Elastic response Youngs modulus

The simplest mechanical properties are those of homogeneous isotropic and purely elastic materials their mechanical response can be defined by only two constants, e.g. the Young modulus E and the Poisson ratio v. For anisotropic, oriented-amorphous, crystalline and oriented-crystalline materials more constants are required to describe the mechanical behaviour. 

The results indicate that the Young s modulus (E) determined perpendicularly to the surface by instrumented nano-indentation on solid steel and Ta specimens were in good agreement (within 10%) with the values obtained by dynamic bending and by tensile testing. Such a result was corroborated hy measurements obtained from other materials presenting different elastic responses. 

For isotropic elastic solids there are only two independent elastic constants, or compliances. While Young s modulus E and the shear modulus // are the most widely used, we shall choose as the two physically independent pair of moduli the shear modulus /i and the bulk modulus K, where the first gauges the shear response and the second the bulk or volumetric response. However, in stating the linear elastic response in the equations below we still choose the more compact pair of E and //. Thus, for the six strain elements we have... 

In practice, it is essential to match the type of measurement with the actual performance of the material (tension, flexure or compression), in spite of the popularity of the tensile test. Sometimes, it is desirable to follow the response to shear stresses. The relationship between the modulus of elasticity E (Young s) and the shear modulus G, is given by ... 

The Xm of an E-M material is directly related to the displacement generated in an actuator. For most piezoelectric ceramics and polymers, Xm is about 0.1-0.2 %, while the newly developed E-M polymers exhibit a strain response of more than 5%, in some cases achieving as much as 100 %. This makes it possible to create actuators that exhibit a giant displacement. measures the maximum force needed to maintain the zero displacement when the material is under electric field. For an E-M polymer with a linear elastic response, Fb can be expressed as Fb=Fxj . Due to their low Young s modulus, E-M polymers usually exhibit a small block force compared to E-M ceramics. 

Figure 2. (a) Variation of the dynamic elastic response, d /z, measured by local force modulation as a function of the bulk elastic modulus, (b) Comparison between the surface Young s modulus deduced from force modulation experiments and the volume modulus calculated using Hertz model. 

Here E, the constant of proportionality between the stress a (stimulus) and strain e (response), is called the modulus of elasticity, or Young s modulus. is a property of the material that indicates how stiff it is. Young s modulus for polysilicon is approximately 160 GPa. Young s modulus for single crystal silicon depends on the crystallography. E is... 

The essential difference in character between the elastic low-temperature behavior and the viscous high-temperature response is shown by plotting the extension as a function of time (see Fig. 5.9). On a log-log scale an elastic response should have an asymptotic slope of 1 at low strains for a system with a well defined Young s modulus, E,... 

Thus, it is apparent that the composite longitudinal Young s modulus and major Poisson s ratio are strongly influenced by the fiber elastic response whereas the composite transverse Young s modulus and shear modulus behavior is dominated by the matrix elastic response, except at large fiber volume fractions. 

We have written Eq. (5.4) with variables grouped as they are in order to define two very important quantities. The first quantity in parentheses is called the modulus—or in this case, the tensile modulus, E, since a tensile force is being applied. The tensile modulus is sometimes called Young s modulus, elastic modulus, or modulus of elasticity, since it describes the elastic, or recoverable, response to the applied force, as represented by the springs. The second set of parentheses in Eq. (5.4) represents the tensile strain, which is indicated by the Greek lowercase epsilon, e. The strain is defined as the displacement, r — rp, relative to the initial position, rp, so that it is an indication of relative displacement and not absolute displacement. This allows comparisons to be made between tensile test performed at a variety of length scales. Equation (5.4) thus becomes... 

Dynamic mechanical tests measure the response of a material to a periodic force or its deformation by such a force. One obtains simultaneously an elastic modulus (shear, Young s, or bulk) and a mechanical damping. Polymeric materials are viscoelastic-i.e., they have some of the characteristics of both perfectly elastic solids and viscous liquids. When a polymer is deformed, some of the energy is stored as potential energy, and some is dissipated as heat. It is the latter which corresponds to mechanical damping. 

Elasticity of solids determines their strain response to stress. Small elastic changes produce proportional, recoverable strains. The coefficient of proportionality is the modulus of elasticity, which varies with the mode of deformation. In axial tension, E is Young s modulus for changes in shape, G is the shear modulus for changes in volume, B is the bulk modulus. For isotropic solids, the three moduli are interrelated by Poisson s ratio, the ratio of traverse to longitudinal strain under axial load. 

Since the force-displacement curve contains information about the whole indentation process, the elastic deformation of the sample can be measured and used to calculate the stiffness S=dFldh at h=hmax, where F is the force and h is the indentation. As already explained in Sect. 3.1.1., in order to relate the stiffness to the Young s modulus, it is necessary to make assumptions about the contact area. The depth of the permanent indentation (plastic deformation), i.e. the depth DFdi shown in Fig. 26b, and the maximum indentation (sum of the plastic and of the elastic deformation) can be used to calculate a parameter that describes the relative weight of the elastic and of the plastic response. 

Any two complex, freguency dependent, elastic moduli are sufficient to describe the mechanical response of a viscoelastic polymer. The two moduli which are most freguently measured are the bulk modulus [41], and the Young s modulus E (o)) = E (o)) + iE"(o)) [42]. Poisson s... 

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