On an inhomogeneous deformation of a generalized Neo-Hookean material

We study the inhomogeneous deformation of a wedge of an incompressible generalized power-law Neo-Hookean material. We find solutions which have a boundary layer structure, in the sense that adjacent to the boundary the solution is inhomogeneous, while in the core region the solution is homogeneous. It is found that such solutions have an associated pressure field that is bounded. Inhomogeneous solutions are also possible when the pressure varies logarithmically with the radial coordinate. We also establish explicit exact solutions for specific values of the parameter. The results reduce to the Neo-Hookean solution when the power law exponent is set to unity.     

A direct representation of the rotation tensor in polar decomposition of deformation gradient and its applications

Through a variational approach, an explicit connection between the additive and the polar decompositions of deformation gradient has been established. An exact formula for determining the rotation tensor in polar decomposition is obtained. The formula is fundamental in continuum mechanics and can be used to separate the rotation and the pure strain in deformation, by which various approximate expressions can be easily obtained.     

Incremental deformation model for a shell

A nonlinear deformation model for a shell with rigid transverse fibers is proposed. A complete system of incremental equations, a variational equation equivalent to this system, and a particular equation of virtual work are formulated. Numerical analysis of the nonlinear deformation of a spherical dome is performed using the complete equation. Institute of Computer Modeling, Siberian Division, Russian Academy of Sciences, Krasnoyarsk 660036. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 202–207, September–October, 1999.     

On the deformation of a ribbed cylindrical shell with initial deflection under dynamic loading

The finite-difference method with explicit time coordinate is used to study the dynamic stress-strain state of a cylindrical shell with ribs and initial deflections. A specific numerical example is considered to trace the time evolution of longitudinal and transverse displacements for different parameters of the initial deflection __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 61–66, December 2005.     

Physically nonlinear deformation of an orthotropic cylindrical shell with a circular hole

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 10, pp. 41–46, October, 1988.     

Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua

This work elaborates upon two robust models of gradient elasticity and gradient plasticity, and one gradient model of heat transfer, as originally advocated by the second author in the 1980’s. The objective is, after recalling the links between these models and existing generalized continuum theories as developed in the 1960’s and subsequently, to apply the same methodology to the case of diffusion with a view to establishing generalized transport equations. Aifantis double diffusivity and conductivity theory that provides generalized mass or heat transfer equations is compared to micromorphic-type hyper-temperature and micro-entropy proposals. The double temperature and the micromorphic thermal models are shown to lead to equations more general that Cattaneo’s. The sign of the coefficient of the second time-derivative of temperature is found to differ according to both approaches. The double temperature model contains a fourth space derivative term not present in the micromorphic models. Such generalized equations can be useful, for example, in the interpretation of recent femtosecond laser experiments on metals.     

圆柱壳的塑性膨胀和变形

根据材料不可压缩假设,适当选取材料塑性本构关系,将圆柱壳塑性膨胀运动简化为以圆柱壳内 半径为因变量的常微分方程初值问题。对3种初始内外半径比纯铜厚壁圆柱壳膨胀过程的计算表明,应变率 硬化阻碍圆柱壳的塑性膨胀运动,应变硬化和温度效应的影响可以忽略,而与圆柱壳的厚度无关。     

On the derivatives of the stretch and rotation with respect to the deformation gradient

In this paper, a result involving the eigenprojections of the right stretch and its derivative with respect to the deformation gradient is derived, and a related result is found for the rotation. As an application, the form of the constitutive law for an isotropic hyperelastic material in the case when the strain energy function is expressed in terms of the right stretch, is shown to follow at once.     

On a mechanical analogy in the ideal plasticity theory

The Cauchy problem of propagation of plastic state zones in a boundless medium from the boundary of a convex surface, along which normal pressure and shear forces act, is considered. In the case of complete plasticity, the Tresca system of quasi-static equations of ideal plasticity, which describes the stress-strain state of the medium, is known to be hyperbolic and to be similar to a system that describes a steady-state flow of an ideal incompressible fluid. This system is numerically solved with the use of a difference scheme applied for hyperbolic systems of conservation laws. Results of numerical calculations are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 74–80, July–August, 2008.     

Ultimate deformation behavior of a laminated spherical shell

Summary A solution method is developed for a laminated spherical shell, subject to uniform normal tractions or constant radial displacements on the boundary. Displacements and stresses in the shell can be obtained from the solution formulas, which are as simple as those for homogeneous spherical shells. These formulas are accurate when the thicknesses of the constituent spherical layers are small in comparison with the radii of the shell, and become exact in case these thicknesses tend to zero under a fixed overall shell thickness. The solution is obtained mainly by exactly treating the product of an infinitely large number of matrices which link the displacements and stresses in different constituent spherical layers. A continuous analysis and a number of numerical results are provided to validate this development. Received 4 December 1996; accepted for publication 12 September 1997     

Incompatible deformation field and Riemann curvature tensor

Compatibility conditions of a deformation field in continuum mechanics have been revisited via two different routes. One is to use the deformation gradient, and the other is a pure geometric one. Variations of the displacement vector and the displacement density tensor are obtained explicitly in terms of the Riemannian curvature tensor. The explicit relations reconfirm that the compatibility condition is equivalent to the vanishing of the Riemann curvature tensor and reveals the non-Euclidean nature of the space in which the dislocated continuum is imbedded. Comparisons with the theory of Kr¨oner and Le-Stumpf are provided.     

On the holographic interferometry fringe gradient method for complete surface deformation measurements

A fringe gradient method for complete surface gradient deformation measurements using five holograms is presented. Feasibility of the method is studied by combining the experimental results with the computer simulations. It is shown that the goveming equations relating the fringe gradient to the displacement vector and the displacement gradient are not well conditioned. Accuracy of the experimental data required to make the method successful is analyzed.