Elastic bulk modulus

Chemical hardness is an energy parameter that measures the stabilities of molecules—atoms (Pearson, 1997).This is fine for measuring molecular stability, but energy alone is inadequate for solids because they have two types of stability size and shape. The elastic bulk modulus measures the size stability, while the elastic shear modulus measures the shape stability. The less symmetric solids require the full set of elastic tensor coefficients to describe their stabilities. Therefore, solid structures of high symmetry require at least two parameters to describe their stability. 

Fig. 42. Depth variation of the elastic modulus for rubber, two different polyurethanes, PVC and PS as a function of the indentation depth. Bars demonstrate the range of elastic bulk modulus variation for a specific material. Reproduced from [415]...

The dependence of the energy on volume, if interpolated by a suitable analytical formula, yields all relevant information on the equations of state of magnesite at 0 K. The Murnaghan equation[26], widely used in solid state thermodynamics, is based on the assumption of a linear dependence of the elastic bulk modulus on pressure B = Bo + B p) ... 

For both magnesite and calcite, the elastic bulk modulus Bq was computed straightforwardly by the Murnaghan interpolation formula, while of the elasticity tensor only the C33 component and the C + C 2 linear combination could be calculated in a simple way. The relations used are C = (l/Vo)c (d L /crystal structure. To derive other elastic constants, the symmetry must be lowered with a consequent need of complex calculations for structural relaxation. A detailed account of how to compute the Ml tensor of crystal elasticity by use of simple lattice strains and structure relaxation was given previously[10, 11]. For the present deformations only the c-o ( ) relaxation need be considered. The results are reported in Table 6, together with the corresponding values extrapolated to 0 K from experimental data (Table 2). For calcite, the mea-... 

The calculated Young s modulus for rubier is 2.9 0.6 MPa and 2.4 0.5 MPa as determined by the Hertzian model and JKR models, respectively. Both values are very close to elastic bulk modulus for polyisoprene measured from tensile experiment (1.6 -2.5 MPa). For further calculations, we selected the Hertzian model, which is relatively simple, gives reliable results, and does not require additional speculations or measurements of interfacial energies needed to be known in the JKR theory. 

Figure 5. Summarized data for depth variation of elastic modulus for rubber, two different PUs, PVC, and PS. Bars demonstrate values of elastic bulk modulus measured from tensile experiments.

Calculated dependences of the modulus of elasticity, bulk modulus of nanoparticles of cesium on their diameter was obtained. Curves were obtained on the basis of matching the solutions of the molecular djmamics and the theory of elasticity, carried by vectors of displacements in points coinciding with the position of the atoms of the nanoparticle. The critical... 

Fig. 9.3 Variation of the radial stress and hoop stress with position in the viscoelastic reinforced cylinder loaded with a step input of internal pressure. Parameters used are K /Go=3, t=1000, where the viscoelastic cylinder has an elastic bulk modulus and is a single Maxwell element in shear modulus. Response parameterized with time from the initial application of load at t=0 to asymptotic response at long times.

Using the same assumptions of the example solved in the Laplace domain (step input in pressure, elastic bulk modulus and Maxwell behavior in shear) with Eq. 9.58, the solution of the integral equation (Eq. 9.58) will yield the same results. (See homework problem 9.4). Since polymers are such that many Maxwell or Kelvin elements are needed to represent actual behavior, this example shown here is simplistic. However, such simple solutions can show trends in behavior and may give insight to the differences between thermosets and thermoplastics. The next section discusses briefly use of broadband material response functions for more physically realistic... 

To and Fq are the references at the ambient conditions. Ay is the energy perturbation and 8y the strain caused by the applied stimuli. The summation and the production are preceded over all the J stimuli of (z, s, P, T). The a(f) is the temperature-dependent thermal expansion coefficient (TEC), p = —dv/ vdp) is the compressibility (p <0 compressive stress) or extensibility (p > 0 tensile stress) that is proportional to the inverse of elastic bulk modulus. The k(s) is the effective force constant. rj(t) is the T-dependent specific heat of the representative bond, which approximates Debye approximation, C iTIdi)), for a z-coordinated atom. Generally, the thermal measurement is conducted under constant pressure and the i/(f) is related to the Cp. However, there is only a few percent difference between the Cp and Cy [1]. 

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