Stress, strain, and modulus

Note that these stress, strain and modulus equations are given for illustration purposes. They apply to three-point bending as shown in Fig. 2.3. Other types of bending can occur (e.g. four-point bending, cantilever, etc.) and different equations will apply. Some of these are illustrated in the Worked Examples later in this chapter and the reader is referred to Benham et al. for a greater variety of bending equations. 

From the relationship between stress, strain and modulus... 

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

Figure 7.2. Definitions of stress, strain, and modulus. Stress is defined as force per unit area, and strain is the change in length divided by the original length. When stress is plotted versus strain, then the slope is the modulus (A). When the load is removed, any strain remaining is called permanent or plastic deformation (B). When elastic materials are loaded, they are characterized by a constant strain as a function of time, whereas viscoelastic materials have strains that increase with time (C).

FIGURE 2.41 Stress-strain and modulus-strain curves for steam-treated drawn nylon-6 filaments. (Solid curve.) Untreated filaments (a) constant filament length (b) free shrinkage. (From Schultz-Gebhart, F., Faserf. Textiltechn., 1911, 28, 467. With permission.)... 

Figure 4.17 Stress, strain and modulus at different strain rates.

The main purpose of the tensile tests was to determine the stress-strain and modulus curves in the warp and weft directions for the material in its molten state, for later use in the construction of a numerical model in PAMFORM . Experiments indicate that the elastic behavior of the dry fabric is very similar to that of the molten composite material at reasonably low forming rates (< 100 trun/min), which implies that viscous effects have a minimal impact on the stress-strain behavior of the molten composite under these conditions. 

When we consider the mechanical properties of polymeric materials, and in particular when we design methods of testing them, the parameters most generally considered are stress, strain, and Young s modulus. Stress is defined as the force applied per unit cross sectional area, and has the basic dimensions of N m in SI units. These units are alternatively combined into the derived unit of Pascals (abbreviated Pa). In practice they are extremely small, so that real materials need to be tested with a very large number of Pa... 

Compression Test. Compression tests similar to that described in (5) were conducted for yield stress C and modulus E measurement. Rectangular neat resin specimens (1.27 cm x 1.27 cm x 2.54 cm) cut from the cast resin plates were tested under compression, as shown in Figure 1, in an universal testing machine at a loading rate of 0.05 cm/min. For each resin system studied, tests were conducted at several temperature levels between -60 and 60 degree C. All specimens were instrumented with strain gages for... 

In the category of deformation properties the phenomena of stress-strain behaviour, modulus and yield, stress relaxation and creep have been discussed already in Chap. 13. Here we want to give special attention to the long-term deformation properties. For a good design we need sufficiently reliable creep data (or stress-strain curves as a function of time and temperature). 

Dynamic or oscillatory rheometers measure viscous and elastic modulus in shear or tension. Energy dissipation produces a phase difference, so stress, strain, and phase angle can be used to characterize complex viscosity behavior. 

The two main elastic properties are modulus of elasticity, which describes the relationship of load (stress) to deformation (strain), and modulus of rigidity or shear modulus, which describes the internal distribution of shearing stress to shear strain or, more precisely, angular displacement within a material. 

When we consider the mechanical properties of polymeric materials, and in particular when we design methods of testing them, the parameters most generally considered are stress, strain, and Young s modulus. Stress is defined as the force applied per unit... 

Stress-Strain Relationship, Modulus of Elasticity and Ductility... 

The stress-strain behavior of deep-sea clay is not fimdamentally different than terrestrial counterparts but some important differences in parameters, such as elastic and shear modulus, may occur due to the high void ratios and fineness of materials. The effects of the removal of water pressure (over 60 MPa in ocean basins) on stress-strain and strength properties of deep-sea sediments is still a matter of discussion and there is very little reliable data for comparison of laboratory and in-situ properhes. The present approach is to generally ignore these possible effects but to make correchons to results along the lines developed for terrestrial soils. 

We now turn our attention from considerations of stiffness to stress-strain and tensile behavior. Variations in strength and modulus as a function of direction (Broutman and Krock, 1967, Chapter 12) have been treated by several investigators for example, Tsai (1965) and Brody and Ward (1971). Even though the polymer matrix typically has such a low modulus that it does not contribute much overall to the composite modulus, the matrix can by no means be neglected, because failure often involves catastrophic crack growth in the matrix (see below). Stress-strain curves for unidirectional composites are typically fairly linear up to failure for loading in the direction of the fibers (Broutman and Krock, 1967, p. 370), but quite nonlinear transverse to the fiber direction. The stress to rupture is also very low in the latter case, presumably due to a high concentration of stress in the matrix. 

Fig. 3. Complex-plane diagram showing vector relationships among stress, strain, = complex modulus, = Mreal = storage modulus, M2 = M maginary modulus, and (j) = loss angle = internal friction. Adapted from Nowick and Berry.

The E-modulus in the isotropic phase can be determined from both stress-strain and thermoelastic measurements and Me can be calculated according to Me = 3 when the density p of the elastomer is known. The degree of... 

The flexural properties in terms of flexural stress, strain and chord modulus (between 0.25% and 0.75% of the yield point) of the flax yam reinforced composites with SPI, PH2-SPI and PH4-SPI composites are presented in Fig. 9.8. As in the case of tensile properties, flax yarn reinforced PH2-SPI composites showed significantly higher chord modulus and flexural stress, 7.8 GPa and 105 MPa, respectively, than the SPI (2.8 GPa and 48.9 MPa)... 

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