Materials with specific microstructural characteristics and composite structures are able to exhibit negative Poisson's ratio. This result has been proved for continuum materials by analytical methods in previous works of the first author, among others . Furthermore, it also has been shown to be valid for certain mechanisms involving beams or rigid levers, springs or sliding collars frameworks and, in general, composites with voids having a nonconvex microstructure.Recently microstructures optimally designed by the homogenization approach have been verified. For microstructures composed of beams, it has been postulated that nonconvex shapes with re-entrant corners are responsible for this effect . In this paper, it is numerically shown that mainly the shape of the re-entrant corner of a non-convex, star-shaped, microstructure influences the apparent (phenomenological) Poisson's ratio. The same is valid for continua with voids or for composities with irregular shapes of inclusions, even if the individual constituents are quite usual materials. Elements of the numerical homogenization theory are reviewed and used for the numerical investigation.
An exact axisymmetric piezothermoelastic solution is presented for a simply-supported hybrid cylindrical shell made of cross-ply composite laminate and piezoelectric layers. Numerical results for hybrid shells are presented for sinusoidal and central band thermal and electrical loads. The effect of the loading, the radius-to-thickness ratio, the span-to-radius ratio and the number of layers of the substrate on the response is investigated. The interface between the substrate and the actuated piezoelectric layer has been found to be subjected to high shear stress. It has been shown that the maximum values of the deflection and the stresses, due to thermal load, can be appreciably reduced by appropriate application of actuation potential.
A formulation of the equilibrium problem for nonlinear elastic networks is presented. Explicit necessary and sufficient conditions for minimum-energy configurations are derived. These are used to generate a relaxed formulation of the theory in which fibre slackening is accounted for automatically. For the relaxed problem, minimum-energy and uniqueness theorems are proved and used as the basis of a numerical method in which equilibrium configurations are recovered asymptotically in the long-time limit of an artificial dynamical problem. Such an approach is particularly useful for networks, as stiffness-based equilibrium formulations are known to suffer from ill-conditioning in a wide variety of applications. Several illustrative examples are discussed.
Nonrotating half-planes in contact under oblique loading are investigated in this paper. The solution is based on the influence integrals of the Flamant solution. The problem is determined by two integral equations for the normal and tangential stresses, which are uncoupled for special cases, as bodies of similar material in contact. In order to simplify the singular integrals, the method of superposition of flat punches is used. The result for the symmetric case is almost identical with the axisymmetric solution. For polynomial profiles of the form x s , the Muskhelishvili potentials can be written in terms of a complex hypergeometric function. The interior stress field is illustrated for an example.
A new model for the description of micro- and macro-dynamics of linear elastic solid body with micro-periodic nonconnected fluid filled inclusions is proposed. Using the model we are able to account for the microstructural length-scale effect on the global dynamic behavior of the body. The effect falls out of the scope of the known homogenized models obtained by asymptotic approaches. The general equations obtained in the model proposed here are applied in order to investigate dispersion phenomena in wave propagation problems. The obtained results are compared with those for the asymptotic model when the length-scale effect vanishes. It is shown that this effect plays an important role in the analysis of dynamic problems.
A phenomenological constitutive model for characterization of creep and damage processes in metals is applied to the simulation of mechanical behaviour of thin-walled shells and plates. Basic equations of the shell theory are formulated with geometrical nonlinearities at finite time-dependent deflections of shells and plates in moderate bending. Numerical solutions of initial/boundary-value problems have been obtained for rectangular thin plates (two-dimensional case) and axisymmetrically loaded shells of revolution (one-dimensional case). Based on the numerical examples for the two problems, the influence of geometrical nonlinearities on the creep deformation and damage evolution in shells and plates is discussed.
Response of structures to earthquake excitations and response of vehicles to road undulations are two typical evolutionary random problems in engineering. Both kinds of the evolutionary random excitations can be regarded as evolved from stationary random excitations, though through two utterly different ways. The former one may be obtained by filtering a stationary random process through a linear time-dependent system, while the latter one may result from nonlinear transformations of the argument of a stationary random process. However, the response problems due to both types of excitations have much in common. By introducing the concept of “evolutionary frequency response”, the expressions of the response evolutionary spectra for both problems can be obtained in a unified, concise way, similar to the input/output PSD relationship in a stationary random problem. For both the evolutionary random problems, the solution procedures are all the same, but the expressions for evolutionary frequency responses are different from each other. Moreover, the evolutionary frequency responses may be interpreted as transient responses of the system subject to certain deterministic evolutionary harmonic excitations. In this sense, an evolutionary random response problem can be reduced to a deterministic response problem. Based on the complex modal analysis, a unified approach to these two response problems is derived here. The method can be applied to any linear time-invariant systems, whether they are symmetrical or not, and whether they are classically damped or not. And the method might be hopefully applied to nonlinear systems, if the statistical linearization technique is accompanied. To the knowledge of the authors, this unified approach to two types of evolutionary random response problems is the first time reported in literature.
This paper presents the exact relationships between the deflections and stress resultants of Timoshenko curved beams and that of the corresponding Euler-Bernoulli curved beams. The curved beams considered are of rectangular cross sections and constant radius of curvature. They may have any combinations of classical boundary conditions, and are subjected to any loading distribution that acts normal to the curved beam centreline. These relationships allow engineering designers to directly obtain the bending solutions of Timoshenko curved beams from the familiar Euler-Bernoulli solutions without having to perform the more complicated shear deformation analysis.
The effect of surface mass flux on the non-Darcy natural convection over a horizontal flat plate in a saturated porous medium is studied using similarity solution technique. Forchheimer extension is considered in the flow equations. The suction/injection velocity distribution has been assumed to have power function form Bx l , similar to that of the wall temperature distribution Ax n , where x is the distance from the leading edge. The thermal diffusivity coefficient has been assumed to be the sum of the molecular diffusivity and the dynamic diffusivity due to mechanical dispersion. The dynamic diffusivity is assumed to vary linearly with the velocity component in the x direction, i.e. along the hot wall. For the problem of constant heat flux from the surface (n=1/2), similarity solution is possible when the exponent l takes the value −1/2. Results indicate that the boundary layer thickness decreases whereas the heat transfer rate increases as the mass flux parameter passes from the injection domain to the suction domain. The increase in the thermal dispersion parameter is observed to favor the heat transfer by reducing the boundary layer thickness. The combined effect of thermal dispersion and fluid suction/injection on the heat transfer rate is discussed.
A Theoretical analysis is carried out to study the boundary-layer flow over a continuously moving surface through an otherwise quiescent micropolar fluid. The transformed boundary-layer equations are solved numerically for a power-law surface velocity using the Keller-box method. The effects of the micropolar K and exponent m parameters on the velocity and microrotation field as well as on the skin-friction group are discussed in a detailed manner. It is shown that there is a near-similarity solution of this problem. The accuracy of the present solution is also discussed.
The stability of a spinning liquid-filled spacecraft has been investigated in the present paper. Using Galerkin's method, the attitude dynamic equations have been given. The Liapunov direct method was employed to obtain a sufficient condition for stability. Three kinds of characteristic modals were investigated: free motion of inviscid fluid, slosh motion and non-slosh motion. All characteristic problems can be solved numerically by the Finite Element Method or the Boundary Element Method. It has been demonstrated that the viscosity of the fluid has a dissipative effect at large Reynolds number, while the slosh motion plays a destabilizing role. The non-slosh model of fluid does not affect the stability criterion.
Three-dimensional axisymmetric solution is presented for a simply supported piezoelectric cylindrical shell. The variables are expanded in Fourier series to satisfy the boundary conditions at the ends. The solution of the governing differential equations with variable coefficients is constructed as a product of an exponential function and a power series. The coefficients of terms of all degrees in the governing equations are set to zero, yielding a characteristic equation for the exponent and recursive relations for the coefficients of the power series. Results are presented illustrating the effect of thickness parameter of the shell. An inverse problem of inferring the applied temperature from the measured potential difference has been solved.
A transient contact problem with frictional heating and wear for two nonuniform sliding half-spaces is considered. One of the two half-spaces is assumed to be slightly curved to give a Hertzian initial pressure distribution: the other is a rigid nonconductor. Under the assumption that the contact pressure distribution could be described by Hertz formulas during all the process of interaction, the problem is formulated in terms of one integral equation of Volterra type with unknown radius of contact area. A numerical solution of this equation is obtained using a piecewise-constant presentation of an unknown function. The influence of operating parameters on the contact temperature and the radius of the contact area is studied.
Further investigation of subinterface cracks in bimaterial solids is presented. The traction method is proposed permitting easy calculation of the stress intensity factors of the cracks. The elastic T-terms of the cracks are determined. The J 1 and J 2 integrals are analyzed for a contour enclosing all the cracks in the global coordinate system x,y, where the x-axis is parallel to the interface. Numerical examples are given, and the results are presented for two kinds of material combinations, Cu/Al2O3 and Ni/MgO.
The nonlinear bending theory for symmetrically laminated elliptical plates exhibiting rectilinear orthotropy with transverse shear deformation is developed. Using Galerkin's method, the paper solves the problem of large deflections for plates under uniform lateral pressure. The special case of symmetrically laminated rectilinearly orthotropic circular plates is also discussed. Analytical solutions obtained may be applied directly to the design of engineering structures.
An integral equation formulation for the dynamic biaxial response of slightly curved elastic-viscoplastic beams is presented in the context of a multiple field analysis, which takes into account the geometrically nonlinear influence of moderately large deflections. Materials are considered in the regime of rate-dependent plasticity and are subjected to accumulated ductile damage. The latter is modeled by the growth of voids in the plastic zones of an initially porous elastic material. Inelastic defects of the material are considered in the linear elastic background beam by a second imposed strain field (eigenstrains). Geometrically nonlinear effects of large deflections under conditions of immovable supports are approximately taken into account. By inspection, they render another “strain field” to be imposed on the linear background beam. Superposition applies in the linear elastic background in an incremental formulation. Linear methods, as those based on Green's functions and Duhamel's integral, are used to account for the given loads as well as for the resultants of the imposed strain fields. The intensity and the distribution of the imposed strain fields are calculated incrementally in a time-stepping procedure. They are determined by the constitutive law and by application of the nonlinear geometric relations. The numerical procedure resulting from the multiple fields in the elastic background is illustrated for two cases: (1) a preloaded viscoplastic beam of rectangular cross section is subjected to oblique flexural vibrations when forced by a sinusoidal load, and (2) an I-beam with a prescribed initial curvature is severely impacted and thus driven into the plastic regime.
A criterion for ductile fracture is introduced in the finite element simulation of sheet metal forming. From the calculated histories of stress and strain in each element, the fracture initiation site and the critical stroke are predicted by means of the ductile fracture criterion. The calculations are carried out for axisymmetric stretch forming of various aluminium alloy sheets and their laminates clad by mild steel sheets. The predictions so obtained are compared with experimental observations. The results show that the combination of the finite element simulation and the ductile fracture criterion enables the prediction of forming limit in a wide range of sheet metals.
Stability of a heavy rotating rod with a variable cross section is studied by energy method. Bifurcation points for the system of equilibrium equations are analyzed. It is shown that for the case when the rotation speed exceeds the critical one, the trivial solution ceases to be the minimizer of the potential energy, so that rod loses stability, according to the energy criteria. Also, a new estimate of the maximal rod deflection in the post-critical state is obtained.
In this paper, the curved-crack problem for an infinite plate containing an elastic inclusion is considered. A fundamental solution is proposed, which corresponds to the stress field caused by a point dislocation in an infinite plate containing an elastic inclusion. By placing the distributed dislocation along the prospective site of the crack, and by using the resultant force function as the right-hand term in the equation, a weaker singular integral equation is obtainable. The equation is solved numerically, and the stress intensity factors at the crack tips are evaluated. Interaction between the curved crack and the elastic inclusion is analyzed.