Asymptotic Stress Field in a Degenerate Orthotropic Material Containing a Cohesive Zone ahead of a Crack Tip

Complex variable and eigenfunction expansion methods are used to derive elastic stress fields near the cohesive zone of a crack aligned with principal axes of a degenerate orthotropic material. Asymptotic field terms are obtained using coordinate rescaling and transformation techniques. The stress fields do not have stress singularity. Variations of stress components associated with the first-order or dominant stress terms are discussed in detail.      

Natural convection in a horizontal porous layer with anisotropic thermal diffusivity

The present paper is concerned with free convection in a horizontal porous layer with anisotropic thermal diffusivity. It is assumed that the diffusivity has rotational symmetry, with a symmetry axis making an arbitrary angle against the vertical. The critical Rayleigh number and wave number at marginal stability are calculated and the steady motion occurring at convection onset is examined. It is found that there are two different types of convection cells, depending on whether the longitudinal diffusivity is larger than the transverse diffusivity or not. In the former case, the convection cells are rectangular with vertical lateral walls. In the latter case, however, the lateral cell walls are tilted as well as curved.     

Surface Pile-Up Patterns in Indentation Testing of Cu Single Crystals

Nano- and micro-indentation of Cu single crystals is performed in directions not aligned with crystallographic axes. Such tests correspond to mechanical characterization of incidentally oriented grains in a polycrystalline or composite material. Orientation and size dependence of complex patterns of surface piling-up and sinking-in around the imprint are investigated. Experimental observations are compared with finite element simulations based on the large deformation crystal plasticity theory.     

Comparison of Two Mathematical Models for 3D Groundwater Flow: Block-Centered Heads and Edge-Based Stream Functions

Traditionally, groundwater flow models, as well as oil reservoir models, are based on the block-centered finite difference method. Well-known models based on this approach are MODFLOW (groundwater) and ECLIPSE (oil and gas). Such models are well proven and robust; their underlying principles are well understood by hydrologists and petroleum reservoir engineers. Nevertheless, the desire to improve the block-centered finite difference paradigm has always been alive, for instance, to be able to apply deformed grid blocks, or to model anisotropy that is not aligned along the coordinate axes. This article introduces the edge-based stream function as a potential alternative to the paradigmatic model, not only to mitigate the above mentioned limitations, but especially for its promise to inverse modeling. Computer programs have been developed for the discrete analog equations of the stream function method and the conventional method. The two methods are tested by using synthetic forward modeling problems of uniform and radial flow. The theoretical formulation and the numerical results show that the two methods are algebraically equivalent and yield the same flux output. However, for rectangular grid blocks and anisotropy aligned along the coordinate axes, the block-centered method is shown to be computationally more efficient than the edge-based stream function method. The major advantage of the stream function method is that it is linear in the resistivities, proving it an ideal candidate for direct inverse modeling. Moreover, any arbitrary specification of stream functions yields a solution that satisfies the mass balance.     

Stability of thermal convection between non-uniformly heated plates

The effect of horizontal as well as vertical temperature gradients on the stability of natural convection in a thin horizontal layer of viscous, incompressible fluid is studied on the basis of linear theory. The boundaries are taken to be rigid, perfectly thermally conducting, having prescribed temperatures and the horizontal temperature gradient is assumed to be small. It is found that for Prandtl number greater than 0.13, the critical Rayleigh number is always larger than that for the corresponding Benard problem. The preferred mode of disturbance is stationary and will be a transverse roll (having axes normal to the basic flow) or a longitudinal roll (having axes aligned in the direction of the basic flow) depending on whether the Prandtl number is less or larger than 1.7. Finally, it is shown that the instability is of thermal origin.     

Determining the stress state of flexible orthotropic shells of revolution in magnetic field

A two-dimensional nonlinear magnetoelastic model of a current-carrying orthotropic shell of revolution is constructed taking into account finite orthotropic conductivity, permeability, and permittivity. It is assumed that the principal axes of orthotropy are aligned with the coordinate axes and that the orthotropic body is magnetically and electrically linear. The coupled nonlinear differential equations derived describe the stress-strain state of flexible current-carrying orthotropic shells of revolution that have an arbitrarily shaped meridian and orthotropic conductivity and are in nonstationary mechanical and electromagnetic fields. A method to solve this class of problems is proposed Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 64–76, August 2008.     

The hyperbolic heat conduction equation in an anisotropic material

In this work, the phase-lag concept in the wave theory of heat conduction is extended to describe the thermal behavior of anisotropic material. This is achieved by assuming that there are phase-lags of different magnitudes between each component of the heat flux vector and the summation of temperature gradients in all directions of the orthogonal coordinate system. Also, expressions are provided to specify the locations of the principal coordinate axes, the principal thermal conductivities and the principal thermal relaxation times. Received on 26 March 1999     

Two-dimensional transient dynamic response of orthotropic layered media

In this study, two-dimensional transient dynamic response of orthotropic plane layered media is investigated. The plane multilayered media consist of N different generally orthotropic, homogeneous and linearly elastic layers with different ply angles. In the generally orthotropic layer, representing a ply reinforced by unidirectional fibers with an arbitrary orientation angle, the principal material directions do not coincide with body coordinate axes. The solution is obtained by employing a numerical technique which combines the use of Fourier transform with the method of characteristics. The numerical results are displayed in curves denoting the variations of stress and displacement components with time at different locations. These curves clearly reveal, in wave profiles, the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers, and also the effects of anisotropy caused by fiber orientation angle. The curves properly predict the sharp variations in the response at the neighborhood of the wave fronts, which shows the power of the numerical technique employed in the study. By suitably adjusting the elastic constants, the results for multilayered media with transversely isotropic layers, or layers with cubic symmetry, or isotropic layers can easily be obtained from the general formulation. Furthermore, solutions for some special cases, including Lamb’s problem for an elastic half-space, are obtained and compared with the available solutions in the literature and very good agreement is found. Preliminary version presented at the Second International Congress on Mechatronics (MECH2K3), Graz, Austria, July 14-17, 2003.     

Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer

The stability of a fluid-saturated horizontal rotating porous layer subjected to time-periodic temperature modulation is investigated when the condition for the principle of exchange of stabilities is valid. The linear stability analysis is used to study the effect of infinitesimal disturbances. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Darcy–Rayleigh number and wavenumber. The shift in critical Darcy–Rayleigh number is calculated as a function of frequency of modulation, Taylor number, and Darcy–Prandtl number. It is established that the convection can be advanced by the low frequency in-phase and lower-wall temperature modulation, where as delayed by the out-of-phase modulation. The effect of Taylor number and Darcy–Prandtl number on the stability of the system is also discussed. We found that by proper tuning of modulation frequency, Taylor number, and Darcy–Prandtl number it is possible to advance or delay the onset of convection.     

Entropy generation of radiation effect on laminar-mixed convection along a wavy surface

The effects of thermal radiation on laminar-forced and free convection along the wavy surface are studied. The optically thick limit approximation for the radiation flux is assumed. A modified form for the entropy generation equation is derived. The effect of geometry (e.g. flat surface, wavy surface), fluid friction and heat transfer (e.g. convection and radiation effects) are all included in the modified entropy generation form. Prandtl’s transposition theorem is used to stretch the ordinary coordinate system in certain directions. The wavy surface can be transformed into a calculable planar coordinate system. The governing equations are derived from the complete Navier–Stokes equations. A simple transformation is proposed to transform the governing equations into boundary layer equations for solution by the cubic spline collocation method.     

Stability of axially compressed cylindrical shells made of reinforced materials with specific fiber orientation within each layer

A technique for stability analysis of anisotropic cylindrical shells is developed. It permits us to examine the cases of reinforcement where the elastic axes of layers do not coincide with the coordinate axes of the shell. The solution is obtained using the mixed equations of the Donnell-Mushtari-Vlasov theory of shells. The deflection and force functions are approximated by trigonometric series. Single-layer and multilayer cylindrical shells with fiber orientation of two types are analyzed for stability. It is revealed that when layers are few, failure to incorporate the direction of fibers in layers into the design model results in highly inaccurate values of critical loads __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 80–88, March 2006.     

The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Periodic and Localized Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated with the aid of a computer package. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. Previous work on the case of non-periodic global variation is now extended to the case of either periodic variation or localized variation.     

Calculation of nonaxisymmetric thermally stressed state of discrete homogeneous bodies of revolution with orthotropic layers

The nonaxisymmetric thermally stressed state of laminar bodies of revolution made of isotropic and orthotropic materials is considered. The procedure used is based on the semi-analytical finite-element method and the method of successive approximations. The results from calculations for the stressed state of a structural element are reported. The structure has layers of isotropic and orthotropic materials with principal axes of anisotropy that coincide with the directions of the axes of the cylindrical and Cartesian coordinate systems.Translated from Prikladnaya Mekhanika, Vol. 32, No. 2, pp. 53–58, February, 1996.     

Stability of cylindrical shells with one plane of elastic symmetry under axial compression and torsion

A numerical technique is developed for stability analysis of laminated cylindrical shells with one plane of symmetry subject to axial compression and torsion. Shells filament-wound in directions other than the coordinate axes are considered as a special case. These shells are analyzed for stability under axial compression, external pressure, and torsion. It is shown that shells with a great number of plies and different ply angles may be considered orthotropic __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 115–121, August 2006.     

Convective instability in a rapidly rotating fluid layer in the presence of a non-uniform magnetic field

Magnetoconvective instabilities in a rapidly rotating, electrically conducting fluid layer heated from below in the presence of a non-uniform, horizontal magnetic field are investigated. It was first shown by Chandrasekhar that an overall minimum of the Rayleigh number may be reached at the onset of magnetoconvection when a uniform basic magnetic field is imposed. In this paper, we show that the properties of instability can be quite different when a non-uniform basic magnetic field is applied. It is shown that there is an optimum value of the Elsasser number provided that the basic magnetic field is a monotonically decreasing or increasing function of the vertical coordinate. However, there exist no optimum values of the Elsasser number that can give rise to an overall minimum of the Rayleigh number at the onset of magnetoconvection if the imposed basic magnetic field has an inflexion point. The project supported by the National Natural Science Foundation of China (40174026 and 40074041)     

Measurement of residual stress using interferometric moiré: A new insight

The use of interferometric moiré and hole drilling to determine residual stress has been well reported and accepted for stress fields whose principal directions can be predicted well enough to permit the moiré grids to be aligned with the principal strain axes. When the principal strains do not align themselves with the grid axes, a third strain component can be obtained by working with the diagonal pitch of the moiré grid, but this requires resetting the optical bench to the lower frequency. Diffraction efficiency is lost, with an additional loss in sensitivity. In this paper, the authors determine the shear strain component by observing the rotation of the moiré fringes in close proximity to the hole. The results of experiments on a specimen containing a model residual stress distribution are presented and compared with the theoretical prediction. Finally, the isothetic contours, based on elastic theory, were computed and plotted for several cases to verify this proposition. These results and the expected residual stress distribution are also compared to the experimentally obtained moiré fringes.     

The Effect of Strong Heterogeneity and Strong Throughflow on the Onset of Convection in a Porous Medium: Non-Periodic Global Variation

The effect of strong throughflow and strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been extended to deal with the case of vertical throughflow.     

Identifying Symmetry Classes of Elasticity Tensors Using Monoclinic Distance Function

We present a method to identify the symmetry class of an elasticity tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the concept of distance in the space of tensors, and relies on the monoclinic or transversely isotropic distance function. Since the orientation of a monoclinic or transversely isotropic tensor depends on two Euler angles only, we can plot the corresponding distance functions on the unit sphere in ℝ³ and observe the symmetry pattern of the plot. In particular, the monoclinic distance function vanishes in the directions of the normals of the mirror planes, so the number and location of the zeros allows us to identify the symmetry class and the orientation of the natural coordinate system. Observing the approximate locations of the zeros on the plot, we can constrain a numerical algorithm for finding the exact orientation of the natural coordinate system.