The large deflection of an axially extensible curved beam with a rectangular cross-section is investigated. The elastic beam is assumed to satisfy the Euler-Bernoulli postulation and be made of the Ludwick type material. Through reasonably simplified integration, the strain and curvature of the axis of the beam are presented in implicit formulations. The governing equations involving both geometric and material nonlinearities of the curved beam are derived and solved by the shooting method. When the initial curvature of the beam is zero, the curved beam is degenerated into a straight beam, and the predicted results obtained by the present model are consistent with those in the open literature. Numerical examples are further given for curved cantilever and simply supported beams, and the couplings between elongation and bending are found for the curved beams.

The materials with different moduli in tension and compression are called bi-modulus materials. Graphene is such a kind of materials with the highest strength and the thinnest thickness. In this paper, the mechanical response of the bi-modulus beam subjected to the temperature effect and placed on the Winkler foundation is studied. The differential equations about the neutral axis position and undetermined parameters of the normal strain of the bi-modulus foundation beam are established. Then, the analytical expressions of the normal stress, bending moment, and displacement of the foundation beam are derived. Simultaneously, a calculation procedure based on the finite element method (FEM) is developed to obtain the temperature stress of the bi-modulus structures. It is shown that the obtained bi-modulus solutions can recover the classical modulus solution, and the results obtained by the analytical expressions, the present FEM procedure, and the traditional FEM software are consistent, which verifies the accuracy and reliability of the present analytical model and procedure. Finally, the difference between the bi-modulus results and the classical same modulus results is discussed, and several reasonable suggestions for calculating and optimizing the certain bi-modulus member in practical engineering are presented.

The plane elastic problem for a semi-strip with a transverse crack is investigated. The initial problem is reduced to a one-dimensional continuous problem by use of an integral transformation method with a generalized scheme. The one-dimensional problem is first formulated as a vector boundary problem, and then reduced to a system of three singular integral equations (SIEs). The system is solved by use of an orthogonal polynomial method and a special generalized method. The contribution of this work is the consideration of kernel fixed singularities in solving the system. The crack length and its location relative to the semi-strip's lateral sides are investigated to simplify the problem's statement. This simplification reduces the initial problem to a system of two SIEs.

The incompressible flow of a non-Newtonian fluid with mixed convection along a stretching sheet is analyzed. The heat transfer phenomenon is discussed through thermal radiation. The effects of the melting heat transfer and heat generation/absorption are also taken. Suitable transformations are utilized to attain the nonlinear ordinary differential expressions. The convergent series solutions are presented. The fluid flow, temperature, and surface heat transfer rate are examined graphically. It is observed that the velocity decreases when the relaxation time increases while increases when the retardation time is constant. The results also reveal that the temperature distribution reduces when the radiation parameter increases.

The effect of non-linear convection in a laminar three-dimensional Oldroyd-B fluid flow is addressed. The heat transfer phenomenon is explored by considering the non-linear thermal radiation and heat generation/absorption. The boundary layer assumptions are taken into account to govern the mathematical model of the flow analysis. Some suitable similarity variables are introduced to transform the partial differential equations into ordinary differential systems. The Runge-Kutta-Fehlberg fourth-and fifth-order techniques with the shooting method are used to obtain the solutions of the dimensionless velocities and temperature. The effects of various physical parameters on the fluid velocities and temperature are plotted and examined. A comparison with the exact and homotopy perturbation solutions is made for the viscous fluid case, and an excellent match is noted. The numerical values of the wall shear stresses and the heat transfer rate at the wall are tabulated and investigated. The enhancement in the values of the Deborah number shows a reverse behavior on the liquid velocities. The results show that the temperature and the thermal boundary layer are reduced when the nonlinear convection parameter increases. The values of the Nusselt number are higher in the non-linear radiation situation than those in the linear radiation situation.

The nth-order expansion of the parabolized stability equation (EPSEn) is obtained from the Taylor expansion of the linear parabolized stability equation (LPSE) in the streamwise direction. The EPSE together with the homogeneous boundary conditions forms a local eigenvalue problem, in which the streamwise variations of the mean flow and the disturbance shape function are considered. The first-order EPSE (EPSE1) and the second-order EPSE (EPSE2) are used to study the crossflow instability in the swept NLF(2)-0415 wing boundary layer. The non-parallelism degree of the boundary layer is strong. Compared with the growth rates predicted by the linear stability theory (LST), the results given by the EPSE1 and EPSE2 agree well with those given by the LPSE. In particular, the results given by the EPSE2 are almost the same as those given by the LPSE. The prediction of the EPSE1 is more accurate than the prediction of the LST, and is more efficient than the predictions of the EPSE2 and LPSE. Therefore, the EPSE1 is an efficient eN prediction tool for the crossflow instability in swept-wing boundary-layer flows.

A three-dimensional inner stereocilium model is established by PATRAN. According to the relevant data, the corresponding pressure is applied to one side of the inner stereocilia. The top displacement of the inner stereocilia along the cross section of the basilar membrane (the x-displacement) is similar to the available data in the literature, which verifies the correctness of the model. Based on Castigliano's theorem, the displacement of a single stereocilium is achieved under the inverted triangle force. The results are in good agreement with the data obtained from the finite element (FE) model, which confirms the validity of the formula. With the FE model, the effects of the movement of the hair cells and fluid in the cochlear duct on the x-displacements of the inner stereocilia are studied. The results show that the movement of the hair cells affects the x-displacements of the inner stereocilia, especially for the shortest stereocilium, and the fluid in the cochlear duct affects the x-displacements of the inner stereocilia, especially for the middle stereocilium. Moreover, compared with the effects of the hair cells on the stereocilia, the effect of the cochlear duct fluid is greater.

In order to extend the eN method to general three-dimensional boundary layers, the conservation law of the imaginary parts for the wave parameters with a fixed wave vector is deduced. The compatibility relationship (CR) and the general theory of ray tracing (RT), which have been extensively used in conservative systems, are applied to a general three-dimensional boundary layer belonging to non-conservative systems. Two kinds of eN methods, i.e., the eN-CR method and the eN-RT method, are established. Both the two kinds of methods can be used to predict the evolutions of the spanwise wavenumber and the amplitude of the disturbances in general three-dimensional boundary layers. The reliability of the proposed methods is verified and validated by performing a direct numerical simulation (DNS) in a hypersonic general three-dimensional boundary layer over an aircraft model. The results are also compared with those obtained by other eN methods, indicating that the proposed methods have great potential applications in improving the transition prediction accuracy in general three-dimensional boundary layers.

A mechanical-piezoelectric system is explored to reduce vibration and to harvest energy. The system consists of a piezoelectric device and a nonlinear energy sink (NES), which is a nonlinear oscillator without linear stiffness. The NES-piezoelectric system is attached to a 2-degree-of-freedom primary system subjected to a shock load. This mechanical-piezoelectric system is investigated based on the concepts of the percentages of energy transition and energy transition measure. The strong target energy transfer occurs for some certain transient excitation amplitude and NES nonlinear stiffness. The plots of wavelet transforms are used to indicate that the nonlinear beats initiate energy transitions between the NES-piezoelectric system and the primary system in the transient vibration, and a 1:1 transient resonance capture occurs between two subsystems. The investigation demonstrates that the integrated NES-piezoelectric mechanism can reduce vibration and harvest some vibration energy.

A new boundary extension technique based on the Lagrange interpolating polynomial is proposed and used to solve the function approximation defined on an interval by a series of scaling Coiflet functions, where the coefficients are used as the single-point samplings. The obtained approximation formula can exactly represent any polynomials defined on the interval with the order up to one third of the length of the compact support of the adopted Coiflet function. Based on the Galerkin method, a Coiflet-based solution procedure is established for general two-dimensional p-Laplacian equations, following which the equations can be discretized into a concise matrix form. As examples of applications, the proposed modified wavelet Galerkin method is applied to three typical p-Laplacian equations with strong nonlinearity. The numerical results justify the efficiency and accuracy of the method.