Elastic stress definition

To evaluate the second term of Eq. (10-10), we need to obtain N. Using the definition of N in Eq. (10-12) along with Eq. (10-3), which is valid in the absence of Frank elastic stresses, we get... 

Yield stress may be regarded most simply as the minimum stress at which permanent strain is produced when the stress is subsequently removed. Although this deformation is satisfactory for metals, where there is a clear distinction between elastic recoverable definition and plastic irrecoverable deformation, in polymers the distinction is not so straightforward. In many cases, such as the tensile tests discussed above, yield coincides with the observation of a maximum load in the load-elongation curve. The yield stress then can be defined as the true stress at the maximum observed load (Figure 11.8(a)). Because this stress is achieved at a comparatively low elongation of the sample, it is often adequate to use the engineering definition of the yield stress as the maximum observed load divided by the initial cross-sectional area. 

If the adherends have different elastic properties, then a rigorous definition of the nominal phase angle should include a length scale to account for the oscillatory nature of the elastic stresses at the crack tip. 

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

Fracture Mechanics. Linear elastic fracture mechanics (qv) (LEFM) can be appHed only to the propagation and fracture stages of fatigue failure. LEFM is based on a definition of the stress close to a crack tip in terms of a stress intensification factor K, for which the simplest general relationship is... 

C = consumption rate (tonne year ) r - fractional growth rate (% year ) t = time. Chapter 3 Definition of Stress, Strain, Poisson s Ratio, Elastic Moduli... 

Resilin and elastin, unlike other structural proteins, fulfill both definitions of an elastic material. Colloquially speaking, resilin and elastin are stretchy or flexible. They also fulfill the strict definition of an elastic material, i.e., the ability to deform in proportion to the magnitude of an applied stress without a loss of energy, and the recovery of the material to its original state when that stress is removed. Resilin and elastin are alone in the category of structural proteins (e.g., collagen, silk, etc.) in that they have the correct blend of physical properties that allow the proteins to fulfill both definitions of elasticity. Both proteins have high extensibility and combine that property with remarkable resilience [208]. 

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

The term elastic limit is mainly a definition. It describes a stress which, if exceeded, will influence plastic deformation. Experimentally, the elastic limit is practically unattainable because it is a limit. Either it has not been reached or it is overreached. Ideally, the elastic limit and proportional limit are the same. 

Here t is the resulting shear stress, 6 is the phase shift often represented as tan(d), and (O is the frequency. The term 6 is often referred to as the loss angle. The in-phase elastic portion of the stress is To(cosd)sin(wt), and the out-of-phase viscous portion of the stress is To(sind)cos(complex modulus and viscosity, which can be used to extend the range of the data using the cone and plate rheometer [6] ... 

Actually, crosslinks control the molecular packing and indeed significantly affect the elastic modulus of the material. As the intermolecular energy of kink formation is also determined by elastic modulus, the yield stress will definitely vary with modulus and thus the cross -linking density. In other words, crosslinks may not seriously affect the activation segment configuration in the molecular chain but will indirectly control the yield stress. 

When the helix amount increases the medium changes from a viscous liquid (sol) to an elastic solid (gel). The kinetics of gelation depends strongly on the quenching temperature. The rheological measurements that we performed are particularly focused on the sol-gel transition and on the definition of the "gel point". The greatest difficulty encountered is due to the weakness of the bonds which can easily be destroyed by the mechanical stress. 

For a given deformation or flow, the resulting stress depends on the material. However, the stress tensor does take particular general forms for experimentally used deformations (see section 2). The definitions apply to elastic solids, and viscoelastic liquids and solids. 

A measure of the stiffness of a polymer is the modulus of elasticity (Young s modulus) E. It can be calculated fi om the stress-strain curve as the slope in the linear region of Hooke s law. It should be considered that due to the definition E = o/e for rubberlike materials which show a rather large extension e at quite... 

E. Winkler, F. Grashof, H. Hertz,8 etc., have studied the stresses which are set up when two elastic isotropic bodies are in contact over a portion of their surface, when the surfaces of contact are perfectly smooth, and when the press, exerted between the surfaces is normal to the plane of contact. H. Hertz showed that there is a definite point in such a surface representing the hardness defined as the strength of a body relative to the kind of deformation which corresponds to contact with a circular surface of press. and that the hardness of a body may be measured by the normal press, per unit area which must act at the centre of a circular surface of press, in order that in some point of the body the stress may first reach the limit consistent with perfect elasticity. If H be the hardness of a body in contact with another body of a greater hardness than H, then for a circular surface of pressure of diameter d press. p radius of curvature of the line p and the modulus of penetration E,... 

In performing such experiments on isotropic materials, one is accustomed to express the elastic stiffness parameters in the experimentally more readily accessible technical parameters E (Young s modulus) and v (Poisson ratio). The relative change in length, in the direction of the tensile stress a is, by definition, given by (Al/t)i — a/E, whereas v = (Af/ )x/( A / )u. For several magnetostrictive films and substrates, E and v values are listed in table 1. Some useful relations are ... 

If a crystal is exposed to stress in such a way that the strain is kept constant, the stress will decrease with time as shown in Figure 14-4. One concludes that stress relaxation has occurred. Conversely, strain does not remain constant under constant load. Time dependent (i.e., plastic) strain in stressed crystals is called creep. It was already mentioned that elastic strain due to the applied stress is usually less than 1%. Plastic strain definitely dominates beyond the elastic limit which, to a large extent, is due to dislocation formation and motion. Since the crystal lattice is conserved during this... 

PLASTICITY. A rheological property of solid or semisolid materials expressed as the degree to which they will flow or deform under applied stress and retain the shape so induced, either permanently of for a definite tune interval. It may be considered the reverse of elasticity. Application of heat and/or special additives is usually required for optimum results. 

An elastic solid has a definite shape. When an external force is applied, the elastic solid instantaneously changes its shape, but it will return instantaneously to its original shape after removal of the force. For ideal elastic solids, Hooke s Law implies that the shear stress (o force per area) is directly proportional to the shear strain (7 Figure H3.2.1A) ... 

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

The purpose of this chapter is to remind the reader of the basis of the theory of elasticity, to outline some of its principal results and to discuss to what extent the classical theory can be applied to polymeric systems. We shall begin by reviewing the definitions of stress and strain and the compliance and stiffness matrices for linear elastic bodies at small strains. We shall then state several important exact solutions of these equations under idealised loading conditions and briefly discuss the changes introduced if realistic loading conditions are considered. We shall then move on to a discussion of viscoelasticity and its application to real materials. 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

By definition, yield stress demarcates the point beyond which the material looses its ability to elastically return as shown in Fig. 1(a). Thus, beyond this point the material is plastically deformed. From the electronic bonding point of view, this demarcation correspond to the upper limit of the energy potential well where the energy curve begins to deviate from symmetric shape (see Fig. 3). It must be emphasized that it is not the mechanical strength required to break the covalent bond completely but rather the stress required to move beyond the symmetric portion of the potential curve. 

According with their definition the elastic modulus take into account the stress necessary to produce an unitary lengthening and the docility the strain produced due to the application of a unity stress [1-7]. 

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