Solutions of the Saint Venant problem for a cylinder with helical anisotropy

Ustinov, Yu. A.

doi:10.1016/s0021-8928(03)00020-0

Cited by 8 publications

(5 citation statements)

References 2 publications

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“…In [38,41,47,[54][55][56][57], using the method of homogeneous solutions and spectral theory of operators, was shown that the general solution may be represented as linear combinations of the twelve elementary solutions, corresponding to such problems as extension-torsion, pure bending and bending of the shear force, each of them satisfies the equilibrium equations (5) and the boundary conditions (6) on the side.…”

Section: Fundamental Equationsmentioning

confidence: 99%

“…The solution u b of the bending problems [56,57] is a linear combination of elementary solutions corresponding to the eigenvalues γ ± 1 = ±iτ , and can be written as …”

Section: The Bending Problemmentioning

confidence: 99%

“…The Saint-Venant problem [49] and method to solve this problem for inhomogeneous bodies was proposed by Dorin Ieşan [33,34] most fully and consistently, see also [35]. The solution of Saint-Venant problem for cylinders with helical anisotropy was obtained using a unified approach [57] based on modification of the method of homogeneous solutions and the theory of nonself-conjugated operators.…”

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confidence: 99%

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Saint-Venant problem for solids with helical anisotropy

Kurbatova

Ustinov

2015

Continuum Mech. Thermodyn.

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We discuss the solution of Saint-Venant's problem for solids with helical anisotropy. Here the governing relations of the theory of elasticity in terms of displacements were presented using the helical coordinate system. We proposed an approach to construct elementary Saint-Venant solutions using integration of ordinary differential equations with variable coefficients in the case of a circular cylinder with helical anisotropy. Elementary solutions correspond to problems of extension, of torsion, of pure bending and of bending of shear force. The solution of the problem is obtained using small parameter method for small values of twist angle and numerically for arbitrary values. Numeric results correspond to problems of extensiontorsion. Dependencies of the stiffness matrix (in dimensionless form) on angle between the tangent to the helical coil and the axis of the cylinder corresponding to stiffness on stretching and torsion are illustrated graphically for different values of material and geometrical parameters.

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Section: Fundamental Equationsmentioning

confidence: 99%

“…The solution u b of the bending problems [56,57] is a linear combination of elementary solutions corresponding to the eigenvalues γ ± 1 = ±iτ , and can be written as …”

Section: The Bending Problemmentioning

confidence: 99%

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confidence: 99%

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Saint-Venant problem for solids with helical anisotropy

Kurbatova

Ustinov

2015

Continuum Mech. Thermodyn.

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“…It was shown [1,3,4] that γ 0 = 0 and γ ± 1 = ±iτ are quadruple eigenvalues, and there are no purely imaginary eigenvalues except for γ ± 1 . 108…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…The Saint-Venant solution of the tension-torsion problem [3,4] is a linear combination of elementary solutions corresponding to the quadruple eigenvalue γ 0 = 0 and can be presented as…”

Section: Elementary Saint-venant Solutions Of the Tension-torsion Promentioning

confidence: 99%

Saint-venant problem for solids with helical rhombohedral anisotropy. Tension-torsion problems

Vatulyan

Ustinov

2010

J Appl Mech Tech Phy

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By methods of homogeneous solutions and the spectral theory of operators, the construction of solutions of the Saint-Venant problems of tension-torsion of a cylindrical tube with helical anisotropy is reduced to integration of boundary-value problems for ordinary differential equations with variable coefficients. The solutions are constructed by analytical and numerical methods. Elements of the stiffness matrix and the stress-strain state are analyzed, depending on the problem parameter.The present paper describes the construction of solutions of the Saint-Venant problem of tension and torsion of a cylinder possessing helical rhombohedral anisotropy. By methods of the spectral theory of operators [1-6], the problems are reduced to integration of boundary-value problems for ordinary differential equations. An analytical solution is obtained for a particular case of cylindrical rhombohedral anisotropy. The solutions of problems for a cylinder with helical rhombohedral anisotropy are constructed by two methods: for small values of the dimensionless parameter τ 0 = aτ , where a is the cylinder radius and τ is the "torsion" (characteristic of helical anisotropy), the analytical solution is constructed by the method of the small parameter; for arbitrary values of τ , the solution is obtained by means of numerical integration of appropriate boundary-value problems. CONSTITUTIVE RELATIONS OF THE ELASTICITY THEORY AND FORMULATION OF BOUNDARY-VALUE PROBLEMS Formulation of the Problem. Let us consider a cylindrical solid occupying the volume V = S ×[0, L](S is the cylinder cross section and L is its length). We denote the side surface by Γ = ∂S × [0, L], where ∂S is the boundary of the cross section S. We align the origin of the Cartesian coordinate system Ox 1 x 2 x 3 with the geometric center of gravity of one of the end faces of the cylinder and direct the Ox 3 axis along the cylinder centerline. This coordinate system will be called the basic coordinate system. To describe helical anisotropy, we introduce a helical cylindrical coordinate system (r, θ, z) related to the basic coordinate system by the expressionswhere τ = const. The transition to the cylindrical coordinate system is made because the main attention will be paid below to solving the problem of a cylinder with a ring-shaped cross section S = [r 1 , r 2 ] × [0, 2π] (r 1 and r 2 are the inner and outer radii of the cylinder, respectively).At r = const and θ = const, relations (1.1) are parametric equations of the helical line, with τ = 2π/h (h is the helical pitch). The radius-vector of the points of the helical line is presented in the form R = re 1 + ze 3 ,

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State space elastostatics of prismatic structures

Stephen

2004

International Journal of Mechanical Sciences

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Solutions of the Saint Venant problem for a cylinder with helical anisotropy

Cited by 8 publications

References 2 publications

Saint-Venant problem for solids with helical anisotropy

Saint-Venant problem for solids with helical anisotropy

Saint-venant problem for solids with helical rhombohedral anisotropy. Tension-torsion problems

State space elastostatics of prismatic structures

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