Retardation trace in tension with subsequent compression in the transverse direction

The local strains theory is employed to determine the strain components ₁ and ₂ in five-dimensional space in the case of tension with subsequent compression at nonlinearities n=3 and n=5. The relations between the vectors and S and the tangents to the loading and deformation trajectories and between the moduli of the strain for complex and simple loading are considered.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 597–602, July–August, 1969.     

Effect of loading history on changes in the characteristic Angles and the length of the retardation trace

The local strains theory and the theory of small elastoplastic strains are used to determine the values of the components of the plastic strain vector {₁, ₂} for nonlinearity n=3 in five-dimensional Euclidean space in the case of complex loading along a two-segment broken line, when both tensile and compressive stresses are present in each loading stage. The relation between the vectors , and S and the tangents to the deformation and loading trajectories are examined. Values of the retardation trace are obtained in terms of the loading history. Numerical results have been derived with the aid of a BÉSM-3M computer.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 92–97, January–February, 1971.     

Study of the two-dimensional deformation of polymers

The effect of two-dimensional deformation on the mechanical properties of films made from the hydrochloride of polyisoprene rubber is studied. The laws determining the reversible deformation of films when _y=0 are investigated. The results obtained are discussed in relation to supramolecular structures in polymers.Mekhanika Polimerov, Vol. 1, No. 2, pp. 21–26, 1965     

Deformation properties of demineralized human compact bone tissue upon stretching

Conclusions 1. Demineralized human compact bone is a physically nonlinear material with a nonuniform distribution of deformation parameters over the bone cross section.2. At the initial moment of stressing (₁₁=0), the modulus of elasticity of demineralized tissue is 380 times less than that of normal tissue. The minimum value of this parameter is in zone 6 (3.32 kgf/mm²) and the maximum value is in zone 1 (7.12 kgf/mm²). With increasing stress, the tangential modulus of elasticity increases.3. The specific energy of deformation under the same stressing conditions ₁₁/*₁₁ expended is greater for demineralized bone than for normal bone.4. The parameter of cross-sectional deformation ₁₂ upon stressing (at ₁₁/*₁₁ from 0.1 to 0.3) increases for demineralized tissue by a factor of 1.56, while for normal tissue this increase is only by a factor of 1.01. The average values of ₁₂ ^0·1 and ₁₂ ^0·3 for demineralized tissue are 3.3 and 2.2 times, respectively, higher than for normal tissue.5. The greatest correlations for the deformation parameters were found between the internal and external layers of the cross-sectional zones for E ₁ ⁰ (0.94) and ₁₂ ^0·3 (0.87), though the discrepancies in the zones between the internal and external layers are insignificant (p>0.05).6. The changes in the deformation parameters over the cross section have a positive correlation (with the exception of U ₁ ^0.3 ) with the amount of ground substance in normal tissue (determined relative to hexosamine).7. The deformation parameters of demineralized tissue may be described by a multi-dimensional linear equation using quantitative indices of the major biochemical components (hydroxyproline, phosphorus, and hexosamine) of normal tissue, which characterize the amounts of collagen, mineral substance, and ground substance, respectively.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 101–108, January–February, 1978.     

Relation between fracture and deformation in oriented polymers

The development of submicroscopic cracking under load has been studied (by a small-angle X-ray scattering technique) in such oriented polymers as Kapron (Nylon 6), polypropylene, etc. The increase in crack density (N_cr) is compared with the variation of the tensile strain (). The observed correlation between and N_cr indicates that the processes of fracture and deformation of oriented polymers are closely connected. The presence of a similar correlation in loading-relaxation-repeat loading experiments establishes fracture as the primary process.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 43–47, January–February, 1970.     

Collision Measure and Collision Local Time for (α, d, β) Superprocesses

Existence and uniqueness of the solutions for some nonlinear evolution equations with measure-valued boundary conditions is established. This gives the existence of the collision local time and the collision measure for two independent (₁, d, ₁) and (₂, d, ₂) superprocesses without using any moment conditions on the mass processes. We obtain expressions for the Laplace transforms of the collision local time and the collision measure.     

Effect of loading on intensity of fracture and deformation of polymers

An expression is proposed for the activation energy U = U₀ –a + b², by means of which it is possible qualitatively to describe the various cases of the kinetics of deformation and rupture in polymers.Scientific-Research Institute of Plastics, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 366–368, March–April, 1972.     

Effect of the rate of deformation on the mechanical properties of compact bone tissue

Conclusions The results of this study show that the ultimate stress increases with increasing deformation rate from 10^–5 to 1 sec^–1 but the initial elasticity modulus remains virtually constant. A characteristic feature of bone tissue is the significant increase or even maximum in specific deformation energy for destruction in the range of deformation rates corresponding to normal physiological conditions for bone function. The deformation diagrams of human bone tissue for the same values of moisture content and deformation rate in tensile testing do not differ from the analogous curves for the bone tissue of cattle. Quantitatively, the ultimate stress ₁₁ and the initial elasticity modulus E₁ are 5–10% and 15–25% greater, respectively, than for cattle bone tissue, while the ultimate deformation eu is virtually the same. An increase in the moisture content of bone tissue leads to a more pronounced dependence of the mechanical parameters on the deformation rate.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 512–517, May–June, 1982.     

Effect of submicroscopic crack formation and relaxation phenomena in polymers

The development of submicrocracks and the changes taking place in their concentration during loading and unloading cycles were studied by the low-angle x-ray scattering method in oriented amorphous-crystalline polymers (capron and polypropylene). The deformation () of the polymers was compared with the changes in submicrocrack concentration (N_cr).A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 3, pp. 387–392, May–June, 1970.     

Cohomologie lp et produits amalgamés

The first l_p-cohomology group of a hyperbolic groupe induces on the boundary of equivalence relations which are invariant by quasi-isometries. We study these equivalence relations in case of some amalgamated products =A _{* c} B.     

Versal deformations and versality in central extensions of Jacobi schemes

Let be the scheme of the laws defined by the Jacobi identities on with a field. A deformation of , parametrized by a local ring A, is a local morphism from the local ring of at ϕ _m to A. The problem of classifying all the deformation equivalence classes of a Lie algebra with given base is solved by “versal” deformations. First, we give an algorithm for computing versal deformations. Second, we prove there is a bijection between the deformation equivalence classes of an algebraic Lie algebra ϕ _m = R ⋉ φ _n in and its nilpotent radical φ _n in the R-invariant scheme with reductive part R, under some conditions. So the versal deformations of ϕ _m in are deduced from those of φ _n in , which is a more simple problem. Third, we study versality in central extensions of Lie algebras. Finally, we calculate versal deformations of some Lie algebras. Supported by the EC project Liegrits MCRTN 505078.     

Comparison of the shapes of macro- and submicrocracks in loaded polymers

Small-angle x-ray scattering has been used to estimate the longitudinal (L) and transverse (L) dimensions of the submicrocracks formed in loaded polymers. The growth under load of macrocracks, initiated in the same polymers by a slit or a circular hole, has also been observed. The ratios L/L for the macro- and submicrocracks are found to coincide.A. F. Ioffe Physico-Technical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 6, pp. 1013–1018, November–December, 1971.     

Flache T1-Stabilität in der Strati-fizierung verseller Entfaltungen

Let 0 be the local ring of a simple singularity defined over the complex numbers and the dimension of its versal deformation space. Than it is well known that any nearby singularity in this space is also simple and has smaller unfolding dimension in the hierarchy of simple singularities. In particular this implies that the =max-stratum consists just of one point namely the given singularity. We want to generalize this concept as we are interested in families of varieties with formal unchanged singularities. For this we introduce in quite generality the notion of flat T¹-stabi1ity which may be checked for any k- algebra 0 where k is for simplicity an algebraically closed field of à priori arbitrary characteristics. We call 0 formal flat T¹ stable or for short flat T¹-stable if the following is true: if R is any deformation of 0 over an Artin local finite k-algebra A and if T¹(R/A,R) is A-flat than R is isomorphic to the trivial deformation . T¹(R/A,R) is the first cotangent module of R over A with values in R. Obviously the simple singularities A_k, D_k, E₆, E₇, E₈ fulfill this criterion over C but we look also at fibres of arbitrary stable map germs, generic singularities of algebraic varieties where we have to modify this notion in order to deal with wild ramification and to quasihomo-genous hypersurface singularities where it functorializes because in this case T¹ commutes with arbitrary base change. The notion of flat T¹-stable singularities is closely related to questions of existence of equisingular families and is used in[12] and [5], [6] to stratify certain Hilbert schemes.     

The packing measure of a general subordinator

Summary Precise conditions are obtained for the packing measure of an arbitrary subordinator to be zero, positive and finite, or infinite. It develops that the packing measure problem for a subordinatorX(t) is equivalent to the upper local growth problem forY(t)=min (Y ₁ (t), Y ₂ (t)), whereY ₁ andY ₂ are independent copies ofX. A finite and positive packing measure is possible for subordinators close to Cauchy; for such a subordinator there is non-random concave upwards function that exactly describes the upper local growth ofY (although, as is well known, there is no such function for the subordinatorX itself).Research supported in part by NSF under contracts (1) DMS 87-01866, and (2) DMS 87-01212     

Nonparametric comparison of regression functions

In this work, we provide a new methodology for comparing regression functions m₁ and m₂ from two samples. Since apart from smoothness no other (parametric) assumptions are required, our approach is based on a comparison of nonparametric estimators and of m₁ and m₂, respectively. The test statistics incorporate weighted differences of and computed at selected points. Since the design variables may come from different distributions, a crucial question is where to compare the two estimators. As our main results we obtain the limit distribution of (properly standardized) under the null hypothesis H₀:m₁=m₂ and under local and global alternatives. We are also able to choose the weight function so as to maximize the power. Furthermore, the tests are asymptotically distribution free under H₀ and both shift and scale invariant. Several such ’s may then be combined to get Maximin tests when the dimension of the local alternative is finite. In a simulation study we found out that our tests achieve the nominal level and already have excellent power for small to moderate sample sizes.     

Cohomology and deformations of the infinite-dimensional filiform Lie algebra

Denote the infinite-dimensional -graded Lie algebra defined by basis e_i, i1, and relations [e₁,e_i]=e_i+1 for all i2. We compute in this article the bracket structure on , and in relation to this, we establish that there are only finitely many true deformations of in each non-positive weight by constructing them explicitly. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.     

Divisible designs and groups

We study (s, k, ₁, ₂)-translation divisible designs with ₁0 in the singular and semi-regular case. Precisely, we describe singular (s, k, ₁, ₂)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, ₁, ₂)-TDD's (and, more general, for the case ₂>₁) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.     

Analysis of possible combinations of local stresses in local strain theory

In determining the strain tensor e_ij instead of all the components of the local stress deviator, it is possible to use only the shear stress _xz acting on a small local area. This fact makes it easy to establish the incremental loading conditions in solving complex loading problems for a plastic material. It is shown that in the local strain theory, distinct from the deformation theory, at degrees of nonlinearity n>3 the effect of the third, as well as the second, invariant of the stress deviator is taken into account.Mekhanika Polimerov, Vol. 3, No. 4, pp. 636–644, 1967