Elasticity theoretical

Landau, L.D., Lifshitz, E.M. (1986), Theory of Elasticity. Theoretical Physics, vol. 6, 3rd Edition (co-authored with Kosevich, A.M., and Pitaevskii, L.P.), Butterworth-Heinemann, Oxford. 

In a recent case study (see Svendsen et al, 2007 and also Problem 6.1), in collaboration with a paint company, the adhesion of six different epoxies-silicon systems has been studied. These paints are used in marine coating systems. Some epoxies showed adhesion problems in practice while others did not. The purpose of the study was to understand the origin of these problems and whether adhesion could be described/ correlated to surface characteristics, e.g. surface tensions. An extensive experimental study has been carried out including both surface analysis (contact angle measurements on the six epoxies, surface tension of silicon at various temperatures, atomic force microscopy (AFM) studies of the epoxies), as well as measurements of bulk properties (pull-off adhesion tests and modulus of elasticity). Theoretical analysis included both estimation of Zisman s critical surface tensions and surface characterization using the van Oss-Good theory. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

Landau L D and Lifshitz E M 1970 Theory of Elasticity (Course of Theoretical Physics vol 7) (Oxford Pergamon)... 

The more quickly and completely a fiber recovers from an imposed strain, the more nearly perfectly elastic it is. The ratio of the instantaneous elastic deformation to the total deformation may be used as a criterion of elasticity (62). The integrated divergences from a theoretical graph of perfect elasticity versus elongation is also used as a criterion for determination of the elasticity index. 

Mechanical Properties. Measuremeat of the mechanical properties of diamoad is compHcated, and references should be consulted for the vahous qualifications (7,34). Table 1 compares the theoretical and experimental bulk modulus of diamond to that for cubic BN and for SiC (29) and compares the compressive strength of diamond to that for cemented WC, and the values for the modulus of elasticity E to those for cemented WC and cubic BN. 

Much of the experimental work in chemistry deals with predicting or inferring properties of objects from measurements that are only indirectly related to the properties. For example, spectroscopic methods do not provide a measure of molecular stmcture directly, but, rather, indirecdy as a result of the effect of the relative location of atoms on the electronic environment in the molecule. That is, stmctural information is inferred from frequency shifts, band intensities, and fine stmcture. Many other types of properties are also studied by this indirect observation, eg, reactivity, elasticity, and permeabiHty, for which a priori theoretical models are unknown, imperfect, or too compHcated for practical use. Also, it is often desirable to predict a property even though that property is actually measurable. Examples are predicting the performance of a mechanical part by means of nondestmctive testing (qv) methods and predicting the biological activity of a pharmaceutical before it is synthesized. 

A partial answer to the first question has been provided by a theoretical treatment (1,2) that examines the conditions under which a matrix crack will deflect along the iaterface betweea the matrix and the reinforcement. This fracture—mechanics analysis links the condition for crack deflection to both the relative fracture resistance of the iaterface and the bridge and to the relative elastic mismatch between the reinforcement and the matrix. The calculations iadicate that, for any elastic mismatch, iaterface failure will occur whea the fracture resistance of the bridge is at least four times greater than that of the iaterface. For specific degrees of elastic mismatch, this coaditioa can be a conservative lower estimate. This condition provides a guide for iaterfacial desiga of ceramic matrix composites. 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

Figure 6 Apparent elastic incoherent structure factor A q(Q) for ( ) denatured and ( ) native phosphoglycerate kinase. The solid line represents the fit of a theoretical model in which a fraction of the hydrogens of the protein execute only vihrational motion (this fraction is given by the dotted line) and the rest undergo diffusion in a sphere. For more details see Ref. 25.

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

The measured stiffnesses for two- and three-layered special cross-ply laminates are shown with symbols in Figure 4-28, and the theoretical results are shown with solid lines. In all cases, the load was kept so low that no strain exceeded SOOp. Thus, the behavior was linear and elastic. The agreement between theory and experiment is quite good. Both the qualitative and the quantitative aspects of the theory are verified. Thus, the capability to predict cross-ply laminate stiffnesses exists and is quite accurate. 

S. G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Government Publishing House for Technical-Theoretical Works, Moscow and Leningrad, 1950. Also P. Fern (Translator), Holden-Day, San Francisco, 1963. 

Precipitate microstructures are important for the strength and hardness of many alloys [196-210]. A number of experimental [211-228] and theoretical [220,229-247] investigations have shown that the development of precipitate morphologies is influenced by elastic interactions (El) resulting from a lattice misfit between matrix and precipitates and from an externally applied elastic strain. 

Pd4oCu4oP2o, Pd5oCu3oP2o, and Pd6oCu2oP20 alloys were measured by resonant ultrasound spectroscopy (RUS). In this technique, the spectrum of mechanical resonances for a parallelepiped sample is measured and compared with a theoretical spectrum calculated for a given set of elastic constants. The true set of elastic constants is calculated by a recursive regression method that matches the two spectra [28,29]. 

Table I. Experimental and calculated lattice constants a (in A), elastic constants, bulk and shear moduli (in units of 10 ) for the M3X (X = Mn, Al, Ga, Ge, Si) intermetallic series. Also listed are values of the anisotropy factor A and Poisson s ratio V. The experimental data for a are from Ref. . The experimental data for B, the elastic constants, A and v are taken from Ref. . The theoretical values for NiaSi are from Ref.. Also listed in the table are values of the polycrystalline elastic quantities-shear moduli G, Yoimg moduli (in units of and the ratio The experimental data for these quantities are from Ref. ...

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