Plane-stress creep of a glass-reinforced plastic

The creep behavior of an orthotropic glass-reinforced plastic based on Polimal'-109 resin and 19-S4 glass fabric is described. A special device for testing tubular specimens in plane-stress creep has been designed. The loading conditions include combined torsion and tension, internal pressure, and loading along the principal directions of anisotropy. Constitutive equations describing creep and recovery are proposed for the two-dimensional case.Warsaw. Translated from Mekhanika Polimerov, No. 3, pp. 398–410, May–June, 1971.     

On the motion of a mechanical system inside a rolling ball

We consider a mechanical system inside a rolling ball and show that if the constraints have spherical symmetry, the equations of motion have Lagrangian form. Without symmetry, this is not true.     

Connections induced by the rolling of a ball on a surface

We consider the rolling of a ball on a surface and establish new connections induced by this rolling. We state and prove two theorems about the dependence of these connections on invariants of the intrinsic geometry of the surface.     

Modeling the creep stability of plastic shells

The problem of modeling the creep stability of shells is investigated on the basis of the strain compatibility, equilibrium and physical equations, and the boundary conditions. The mechanical similarity of physically nonlinear elastic and elasto-hereditary materials is examined in detail. The concept of an elastic potential and Volterra-type integral relations with increasing multiplicity are employed.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1059–1063, November–December, 1969.     

Invariant measure in the problem of a symmetrical ball rolling on a surface

The problem of a symmetrical ball, rolling without slip on a fixed surface, is considered, along with some generalizations. It is proved that, under fairly general assumptions as to the nature of the external forces. the system admits of a smooth invariant measure.     

Asymptotic estimates for the principal eigenvalue of the laplacian in a geodesic ball

LetM be a compact Riemannian manifold and letB _ε be a geodesic ball of radiusε with center0 ∈ M. We investigate the asymptotic behavior ofλ _ε , the principal eigenvalue of the Laplace-Beltrami operator on \(M/\bar B_\varepsilon\) with homogeneous Dirichlet boundary conditions. We prove thatλ _ε ~Cφ _n (ε) wheren = dimM, φ ₂ (ε)=(logε ^?1)^?1 andφ _n (ε) = (n-2)ε ^n-2 (n>2). In the case whereM is a model the constantC is explicitly evaluated.     

High frequency behavior of a rolling ball and simplification of the separation equation

The Chaplygin separation equation for a rolling axisymmetric ball has an algebraic expression for the effective potential V (z = cosθ, D, λ) that is difficult to analyze. We simplify this expression for the potential and find a 2-parameter family for when the potential becomes a rational function of z = cosθ. Then this separation equation becomes similar to the separation equation for the heavy symmetric top. For nutational solutions of a rolling sphere, we study a high frequency ω ₃-dependence of the width of the nutational band, the depth of motion above V(z _min,D, λ) and the ω ₃-dependence of nutational frequency $\tfrac{{2\pi }} {T} $ .     

Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: u_t=Δu+u^p+u^q(x^*,t) in B(R) where p,q>0,B(R)={x∈R^N:|x|<R} and x^*≠0. If max(p,q)>1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.

(i)

If or p=q>2, then single point blow-up occurs whenever solutions blow up.

(ii)

If 1<p<q, both phenomena, total blow-up and single point blow-up, occur depending on the initial data.

(iii)

If p?1<q, total blow-up occurs whenever solutions blow up.

(iv)

If max(p,q)?1, every solution exists globally in time.

     

Topological analysis of an integrable system related to the rolling of a ball on a sphere

A new integrable system describing the rolling of a rigid body with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.     

The rolling motion of a truncated ball without slipping and spinning on a plane

This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.     

The problem of drift and recurrence for the rolling Chaplygin ball

We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.     

Experimental determination of the transverse shear modulus of a wound glass-reinforced plastic in creep

It is proposed to determine the interlayer shear modulus in creep starting from the results of bending tests on simply supported beams with a concentrated load. It is assumed that the increase in deflection with time is caused by interlayer creep. The data of creep tests on wound glass-reinforced plastic beams are presented and used to determine the long-time interlayer shear modulus.Leningrad. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 370–372, March–April, 1969.     

Nonlinear creep of a glass fabric-reinforced plastic in several complex states of stress

Tensile, compressive, torsional, and combined creep tests have been performed on tubular specimens of a glass fabric-reinforced plastic. The sign of the normal stress has been found to have an important influence on the development of the shear creep strains. The rheological parameters of the material necessary for approximating the test curves have been determined and the results of the calculations compared with experiment.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 212–218, March–April, 1973.     

Static and vibrational creep of a fabric-reinforced plastic in shear in the plane of the reinforcement

The results of a comparative experimental investigation of the short-term static and vibrational creep of a fabric-reinforced plastic in shear in the plane of the reinforcement are presented. The experimental procedure is described in detail. It is shown that on the investigated ranges of temperature, stress, and amplitude-frequency parameters the effect of an additional vibrational load on the creep process is unimportant.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 605–610, July–August, 1971.     

Investigation of the principal dense-winding conditions for wound glass-reinforced plastic pipe

The dense-winding conditions have been found for glass-reinforced plastic pipe wound simultaneously with a large number of strands [conditions (15), (17), and (18)] and relations have been obtained for calculating the winding mechanism kinematics. Kinematic equations have been derived for dense winding by displacement of the strands in the pipe cross section, along the pipe axis, and in both directions at once.Glass-Reinforced Plastics Division of the All-Union Lenin Electrical Engineering Institute, Khar'kov. Translated from Mekhanika Polimerov, No. 6, pp. 1057–1064, November–December, 1970.