Elastic region

Once a rubberband is stretched beyond its elastic region, it becomes much harder to stretch and soon breaks. At this point, the polymer chains are linear and more energy must be applied to slide chains past one another and break bonds. Thus, determining the energy required to break the material requires a different type of simulation. 

The resistance to plastic flow can be schematically illustrated by dashpots with characteristic viscosities. The resistance to deformations within the elastic regions can be characterized by elastic springs and spring force constants. In real fibers, in contrast to ideal fibers, the mechanical behavior is best characterized by simultaneous elastic and plastic deformations. Materials that undergo simultaneous elastic and plastic effects are said to be viscoelastic. Several models describing viscoelasticity in terms of springs and dashpots in various series and parallel combinations have been proposed. The concepts of elasticity, plasticity, and viscoelasticity have been the subjects of several excellent reviews (21,22). 

It is important to differentiate between brittie and plastic deformations within materials. With brittie materials, the behavior is predominantiy elastic until the yield point is reached, at which breakage occurs. When fracture occurs as a result of a time-dependent strain, the material behaves in an inelastic manner. Most materials tend to be inelastic. Figure 1 shows a typical stress—strain diagram. The section A—B is the elastic region where the material obeys Hooke s law, and the slope of the line is Young s modulus. C is the yield point, where plastic deformation begins. The difference in strain between the yield point C and the ultimate yield point D gives a measure of the brittieness of the material, ie, the less difference in strain, the more brittie the material. 

Yield strength or tensile proof stress the maximum stress that can be applied without permanent deformation of the test specimen. For the materials that have an elastic limit (some materials may not have an elastic region) this may be expressed as the value of the stress on... 

Unloading. The strain lies on the elastic limit surface = 0, but the tangent to the strain history points inward into the elastic region < 0. It is assumed that k — 0. The material is said to be unloading and the elastic limit surface is stationary. 

Elastic. The stress lies within the elastic region and /< 0. Then k = 0 and the elastic limit surface is stationary. 

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 y-cut crystals showed little, if any, output signal under the same conditions for which the z-cut crystals were studied. In this case it should be observed that the y-cut crystals exhibit higher elastic limits and much higher piezoelectric polarizations than the z-cut crystals. These conditions result in much higher electric fields in the elastic region, and these fields are apparently sufficiently large that the crystals were completely conductive internally in the region between the elastic and plastic waves. 

The presence of hydrogen in pre-exposed specimens was revealed by straining specimens in vacuo. Hydrogen evolution occurred in the elastic region of the stress/strain curve, an effect that had been shown to be very much reduced by electropolishing pre-exposed specimens prior to testing... 

Beck, et al. have used the permeation technique to study the effect of uniaxial tensile stresses in the elastic region on hydrogen permeation through pure iron, and have shown that it increases with increase in stress. The partial molar volume of hydrogen (cubic centimetres of hydrogen per mole of iron) in ferrous alloys can be evaluated from the variation of permeation with applied stress, and from the relationship... 

When an engineering plastic is used with the structural foam process, the material produced exhibits behavior that is easily predictable over a large range of temperatures. Its stress-strain curve shows a significantly linearly elastic region like other Hookean materials, up to its proportional limit. However, since thermoplastics are viscoelastic in nature, their properties are dependent on time, temperature, and the strain rate. The ratio of stress and strain is linear at low strain levels of 1 to 2%, and standard elastic design... 

The mathematical dependence applies only in the linear visco-elastic region. In this region the sample can be deformed up to a maximum deformation (yo, ax) without destroying the structure of the sample. 

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. 

For steel, the modulus of elasticity is the same in the elastic region and yield plateau for static and dynamic response. In the strain hardening region the slope of the stress-strain curve is different for static and dynamic response, although this difference is not important for most structural design applications. 

Elastic Region - The deformation range from zero up to the formation of the first plastic hinge., ... 

Equation (5.5) is known as Hooke s Law and simply states that in the elastic region, the stress and strain are related through a proportionality constant, E. Note the similarity in form to Newton s Law of Viscosity [Eq. (4.3)], where the shear stress, r, is proportional to the strain rate, y. The primary differences are that we are now describing a solid, not a fluid, the response is to a tensile force, not a shear force, and we do not (yet) consider time dependency in our tensile stress or strain. 

We also know that to a first approximation, Hooke s Law applies to the elastic region, which for brittle materials is effectively up to the point of cleavage, so that the stress at cleavage, Oc, is given by Hooke s Law as... 

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