Finite element approximation of field dislocation mechanics

A tool for studying links between continuum plasticity and dislocation theory within a field framework is presented. A finite element implementation of the geometrically linear version of a recently proposed theory of field dislocation mechanics (J. Mech. Phys. Solids 49 (2001) 761; Proc. Roy. Soc. 459 (2003) 1343; J. Mech. Phys. Solids 52 (2004) 301) represents the main idea behind the tool. The constitutive ingredients of the theory under consideration are simply elasticity and a specification of dislocation velocity and nucleation. The set of equations to be approximated are non-standard in the context of solid mechanics applications. It comprises the standard second-order equilibrium equations, a first-order div-curl system for the elastic incompatibility, and a first-order, wave-propagative system for the evolution of dislocation density. The latter two sets of equations require special treatment as the standard Galerkin method is not adequate, and are solved utilizing a least-squares finite element strategy. The implementation is validated against analytical results of the classical elastic theory of dislocations and analytical results of the theory itself. Elastic stress fields of dislocation distributions in generally anisotropic media of finite extent, deviation from elastic response, yield-drop, and back-stress are shown to be natural consequences of the model. The development of inhomogeneity, from homogeneous initial conditions and boundary conditions corresponding to homogeneous deformation in conventional plasticity, is also demonstrated. To our knowledge, this work represents the first computational implementation of a theory of dislocation mechanics where no analytical results, singular solutions in particular, are required to formulate the implementation. In particular, a part of the work is the first finite element implementation of Kröner's linear elastic theory of continuously distributed dislocations in its full generality.     

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R→∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified.     

Quantitative surface-enhanced Raman measurements with embedded internal reference

Analytical applications of SERS are often more associated with qualitative than quantitative analysis, because of the difficulty in obtaining quantitative SERS results. In this paper we introduce a new strategy to quantitatively measure the SERS signals of analytes based on Au-core/Ag-shell nanoparticles with embedded 4-aminothiophenol as the internal reference. Successful detections of two analytes, Toluidine Blue O in aqueous solution (detection limit of 0.1 μM) and melamine in milk (detection limit of ∼5 μM), are demonstrated. The improvement in the linear fitting illustrates that the use of internal reference significantly improves the accuracy of the quantitative SERS measurements. The successful detection of melamine in milk illustrates the versatility of this detection scheme for a wide variety of analytes.     

A Variational Principle for the Revised Goodman–Cowin Theory with Internal Length

In the present study a variational principle is proposed for the revised Goodman–Cowin theory with internal length for cohesionless granular materials (Fang et al. in Continuum Mech Thermodyn in press). The balance equations of the internal variables employed in the theory in equilibrium states, the equilibrium expressions of the constitutive variables and the corresponding natural boundary conditions are derived by use of the proposed variational principle for both cases of compressible and incompressible grains. It is demonstrated that the derived results coincide with those obtained by use of the thermodynamic analysis. The current work serves as a supplementary variational verification of the constitutive theory proposed in Fang et al. (in Continuum Mech Thermodyn in press).     

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.     

气相色谱-质谱法测定富马酸二甲酯时内标物的选择

探讨了当采用气相色谱-质谱法测定富马酸二甲酯(DMFU)时内标物的选择问题。采用HP-5MS毛细管色谱柱,在50℃～300℃范围内程序升温分离及采用在50～400amu范围内全扫描方式和选择离子监测模式检测。试验结果表明:以丙二酸二甲酯(DMM)、丙二酸二乙酯(DEM)和戊二酸二甲酯(DMG)作为内标物,既能保持DMFU与3种内标物有良好的分离度,又能保持长久的稳定性,且3种内标物与DMFU的保留时间差均小于2.8min。DMFU的线性范围均为0.005～5.0mg·L~(-1)。方法用于实际样品中DMFU的测定,以上3种物质作为内标,所得结果无显著差别,因此DMM、DEM、DMG均可作为比较理想的内标物用于日常用品中DMFU的检测。     

Mass and momentum conservation of the least-squares spectral collocation method for the Navier-Stokes equations

From the literature, it is known that the Least-Squares Spectral Element Method (LSSEM) for the stationary Stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum. Furthermore, it is known that the Least-Squares Spectral Collocation Method (LSSCM) leads to superior conservation of mass and momentum for the stationary Stokes equations. In the present paper, we consider mass and momentum conservation of the LSSCM for time-dependent Stokes and Navier-Stokes equations. We observe that the LSSCM leads to improved conservation of mass (and momentum) for these problems. Furthermore, the LSSCM leads to the well-known time-dependent profiles for the velocity and the pressure profiles. To obtain these results, we use only a few elements, each with high polynomial degree, avoid normal equations for solving the overdetermined linear systems of equations and introduce the Clenshaw-Curtis quadrature rule for imposing the average pressure to be zero. Furthermore, we combined the transformation of Gordon and Hall (transfinite mapping) with the least-squares spectral collocation scheme to discretize the internal flow problems.     

High-throughput and cost-effective global DNA methylation assay by liquid chromatography-mass spectrometry

An affordable and fast liquid chromatography-tandem mass spectrometry (LC-MS/MS) method was developed for the accurate and precise determination of global DNA methylation levels in peripheral blood. Global DNA methylation extent was expressed as the ratio of methylated 2′-deoxycytidine (5MedC) to 2′-deoxyguanosine (dG), which were obtained after DNA extraction and hydrolysis and determined by positive electrospray LC–ESI-MS/MS. The cost-effective internal standards ¹⁵N₃-dC and ¹⁵N₅-dG were incorporated for the accurate quantification of 5MedC and dG, respectively. The desired nucleoside analytes were separated and eluted by LC within 2.5 min on a reverse phase column with a limit of detection of 1.4 femtomole on column for 5MedC. Sample preparation in 96-well format has significantly increased the assay throughput and filtration was found to be a necessary step to assure precision. Precision was performed with repeated analysis of four DNA QC sample over 12 days, with mean intra- and inter-day CVs of 6% and 11%, respectively. Accuracy was evaluated by comparison with a previously reported method showing a mean CV of 4% for 5 subjects analyzed. Furthermore, application of the assay using a benchtop orbitrap LCMS in exact mass full scan mode showed comparable sensitivity to tandem LCMS using multiple reaction monitoring.     

Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems

The non-linear modal properties of a vibrating 2-DOF system with non-smooth (piecewise linear) characteristics are investigated; this oscillator can suitably model beams with a breathing crack or systems colliding with an elastic obstacle. The system having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of non-linear normal modes (NNMs) that are greater than the degrees of freedom. Since the non-linearities are concentrated at the origin, its non-linear frequencies are independent of the energy level and uniquely depend on the damage parameter. An analysis of the NNMs has been performed for a wide range of damage parameter by employing numerical procedures and Poincaré maps. The influence of damage on the non-linear frequencies has been investigated and bifurcations characterized by the onset of superabundant modes in internal resonance, with a significantly different shape than that of modes on fundamental branch, have been revealed.     

Complementary mild-slope equations in a two-layer fluid

Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. By using a streamfunction formulation instead of a velocity potential one, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact linear theory compared to other MS-type equations. The main goal of this work is to extend the CMSE model for solving two-layer flow with a free-surface. In order to allow for an exact reference, an analytical solution for a two-layer fluid over a sloping plane beach is derived. This analytical solution is used for validating the results of the approximated MS-type models. It is found that the two-layer CMSE model performs better than the potential based one. In addition, the new model is used for investigating the scattering of linear surface water waves and interfacial ones over variable bathymetry.