Solid material linear strain

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... 

Figure 5.3 Schematic diagrams showing potential energy top) and applied stress (bottom) versus linear strain for crystalline solids with (a) strong bonds and (b) weak bonds. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission John Wiley Sons, Inc.

Large deformation tests, in which a solid sample is strained to well beyond its linear limit, and often to fracture, are designed to obtain a quantitative measure of a product s functionality in end use. Many large deformation tests are empirical or imitative, and do not yield fundamental rheological or fracture data. However, such tests can, with some materials, be set up and performed in such a way that fundamental information is obtained (McCarthy, 1987). 

Dimensional stability is one of the most important properties of solid materials, but few materials are perfect in this respect. Creep is the time-dependent relative deformation under a constant force (tension, shear or compression). Hence, creep is a function of time and stress. For small stresses the strain is linear, which means that the strain increases linearly with the applied stress. For higher stresses creep becomes non-linear. In Fig. 13.44 typical creep behaviour of a glassy amorphous polymer is shown for low stresses creep seems to be linear. As long as creep is linear, time-dependence and stress-dependence are separable this is not possible at higher stresses. The two possibilities are expressed as (Haward, 1973)... 

In Chapter 4, we discussed Hooke s law (Eq. (4.6)) for a small strain in one dimension. As long as the applied strain is very small, Eq. (4.6) is valid for a solid material. Here, we generalize it in three dimensions. It appears logical that the stress tensor is linearly proportional to the deformation tensor, that is. 

Adhesives, as all plastics, are viscoelastic materials combining characteristics of both solid materials like metals and viscose substrates like liquids. Typically, the adhesive shear stress vs. shear strain curve is non-linear. This behaviour is characteristic especially for thermoplastic adhesives and modified thermosetting adhesives. Thermosetting adhesives are, by their basic nature, more brittle than thermoplastic adhesives but, as discussed earlier, are often modified for more ductile material behaviour. 

When a solid material is subjected to a small stress (i.e., tension, compression, or torsion), the resulting strain is proportional to the applied stress and the proportional factor is called an elastic modulus, and this simple linear relationship between stress and strain is called Hooke s law. 

Rubber as an engineering material is unique in its physical behaviour. It exhibits physical properties that lie mid-way between a solid and liquid, giving the appearance of solidity, while possessing the ability to deform substantially. Most solid materials have an extensibility of only a few percent strain and only a portion of that is elastic, being typically Hookean in character, exhibiting a linear stress-strain relationship. Rubbers, however, may be extensible up to over 1000% strain, most of which is... 

Theory and Physics of Piezoelectricity. The discussion that follows constitutes a very brief introduction to the theoretical formulation of the physical properties of crystals. If a solid is piezoelectric (and therefore also anisotropic), acoustic displacement and strain will result in electrical polarization of the solid material along certain of its dimensions. The nature and extent of the changes are related to the relationships between the electric field (E) and electric polarization (P). which are treated as vectors, and such elastic factors as stress Tand strain (S), which are treated as tensors. In piezoelectric crystals an applied stress produces an electric polarization. Assuming Ihe dependence is linear, the direct piezoelectric effect can be described by the equation ... 

Fig. 4.4. Axial stress and strain in the notched cross section for two different materials Linear-elastic (dashed line) and elastic-plastic (solid line). The figure shows the nominal stresses and strains (Tnss, nss) and the maximum values in the linear-elastic (o-max,el, Smax.el) and elastic-plastic case (Umax, Smax)...

In the creep test, a constant stress is suddenly applied to the material at time 0, and the time-varying strain or deformation resulting from the stress is measured. Refer to the first column in Fig. 32. The diagram characteristics of the vtirious behaviors (Newtonian, Hookean solid, and linear viscoelastic) are shown. If the material exhibits linear properties or if the deformation is small enough to justify the linearity, the deformation or strain will be proportional to the stress ... 

When a stress o produces a linear strain e in a Hookean solid of volume F, the strain energy in the solid is aeF/2, which is equal to o Vl2E, where E is Young s modulus of the material. If y is the specific surface free energy of the material (see Chapter 7), the total surface energy of the surfaces created by the fracture process is lay. 

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. 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

The terms are arranged into sections dealing with basic definitions of stress and strain, deformations used experimentally, stresses observed experimentally, quantities relating stress and deformation, linear viscoelastic behaviour, and oscillatory deformations and stresses used experimentally for solids. The terms which have been selected are those met in the conventional mechanical characterization of polymeric materials. 

Ferroelectrics. Among the 32 crystal classes, 11 possess a centre of symmetry and are centrosymmetric and therefore do not possess polar properties. Of the 21 noncentrosymmetric classes, 20 of them exhibit electric polarity when subjected to a stress and are called piezoelectric one of the noncentrosymmetric classes (cubic 432) has other symmetry elements which combine to exclude piezoelectric character. Piezoelectric crystals obey a linear relationship P,- = gijFj between polarization P and force F, where is the piezoelectric coefficient. An inverse piezoelectric effect leads to mechanical deformation or strain under the influence of an electric field. Ten of the 20 piezoelectric classes possess a unique polar axis. In nonconducting crystals, a change in polarization can be observed by a change in temperature, and they are referred to as pyroelectric crystals. If the polarity of a pyroelectric crystal can be reversed by the application on an electric field, we call such a crystal a ferroelectric. A knowledge of the crystal class is therefore sufficient to establish the piezoelectric or the pyroelectric nature of a solid, but reversible polarization is a necessary condition for ferroelectricity. While all ferroelectric materials are also piezoelectric, the converse is not true for example, quartz is piezoelectric, but not ferroelectric. 

Viscoelastic materials are those which exhibit both viscous and elastic characterists. Viscoelasticity is also known as anelasticity, which is present in systems when undergoing deformation. Viscous materials, like honey, polymer melt etc, resist shear flow (shear flow is in a solid body, the gradient of a shear stress force through the body) and strain, i.e. the deformation of materials caused by stress, is linearly with time when a stress is applied [1-4]. Shear stress is a stress state where the stress is parallel or tangencial to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. The variable used to denote shear stress is r which is defined as ... 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

The strength properties of solids are most simply illustrated by the stress-strain diagram, which describes the behaviour of homogeneous brittle and ductile specimens of uniform cross section subjected to uniaxial tension (see Fig. 13.60). Within the linear region the strain is proportional to the stress and the deformation is reversible. If the material fails and ruptures at a certain tension and a certain small elongation it is called brittle. If permanent or plastic deformation sets in after elastic deformation at some critical stress, the material is called ductile. 

It has been experimentally observed that for small deformations, the strain in a body is linearly proportional to the applied stress. In one dimension this is known as Hooke s law, relating the elongation of a spring or elastic material to the tensile force. A principle such as this, which relates stress to strain, is known as a constitutive relation, and can be generalized to three-dimensional, non-piezoelectric solids [1] ... 

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