Normal Stresses and Strains

Rather than talk in terms of force, polymer scientists use the concept of stress. This concept allows us to talk about the applied force independent of the size of the object. A stress is the force applied to a system over some area. The area in question is the cross-sectional area over which the deformation occurs. This concept is illustrated in Fig. 6.1. Here we see a piece of plastic being deformed by a force and the cross sectional area of deformation, which is 

The strain is the magnitude of the deformation of a material in the direction of the applied stress as related to its length in that direction, as in Eq. 6.1 and Fig. 6.2. 

In Fig. 6.2, the direction of the stress, as well as the directionality of the deformation, is indicated. It is important to note that the directionality of the deformation is perpendicular to the area over which the stress is applied, making this an example of the effect of a normal stress on a material. 

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Viscoelasticity is the phenomenon of time-dependent strain. Often, it is also referied to as anelasticity. Glassy materials, above the glass transition temperature show Newtonian viscosity, i.e. the stress is proportional to the strain rate. This property is exploited in the drawing of fiber and sheet forms. We can write, in terms of normal stresses and strains. 

Strong and solid to meet normal stresses and strains... 

The numbers 1 to 3 are associated with normal stresses and strains, 4 to 6 with the shear components. It is useful to note that the numbers run down the diagonal of the stress and strain tensors and circle back up the third column and along the top row to the starting position. This new notation also removes the difference between simple and pure shear strains discussed earlier. With the new notation, Eq. (2.50) becomes... 

In the structure of the constitutive equation for the mechanically and electrostatically orthotropic piezoelectric material of Eq. (4.17), the partial coupling needs to be noted. The normal stresses and strains in all three directions are solely connected to flux density and field strength along the polarization direction ... 

A comprehensive approach to model the stress transfer requires simultaneous treatment of all the above-mentioned effects elastic shear transfer, frictional slip, debonding, and normal stresses and strains. Unfortunately, such a unified approach is complex. Therefore, in this chapter, each of these effects will first be discussed separately, based on models developed for fibres of a simple shape, usually straight fibres with a circular cross section. The stress transfer in uncracked and in cracked composites will also be dealt with separately. 

As an example of how the LSM can be applied to nanorod polymer composites, we take the morphological information shown in Figure 1 and use this as the input into this mechanical model. Figure 2 shows both the normal stress and strain fields for the randomly dispersed rods in Figure 1(a) and the... 

Let C i be a bounded domain with a smooth boundary L, and n = (ni,n2,n3) be a unit outward normal vector to L. Introduce the stress and strain tensors of linear elasticity (see Section 1.1.1),... 

At low strains there is an elastic region whereas at high strains there is a nonlinear relationship between stress and strain and there is a permanent element to the strain. In the absence of any specific information for a particular plastic, design strains should normally be limited to 1%. Lower values ( 0.5%) are recommended for the more brittle thermoplastics such as acrylic, polystyrene and values of 0.2-0.3% should be used for thermosets. 

The convention normally used is that direct stresses and strains have one suffix to indicate the direction of the stress or strain. Shear stresses and strains have two suffices. The first suffix indicates the direction of the normal to the plane on which the stress acts and the second suffix indicates the direction of the stress (or strain). Poisson s Ratio has two suffices. Thus, vi2 is the negative ratio of the strain in the 2-direction to the strain in the 1-direction for a stress applied in the 1-direction (V 2 = — il for an applied a ). v 2 is sometimes referred to as the major Poisson s Ratio and U2i is the minor Poisson s Ratio. In an isotropic material where V21 = i 2i. then the suffices are not needed and normally are not used. 

The overall distribution of stresses and strains in the local and global directions is shown in Fig. 3.23. If both the normal stress and the bending are applied together then it is necessary to add the effects of each separate condition. That is, direct superposition can be used to determine the overall stresses. 

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

Other anisotropic elasticity relations are used to define Chentsov coefficients that are to shearing stresses and shearing strains what Poisson s ratios are to normal stresses and normal strains. However, the Chentsov coefficients do not affect the in-plane behavior of laminaeS under plane stress because the coefficients are related to S45, S46, Equation (2.18). The Chentsov coefficients are defined as... 

A key element in the experimental determination of the stiffness and strength characteristics of a lamina is the imposition of a uniform stress state in the specimen. Such loading is relatively easy for isotropic materials. However, for composite materials, the orthotropy introduces coupling between normal stresses and shear strains and between shear stresses and normal and shear strains when loaded in non-principal material coordinates for which the stress-strain relations are given in Equation (2.88). Thus, special care must be taken to ensure obtaining... 

The receptacle and any associated packaging must be designed, constructed, maintained and closed so as to prevent the escape of any of the contents of the receptacle when subjected to the stresses and strains of normal handling. A suitable safety device (e.g. pressure relief valve) may be fitted. 

In the context of elastic deformation two parameters, known as stress and strain respectively, are very relevant. Stress is an internal distributed force which is the resultant of all the interatomic forces that come into play during deformation. In the case of the solid bar loaded axially in tension, let the cross sectional area normal to the axial direction be A0. From a macroscopic point of view the stress may be considered to be uniformly distributed on any plane normal to the axis and to be given by o A0 where o is known as the normal stress. The stress has to balance the applied load, F, and one must, therefore have o Aq = F or o = F/Aq. The units of stress are those of force per unit area, i.e., newtons per square... 

At high stresses and strains, non-linearity is observed. Strain hardening (an increasing modulus with increasing strain up to fracture) is normally observed with polymeric networks. Strain softening is observed with some metals and colloids until yield is observed. 

Fig. F.l. Normal stress and normal strain, (a) The normal force per unit area in the X direction is the x component of the stress tensor, (b) The normal stress causes an elongation in the x direction and a contraction in the y, and z directions.

The relation between shear stress and shear strain can be established based on the relation between normal stress and normal strain. Equations (F.3) and (F.4). Actually, by rotating the coordinate system 45°, it becomes a problem of normal stress and normal strain. Using geometrical arguments, it can be shown that (see, for example, Timishenko and Goodier, 1970) ... 

Detonation Wave, Plastic. These waves are complicated by the fact that there is no longer a linear relation between stress and strain. A plastic wave does not maintain its form as it progresses but rather the front of increasing stress tends to become longer and longer, at least in normal cases. The reflection at a discontinuity resembles generally the reflection of an elastic wave. Reflection of stress wave at a fixed end in an elastic member gives rise to stresses strains that ass exactly double those in the incident wave... 

Generalized Hooke s Law. The discussion in the previous section was a simplified one insofar as the relationship between stress and strain was considered in only one direction along the applied stress. In reality, a stress applied to a volnme will have not only the normal forces, or forces perpendicular to the surface to which the force is applied, but also shear stresses in the plane of the surface. Thus there are a total of nine components to the applied stress, one normal and two shear along each of three directions (see Eigure 5.4). Recall from the beginning of Chapter 4 that for shear stresses, the first subscript indicates the direction of the applied force (ontward normal to the surface), and the second subscript indicates the direction of the resnlting stress. Thus, % is the shear stress of x-directed force in the y direction. Since this notation for normal forces is somewhat redundant—that is, the x component of an... 

The temperature, fiber tension, stresses, and strains vary only in the radial directions. An elasticity solution is employed to calculate the six components of the stresses and strains. The solution procedure follows the established techniques of elasticity solutions. A displacement field is assumed that satisfies the equilibrium equations and the compatibility conditions. The latter requires that at each interface the displacements and the normal stresses in adjacent... 

The uniaxial failure envelope developed by Smith (95) is one of the most useful devices for the simple failure characterization of many viscoelastic materials. This envelope normally consists of a log-log plot of temperature-reduced failure stress vs. the strain at break. Figure 22 is a schematic of the Smith failure envelope. Such curves may be generated by plotting the rupture stress and strain values from tests conducted over a range of temperatures and strain rates. The rupture locus moves counterclockwise around the envelope as the temperature is lowered or the strain rate is increased. Constant strain, constant strain rate, and constant load tests on amorphous unfilled polymers (96) have shown the general path independence of the failure envelope. Studies by Smith (97) and Fishman (29) have shown a path dependence of the rupture envelope, however, for solid propellants. 

Consider first the trivial case of a static fluid. Here there can only be normal forces on a fluid element and they must be in equilibrium. If this were not the case, then the fluid would move and deform. Certainly any valid relationship between stress and strain rate must accommodate the behavior of a static fluid. Hence, for a static fluid the strain-rate tensor must be exactly zero e(/- = 0 and the stress tensor must reduce to... 

With dumb-bells, it is assumed that stress and strain are uniform throughout the gauge length and, hence, the calculation of stress presents no difficulties. Modulus as such is not normally measured but the stress quoted for a given elongation. It is sometimes debated whether the mean or the minimum cross sectional area should be used for ultimate stress but whatever the arguments in favour of the minimum, it is rather difficult to measure this and the mean is normally used. 

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