Elastic, defined moduli, shear

Considering a mass of ceramic powder about to be molded or pressed into shape, the forces necessary and the speeds possible are determined by mechanical properties of the diy powder, paste, or suspension. For any material, the elastic moduli for tension (Young s modulus), shear, and bulk compression are the mechanical properties of interest. These mechanical properties are schematically shown in Figure 12.1 with their defining equations. These moduli are mechanical characteristics of elastic materials in general and are applicable at relatively low applied forces for ceramic powders. At higher applied forces, nonlinear behavior results, comprising the flow of the ceramic powder particles over one another, plastic deformation of the particles, and rupture of... 

If the material between the surfaces is a perfectly elastic solid, the shear stress <7 and shear strain 7 are proportional, with the constant of proportionality defining the shear modulus G ... 

When a constant stress is imposed (its time derivative cr = 0), this equation describes the ideal Newtonian fluid under steady shear flow. When 77 —> 00, this equation describes the ideal elastic solid. The instantaneous response of the solid to an imposed stress is elastic, and the shear modulus E corresponds to the modulus 00 at high frequency. Consequently, the shear stress will relax down to zero exponentially. Under the condition of y = 0, the exponential function (6.18) can be solved from (6.24), which defines the characteristic relaxation time as... 

Metals are normally homogeneous and isotropic in nature and their reaction to an applied load can be defined by knowing two of the three basic elastic constants (the modulus of elasticity E, the shear modulus G and Poisson s ratio v). A basic uni-... 

Where a is the longitudinal stress, e is corresponding strain, and E is called Young s modulus (or the modulus of elasticity). Similarly, in shear deformation, the modulus is called the shear modulus or the modulus of rigidity (G). When a hydrostatic force is applied, a third elastic modulus is used the modulus of compressibility or bulk modulus (K). It is defined as the ratio of hydrostatic pressure to volume strain. A deformation (elongation or compression) caused by an axial force is always associated with an opposite deformation (contraction or expansion) in the lateral direction. The ratio of the lateral strain to the longitudinal strain is the fourth elastic constant called Poisson s ratio (v). For a small deformation, elastic parameters can be correlated in the following way ... 

At this point it is useful to consider two idealized substances the elastic solid and the viscous fluid. Each represents fairly accurately the rheological behavior of a class of real materials under specified conditions. Under steady shear stress, the elastic solid will take on a uniform shear strain. We define the shear modulus G as the ratio of shear stress to shear strain ... 

This material function thus defines the shear modulus, which then serves to characterize an elastic solid. 

With the assumption that the deformations in both the adhesive and the adherend are elastic, the interphase shear modulus can be defined as ... 

Rigidity, modulus of Modulus of Rigidity (or Shear Modulus) is the coefficient of elasticity for a shearing fmce. It is defined as the ratio of shear stress to the displacement per unit sample length (shear strain) . 

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. 

For isotropic materials, certain relations between the engineering constants must be satisfied. For example, the shear modulus is defined in terms of the elastic modulus, E, and Poisson s ratio, v, as... 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

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 constants s and c ( = 1 /s) are known as the elastic compliance constant and the elastic stiffness constant, respectively. The elastic stiffness constant is the elastic modulus, which is seen to be the ratio of stress to strain. In the case of normal stress-normal strain (Fig. 10.3a) the ratio is the Young s modulus, whereas for shear stress-shear strain the ratio is called the rigidity, or shear, modulus (Fig. 10.36). The Young s modulus and rigidity modulus are the slopes of the stress-strain curves and for nonHookean bodies they may be defined alternatively as da-/ds. They are requited to be positive quantities. Note that the higher the strain, for a given stress, the lower the modulus. 

Tensile and shear forces are not the only types of loads that can result in deformation. Compressive forces may as well. For example, if a body is subjected to hydrostatic pressure, which exists at any place in a body of fluid (e.g. air, water) owing to the weight of the fluid above, the elastic response of the body would be a change in volume, but not shape. This behavior is quantified by the bulk modulus, B, which is the resistance to volume change, or the specific incompressibihty, of a material. A related, but not identical property, is hardness, H, which is defined as the resistance offered by a material to external mechanical action (plastic deformation). A material may have a high bulk modulus but low hardness (tungsten carbide, B = 439 GPa, hardness = 30 GPa). 

Equations 1.2 to 1.4 represent material functions under large deformations (e.g., continuous shear of a fluid). One may recall a simple experiment in an introductory physics course where a stress (a) is applied to a rod of length Z, in a tension mode and that results in a small deformation AL. The linear relationship between stress (ct) and strain (j/) (also relative deformation, y = AL/L) is used to define the Young s modulus of elasticity E (Pa) ... 

In the ideal case of a Hookean body, the relationship between stress and strain is fully linear, and the body returns to its original shape and size, after the stress applied has been relieved. The proportionality between stress and strain is quantified by the modulus of elasticity (unit Pa). The proportionality factor under conditions of normal stress is called modulus of elasticity in tension or Young s modulus E), whereas that in pure shear is called modulus of elasticity in shear or modulus of rigidity (G). The relationships between E, G, shear stress, and strain are defined by ... 

The compliance is defined as the reciprocal of the modulus of elasticity (unit Pa 7 Hence, shear compliance is defined as / = 1/G, while tensile compliance is defined by D = /E. 

Above relation (1) between cr and y is exact in linear response, where nonlinear contributions in 7 are neglected in the stress. The linear response modulus (to be denoted as g (f)) itself is defined in the quiescent system and describes the small shear-stress fluctuations always present in thermal equilibrium [1, 3]. Often, oscillatory deformations at fixed frequency co are applied and the frequency dependent storage- (G (m)) and loss- (G"((u)) shear moduli are measured in or out of phase, respectively. The former captures elastic while the latter captures dissipative contributions. Both moduli result from Fourier-transformations of the linear response shear modulus g (f), and are thus connected via Kramers-Kronig relations. 

Conventionally elastic properties of solids are described using moduli called Young s modulus, E, shear modulus, G, bulk modulus, K and the Poisson s ratio, v (due to the fact that mechanical properties were studied more extensively by engineers).Young s modulus is defined by the relation... 

For the case where cr, = cTj = cTj which is known as hydrostatic stress (the situation when pressure is applied on a glass embedded in a material of low elastic constants - glass piece in steatite or in AgCl in a high pressure cell or simply embedded in a liquid) - then there are no shear strains. The hydrostatic stress is simply the pressure, P and the volumetric strain is Cm so that bulk modulus is also defined as... 

Through use of classical network theories of macromolecules, G has been shown to be proportional to crosslink density by G = nKT -i- Gen, where n is the nnmber density of crosslinkers, K is the Boltzmann s constant, T is the absolnte temperature, and Gen is the contribution to the modulus because of polymer chain entanglement (Knoll and Prud Homme, 1987). The loss modulus (G") gives information abont the viscous properties of the fluid. The stress response for a viscous Newtonian fluid would be 90 degrees out-of-phase with the displacement but in-phase with the shear rate. So, for an elastic material, all the information is in the storage modulus, G, and for a viscous material, aU the information is in the loss modulus, G". Refer to Eigure 6.2, the dynamic viscosities p and iT are defined as... 

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