Material deformation elastic moduli

Elasticity. Glasses, like other britde materials, deform elastically until they break in direct proportion to the appHed stress. The Young s modulus E is the constant of proportionaUty between the appHed stress and the resulting strain. It is about 70 GPa (10 psi) [(0.07 MPa stress per )Tm/m strain = (0.07 MPa-m) / Tm)] for a typical glass. 

The velocity is therefore determined by two fundamental physical properties of a material its elastic modulus and density. The less dense a material or the more resistant it is to deformation the faster an ultrasonic wave propagates. Usually, differences in the moduli of materials are greater than those in density and so the ultrasonic velocity is determined more by the elastic moduli than by the density. This explains why the ultrasonic velocity of solids is greater than that of fluids, even though fluids are less dense [1],... 

Rigidity r9- ji-d9-te (1624) n. The ability of a structure to resist deformation under load. It is a function of both the material s modulus of elasticity and, often more critically, of the geometry of the structure. In a loaded beam, whatever the load distribution or type of beam supports, the maximum deflection is inversely proportional to the product, El, of the material s elastic modulus and the moment of inertia of the beam s cross-section about its neutral axis. The term rigidity is often applied loosely to materials themselves without reference to a particular structure when what the speaker actually has in mind is the elastic modulus. See also section modulus. 

Figure 30 shows the typical stress-strain curves for pure PI film (without particles containing healing resin). The diagram is typical for plastic materials. Initially, the material deformed elastically, and yielding at strain more than 5 % was observed. The Yoimg modulus of fracture deformation of samples was very high, around 130 %. 

In any given material, the relaxation modulus will reflect the response of the material on different timescales. To make a measurement, materials are deformed under a periodic load with frequency w. Then, G and G are measured across a wide range of frequencies (typically three to four decades). Measurements of G and G" can be used to characterize the mechanical properties of soft materials, including polymer networks and colloidal systems. The technique is also known as mechanical spectroscopy. In a viscoelastic material, the elastic modulus will cross over the viscous modulus at the transition point from viscous to elastic bulk behavior and indicates a possible sol-gel transition or the onset of rubbery behavior in a polymer network. 

Hardness is a measure of a material s resistance to deformation. In this article hardness is taken to be the measure of a material s resistance to indentation by a tool or indenter harder than itself This seems a relatively simple concept until mathematical analysis is attempted the elastic, plastic, and elastic recovery properties of a material are involved, making the relationship quite complex. Further complications are introduced by variations in elastic modulus and frictional coefficients. 

Dynamic mechanical tests measure the response or deformation of a material to periodic or varying forces. Generally an applied force and its resulting deformation both vary sinusoidally with time. From such tests it is possible to obtain simultaneously an elastic modulus and mechanical damping, the latter of which gives the amount of energy dissipated as heat during the deformation of the material. 

Currently, there is a trend of low dielectric constant (low-k) interlevel dielectrics materials to replace Si02 for better mechanical character, thermal stability, and thermal conductivity [37,63,64]. The lower the k value is, the softer the material is, and therefore, there will be a big difference between the elastic modulus of metal and that of the low-k material. The dehiscence between the surfaces of copper and low-k material, the deformation and the rupture of copper wire will take place during CMP as shown in Fig. 28 [65]. 

Even in cases where the rigid polymer forms the continuous phase, the elastic modulus is less than that of the pure matrix material. Thus two-phase systems have a greater creep compliance than does the pure rigid phase. Many of these materials craze badly near their yield points. When crazing occurs, the creep rate becomes much greater, and stress relaxes rapidly if the deformation is held constant. 

The term hardness is a relative term. Hardness is the resistance to local deformation that is often measured as the ease or difficulty for a material to be scratched, indented, marred, cut, drilled, or abraded. It involves a number of interrelated properties such as yield strength and elastic modulus. Because polymers present such a range of behavior, as they are viscoelastic... 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

Hence, the elastic modulus corresponds in principle to the force per square millimeter that is necessary to extend a rod by its own length. Materials with low elastic modulus experience a large extension at quite low stress (e.g., rubber, = 1 N/mm ). On the other hand, materials with high elastic modulus (e.g., polyoxymethylene, s 3500 N/mm ) are only slightly deformed under stress. Different kinds of elastic modulus are distinguished according to the nature of the stress applied. For tension, compression, and bending, one speaks of the intrinsic elastic modulus ( modulus). For shear stress (torsion), a torsion modulus (G modulus) can be similarly defined, whose relationship to the modulus is described in the literature. 

The tensile test is typically destructive that is, the sample is extended until it plasticly deforms or breaks, though this need not be the case if only elastic modulus determinations are desired. As described in the previous section, ductile materials past their yield point undergo plastic deformation and, in doing so, exhibit a reduction in the cross-sectional area in a phenomenon known as necking. 

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