Stress/strain relationships

The use of eqn 3.29 will be demonstrated for a specimen deformed in tension—the simplest case. The forces for ai more complex deformation may be derived a parallel argument (Problems 3.14, IS, 16). 

Let the specimen deformed along the z axis from initial length Lj to a final length L, by an applied force F (see Hgure 3.6). The usual assumption of constant volume deformation will be made so that the volume V is 

341 A specimen of initial length i. is deformed by a tensile force F to length L The cross-sectional area changes from A to A. 

It then follows that the total work done (eqn 3.29) is 

The specimen of Example 3.1 is of initial length 0.10 m. What is the force required to double its length  

The dependence of the true stress on X may also be obtained. The true stress is the applied load F divided by the instantaneous area of the cross-section A 

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By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

Modified ETEE is less dense, tougher, and stiffer and exhibits a higher tensile strength and creep resistance than PTEE, PEA, or EEP resins. It is ductile, and displays in various compositions the characteristic of a nonlinear stress—strain relationship. Typical physical properties of Tef2el products are shown in Table 1 (24,25). Properties such as elongation and flex life depend on crystallinity, which is affected by the rate of crysta11i2ation values depend on fabrication conditions and melt cooling rates. 

Cord materials such as nylon, polyester, and steel wire conventionally used in tires are twisted and therefore exhibit a nonlinear stress—strain relationship. The cord is twisted to provide reduced bending stiffness and achieve high fatigue performance for cord—mbber composite stmcture. The detrimental effect of cord twist is reduced tensile strength. Analytical studies on the deformation of twisted cords and steel wire cables are available (22,56—59). The tensile modulus E of the twisted cord having diameter D and pitchp is expressed as follows (60) ... 

This stress—strain relationship is often written as (1)... 

The stress—strain relationship is used in conjunction with the rules for determining the stress and strain components with respect to some angle 9 relative to the fiber direction to obtain the stress—strain relationship for a lamina loaded under plane strain conditions where the fibers are at an angle 9 to the loading axis. When the material axes and loading axes are not coincident, then coupling between shear and extension occurs and... 

Proportional limit the point on the stress-strain curve at which will commence the deviation in the stress-strain relationship from a straight line to a parabolic curve (Figure 30.1). 

The stress-strain behavior of plastics in flexure generally follows from the behavior observed in tension and compression for either unreinforced or reinforced plastics. The flexural modulus of elasticity is nominally the average between the tension and compression moduli. The flexural yield point is generally that which is observed in tension, but this is not easily discerned, because the strain gradient in the flexural RP sample essentially eliminates any abrupt change in the flexural stress-strain relationship when the extreme fibers start to yield. 

Equation (13) is valid for r/Nlp < 0.25 (Fig. 3). At much higher extension ratios, the force must increase indefinitely since the molecule is almost straightened out. The thermodynamic approach to the problem of coil stretching for a freely-jointed chain was considered by Treloar [32], who obtained the following expression for the stress-strain relationship when the two chain ends are kept a distance r apart ... 

The decrease in the fiber diameter of fabric resulted in a decrease in porosity and pore size, but an increase in fiber density and mechanical strength. The microfiber fabric made of PCLA (1 1 mole ratio) was elastomeric with a low Young s modulus and an almost linear stress-strain relationship under the maximal stain (500%) in this measurement. 

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... 

The degree of vulcanisation of a rubber compound is assessed technically by the indefinite terms of undercure, correct cure, optimum cure and overcure. It may be given precision by (a) measurement of stress-strain relationship of a range of cures, (b) measurement of the modulus at 100% elongation, (c) measurement of the volume swelling in benzene, or (d) by the use of instruments such as the oscillating disc rheometer and the moving die rheometer. 

In general, during the initial stages of deformation, a material is deformed elastically. That is to say, any change in shape caused by the applied stress is completely reversible, and the specimen will return to its original shape upon release of the applied stress. During elastic deformation, the stress-strain relationship for a specimen is described by Hooke s law ... 

It is important to place the first near-wall grid node far enough away from the wall at yP to be in the fully turbulent inner region, where the log law-of-the-wall is valid. This usually means that we need y > 30-60 for the wall-adjacent cells, for the use of wall functions to be valid. If the first mesh point is unavoidably located in the viscous sublayer, then one simple approach (Fluent, 2003) is to extend the log-law region down to y — 11.225 and to apply the laminar stress-strain relationship U — y for y < 11.225. Results from near-wall meshes that are very fine using wall functions are not reliable. 

Response of a material under static or dynamic load is governed by the stress-strain relationship. A typical stress-strain diagram for concrete is shown in Figure 5.3. As the fibers of a material are deformed, stress in the material is changed in accordance with its stress-strain diagram. In the elastic region, stress increases linearly with increasing strain for most steels. This relation is quantified by the modulus of elasticity of the material. 

Concret does not have well defined elastic and plastic regions due to its brittle nature. A maximum compressive stress value is reached at relatively low strains and is maintained for small deformations until crushing occurs. The stress-strain relationship for concrete is a nonlinear curve. Thus, the elastic modulus varies continuously with strain. The secant modulus at service load is normally used to define a single value for the modulus of elasticity. This procedure is given in most concrete texts. Masonry lias a stress-strain diagram similar to concrete but is typically of lower compressive strength and modulus of elasticity. 

Stress-strain relationships for soil are difficult to model due to their complexity. In normal practice, response of soil consists of analyzing compression and shear stresses produced by the structure, applied as static loads. Change in soil strength with deformation is usually disregarded. Clay soils will exhibit some elastic response and are capable of absorbing blast-energy however, there may be insufficient test data to define this response quantitatively. Soil has a very low tensile capacity thus the stress-strain relationship is radically different in the tension region than in compression. 

Stress-strain relationships 5-9 Structural detail 8-1—8-7 Structural failure 7-2 Structural strengthening See also Blast resistant design... 

Firstly, it helps to provide a cross-check on whether the response of the material is linear or can be treated as such. Sometimes a material is so fragile that it is not possible to apply a low enough strain or stress to obtain a linear response. However, it is also possible to find non-linear responses with a stress/strain relationship that will allow satisfactory application of some of the basic features of linear viscoelasticity. Comparison between the transformed data and the experiment will indicate the validity of the application of linear models. 

In the [ 45]j tensile test (ASTM D 3518,1991) shown in Fig 3.22, a uniaxial tension is applied to a ( 45°) laminate symmetric about the mid-plane to measure the strains in the longitudinal and transverse directions, and Ey. This can be accomplished by instrumenting the specimen with longitudinal and transverse element strain gauges. Therefore, the shear stress-strain relationships can be calculated from the tabulated values of and Ey, corresponding to particular values of longitudinal load, (or stress relations derived from laminated plate theory (Petit, 1969 Rosen, 1972) ... 

A simple bubble machine is devised and successfully applied in characterising lightly crosslinked PE resins for foam expansion. The biaxial stress-strain relationship is deduced from the air injection rate and pressure. The effects of strain rate, temperatnre and crosslinker level on the stress-strain behavionr are investigated. Uniaxial extension experiments are also performed and compared with biaxial extension data. 5 refs. 

The main reasons for studying rheology (stress-strain relationships) in this chapter are (1) to determine the suitability of materials to serve specific applications and (2) to relate the results to polymer structure and form. Studying structure-property relationships allows a better understanding of the observed results on a molecular level resulting in a more knowledgeable approach to the design of materials. 

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