Muskhelishvili

Problems of inclusions in solids are also treated by exact elasticity approaches such as Muskhelishvili s complex-variable-mapping techniques [3-9]. In addition, numerical solution techniques such as finite elements and finite differences have been used extensively. 

N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, The Netherlands, 1953. 

Fig. 7.2. Dugdale-Muskhelishvili-model of plastic zones at the ends of a loaded elliptical hole [81].

Ikada Y. Surface modification of polymers for medical application. Biomaterials, 1994, 15, 725-736. James SJ, Pogribna M, Miller BJ, Bolon B, and Muskhelishvili L. Characterization of cellular response to silicone implants in rats Implications for foreign-body carcinogenesis. Biomaterials, 1997, 18, 667-675. 

Human CYPIBI transgene expressed in mouse brain underlet promoter (Hwang et al., 2001). In situ hybridization and IHC reveal CYPIBI mRNA and protein in brain parenchymal and stromal tissue CYPIBI protein was nuclear (Muskhelishvili et al., 2001). 

Muskhelishvili L, Thompson PA, Kusewitt DF, Wang C, Kadlubar FF. 2001. In situ hybridization and immuno-histochemical analysis of cytochrome P450 IBl expression in human normal tissues. J Histochem Cytochem 49 229-236. 

ScaUet AC, Muskhelishvili L, Slikker W Jr, Kadlubar FF. 2005. Sex differences in cytochrome P450 IBl, an estrogen-metabolizing enzyme, in the rhesus monkey telencephalon. J Chem Neuroanat 29 71-80. 

Pogribny, I.P., Tryndyak, V.P., Muskhelishvili, L., Rusyn, I. and Ross, S.A. (2007) Methyl deficiency, alterations in global histone modifications, and carcinogenesis. The Journal of Nutrition, 137, 2168-222. 

Muskhelishvili NJ (1953) Some basic problems of the mathematical theory of elasticity. Groningen, Noordhoff, p 340... 

Muskhelishvili, N. I., Singular Integral Equations, Mineola, NY Dover Publications, 1991. 

Here, Eg is the plane strain modulus of the isotropic elastic substrate and the singular integral is defined in the sense of its Cauchy principal value (Muskhelishvili 1953). 

Savkoor, A.R. Briggs, G.A.D. Proc. Rov. Soc. A. 1977, 356, 103-114. Muskhelishvili, N.I. Some basic probhms of the mathematical theory of elasticity, Noordhoff International biishing Leyden, 1975. 

Singular integral equation analysis. A closed form analytical solution can be obtained. We use results from thin airfoil theory (e.g., Ashley and Landahl, 1965) and singular integral equations (Muskhelishvili, 1953 Gakhov, 1966 Carrier, Krook and Pearson, 1966). Now the standard log r source... 

The stresses may be calculated by differentiating (2.8.5). One finally obtains the viscoelastic generalization of the Kolosov-Muskhelishvili equations which we collect together as follows ... 

Our method of attacking plane, non-inertial problems will be, in the first instance, to reduce (2.8.9) to a Hilbert problem, in precisely the manner developed by Muskhelishvili (1963), and then to handle the specifically viscoelastic aspects, essentially by the methods outlined in Sects. 2.4-6. We remark that an alternative way of approaching the first stage is the dual integral equation method originally used in this context by Sneddon (1951) but with a long history of mathematical development summarized by Gladwell (1980). 

Michell (1899) showed that for a single elastic body, condition (2.10.1), valid for all closed contours together with the restriction to stress boundary conditions, implies that the stresses are independent of the material constants. A simple dimensional argument shows that they could at most depend upon Poisson s ratio. This theorem eliminates even that dependence. An intuitive demonstration of the result would proceed as follows. Since the first two equations of (2.8.7) have no dependence on the material parameters, the stresses will depend on these quantities if the complex potentials do. Given the nature of the boundary conditions, the only way such dependence on the complex moduli can enter is through a constraint that the displacements and stresses be single-valued. This condition forces certain properties on the complex potentials as shown in the literature quoted in Sect. 8, and can introduce a dependence on the moduli. However, an examination of how this occurs in Muskhelishvili (1963), and the other standard references, indicates that the moduli enter only if contours exist such that (2.10.1) does not hold. 

Equations (2.8.9) extend to viscoelasticity the Kolosov-Muskhelishvili solution of plane elasticity, which is widely used in Chaps. 3, 4. 

Kolosoy-Muskhelishvili Equations Adapted to the Half-Plane... 

Our object is to apply (2.8.9) to problems involving loads on viscoelastic halfspaces. For such problems, it is desirable to re-express these equations in an alternative form [Muskhelishvili (1963)] which facilitates reduction to a Hilbert problem. Let the material occupy the upper half-plane > >0 so that (piz, t) is analytic in this region. It is convenient to extend the region of analyticity of (p(z, t) to the lower half-plane also. Then, as we shall see, it is possible to explore the discontinuities in this function across the real axis, which gives a Hilbert problem. Another approach, possibly more direct, is that of Galin, mentioned previously, which leads to problems of the Riemann-Hilbert type. These however are somewhat more difficult to deal with, from a mathematical point of view. 

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