State space solution of non-axisymmetric Biot consolidation problem for multilayered porous media

In this paper, the non-axisymmetric Biot consolidation problem for multilayered porous media is studied. Taking stresses, pore pressure and displacements at layer interfaces as basic unknown functions, two sets of partial differential equations, which are independent each other, are formulated. Using Fourier expansion, Laplace transforms and Hankel transforms with respect to the circumferential, time and radial coordinates, respectively, the partial differential equations presented are reduced to the ordinary differential equations. Transfer matrices describing the transfer relation between the state vectors for a finite layer are derived explicitly in the transform space. Using the transfer matrices presented, three cases are studied for the lower surface: (1) permeable rough rigid base, (2) impermeable rough rigid base, and (3) poroelastic half space. The explicit solution in the transform space is presented. Considering the continuity condition at layer interfaces, the solutions of the non-axisymmetric Biot consolidation problems for multilayered semi-infinite porous media are presented in the integral form. The time histories of displacements, stresses and pore pressure are obtained by solving a linear equation system for discrete values of Laplace-Hankel transform inversions.     

Transfer matrix solutions for three-dimensional consolidation of a multi-layered soil with compressible constituents

In this paper, transfer matrix solutions for three-dimensional consolidation of a multi-layered soil considering the compressibility of pore fluid are presented. The derivation of the solutions starts with the fundamental differential equations of Biot’s three-dimensional consolidation theory, takes into account the compressibility of pore fluid in the Cartesian coordinate system, and introduces the extended displacement functions. The relationship of displacements, stresses, excess pore water pressure, and flux between the ground surface (z = 0) and an arbitrary depth z is established for Biot’s three-dimensional consolidation problem of a finite soil layer with compressible pore fluid by taking the Laplace transform with respect to t and the double Fourier transform with respect to x and y, respectively. Based on this relationship of the transfer matrix, the continuity between layers, and the boundary conditions, the solutions for Biot’s three-dimensional consolidation problem of a multi-layered soil with compressible constituents in a Laplace-Fourier transform domain is obtained. The final solutions in the physical domain are obtained by inverting the Laplace-Fourier transforms. Numerical analysis is carried out by using a corresponding program based on the solutions developed in this study. This analysis demonstrates that the compressibility of pore fluid has a remarkable effect on the process of consolidation.     

Free vibration analysis of non-axisymmetric and axisymmetric structures by the dual reciprocity BEM

The dual reciprocity boundary element method (DR/BEM) is employed for the free vibration analysis of three-dimensional non-axisymmetric and axisymmetric elastic solids. The method uses the elastostatic fundamental solution in the integral formulation of elastodynamics and as a result of that, an inertial volume integral is created in addition to the boundary ones. This volume integral is transformed into a surface integral by invoking the reciprocal theorem and expanding of the displacement field into a series involving seven different approximation functions. The approximation functions used are local radial basis functions (RBFs) and are applied in combination (or not) with global basis functions (augmentation). All these functions are compared in terms of the accuracy they provide. The axisymmetric case is efficiently treated with the aid of the fast Fourier transform (FFT) algorithm in order to provide even non-axisymmetric vibration modes. Two representative numerical examples involving the determination of natural frequencies and modal shapes of an elastic cube and an elastic cylinder serve to investigate in detail the potentiality of each of the seven approximation functions tested to provide results of high accuracy and to reach useful practical conclusions.     

State space solution to three-dimensional consolidation of multi-layered soils

From the governing equations of a saturated poro-elastic soil in the Cartesian coordinate system, the state space equations of Biot’s three-dimensional consolidation problems are obtained by the Laplace transform and the double Fourier transform. Transfer matrix describing the transfer relation between the state vectors for a finite layer is derived explicitly in the transform space. Based on the continuity conditions between adjacent layers and the boundary conditions, the solution for three-dimensional consolidation of multi-layered soils is derived in a transformed domain. This solution is then transferred to that in a physical domain by the inversion of the double Fourier transform and the Laplace transform. Numerical analysis is carried out for three-dimensional consolidation of single, two, and multi-layered soils. The results for single and two-layered soils are compared well with those by the finite layer method in the literature. Numerical results for three-dimensional consolidation of five-layered soils are presented in this paper as an example to illustrate the use of the solution in this paper for three-dimensional consolidation of more than two-layered soils.     

Analytical solutions to the planar non-axisymmetric elasticity and thermoelasticity problems for homogeneous and inhomogeneous annular domains

This paper presents an analytical approach to solving the plane non-axisymmetric elasticity and thermoelasticity problems in terms of stresses for isotropic, homogeneous or inhomogeneous annular domains. The key feature of this approach is integration of the equilibrium equations in order to: a) express all the stress-tensor components in terms of a governing stress; b) deduce the integral equilibrium conditions, which are vital for the solution. Because the equilibrium equations are insensitive of material properties, the obtained expressions and integral conditions fit both homogeneous and inhomogeneous cases. The governing stress is derived out of the compatibility equation. Regarding complete construction of the solution, the integral compatibility conditions are deduced by integrating the strain-displacement relations. In the case of inhomogeneous material, the governing compatibility equation is reduced to Volterra type integral equation which then is solved by simple iteration method. The rapid convergence of the iterative procedure is established.     

Extended Sneddon and Muki solutions for multilayered elastic materials

This paper extends the Sneddon and Muki solutions to solve elastostatic problems in multilayered elastic materials. Using Hankel transforms, Sneddon and Muki developed general solutions of elastostatic problems in elastic materials occupying either halfspace or one layer regions. Based on the Sneddon and Muki solutions and a transfer matrix method, solutions are presented for elastostatic problems in multilayered elastic materials subject to both external and internal loading. A special technique is adopted to eliminate the ill-conditioned matrices associated with the conventional transfer matrix method for multilayered materials. The extended Sneddon and Muki solutions can be easily calculated using a personal computer. Numerical examples given in the paper show that the extended Sneddon and Muki solution generates results with high accuracy and efficiency. The extended Sneddon and Muki solution has wider applications to engineering science.     

Analytical solutions of functionally graded piezoelectric circular plates subjected to axisymmetric loads

Based on the three-dimensional theory of piezoelectricity, this work analytically investigates the axisymmetric bending of circular plates whose material properties vary along the thickness. The transverse loads are expanded in terms of the Fourier–Bessel series, and the solutions corresponding to each item of the series are derived by a semi-inverse method. The overall results are then obtained through the superposition principle. The exact solutions are obtained for two unusual boundary conditions if the material properties obey an exponential law. Meanwhile, a layerwise model is employed for the case of arbitrary and independent distribution of each component of material properties and the approximate solutions are proposed for simply supported and clamped boundaries. Finally, the numerical examples are illustrated, and the results are compared with those of the finite element method to demonstrate the present method.     

The method of fundamental solutions for axisymmetric potential problems

In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two classes of axisymmetric potential problems. In the first, the boundary conditions as well as the domain of the problem, are axisymmetric, and in the second, the boundary conditions are arbitrary. In both cases, the fundamental solutions of the governing equations and their normal derivatives, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals. The method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Modeling and simulation of two-dimensional consolidation for thermoset matrix composites

A finite element method is developed to solve two-dimensional consolidation problems for composites manufacturing. The consolidation governing equations, one for solid stress and one for fluid pressure, are derived using a local volume averaging approach, and the two equations are strongly coupled. A special anisotropic, hyperelastic constitutive equation is developed for the solid stress. This equation matches Gutowski's model for consolidation transverse to the fibers, and has a high stiffness parallel to the fibers. An updated Lagrangian method is used to solve the equations, using implicit time integration and a successive substitution method. The code is applied to several case studies to explore two-dimensional consolidation effects. A free edge affects the thickness profile during consolidation, but the final thickness can still be uniform. This effect is substantial in the region close to the edge, and it propagates progressively from the edge toward the center. Simulations were also performed for laminates that bend to form a corner. The corner is thicker than the flat region after consolidation. Wiggles, similar to fiber buckling, arise at low values of shear modulus when using a male mold. Large values of the solid shear modulus cause the corner effect to extend far into the adjacent flat region. The length of the flat region also affects the consolidation of the corner.     

Modeling of continuous ultrasonic impregnation and consolidation of thermoplastic matrix composites

Ultrasonic propagation was used to provide heat and pressure in order to perform impregnation and consolidation during production of thermoplastic matrix composites. For this purpose, a new experimental set-up, integrating a laboratory filament winding machine with a horn and a compaction roller, was developed.The heat transfer phenomena occurring during continuous impregnation and consolidation were simulated solving by finite element (FE) analysis the energy balance equations in 2D accounting for the heat generated by ultrasonic waves, the melting characteristics of the matrix and the movement of the thermoplastic commingled roving.The temperature distribution in the composite, predicted by the numerical simulations, was validated by temperature measurements during the production of E-glass/polypropylene cylinders, with the optimized parameters obtained by the FE analysis. The ultrasonic consolidated composite cylinders were characterized by low void content and a shear modulus comparable with that obtained by the micromechanical analysis.     

Analysis of vertically loaded piles in multilayered transversely isotropic soils by BEM

This paper presents a boundary element method (BEM) for the static analysis of vertically loaded single piles embedded in multilayered transversely isotropic soils by using an analytical layer-element solution. The solution which is especially suitable for numerical calculation is deduced through the combination of the order reduction approach and the technology of integral transform. A boundary element formula for axially loaded piles with both enlarged tip and head cap is established by taking the solution as the kernel function, and the compressibility of piles is further considered based on a direct algorithm which can avoid the occurrence of non-convergence results from the conventional iterative scheme while the stiffness of piles approaches to that of the soil foundation. Numerical examples including piles in a finite depth soil and multilayered soils of transverse isotropy are carried out to study the influence of soil properties on single pile behaviors.     

The method of particular solutions for solving axisymmetric polyharmonic and poly-Helmholtz equations

In this paper we derive analytical particular solutions for the axisymmetric polyharmonic and poly-Helmholtz partial differential operators using Chebyshev polynomials as basis functions. We further extend the proposed approach to the particular solutions of the product of Helmholtz-type operators. By using this formulation, we can approximate the particular solution when the forcing term of the differential equation is approximated by a truncated series of Chebyshev polynomials. These formulas were further implemented to solve inhomogeneous partial differential equations (PDEs) in which the homogeneous solutions were obtained by the method of fundamental solutions (MFS). Several numerical experiments were carried out to validate our newly derived particular solutions. Due to the exponential convergence of Chebyshev interpolation and the MFS, our numerical results are extremely accurate.     

Microstructural aspects of titanium metal matrix composites in consolidation processing

Foil–fiber–foil (FFF) method together with vacuum hot pressing (VHP) has been used for the development of titanium metal matrix composites (MMCs). Heterogeneous microstructures prior to and following consolidation have been quantified, and the relations to densification behavior investigated. As shown by the results, dramatic variations of the microstructures including equiaxed α, transformed β and Widmanstätten α are obtained during the process according to the fiber distributions. The dependence of microstructures on the consolidation then has been explained in terms of the change in mechanisms such as grain growth and recrystallization that occur with changing levels of inhomogeneity of deformation.     

Ritz solutions for axisymmetric temperature and stress fields in finite elastic cylinders

Variational formulations are given for the stationary heat conduction and thermodelasticity problems of an orthotropic, non-homogeneous, finite length, cylinder. It is demonstrated that a single-element approach based upon Ritz approximations and the method of Lagrange multipliers offers and efficient means for computing the temperature and thermal stress distributions.     

Near-wall turbulence in the axisymmetric flow of weak acqueous polymer solutions

The axisymmetric turbulent boundary layer is computed by using a finite-difference method and its fluctuation characteristics are determined on the basis of a generalized mixing-path hypothesis.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 58, No. 4, pp. 545–550, April, 1990.     

Exact elasticity solutions to rotating pressurized axisymmetric vessels made of functionally graded materials (FGM)

In this study the plane elasticity equations of pressurized and rotating structures, generalized analytical closed-form solutions are presented for hollow cylinders, spheres and thin disks. Three well known material pairs are graded in radial direction by using the power rule for various boundary and load conditions. Stress and displacement solutions are obtained. The effect of a constant rotational velocity is also investigated separately in absence of the internal and external pressures. Crucial results are also presented in tabular form.     

NiAl-B composites with nanocrystalline intermetallic matrix produced by mechanical alloying and consolidation

Powder mixtures with equiatomic Ni–Al stoichiometry and with the addition of 5, 10, 20 and 30 vol% of boron were mechanically alloyed in a high-energy SPEX mill. Differential scanning calorimetry (DSC) was used for examination of the thermal behaviour of the milled powders. The mechanically alloyed powders and powders after DSC examinations were investigated by X-ray diffraction (XRD). For all the powder mixtures, a nanocrystalline NiAl intermetallic phase was formed during milling. With the increase of boron concentration in the mixtures, more intense refinement of the NiAl grain size during mechanical alloying was observed. Scanning electron microscopy (SEM) and energy dispersive X-ray spectroscopy (EDS) examinations showed that the produced powders have composite structure, with boron particles uniformly distributed in the nanocrystalline NiAl intermetallic matrix. The density of the composite powders decreases with the increase of boron content, following the rule of mixture.The produced powders were subjected to consolidation by hot-pressing at 800 °C under the pressure of 7.7 GPa for 180 s. The produced bulk materials were investigated by XRD, SEM and EDS as well as characterised by hardness, density and open porosity measurements. It was found that during applied consolidation process the nanocrystalline structure of the NiAl matrix was maintained. The average hardness of the bulk composite samples is in the range of 10.58–12.6 GPa, depending on boron content, increases with the increase of boron content, and is higher than that of the NiAl intermetallic reference sample (9.53 GPa). The density of the bulk composite samples is the same as that of the corresponding powders after milling, decreases with the increase of boron content and is lower than that of the NiAl reference sample. To the best of our knowledge, the NiAl-B composites with nanocrystalline intermetallic matrix have been produced for the first time.