Elastic constants relation with force constants

The ratio (p/G) has the units of time and is known as the elastic time constant, te, of the material. Little information exists in the published literature on the rheomechanical parameters, p, and G for biomaterials. An exception is red blood cells for which the shear modulus of elasticity and viscosity have been measured by using micro-pipette techniques 166,68,70,72]. The shear modulus of elasticity data is usually given in units of N m and is sometimes compared with the interfacial tension of liquids. However, these properties are not the same. Interfacial tension originates from an imbalance of surface forces whereas the shear modulus of elasticity is an interaction force closely related to the slope of the force-distance plot (Fig. 3). Typical reported values of the shear modulus of elasticity and viscosity of red blood cells are 6 x 10 N m and 10 Pa s respectively 1701. Red blood cells typically have a mean length scale of the order of 7 pm, thus G is of the order of 10 N m and the elastic time constant (p/G) is of the order of 10 s. 

Table 8 presents a survey of the basic elastic constants of a series of polymer fibres and the relation with the various kinds of interchain bonds. As shown by this table, the interchain forces not only determine the elastic shear modulus gy but also the creep rate of the fibre. 

According to the model, a perturbation at one site is transmitted to all the other sites, but the key point is that the propagation occurs via all the other molecules as a collective process as if all the molecules were connected by a network of springs. It can be seen that the model stresses the concept, already discussed above, that chemical processes at high pressure cannot be simply considered mono- or bimolecular processes. The response function X representing the collective excitations of molecules in the lattice may be viewed as an effective mechanical susceptibility of a reaction cavity subjected to the mechanical perturbation produced by a chemical reaction. It can be related to measurable properties such as elastic constants, phonon frequencies, and Debye-Waller factors and therefore can in principle be obtained from the knowledge of the crystal structure of the system of interest. A perturbation of chemical nature introduced at one site in the crystal (product molecules of a reactive process, ionized or excited host molecules, etc.) acts on all the surrounding molecules with a distribution of forces in the reaction cavity that can be described as a chemical pressure. 

The frequency correlation time xm corresponds to the time it takes for a single vibrator to sample all different cavity sizes. The fluctuation-dissipation theorem (144) shows that this time can be found by calculating the time for a vertically excited v = 0 vibrator to reach the minimum in v = 1. This calculation is carried out by assuming that the solvent responds as a viscoelastic continuum to the outward push of the vibrator. At early times, the solvent behaves elastically with a modulus Goo. The push of the vibrator launches sound waves (acoustic phonons) into the solvent, allowing partial expansion of the cavity. This process corresponds to a rapid, inertial solvent motion. At later times, viscous flow of the solvent allows the remaining expansion to occur. The time for this diffusive motion is related to the viscosity rj by Geo and the net force constant at the cavity... 

The units of Eq. 10.51 are Newtons (1 N = 1 J/m). Note that B can be solved by setting Eq. 10.51 to zero, that is when t/ is a minimum. The force constant of the spring (the interatomic bond) is given by A U/Ar. The force constant is related to the elastic modulus since the latter is defined as dtr/ds. The Young s modulus is thus a second derivative of the strain energy with respect to the applied strain. However, because stress is equal to force per unit area (tr = F/tq) and strain, s, is equal to Ar/ro ... 

K is deformability, a proportionality constant relating the maximum compressive force Q to the deformed-contact area, A is a constant with units of L /F) which relates granule volume to impact compression force and Oj- is the tensile strength of the granule bond [see Eq. (20-43)]. The parameters and Tj depend on the deformation mechanism acting within the contact area, with their values bounded by the cases of complete plastic or complete elastic deformation. For plastic deformation, = 1, Tj = 0, and K = 1/H where H is hardness. For elastic, Hertzian deformation, = 3, and K oc where F is... 

Two common properties which can be calculated from the minimum-energy structure are the elastic and dielectric constants. The elastic constant matrix is used to relate the strains of a material to the internal forces, or stresses It is defined as the second derivative of the energy with respect to the strain, normalised by the cell volume. The inverse of the elastic... 

The three resins above were tested by thermomechanical analysis (TMA) on a Met-tler 40 apparatus. Triplicate samples of beech wood alone, and of two beech wood plys each 0.6 mm thick bonded with each resin system were tested. Sample dimensions were 21 mm x 6 mm x 1.2 mm. The samples were tested in non-isothermal mode from 40°C to 220°C at heating rates of 10°C/min, 20°C/min and 40°C/min with a Mettler 40 TMA apparatus in three-point bending on a span of 18 mm. A continuous force cycling between 0.1 N and 0.5 N and back to 0.1 N was applied on the specimens with each force cycle duration being 12 s. The classical mechanics relation between force and deflection E = [L /(4bh )][AF/(Af)] (where L is the sample length, AF the force variation applied and A/ the resulting deflection, b the width and h the thickness of the sample) allows calculation of the modulus of elasticity E for each case tested and to follow its rise as functions of both temperature and time. The deflections A/ obtained and the values of E obtained from them proved to be constant and reproducible. 

The stress theorem determines the stress from the electronic ground state of any quantum system with arbitrary strains and atomic displacements. We derive this theorem in reciprocal space, within the local-density-functional approximation. The evaluation of stress, force and total energy permits, among other things, the determination of complete stress-strain relations including all microscopic internal strains. We describe results of ab-initio calculations for Si, Ge, and GaAs, giving the equilibrium lattice constant, all linear elastic constants Cy and the internal strain parameter t,. 

An important variable is pressure as under pressure interatomic distances in crystals show larger variations than those induced by temperature. At constant temperature T pressure P is related to the rate of energy change with the unit-cell volume V by relation P = — y)j, including the first-order derivative of total energy with respect to the cell volume. Such observables as the bulk modulus, the elastic and force constants depend on the second-order derivatives of the total energy. 

It can be seen that a rather wide range of predicted values is obtained that is partly due to choice of different force constants. The results are also sensitive to the details of the assumed crystal unit cell structure, especially the angle made by the plane of the planar zigzag polyethylene chain with the b-axis of the orthorhombic unit cell. The overall pattern of elastic anisotropy is however clear. The stiffness in the chain axis direction C33 is by far the greatest value, and the shear stiffnesses C44, C55 and Cee are the lowest values. This reflects the major differences between the intramolecular bond stretching and valence bond bending forces and the intermolecular dispersion forces, which determine the shear stiffnesses. The lateral stiffnesses also relate primarily to dispersion forces and are correspondingly low. 

Elastic constants are directly related to the interchain and intrachain force field. A general matrix method for treating elastic constants was reported by Shiro (1968), Shiro and Miyazawa (1971), where the basic formulations of Bom and Huang (1954) were simplified with the use of matrix equations and symmetry considerations. 

The accuracy of the modulus depends on several factors, most important of which is accuracy of the calculated force constant. This is related directly to the accuracy of prediction of molecular vibrations, and errors in the order of 10% are not uncommon. Comparison with experiment is difficult, due to the finite crystallinity of the samples used. Calculated moduli should be regarded as being a guide to the limiting elastic modulus of 100% crystalline polymer. 

Elastic behavior is introduced with the stress and strain tensors and the elastic coefficient matrix and it is shown that for a maferial with cubic symmetry the coefficient matrix reduces to only three elastic coefficients. Relationships between the various elastic moduli are derived. The concept of a simple cenfral force potential such as the Coulomb potential is taken to the limits of its validity and is used to obtain the bulk modulus and to estimate the thermal expansion of ionically bonded materials. The Coulomb potential is then modified to represent a imidirectional potential and used to estimate the elastic coefficients of these materials. The Morse potential is used to relate the elastic constants of metals to their theoretical strengths. 

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