Shakedown of unidirectional composites

The shakedown problem for a composite lamina made of an elastic-plastic matrix and elastic cylindrical fibers is studied. The plastic deformation modes of the lamina are reviewed, and it is concluded that significant shakedown effects can be caused only by the I₁ = 1/2(T₁₁ + T₂₂) and I₂ = T₃₃ components of the remotely applied stress field which are symmetric about the axis x₃ of the fiber; T₁₁ and T₂₂ are the normal composite stresses in the transverse plane. It is shown that the I₁I₂ stress system is needed also to represent thermal loads caused by a uniform change of temperature in the composite.Two methods for evaluation of shakedown limits in the I₁I₂-plane are described. First, the classical approach involving the determination of parametric families of self-stress fields and the solution of mathematical programming problems is used. Results are presented for selected B-Al, Be-Al, B-Ti and B-Mg composites.In the second method, the shakedown problem is related to the recently developed kinematic hardening rules for fibrous composites. It is shown that the composite will shake down for any loading program within a prescribed domain in the I₁I₂-plane, providing that the domain can be contained within a translated initial yield surface. This approach leads to a closed-form evaluation of shakedown limits for any arbitrary combination of mechanical and thermal variable cyclic loads in fibrous composites with temperaturedependent matrix yield strengths.The relationship between shakedown and fatigue in metal matrix composites is discussed.     

Stress Singularity in an Elastic Cylinder of Cylindrically Anisotropic Materials

The issue of stress singularity in an elastic cylinder of cylindrically anisotropic materials is examined in the context of generalized plane strain and generalized torsion. With a viewpoint that the singularity may be attributed to a conflicting definition of anisotropy at r=0, we study the problem through a compound cylinder in which the outer cylinder is cylindrically anisotropic and the core is transversely isotropic. By letting the radius of the core go to zero, the cylinder becomes one with the central axis showing no conflict in the radial and tangential directions. Closed-form solutions are derived for the cylinder under pressure, extension, torsion, rotation and a uniform temperature change. It is found that the stress is bounded everywhere, and singularity does not occur if the anisotropy at r=0 is defined appropriately. This revised version was published online in July 2006 with corrections to the Cover Date.     

The importance of particle type selection and temperature control for on-chip free-flow magnetophoresis

Free-flow magnetophoresis is becoming a popular technique for the separation and manipulation of magnetic particles and materials in microsystems. A wide variety of magnetic particles are commercially available that differ greatly in size and in magnetic properties. To investigate the suitability of different particle types for magnetophoretic operations in microfluidic devices, we compared a range of particles from three manufacturers by pumping them through a microfluidic separation chamber and deflecting them from the direction of laminar flow by applying an external magnetic field. The on-chip deflection of particles was compared to data provided by the manufacturers and magnetisation data obtained from vibrating sample magnetometer (VSM) measurements. Additionally, the extent of deflection was examined over a range of temperatures. Deflection distances were found to increase significantly with increasing temperature. Further to this, a separation of 2.8 and 1.0 μm magnetic particles was performed at different temperatures. Separation resolution was found to improve at higher temperatures. Hence, temperature manipulation provides a simple and effective means for improving a magnetic separation or for controlling the angle at which a particle is deflected from its hydrodynamic flow path.     

A circular elastic cylinder under its own weight

An exact analysis of deformation and stress field in a finite circular elastic cylinder under its own weight is presented, with emphasis on the end effect. The problem is formulated on the basis of the state space formalism for axisymmetric deformation of a transversely isotropic body. Upon delineating the Hamiltonian characteristics of the formulation, a rigorous solution which satisfies the end conditions is determined by using eigenfunction expansion. The results show that the end effect is significant but confined to a local region near the base where the displacement and stress distributions are remarkably different from those according to the simplified solution that gives a uniaxial stress state. It is more pronounced in the cylinder with the bottom plane being perfectly bonded than in smooth contact with a rigid base.     

A refined state space formalism for piezothermoelasticity

The state space formalism for piezothermoelasticity [Tarn, J.Q., 2002c. A state space formalism for piezothermoelasticity. International Journal of Solids and Structures 39, 5173–5184.] is refined by introducing the generalized displacement vector and generalized stress vectors as the fundamental variables in which appropriate electrical variables are included. The basic equations of piezoelectricity with temperature change are formulated neatly into a state equation and an output equation in terms of the generalized displacement vector and generalized stress vectors. The formalism bears a remarkable resemblance to its elastic counterpart. Various problems of piezothermoelasticity can be solved by simple extension of the corresponding solutions of anisotropic elasticity. For illustration, some fundamental problems are studied within the context and exact solutions are obtained in a systematic and self-contained manner.     

Flat Chains in Banach Spaces

We extend the notion of a flat chain with coefficients in a normed abelian group from Euclidean space to an arbitrary Banach space and prove a compactness result. We also remove the condition that a flat chain with arbitrary coefficients have finite mass in order for its support to exist. This research was part of the author’s Ph. D. dissertation at Stanford University.     

Axisymmetric Deformation of a Transversely Isotropic Cylindrical Body: A Hamiltonian State-Space Approach

A state-space approach for exact analysis of axisymmetric deformation and stress distribution in a circular cylindrical body of transversely isotropic material is developed. By means of Hamiltonian variational formulation via Legendre’s transformation, the basic equations in cylindrical coordinates are formulated into a state-space framework in which the unknown state vector comprises the displacements and associated stress components as the dual variables and the system matrix possesses the symplectic characteristics of a Hamiltonian system. Upon delineating the symplecticity of the formulation, a viable solution approach using eigenfunction expansion is developed. For illustration, an exact analysis of a finite thick-walled circular cylinder under internal and external pressures is presented, with emphasis on the end effects.     

On-chip processing of particles and cells via multilaminar flow streams

The processing of particles, cells, and droplets for reactions, analyses, labeling, and coating is an important aspect of many microfluidic workflows. However, performing multi-step processes is typically a laborious and time-consuming endeavor. By exploiting the laminar nature of flow within microchannels, such procedures can benefit in terms of both speed and simplicity. This can be achieved either by manipulating the flow streams around the objects of interest, particularly for the localized perfusion of cells, or by manipulating the objects themselves within the streams via a range of forces. Here, we review the variety of methods that have been employed for performing such “multilaminar flow” procedures on particles, cells, and droplets.