Linear materials, stress-strain relationships

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... 

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. 

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. 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

Figure 10.2. Stress-strain behavior. With elastic (reversible) deformation, stress and strain are linearly proportional in most materials (exceptions include polymers and concrete). With plastic (permanent) deformation, the stress-strain relationship is nonlinear.

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),... 

Modulus of Elasticity. The MOE quantifies a material s resistance to deformation under load. The MOE corresponds to the slope of the linear portion of the stress-strain relationship from zero to the proportional limit (Figure 6). Stiffness is often incorrectly thought to be synonymous with MOE. However, MOE is solely a material property and stiffness depends on the size of the beam. Large and small beams of similar material would have similar MOEs but different stiffnesses. The MOE can be calculated from the stress-strain curve as the change in stress causing a corresponding change in strain. 

Equations have been derived to define the vertical and shear stresses at any depth below and any radial distance from a point load. The best known and probably the most used are the Boussinesq equations, which assume an elastic, isentropic material, a level surface and an infinite surface extension in all directions. Although these conditions cannot be met by soils, the equation for vertical stress is used with reasonable accuracy with soils whose stress-strain relationship is linear. This normally precludes the use of the equation for stresses approaching failure. In its most useful form the equation reduces to ... 

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. 

The material properties of the ACL are mainly determined from the stress-strain relationship of tensile tests. Stress is applied force per unit cross-sectional area, while strain is deformation divided by initial length. The stress-strain relationships of collagenous tissues are usually nonlinear they increase slowly at low strain in the so-called toe region, have constant slope at middle strains in the linear region, and then decrease at high strain in the prefailure region (Fig. 6.1). It is known that... 

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... 

In most cases the material cannot be regarded as one-dimensional, and forces and deformations in all three principal directions must be taken into account. The normal and shear stresses are interrelated by the equations of mechanical equilibrium. The relationship between stresses and deformations can be described by the generalized Hooke s law in the case of linear elastic behavior. The stress-strain relationship of materials showing more complicated behavior can often be described by advanced theories based on the generalized Hooke s law. All models contain constants that must be determined experimentally, on materials equilibrated in moisture content and temperature. 

Fig. 2.13 Schematic view of the stress-strain relationship of a linear viscoelastic material at different times, corresponding to different deformation rates.

Besides linear viscoelastic behavior, elastoplastic behavior is also often encountered for food products. In Fig. 2.14, the stress-strain behavior of an elastoplastic material is shown schematically. Because the stress-strain relationship is not linear and the strain does not recover if the yield stress is exceeded, the equations to describe this behavior are much more complicated. 

Material behavior can be time dependent or independent and the time dependent behavior is often referred to as linear viscoelasticity. This can be explained based on some equations. Suppose ol is the constant load applied to a viscoelastic specimen, then el gives the time dependent strain. After a particular time when the load is removed the specimen tries to recover back and then e2 becomes the time dependence of the strain upon a larger stress o2. If tl and t2 represent the time after loading, el and e2 are linear strains corresponding to the stresses, ol and o2. Here the stress strain relationship can be given by Eq. (1). 

Usually, sealants and adhesive materials for construction applications are evaluated by looking at the engineering side, butnotthe chemistry of the material. As a result, only tests that measure the mechanical properties are used. Most of the studies on the viscoelastic properties use traditional tests such as tensile testing to obtain data, which can be used in complicated mathematical equations to obtain information on the viscoelastic properties of a material. For example, Tock and co-workers studied the viscoelastic properties of stmctural silicone rubber sealants. According to the author, the behavior of silicone mbber materials subjected to uniaxial stress fields carmotbe predicted by classical mechanical theory which is based on linear stress-strain relationship. Nor do theories based on ideal elastomers concepts work well when extensions exceed... 

Hart-Smith (1973) conducted material nonlinear analysis for SLJs using the linearly elastic and perfectly plastic model to describe the adhesive shear stress-strain relationship and the linear material property for the adhesive peel stress and adherends. In his analysis (Hart-Smith 1973), the adhesive layer is divided into elastic and plastic regions as shown in Fig. 24.11. Grant and Teig (1976) considered material nonlinearity of adhesive by dividing the adhesive into multiregions, in which the shear lag model was used. The obtained governing equations were then solved numerically. 

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