Relaxed Energy for Transversely Isotropic Two-Phase Materials

The paper gives a simple derivation of the relaxed energy W ^qc for the quadratic double-well material with equal elastic moduli and analyzes W ^qc in the transversely isotropic case. We observe that the energy W is a sum of a degenerate quadratic quasiconvex function and a function that depends on the strain only through a scalar variable. For such a W, the relaxation reduces to a one-dimensional convexification. W ^qc depends on a constant g defined by a three-dimensional maximum problem. It is shown that in the transversely isotropic case the problem reduces to a maximization of a fraction of two quadratic polynomials over [0,1]. The maximization reveals several regimes and explicit formulas are given in the case of a transversely isotropic, positive definite displacement of the wells. This revised version was published online in June 2006 with corrections to the Cover Date.     

Accurate dynamic analysis of functionally graded beams under a moving point load

Based on the physical neutral surface, an N-node novel weak form quadrature beam element is proposed and the explicit formulas for computing the stiffness and mass matrices are given. The proposed element is then used to analyze the dynamic behavior of the functionally graded material (FGM) beams under a moving point load. Both elasticity modulus and mass density vary exponentially across the thickness. Investigations show that the maximum dynamic magnification factors are independent of the power-law exponent k at a fixed nondimensional parameter α. This finding may be useful in design and engineering applications.     

Failure instability of a debonded interface in the presence of a main crack

The problem of fracture initiating from an edge crack in a nonhomogeneous beam made of two dissimilar linear elastic materials that are partially bonded along a common interface is studied by the strain energy density theory. The beam is subjected to three-point bending and the unbonded part of the interface is symmetrically located with regard to the applied loading. The applied load acts on the stiffer material, while the edge crack lies in the softer material. Fracture initiation from the tip of the edge crack and global instability of the composite beam are studied by considering both the local and global stationary values of the strain energy density function, dW/dV. A length parameter l defined by the relative distance between the maximum of the local and global minima of dW/dV is determined for evaluating the stability of failure initiation by fracture. Predictions on critical loads for fracture initiation from the tip of the edge crack, crack trajectories and fracture instability are made. In the analysis the load, the length of the edge crack and the length and position of the interfacial crack remained unchanged. The influence of the ratio of the moduli of elasticity of the two materials, the position of the edge crack and the width of the stiffer material on the local and global instability of the beam was examined. A general trend is that the critical load for crack initiation and fracture instability is enhanced as the width and the modulus of elasticity of the stiffer material increase. Thus, the stiffer material acts as a barrier in load transfer.     

Normal Modes and Nonlinear Stability Behaviour of Dynamic Phase Boundaries in Elastic Materials

This paper considers an ideal nonthermal elastic medium described by a stored-energy function W. It studies time-dependent configurations with subsonically moving phase boundaries across which, in addition to the jump relations (of Rankine–Hugoniot type) expressing conservation, some kinetic rule g acts as a two-sided boundary condition. The paper establishes a concise version of a normal-modes determinant that characterizes the local-in-time linear and nonlinear (in)stability of such patterns. Specific attention is given to the case where W has two local minimizers U ^A ,U ^B which can coexist via a static planar phase boundary. Being dynamic perturbations of such interesting configurations, this paper shows that the stability behaviour of corresponding almost-static phase boundaries is uniformly controlled by an explicit expression that can be determined from derivatives of W and g at U ^A and U ^B .     

Numerical analysis of large elasto-plastic deflection of constant curvature beam under follower load

This paper describes a method to analyze the elasto-plastic large deflection of a curved beam subjected to a tip concentrated follower load. The load is made to act at an arbitrary inclination with the tip tangent. A moment-curvature based constitutive law is obtained from linearly hardening model. The deflection governing equation obtained is highly non-linear owing to both kinematics and material non-linearity. It is linearized to obtain the incremental differential equation. This in turn is solved using the classical Runge–Kutta 4^th order explicit solver, thereby avoiding iterations. Elastic results are validated with published literature and the new results pertaining to elasto-plastic cases are presented in suitable non-dimensional form. The load to end angle response of the structure is studied by varying normalized material and kinematic parameters. It is found that the response curves overlap at small deflection corresponding to elastic deformation and diverge for difference in plastic property. The divergent response curves intersect with each other at higher deflection. The results presented also show that the approach may be used to obtain desired non-uniformly curved beam by suitably loading a uniform curvature beam.     

Optimal lower bounds on the hydrostatic stress amplification inside random two-phase elastic composites

Composites made from two linear isotropic elastic materials are subjected to a uniform hydrostatic stress. It is assumed that only the volume fraction of each elastic material is known. Lower bounds on all rth moments of the hydrostatic stress field inside each phase are obtained for r?2. A lower bound on the maximum value of the hydrostatic stress field is also obtained. These bounds are given by explicit formulas depending on the volume fractions of the constituent materials and their elastic moduli. All of these bounds are shown to be the best possible as they are attained by the hydrostatic stress field inside the Hashin-Shtrikman coated sphere assemblage. The bounds provide a new opportunity for the assessment of load transfer between macroscopic and microscopic scales for statistically defined microstructures.     

Pressure development in a non-Newtonian flow through a tapered tube

Summary The flow of viscous fluids through a tapered tube is very interesting from the standpoint of blood flow in blood vessels. The taper of the tube is an important factor in the pressure development. In the first place, we have given a brief summary of our theory of the steady convergent flow of non-Newtonian fluids characterized by an arbitrary time-independent flow curve through a slightly tapered tube. Based on our general formula for the flow per unit time, explicit formulae of the pressure gradient are obtained in several cases of non-Newtonian fluids specified by particular flow curves: power law fluid,Bingham body, and the fluid obeyingCassons equation. In all these cases it is shown that the pressure gradient is not constant along the axis but increases with decrease in the radius of the tapered tube. If we neglect quantities of order ² (: angle of taper), then the pressure gradient increases linearly with the distance along the axis of the tube.With 2 figures     

Surface and non-local effects for non-linear analysis of Timoshenko beams

In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beam theory. The proposed formulation also takes into consideration the surface stress effects. Eringen׳s non-local differential model has been used to rewrite the non-local stress resultants in terms of non-local displacements. Geometric non-linearities are taken into account by using the Green–Lagrange strain tensor. A C⁰ beam element with three degrees of freedom has been developed. Numerical solutions are obtained by performing a non-linear analysis for bending and free vibration cases. Simply supported and clamped boundary conditions have been considered in the numerical examples. A parametric study has been performed to understand the effect of non-local parameter and surface stresses on deflection and vibration characteristics of the beam. The solutions are compared with the analytical solutions available in the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam (Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effect of non-local parameter on deflections for other load cases such as uniformly distributed load and sinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for a cantilever beam with concentrated load at free end, there is definitely a dependency on non-local parameter when Timoshenko beam theory is used. Also the effect of local and non-local boundary conditions has been demonstrated in this example. The example has also been worked out for other loading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local or non-local boundary conditions on the end deflection in all these cases has also been brought out.     

Some remarks on the theory of shallow spherical shells

Conclusions It is shown that Vlasov's explicit solutions for shallow spherical shells correspond to a very special boundary condition which does not usually occur in practice.A shell loaded by a normal loading p is fully discussed and it is shown that the discrepancy between these results and those obtained by Vlasov may be considerable. An asymptotic solution of the same problem is also given.Finally it is indicated how Geckeler's approximate equations can be derived from a suitable transformation of the linearized Marguerre equations.This paper was prepared under the support of the Argentine Council for Scientific and Technological Research.     

Analysis of curved beams using a new differential transformation based curved beam element

Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material. The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved beam element. The static and free vibration of axially functionally graded tapered curved beams including shear deformation and rotary inertia are studied through solving several examples. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal beams (both forms—prime and quadratic) with hinged-hinged, hinged-clamped and clamped-clamped and clamped-free end restraints. Three general taper types (depth taper, breadth taper and square taper) for rectangular cross section are studied. Out of plane vibration is studied and the lowest natural frequencies are calculated and compared with the published results. Out of plane buckling is investigated for circular beams due to radial load.     

Maximum-energy-release-rate criterion applied to a tension-compression specimen with crack

The title problem is studied by using the explicit asymptetic analysis presented in the accompanying paper. The asymptotic analysis indicates that the very basic problem is a semi-infinite L-shaped crack governed by a single integral equation. This equation is discretized to a system of complex algebraic equations and solved by a standard HARWELL subroutine. It is found that the maximum-energy-release-rate criterion has two branches, one for tensile loads and one for compressive loads. Our numerical results indicate that the maximum energy-release rate is always associated with maximum K ₁ and K ₂=0, where K ₁ and K ₂ are the stress-intensity factors at the fractured tip. Thus, the well-known K-G relation valid for crack-parallel propagation also holds for non-crack-parallel propagations. This conclusion is, however, purely numerical.Supported by U.S. Army Research Office-Durham under Grant DAAG-29-76-G-0272.     

Transient responses of beams and plates subject to travelling load. Miscellaneous results

The steady-state response of a free and infinite Timoshenko beam is specified analytically in terms of non-dimensional displacements and stresses. The beam is supposed loaded by a travelling concentrated force or a moving step load. By a validated explicit numerical calculation, it is shown how a load travelling on a beam at constant velocity, from defined time and abscissa, generates a response which evolves towards the steady-state solution for a part, and towards a quantified transient solution for another part. Asymptotic values are given for the transient displacements and stresses according to the time and the speed of the loading. The solution is also found for a plate subject to a pressure, which spreads respecting the cylindrical symmetry. It is possible to identify in the response a part which follows the pressure front, and which is comparable with the steady-state response of a beam, and another transient part, which generates displacements and stresses with a much less oscillating character. An asymptotic solution is also presented for the plate.The whole series of the results makes it possible to better understand qualitatively the beginning of the transient response of a beam or of a plate to a moving load, and also makes it possible to estimate the stresses and displacements without needing specialised numerical codes.     

Transverse vibration of a multiple-Timoshenko beam system with intermediate elastic connections due to a moving load

Based on Timoshenko beam theory, the dynamic response of an elastically connected multiple-beam system is investigated. The identical prismatic beams are assumed to be parallel and connected by a finite number of springs. Assuming n parallel Timoshenko beams, the motion of the system is described by a coupled set of 2n partial differential equations. The method involves a change of variables and modal analysis to decouple and to solve the governing differential equations, respectively. A case study is solved in detail to demonstrate the methodology and several plots of the midpoint deflections of beams are given and investigated for different values of moving load velocity and the stiffness of elastic connections. From the numerical results it is observed that the maximum deflection of the multiple Timoshenko beam system is always smaller than one of a single beam.     

On Saint-Venant's Principle in three-dimensional nonlinear elasticity

Consider a long elastic isotropic beam with a convex cross-section and a sufficiently smooth boundary. Suppose that a self-equilibrated load is applied at each end but the sides are stress-free and there are no internal body forces. It is proved in the context of three-dimensional, nonlinear elastostatics that if the first four derivatives of the displacement vector are a priori assumed to be everywhere sufficiently small with respect to the physical constants and the geometry of the cross-section, then the strains at any point decay exponentially with the distance of the point from the nearest end.This result is an extension of known results on Saint-Venant's Principle in linear and two-dimensional nonlinear elasticity.     

A nonlinear mathematical model for large deflection of incompressible saturated poroelastic beams

Nonlinear governing equations are established for large deflection of incom- pressible fluid saturated poroelastic beams under constraint that diffusion of the pore fluid is only in the axial direction of the deformed beams.Then,the nonlinear bend- ing of a saturated poroelastic cantilever beam with fixed end impermeable and free end permeable,subjected to a suddenly applied constant concentrated transverse load at its free end,is examined with the Gaierkin truncation method.The curves of deflections and bending moments of the beam skeleton and the equivalent couples of the pore fluid pressure are shown in figures.The results of the large deflection and the small deflection theories of the cantilever poroelastic beam are compared,and the differences between them are revealed.It is shown that the results of the large deflection theory are less than those of the corresponding small deflection theory,and the times needed to approach its stationary states for the large deflection theory are much less than those of the small deflection theory.     

Stability and optimization of a clamped beam elastically restrained against translation on one end resting on Winkler foundation

In this study, stability and bimodal optimization of clamped beam elastically restrained against translation on one end subjected to a constant axially load are analyzed. The beam is positioned on elastic Winkler type foundation. The Euler method of adjacent equilibrium configuration is used in deriving the nonlinear governing equations. The critical load parameters, axial force and stiffness of foundation, are obtained for beam with the unit cross-sectional area.The shape of the beam stable against buckling that has minimal volume is determined by using Pontryagin’s maximum principle. The optimality conditions for the case of bimodal optimization are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. A first integral (Hamiltonian) is used to monitor accuracy of integration. It is shown that there is the saving in material for the same buckling force.     

Fracture toughness determination for compact tension specimens

An analysis of determining the plane stress fracture toughness based on a beam-on-elastic foundation model for compact tension specimens (CTS) covering a wide range of a/2H and d/W ratios is presented. The solution is achieved by using the Timoshenko beam theory and Pasternak foundation with alternative formulations of the foundation modulus and the shear parameter to reflect more accurately the stress-strain distributions at the crack tip.The solution applicable to a wider range of a/2H and d/W ratios becomes desirable for practical reasons. For instance, the determination of plane-strain fracture toughness from the CTS specimens at higher a/W ratios enables the reduction of loading capacity from a testing machine which may become prohibitively high for medium strength engineering materials. Maximum fatigue crack growth data to be measured from a CTS specimen also becomes possible when the validity of fracture toughness can be ensured at the extended a/W ratios.The computed fracture toughness from the present analysis are compared with those measured experimentally and found to be satisfactory not only for high a/W ratios but also for a wide range of a/2H ratios commonly used in double-cantilever beam specimens.     

Nonlinear motion of a beam

The investigation reported herein analyzes the vibration of a uniform beam with hinged ends which are restrained. The beam is subjected to a linearly-varying distributed load which has a maximum intensity w ₀ at the center and is released from rest when the load is suddenly removed. The motion is found to be inherently nonlinear, even for small vibrations, and there is dynamic mode-coupling. The mode frequencies are functionally related to initial conditions, particularly the amplitudes of all modes.     

Large deflection of a non-linear,elastic, asymmetric Ludwick cantilever beam subjected to horizontal force,vertical force and bending torque at the free end

The investigated cantilever beam is characterized by a constant rectangular cross-section and is subjected to a concentrated constant vertical load, to a concentrated constant horizontal load and to a concentrated constant bending torque at the free end. The same beam is made by an elastic non-linear asymmetric Ludwick type material with different behavior in tension and compression. Namely the constitutive law of the proposed material is characterized by two different elastic moduli and two different strain exponential coefficients. The aim of this study is to describe the deformation of the beam neutral surface and particularly the horizontal and vertical displacements of the free end cross-section. The analysis of large deflection is based on the Euler–Bernoulli bending beam theory, for which cross-sections, after the deformation, remain plain and perpendicular to the neutral surface; furthermore their shape and area do not change. On the stress viewpoint, the shear stress effect and the axial force effect are considered negligible in comparison with the bending effect. The mechanical model deduced from the identified hypotheses includes two kind of non-linearity: the first due to the material and the latter due to large deformations. The mathematical problem associated with the mechanical model, i.e. to compute the bending deformations, consists in solving a non-linear algebraic system and a non-liner second order ordinary differential equation. Thus a numerical algorithm is developed and some examples of specific results are shown in this paper.     

Stresses and deflections in a pseudo-elastic beam under cyclic loads

We consider a cantilever beam loaded by a concentrated transverse force at its free end. We assume that plane cross-sections remain plane in the deformed state and that the material obeys a particular non-linear stress-strain law, proposed by Landau and extended by Falk and Müller, in order to describe the pseudo-elastic behaviour. We find the explicit solution of the problem, and examine the deflection of the axis of the beam under the action of a prescribed slowly varying cyclical load.Received: 10 August 2004, Accepted: 19 August 2004, Published online: 4 March 2005PACS: 62.20.Fe