Explicit Closed-form Solution Research Articles

A continuum model for the static pull-in behavior of graphene nanoribbon electrostatic actuators with interlayer shear and surface energy effects

Based on multi-beam shear model theory, a continuum mechanics model is developed to investigate the pull-in instability of wedged/curved multilayer graphene nanoribbon (MLGNR) cantilever nanobeams subjected to electrostatic and Casimir forces. The first-order fringing-field correction, the interlayer shear between neighboring graphene nanoribbons (GNRs), surface elasticity, and residual surface tension are incorporated into the analytical model. An explicit closed-form analytical solution to the governing fourth-order nonlinear differential equation of variable coefficients is introduced for the static pull-in behavior of electrostatic nanoactuators using a Fredholm integral equation of the first kind. A comparison study for a [001] silver electrostatic nanoactuator indicates that the proposed analytical closed-form solution yields an improved accuracy over other analytical and numerical methods existing in literature. The results indicate that the interfacial slip between GNRs and the surface material parameters play a significant role in static pull-in behavior of MLGNR electrostatic nanoactuators. From the experimental data and atomistic simulations available in the literature, the value of interlayer shear modulus at the graphene/graphene interface is estimated to be in the order of magnitude of 10−1 GPa. The continuum model proposed in this study will be helpful for characterizing the mechanical properties of GNRs and the design of graphene-based nanoelectromechanical system devices.

Kinematic analysis of 7 degrees of freedom upper-limb exoskeleton robot with tilted shoulder abduction

An exoskeleton robot is a hybrid between human and robot where the two are mechanically integrated to function as one intelligent as well as powerful robot. This study presents a kinematic analysis of 7degrees of freedom (DOF) exoskeleton robot with its base rotational axis skewed to avoid singularity within workspace and to maximize ease of movement in all directions. The tilt of the base rotational axis configures the exoskeleton robot into an initial posture to which it is difficult to identify a geometric transformation from a kinematical model with all joint angels set to zero. When the limbs of the exoskeleton robot are configured either vertical or parallel to the ground, the resulting configuration can be initialized and then, ensuing kinematic analysis can be more conveniently performed from this nominal configuration. An analytical relationship defining a geometric transformation from the skewed configuration to the user-defined nominal configuration is developed and an explicit closed-form inverse kinematic solution for the 7 DOF exoskeleton robot is also presented.

Dynamic Approximate Solutions of the HJ Inequality and of the HJB Equation for Input-Affine Nonlinear Systems

The solution of most nonlinear control problems hinges upon the solvability of partial differential equations or inequalities. In particular, disturbance attenuation and optimal control problems for nonlinear systems are generally solved exploiting the solution of the so-called Hamilton-Jacobi (HJ) inequality and the Hamilton-Jacobi-Bellman (HJB) equation, respectively. An explicit closed-form solution of this inequality, or equation, may however be hard or impossible to find in practical situations. Herein we introduce a methodology to circumvent this issue for input-affine nonlinear systems proposing a dynamic, i.e., time-varying, approximate solution of the HJ inequality and of the HJB equation the construction of which does not require solving any partial differential equation or inequality. This is achieved considering the immersion of the underlying nonlinear system into an augmented system defined on an extended state-space in which a (locally) positive definite storage function, or value function, can be explicitly constructed. The result is a methodology to design a dynamic controller to achieve <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -disturbance attenuation or approximate optimality, with asymptotic stability.

Generalised Risk-Sensitive Control with Full and Partial State Observation

This paper generalises the risk-sensitive cost functional by introducing noise dependent penalties on the state and control variables. The optimal control problems for the full and partial state observation are considered. Using a change of probability measure approach, explicit closed-form solutions are found in both cases. This has resulted in a new risk-sensitive regulator and filter, which are generalisations of the well-known classical results.

Effects of heterogeneous elasticity coupled to plasticity on stresses and lattice rotations in bicrystals: A Field Dislocation Mechanics viewpoint

Abstract In the present paper, the static Field Dislocation Mechanics (FDM) theory is first recalled and next formulated for heterogeneous elasticity and plasticity with respect to a reference homogeneous medium. The issue of an infinite bicrystal with planar boundary undergoing piecewise uniform plastic distortions and elastic properties is then considered. The static FDM theory is accordingly used to derive explicit closed-form solutions of stress and lattice misorientation fields in the general context of heterogeneous anisotropic elasticity. These expressions are validated by Crystal Plasticity Finite Element simulations performed on a bicrystal with periodic boundary conditions. As a result of analytical expressions, it is possible to quantify the contribution of the different incompatibility sources, namely the contribution arising from elastic incompatibilities alone, that due to plastic incompatibilities alone and the one due to couplings between elastic and plastic incompatibilities. Besides, residual stresses and lattice misorientations are accordingly calculated in elastic/plastic bicrystals for different orientations of the elastic crystal. Results are compared with the corresponding homogeneous isotropic elasticity approximation. It is shown that elastic–plastic coupling incompatibilities may have consequent effects on residual stresses and lattice misorientations depending on the degree of elastic anisotropy of the material and the relative orientation of both crystals. As a conclusion, the consideration of coupling incompatibilities in multiscale plasticity models when predicting internal stresses and texture developments is discussed.

On the Compound Capacity of a Class of MIMO Channels Subject to Normed Uncertainty

The compound capacity of uncertain multiple-input multiple-output channels is considered, when the channel is modeled by a class described by a (known) nominal channel and a constrained-norm (unknown) uncertainty. Within this framework, two types of classes are investigated with additive and multiplicative uncertainties subject to a spectral norm constraint, using the singular value decomposition and related singular value inequalities as the main tools. The compound capacity is a maxmin mutual information, representing the capacity of the class, in which the minimization is done over the class of channels while the maximization is done over the transmit covariance. Closed-form solutions for the compound capacity of the classes are obtained and several properties related to transmit and receive eigenvectors are presented. It is shown that, under certain conditions, the compound capacity of the class is equal to the worst-case channel capacity, thus establishing a saddle-point property. Explicit closed-form solutions are given for the worst-case channel uncertainty and the capacity-achieving transmit covariance matrix: the best transmission strategy achieving the compound capacity is a multiple beamforming on the nominal (known) channel eigenmodes with the beam power distribution via the water filling at a degraded SNR. As the uncertainty increases, fewer eigenmodes are used until only the strongest one remains active so that transmit beamforming is an optimal robust transmission strategy in this large-uncertainty regime, for which explicit conditions are given. Using these results, upper and lower bounds of the compound capacity are constructed for other bounded uncertainties and some generic properties are pointed out. The results are extended to compound multiple-access and broadcast channels. In all considered cases, the price to pay for channel uncertainty is an SNR loss (or, equivalently, the nominal channel degradation) commensurate with the uncertainty set radius measured by the spectral norm and the optimal signaling strategy is the transmission on the degraded nominal channel.

Edge crack in front of anisotropic wedge interacting with anti-plane singularity

This paper analyzes the anisotropic elastic problem of an edge crack of length a in front of the wedge tip interacting with an anti-plane singularity such as a screw dislocation or anti-plane line force. The derivation is based on complex potential in conjunction with the conformal mapping method. Explicit closed-form analytical solutions are obtained and used to calculate the stress intensity factors and the strain energy density factors of the crack. The results can be reduced to the existing ones when the wedge angle is taken to be limiting case or the materials are assumed to be isotropic or orthotropic. Numerical calculations show that the stress intensity factors can be positive and negative for different applied loadings but the strain energy density factors are all positive. Stress intensity factors are used to discuss the influences of the wedge angle and singularity on the anti-shielding or shielding effects but they are inconvenient to be used as a criterion for predicting the crack extension. The positive definite characteristic of the strain energy density factor makes it suitable to predict the fracture behavior of the cracked structure.

Dynamic Mean-Variance Portfolio Selection with Transaction Costs

This paper investigates a mean-variance portfolio selection problem in continuous time with fixed and proportional transaction costs. Utilizing the dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation is derived, and the explicit closed form solution is obtained. Furthermore, the optimal strategies and efficient frontiers are also proposed for the original mean-variance problem. Numerical experiments present the variation of the transaction region and efficient frontier with the transaction costs change, which demonstrate the proposed method performs effective.

Translation-invariant and positive-homogeneous risk measures and optimal portfolio management in the presence of a riskless component

Risk portfolio optimization, with translation-invariant and positive-homogeneous risk measures, leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions. This problem was recently treated by the authors for the case when the portfolio does not contain a riskless component. When it does, however, the initial covariance matrix Σ becomes singular and the problem becomes more complicated. In the paper we focus on this case and provide an explicit closed-form solution of the minimization problem, and the condition under which this solution exists. The results are illustrated using data of 10 stocks from the NASDAQ Computer Index.

Rotational Elasticity

We consider an infinite three-dimensional elastic continuum whose material points experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis that gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory, we choose the coframe and a density. We write down the general dynamic variational functional for our rotational theory of elasticity, assuming our material to be physically linear but the kinematic model geometrically nonlinear. Allowing geometric nonlinearity is natural when dealing with rotations because rotations in dimension three are inherently nonlinear (rotations about different axes do not commute) and because there is no reason to exclude from our study large rotations such as full turns. The main result of the paper is an explicit construction of a class of time-dependent solutions that we call plane wave solutions; these are travelling waves of rotations. The existence of such explicit closed-form solutions is a non-trivial fact given that our system of Euler–Lagrange equations is highly nonlinear. We also consider a special case of our rotational theory of elasticity which in the stationary setting (harmonic time dependence and arbitrary dependence on spatial coordinates) turns out to be equivalent to a pair of massless Dirac equations.

Explicit local buckling analysis of rotationally-restrained orthotropic plates under uniform shear

The explicit closed-form local buckling solution of in-plane shear-loaded orthotropic plates with two opposite edges simply supported and other two opposite edges either both rotationally restrained or one rotationally restrained and the other free is presented. Based on the boundary condition of the other two opposite edges, two types of plates are considered: the RR (both the edges rotationally restrained) and RF (one edge restrained and the other free) plate elements. Different plate buckled shape functions are proposed, and the approximate explicit expressions for the buckling loads are derived using the Rayleigh–Ritz method for the plate with the generic rotationally-restrained (R) boundary conditions which can be reduced to two extreme cases, i.e., simply supported (S) and clamped (C). The accuracy of the derived explicit solutions is verified by comparing the predictions with the existing solutions and numerical finite element analysis, and excellent agreements are obtained. The effects of material and boundary restraining parameters on the local shear buckling behavior of the plate elements are discussed. The derived explicit formulas for the shear buckling loads are straightforward, efficient and reliable for preliminary engineering design and analysis of composite structures under primarily shear-dominant loading conditions.

Translation-invariant and positive-homogeneous risk measures and optimal portfolio management

The problem of risk portfolio optimization with translation-invariant and positive-homogeneous risk measures, which includes value-at-risk (VaR) and tail conditional expectation (TCE), leads to the problem of minimizing a combination of a linear functional and a square root of a quadratic functional for the case of elliptical multivariate underlying distributions. In this paper, we provide an explicit closed-form solution of this minimization problem, and the condition under which this solution exists. The results are illustrated using the data of 10 stocks from NASDAQ/Computers. The distance between the VaR and TCE optimal portfolios has been investigated.

Tail asymptotics for M/M/c retrial queues with non-persistent customers

In this paper, we consider a variant of the classical M/M/c retrial queue, in which we allow non-persistent customers. When c > 1, this system does not have an explicit closed form solution for the joint stationary distribution of the number of retrial customers in the orbit and the number of busy servers. Our main focus is on the tail asymptotics for the joint probabilities. We first present a matrix-product solution for the joint stationary probability vectors, which is further simplified to a scalar-product form, according to matrix-analytic theory. We then apply the censoring technique, which has been proven an efficient approach for analyzing queueing systems including retrial queues, to obtain the censored equations and the Key Lemma. In terms of these results, we finally prove an exact tail asymptotic result for the stationary probabilities.

Closed-form solutions for elastic stress–strain analysis in unbalanced adhesive single-lap joints considering adherend deformations and bond thickness

A theoretical model is developed for the stress analysis in adhesive-bonded single-lap joints under tension, for which the two adherends could have different thicknesses and consist of different materials. A two-dimensional (2D) elasticity theory is adopted in the analysis, which simultaneously incorporates the complete strain–displacement and the complete stress–strain relationships for the adherends and adhesive. The approach provides a unified treatment for any possible adhesive layer flexibility and capable of satisfying the stress-free condition at the ends of the bondline. An explicit closed-form analytical solution is formulated for upper and lower adherends/adhesive stresses (strains) and tensile, shear and bending loads acting on the adherends along the overlap and then simplified for practical applications, and simple design formulae for adhesive stresses are produced. The results predicted by the present full and simplified solutions were compared with the previously theoretical solution by Bigwood and Crocombe (1989) [35], and the 2D geometrically nonlinear finite element model using MSC/NASTRAN. The agreement validates the present formulation and solutions for unbalanced bonded joints. The effects of the stiffness unbalanced parameters on the adhesive stress distributions were also discussed.

A Closed-Form Solution for the Eshelby Tensor and the Elastic Field Outside an Elliptic Cylindrical Inclusion

From the analytical formulation developed by Ju and Sun [1999, “A Novel Formulation for the Exterior-Point Eshelby’s Tensor of an Ellipsoidal Inclusion,” ASME Trans. J. Appl. Mech., 66, pp. 570–574], it is seen that the exterior point Eshelby tensor for an ellipsoid inclusion possesses a minor symmetry. The solution to an elliptic cylindrical inclusion may be obtained as a special case of Ju and Sun’s solution. It is noted that the closed-form expression for the exterior-point Eshelby tensor by Kim and Lee [2010, “Closed Form Solution of the Exterior-Point Eshelby Tensor for an Elliptic Cylindrical Inclusion,” ASME Trans. J. Appl. Mech., 77, p. 024503] violates the minor symmetry. Due to the importance of the solution in micromechanics-based analysis and plane-elasticity-related problems, in this work, the explicit analytical solution is rederived. Furthermore, the exterior-point Eshelby tensor is used to derive the explicit closed-form solution for the elastic field outside the inclusion, as well as to quantify the elastic field discontinuity across the interface. A benchmark problem is used to demonstrate a valuable application of the present solution in implementing the equivalent inclusion method.

Optimal two-layer approximation for continuous density stratification

Two-layer fluids of finite depth under gravity are the simplest configuration capable of supporting internal wave motion. The extent to which such systems can be used to provide quantitative information on smoothly stratified fluids, in configurations relevant for geophysical applications, is analysed and a model of practical interest derived. The model is based on long-wave asymptotic expansions and on first-principle criteria for an optimal choice of effective two-layer parameters for the incompressible, smoothly stratified Euler equations. The accuracy of the model is extensively tested, via fully resolved numerical computations, on the class of travelling wave solutions supported by smooth stratification systems. It is found that, despite the severe restrictions posed by the discrete two-layer density assumption, solitary wave solutions corresponding to experimentally realizable parametric values can be accurately predicted, in both wave and fluid parcel markers, such as phase speed and density fields, respectively. Thanks to this analysis, explicit closed-form solutions are available for all relevant physical quantities. The agreement between the simple, optimized two-layer model and the parent smooth-stratification Euler system persists up to extreme cases, such as that of internal fronts, and even up to thicknesses of the pycnocline comparable to that of the effective layers.

STUDY ON FIRE RESISTANCE OF CONCRETE FILLED TUBULAR COLUMNS

This study aims to evaluate theoretically the fire resistance capacities of concrete filled tubular columns subjected to fire and axial compression. A strength analysis has been made in which all strains found in heated concrete including stress strain, thermal expansion and transient strain are taken into account. Since, among others, transient strain plays a key role on the behavior of heated columns, a new theoretical procedure to treat this strain is postulated and an explicit closed form solution is derived for the ultimate column strength. With comparison between the solutions and test data, it is found that the strength of a heated column is decreased significantly below its full plastic strength due to transient strain, if the column is made of higher strength concrete, is subjected to larger axial force or if it is of a larger cross section. For all the cases, the solutions are found in good agreement with the test results.

An alternative approach to determining average distance in a class of scale-free modularnetworks

Various real-life networks of current interest are simultaneously scale-free and modular.Here we study analytically the average distance in a class of deterministically growingscale-free modular networks. By virtue of the recursive relations derived from theself-similar structure of the networks, we compute rigorously this important quantity,obtaining an explicit closed-form solution, which recovers the previous result and iscorroborated by extensive numerical calculations. The obtained exact expression showsthat the average distance scales logarithmically with the number of nodes in thenetworks, indicating the existence of small-world behavior. We show that thissmall-world phenomenon comes from the peculiar architecture of the networkfamily.

Calculation of [formula omitted] for a multidirectional composite double cantilever beam on two-parametric elastic foundation

In this research, a Timoshenko beam (TB) resting on two-parametric (or Pasternak) elastic foundation (PEF) is developed for determination of strain energy release rate (SERR) of mode-I delamination in multidirectional double cantilever beam (DCB) specimens. Based upon the compliance approach, a closed-form solution is obtained for SERR versus delamination length, applied load, effective mechanical properties, and geometrical dimensions of DCB specimen. The required effective flexural modulus in this model is determined with three different methodologies, plane stress resultant method, plies arrangement method, and using longitudinal tensile and compressive moduli method. The proposed model is assessed by experimental and numerical results available in the literature for various lay-ups. A comprehensive evaluation is also carried out among various theories which are used to model mode-I interlaminar fracture toughness. Results show that the developed model predicts G I at the onset of delamination growth very well for unidirectional and multidirectional DCB specimens. However, the main advantages of this model are: 1) It gives a simple and accurate explicit closed-form solution for SERR of symmetric unidirectional and multidirectional lay-ups in comparison with the high-order shear deformation beam theories with a very lengthy and tedious procedure. 2) It models both first-order transverse shear deformation and local effects at the crack tip (root rotation) to improve the split beam solution. 3) By simplifying the Timoshenko beam on two-parametric foundation, as a generalized case, special cases like Euler–Bernoulli and Timoshenko beams on the Winkler elastic foundation (WEF) are obtainable.

Closed-form solutions to Sylvester-conjugate matrix equations

Some explicit closed-form solutions of homogeneous and nonhomogeneous Sylvester-conjugate matrix equations are provided in this paper. One of the solutions is expressed in terms of controllability matrices and observability matrices. The proposed approach does not require all the coefficient matrices to be in any canonical forms and the solutions provide a significant degree of freedom which is represented by an arbitrarily chosen parameter matrix. By specifying the solutions of the homogeneous Sylvester-conjugate equation, some new expressions of the solutions of the normal Sylvester, normal Sylvester-conjugate and Sylvester equations are given. This fact reveals that the Sylveter-conjugate matrix equations are a more general class of some previously investigated matrix equations.