain data to determine the performance of a section. The results obtained from the stress analysis are used to determine the functionality of the product and then, combined with the other factors involved to decide on a suitable design. 

As previously described (Chapter 2), the area under short-term stress-strain curves provides a guide to a material s toughness and impact performance (Fig. 7-6). The ability of a TP to absorb energy is a function of its strength and its ductility that tends to be inversely related. The total absorbable energy is proportional to the area within the lines drawn to the appropriate point on the curve from the axis. The material in area A is... 

Fig. 2. Stress-strain behavior of MDI-based siloxane-urea segmented copolymers as a function of siloxane block molecular weights (PSX-Mn g/mole), [1. PSX-1140, 2. PSX-1770, 3. PSX-2420, 4. PSX-3670] and their comparison with unfilled (curve 7) and silica filled (curves 5 and 6) conventional silicone rubbers51 158,358,359)...

The majority of TPEs function as mbber at temperature as low as —40°C or even lower as measured by their brittle point. The upper temperature limit is determined by the maximum temperature at which it can give satisfactory retention of tensile stress-strain and hardness properties. Table 5.14a... 

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

Because of the previously mentioned inadequacy of the function a —l/a, a different value for the parameter %i is required for the set of points (Fig. 135) at each elongation a. These values are —0.90, — 0.73, and —0.56 for a = 1.4, 2.0, and 3.0, respectively. If the function a — l/a were replaced by an empirical representation of the shape of the stress-strain curve, a single value of xi would suffice to represent all of the data within experimental error. This limitation of Eq. (41) relates to an unexplained feature of the stress-strain curve and is... 

Figure 5.18 This figure shows how the properties of a glass polyalkenoate cement change as it ages. S is the compressive strength, E the modulus, a a stress-relaxation function, and c a strain-conversion function from elastic to plastic strain (Paddon Wilson, 1976).

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