Linear materials, stress-strain

The material cannot be described with linear elastic stress-strain relations... 

The designer can use several approaches to prevent hysteresis failure. The first is material selection. The stiffer the material is, the smaller the strain is for a given stress level and the lower the hysteresis loss per cycle. Some materials are additionally fairly linear in stress-strain characteristics and have smaller hysteresis loops. These would be preferred in dynamic loading applications. 

In all expressions the Einstein repeated index summation convention is used. Xi, x2 and x3 will be taken to be synonymous with x, y and z so that o-n = axx etc. The parameter B will be temperature-dependent through an activation energy expression and can be related to microstructural parameters such as grain size, diffusion coefficients, etc., on a case-by-case basis depending on the mechanism of creep involved.1 In addition, the index will depend on the mechanism which is active. In the linear case, n = 1 and B is equal to 1/3t/ where 17 is the linear shear viscosity of the material. Stresses, strains, and material parameters for the fibers will be denoted with a subscript or superscript/, and those for the matrix with a subscript or superscript m. 

Thermosets are polymeric materials which when heated form permanent network structures via the formation of intermolecular crosslinks. Whether the final product has a glass transition temperature, Tg, above or below room temperature, and therefore normally exists as an elastomer or a glass, it is, strictly speaking, a thermo-set. In practice, however, thermosets are identified as highly crosslinked polymers that are glassy and brittle at room temperature. These materials typically exhibit high moduli, near linear elastic stress-strain behavior, and poor resistance to fracture. 

If a material exhibits linear-elastic stress-strain behavior prior to rupture (an ideal behavior approximated by many thermosets), then a simple relationship exists between the material s fracture toughness and its fracture surface energy, J (or G),... 

When the strain amplitude Is relatively large as In the case of tire cord In a running tire, the viscoelastic behavior Is no longer linear. The stress-strain loop Is not elliptic but distorted (Figure 1). The material properties In the nonlinear regime can not be represented with the real and Imaginary moduli. In the present study, we characterize the viscoelastic properties In nonlinear regime by the effective dynamic modulus and mechanical loss.(J )... 

The stress field in the specimen being tested is related to the applied force. When a material is stressed, the deformations are controlled by what is known as the constitutive behavior of the material. For example, some materials respond to stress linear elastically, and others behave elasto-plastically. The linear elastic stress-strain relationship is called Hooke s Law. AE however, are more strongly dependent on the irreversible (nonelastic) deformations in a material. Therefore, this method is only capable of detecting the formation of new cracks and the progression of existing... 

For single crystals with transverse dimensions large enough to permit a plane wave condition to be attained, the results are unambiguous and virtually free from theoretical assumptions. Five independent elastic constants (stiffnesses or compliances) are required to describe the linear elastic stress-strain relations for hexagonal materials. Only three independent constants are required for cubic (y-Ce, Eu, Yb) materials. Since there are no single crystal elastic constant data for the cubic rare earth metals, this discussion will concentrate on the relationships for hexagonal symmetry. 

Equations 9 and 10 can be solved for arbitrary strain (stress) histories to obtain the material stress (strain) response. Solution for the step-deformations requires use of the unit Heaviside function and is discussed in detail by Tschoegl (10) and by Findley, Onaran and co-workers. (21) Wineman and Rajagopal (22) deal with the step-strains, using Riemann-Stieltjes integrals. Other histories are more directly solvable. Also, in the linear theory the limits on the integral can be written from 0 to rather than from -00 to t (9,10,21,22). 

Linearity Two types of linearity are normally assumed Material linearity (Hookean stress-strain behavior) or linear relation between stress and strain Geometric hnearity or small strains and deformation. 

The ratio of stress to strain in the initial linear portion of the stress—strain curve indicates the abiUty of a material to resist deformation and return to its original form. This modulus of elasticity, or Young s modulus, is related to many of the mechanical performance characteristics of textile products. The modulus of elasticity can be affected by drawing, ie, elongating the fiber environment, ie, wet or dry, temperature or other procedures. Values for commercial acetate and triacetate fibers are generally in the 2.2—4.0 N/tex (25—45 gf/den) range. 

This concept is explained by Figure 12 which shows the uniaxial stress— strain curve for a ductile material such as carbon steel. If the stress level is at the yield stress B or above, the problem is no longer a linear one. 

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic behaviour. This is the behaviour characterised by Hooke s Law (Chapter 3). All solids are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1%. The slope of the stress-strain line, which is the same in compression as in tension, is of... 

Fig. 8.1. Stress-strain behaviour for a linear elastic solid. The axes are calibrated for a material such as steel.

Figure 8.2 shows a non-linear elastic solid. Rubbers have a stress-strain curve like this, extending to very large strains (of order 5). The material is still elastic if unloaded, it follows the same path down as it did up, and all the energy stored, per unit volume, during loading is recovered on unloading - that is why catapults can be as lethal as they are. 

Quite often isochronous data is presented on log-log scales. One of the reasons for this is that on linear scales any slight, but possibly important, non-linearity between stress and strain may go unnoticed whereas the use of log-log scales will usually give a straight-line graph, the slope of which is an indication of the linearity of the material. If it is perfectly linear the slope will be 45°. If the material is non-linear the slope will be less than this. 

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle. Basically this principle proposes that for a linear viscoelastic material, the strain response to a complex loading history is simply the algebraic sum of the strains due to each step in load. Implied in this principle is the idea that the behaviour of a plastic is a function of its entire loading history. There are two situations to consider. 

Several experiments will now be described from which the foregoing basic stiffness and strength information can be obtained. For many, but not all, composite materials, the stress-strain behavior is linear from zero load to the ultimate or fracture load. Such linear behavior is typical for glass-epoxy composite materials and is quite reasonable for boron-epoxy and graphite-epoxy composite materials except for the shear behavior that is very nonlinear to fracture. 

Shear-stress-shear-strain curves typical of fiber-reinforced epoxy resins are quite nonlinear, but all other stress-strain curves are essentially linear. Hahn and Tsai [6-48] analyzed lamina behavior with this nonlinear deformation behavior. Hahn [6-49] extended the analysis to laminate behavior. Inelastic effects in micromechanics analyses were examined by Adams [6-50]. Jones and Morgan [6-51] developed an approach to treat nonlinearities in all stress-strain curves for a lamina of a metal-matrix or carbon-carbon composite material. Morgan and Jones extended the lamina analysis to laminate deformation analysis [6-52] and then to buckling of laminated plates [6-53]. 

When the magnitude of deformation is not too great, viscoelastic behavior of plastics is often observed to be linear, i.e., the elastic part of the response is Hookean and the viscous part is Newtonian. Hookean response relates to the modulus of elasticity where the ratio of normal stress to corresponding strain occurs below the proportional limit of the material where it follows Hooke s law. Newtonian response is where the stress-strain curve is a straight line. 

Consequently, changing the temperature or the strain rate of a TP may have a considerable effect on its observed stress-strain behavior. At lower temperatures or higher strain rates, the stress-strain curve of a TP may exhibit a steeper initial slope and a higher yield stress. In the extreme, the stress-strain curve may show the minor deviation from initial linearity and the lower failure strain characteristic of a brittle material. 

At higher temperatures or lower strain rates, the stress-strain curve of the same material may exhibit a more gradual initial slope and a lower yield stress, as well as the drastic deviation from initial linearity and the higher failure stain characteristic of a ductile material. 

The secant modulus measurement is used during the designing of a product in place of a modulus of elasticity for materials where the stress-strain diagram does not demonstrate a linear proportionality of stress to strain or E is difficult to locate. 

Brittleness Brittle materials exhibit tensile stress-strain behavior different from that illustrated in Fig. 2-13. Specimens of such materials fracture without appreciable material yielding. Thus, the tensile stress-strain curves of brittle materials often show relatively little deviation from the initial linearity, relatively low strain at failure, and no point of zero slope. Different materials may exhibit significantly different tensile stress-strain behavior when exposed to different factors such as the same temperature and strain rate or at different temperatures. Tensile stress-strain data obtained per ASTM for several plastics at room temperature are shown in Table 2-3. 

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... 

Flexural modulus is the force required to deform a material in the elastic bending region. It is essentially a way to characterize stiffness. Urethane elastomers and rigid foams are usually tested in flexural mode via three-point bending and tite flexural (or flex ) modulus is obtained from the initial, linear portion of the resultant stress-strain curve. 

Most engineering materials, particularly metals, follow Hooke s law by which it is meant that they exhibit a linear relationship between elastic stress and strain. This linear relationship can be expressed as o = E where E is known as the modulus of elasticity. The value of E, which is given by the slope of the stress-strain plot, is a characteristic of the material being considered and changes from material to material. 

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