Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model

In the present study, we consider a thermoelastic half-space made of a functionally gradient material with an insulated crack, which is subjected to a thermal impact. The memory-dependent heat conduction model is adopted for analysis. By using the Fourier and Laplace transforms, the thermoelastic problem is formulated in terms of singular integral equations which can be solved numerically. Effects of the time delay, kernel function, and nonhomogeneity parameters on the temperature and stress intensity factor are analyzed. Our results are also compared with those based on the Fourier and CV heat conduction models, which can be viewed as two special cases of the present model. In conclusion, the memory-dependent derivative and nonhomogeneity parameters play an essential role in controlling the heat transfer process.     

Arbitrary-order functionally fitted energy-diminishing methods for gradient systems

It is well known that for gradient systems in Euclidean space or on a Riemannian manifold, the energy decreases monotonically along solutions. In this letter we derive and analyse functionally fitted energy-diminishing methods to preserve this key property of gradient systems. It is proved that the novel methods are energy-diminishing and can achieve damping for very stiff gradient systems. We also show that the methods can be of arbitrarily high order and discuss their implementations. A numerical test is reported to illustrate the efficiency of the new methods in comparison with three existing numerical methods in the literature.     

Contact problems of thermoelasticity for a half-space of a functionally gradient material

We propose a method of solution of problems of thermoelasticity for an inhomogeneous half-space. We assume that Poisson's ratio of a material in the half-space is constant, and Young's modulus and the coefficients of linear thermal expansion and thermal conduction vary exponentially with distance from the surface of the half-space. As an example, we consider the contact problem on sliding an inhomogeneous body along the surface of a rigid base with regard for frictional heating. I. Franko L'viv University, L'viv; Warsaw University, Warsaw. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 45–56, April–June, 1998.     

A comparison of strain gradient theories with applications to the functionally graded circular micro-plate

The strain gradient theory of Zhou et al. is re-expressed in a more direct form and the differences with other strain gradient theories are investigated by an application on static and dynamic analyses of FGM circular micro-plate. To facilitate the modeling, the strain gradient theory of Zhou et al. is re-expressed in cylindrical coordinates, and then the governing equation, boundary conditions and initial condition for circular plate are derived with the help of the Hamilton's principle. The present model can degenerate into other models based on the strain gradient theory of Lam et al., the couple stress theory, the modified couple stress theory or even the classical theory, respectively. The static bending and free vibration problems of a simply supported circular plate are solved. The results indicate that the consideration of strain gradients results in an increase in plate stiffness, and leads to a reduction of deflection and an increase in natural frequency. Compared with the reduced models, the present model can predict a stronger size effect since the contribution from all strain gradient components is considered, and the differences of results from all these models are diminishing when the plate thickness is far greater than the material length-sale parameter.     

Size dependent analysis of functionally graded microbeams using strain gradient elasticity incorporated with surface energy

In this study, strain gradient theory is used to show the small scale effects on bending, vibration and stability of microscaled functionally graded (FG) beams. For this purpose, Euler–Bernoulli beam model is used and the numerical results are given for different boundary conditions. Analytical solutions are given for static deflection and buckling loads of the microbeams while generalized differential quadrature (GDQ) method is used to calculate their natural frequencies. The results are compared with classical elasticity ones to show the significance of the material length scale parameter (MLSP) effects and the general trend of the scale dependencies. In addition, it is shown the effect of surface energies relating to the strain gradient elasticity is negligible and can be ignored in vibration and buckling analyses. Combination of the well-known experimental setups with the results given in this paper can be used to find the effective MLSP for metal-ceramic FG microbeams. This helps to predict their accurate scale dependent mechanical behaviors by the introduced theoretical framework.     

Thermal vibration analysis of functionally graded shallow spherical caps by introducing a decoupling analytical approach

Due to many applications of spherical shells on a circular planform such as the nose of the plane and spacecraft and caps of pressurized cylindrical tanks, in this article, free vibration analysis of a thin functionally graded shallow spherical cap under a thermal load is considered. A decoupling technique is employed to analytically solve the equations of motion. Introducing some new auxiliary and potential functions as well as using the separation method of variables, the governing equations of the vibrated functionally graded shallow spherical cap were exactly solved. The superiority of the relations is validated by some comparative studies for various types of boundary conditions. Also, thermal buckling phenomenon is considered. Using new different material models, efficiency of the functionally graded materials is investigated when the shell is subjected to a temperature gradient. The effects of various parameters such as radius of curvature, material grading index and thermal gradient are discussed.     

Ductile failure with gradient plasticity coupled to the phase-field fracture at finite strains

Most metals fail in a ductile fashion, i.e, fracture is preceded by significant plastic deformation. The modeling of failure in ductile metals must account for complex phenomena at micro-scale, such as nucleation, growth and coalescence of micro-voids. In this work, we start with von-Mises plasticity model without considering void generation. The modeling of macroscopic cracks can be achieved in a convenient way by the continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities [1]. This avoids the use of complex discretization methods for crack discontinuities and can account for complex crack patterns. The key aspect of this work is the extension of the energetic and the stress-based phase field driving force function in brittle fracture to account for a coupled elasto-plastic response in line with our recent work [3]. We develop a new theoretical and computational framework for the phase field modeling of ductile fracture in elastic-plastic solids. To account for large strains, the constitutive model is constructed in the logarithmic strain space, which simplify the model equations and results in a formulation similar to small strains. We demonstrate the modeling capabilities and algorithmic performance of the proposed formulation by representative simulations of ductile failure mechanisms in metals. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory

In the present work, attention is focused on the prediction of thermal buckling and post-buckling behaviors of functionally graded materials (FGM) beams based on Euler–Bernoulli, Timoshenko and various higher-order shear deformation beam theories. Two ends of the beam are assumed to be clamped and in-plane boundary conditions are immovable. The beam is subjected to uniform temperature rise and temperature dependency of the constituents is also taken into account. The governing equations are developed relative to neutral plane and mid-plane of the beam. A two-step perturbation method is employed to determine the critical buckling loads and post-buckling equilibrium paths. New results of thermal buckling and post-buckling analysis of the beams are presented and discussed in details, the numerical analysis shows that, for the case of uniform temperature rise loading, the post-buckling equilibrium path for FGM beam with two clamped ends is also of the bifurcation type for any arbitrary value of the power law index and any various displacement fields.     

Dimension of gradient measures

We prove that if pure derivatives of a function on Rⁿ${\mathbb{R}}^n$ are complex measures, then their lower Hausdorff dimension is at least n−1$n-1$. The derivatives with respect to different coordinates may be of different order.     

Pointwise gradient estimates

We survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear, possibly degenerate, elliptic and parabolic equations.     

Differential gradient methods

A class of recently developed differential descent methods for function minimization is presented and discussed, and a number of algorithms are derived which minimize a quadratic function in a finite number of steps and rapidly minimize general functions. The main characteristics of our algorithms are that a more general curvilinear search path is used instead of a ray and that the eigensystem of the Hessian matrix is associated with the function minimization problem. The curvilinear search paths are obtained by solving certain initial-value systems of differential equations, which also suggest the development of modifications of known numerical integration techniques for use in function minimization. Results obtained on testing the algorithms on a number of test functions are also given and possible areas for future research indicated.     

Spinel fibers for ceramics

The process of formation of a fibrous spinel in the Al₂O₃–MgO system is investigated. The dependences of the spinel structure and properties on technological parameters of the process of synthesis is studied. Optimal relations between the fiber constituents for obtaining a stoichiometric spinel are determined and a two-stage mechanism of its formation has been found. Some ceramic materials are obtained from fibrous dispersions, and their physicomechanical characteristics have been estimated.Institute of General and Inorganic Chemistry, Belarus National Academy of Sciences, Minsk, Belarus. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 509–516, July–August, 1999.     

Proper gradient otopies

We prove that the inclusion of the space of proper gradient local maps into the space of proper local maps induces a bijection between the sets of the respective otopy classes of these maps.     

Mechanical stability of functionally graded stiffened cylindrical shells

This paper is concerned with the elastic buckling of stiffened cylindrical shells by rings and stringers made of functionally graded materials subjected to axial compression loading. The shell properties are assumed to vary continuously through the thickness direction. Fundamental relations, the equilibrium and stability equations are derived using the Sander’s assumption. Resulting equations are employed to obtain the closed-form solution for the critical buckling loads. The results show that the inhomogeneity parameter and geometry of shell significantly affect the critical buckling loads. The analytical results are compared and validated using the finite element method.     

Dynamic analysis of multi-directional functionally graded annular plates

Dynamic analysis of multi-directional functionally graded annular plates is achieved in this paper using a semi-analytical numerical method entitled the state space-based differential quadrature method. Based on the three-dimensional elastic theory and assuming the material properties having an exponent-law variation along the thickness, radial direction or both directions, the frequency equations of free vibration of multi-directional functionally graded annular plates are derived under various boundary conditions. Numerical examples are presented to validate the approach and the superiority of this method is also demonstrated. Then free vibration of functionally graded annular plates is studied for different variations of material properties along the thickness, radial direction and both directions, respectively. And the influences of the material property graded variations on the dynamic behavior are also investigated. The multi-directional graded material can likely be designed according to the actual requirement and it is a potential alternative to the unidirectional functionally graded material.