Approximate solutions of the Alekseevskii–Tate model of long-rod penetration

The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretica solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical(exact) solution, and the first-order perturbation solution obtained by Walters et al.(Int. J. Impac Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical(exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetrator/target combinations to compare with the theoretical(exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration.

Theoretical solution of transient heat conduction problem in one-dimensional double-layer composite medium

To make heat conduction equation embody the essence of physical phenomenon under study, dimensionless factors were introduced and the transient heat conduction equation and its boundary conditions were transformed to dimensionless forms. Then, a theoretical solution model of transient heat conduction problem in one-dimensional double-layer composite medium was built utilizing the natural eigenfunction expansion method. In order to verify the validity of the model, the results of the above theoretical solution were compared with those of finite element method. The results by the two methods are in a good agreement. The maximum errors by the two methods appear when τ（τ is nondimensional time） equals 0.1 near the boundaries of ζ =1 （ζ is nondimensional space coordinate） and ζ =4. As τ increases, the error decreases gradually, and when τ =5 the results of both solutions have almost no change with the variation of coordinate 4.

Theoretical study of the effect of topographic height and width on generation of internal tides

Internal tides generated upon two-dimensional Gaussian topographies of different sizes and steepness are investigated theoretically in a numerical methodology.Compared with previous theoretical works,this model is not restricted by weak topography,but provides an opportunity to examine the influence of topography.Ten typical cases are studied using different values of height and/or width of topography.By analyzing the baroclinic velocity fields,as well as their first eight baroclinic modes,it is found that the magnitude of baroclinic velocity increases and the vertical structure becomes increasingly complex as height increases or width decreases.However,when both height and width vary,while parameter s(the ratio of the topographic slope to the characteristic slope of the internal wave ray) remains invariant,the final pattern is influenced primarily by width.The conversion rate is studied and the results indicate that width determines where the conversion rate reaches a peak,and where it is positive or negative,whereas height affects only the magnitude.High and narrow topography is considerably more beneficial to converting energy from barotropic to baroclinic fields than low and wide topography.Furthermore,parameter s,which is an important non-dimensional parameter for internal tide generation,is not the sole parameter by which the baroclinic velocity fields and conversion rate are determined.

Theoretical solution for wave diffraction by wedge or corner with arbitrary reflection characteristics

An exact analytic solution for wave diffraction by wedge or corner with arbitrary angle (rπ) and reflection coefficients (R0 and Rr) is presented in this paper. It is expressed in two forms-series and integral representations, corresponding recurrence relation and asymptotic expressions are also derived. The solution is simplified for some special cases of rπ. For Rr= R0,r= 1/N and Rr≠R0,r = 1/2N, the solution can be reduced to linear superpositions of incident and partially reflected waves, hence a nonlinear solution of forth order for this problem can be obtained by using the author's theory of nonlinear interaction among gravity surface waves. The given solution is related to inhomogeneous Robin boundary conditions, which include the Neumann boundary conditions usually accepted in wave diffraction theory.

THEORETICAL ANALYSIS ON HYDRAULIC TRANSIENT RESULTED BY SUDDEN INCREASE OF INLET PRESSURE FOR LAMINAR PIPELINE FLOW

Hydraulic transient, which is resulted from sudden increase of inlet pressure for laminar pipeline flow, is studied. The partial differential equation, initial and boundary conditions for transient pressure were constructed, and the theoretical solution was obtained by variable-separation method. The partial differential equation, initial and boundary conditions for flow rate were obtained in accordance with the constraint correlation between flow rate and pressure while the transient flow rate distribution was also solved by variable-separation method. The theoretical solution conforms to numerical solution obtained by method of characteristics (MOC) very well.

A simple theoretical approach for analysis of slide-toe-toppling failure

A prevalent kind of failure of rock slopes is toppling instability.In secondary toppling failures,these instabilities are stimulated through some external factors.A type of secondary toppling failure is"slide-toe-toppling failure".In this instability,the upper and toe parts of the slope have the potential of sliding and toppling failures,respectively.This failure has been investigated by an analytical method and experimental tests.In the present study,at first,the literature review of toppling failure is presented.Then a simple theoretical solution is suggested for evaluating this failure.The recommended method is compared with the approach of AMINI et al through a typical example and three physical models.The results indicate that the proposed method is in good agreement with the results of AMINI et al’s approach and experimental models.Therefore,this suggested methodology can be applied to examining the stability of slide-toe-toppling failure.

STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH LARGE OPENINGS

Based on Donnell's shallow shell equation, a new method is given in this paper to analyze theoretical solutions of stress concentrations about cylindrical shells with large openings. With the method of complex variable function, a series' of conformal mapping functions are obtained from different cutouts' boundary curves in the developed plane of a cylindrical shell to the unit circle. And, the general expressions for the equations of a cylindrical shell, including the solutions of stress concentrations meeting the boundary conditions of the large openings' edges, are given in the mapping plane. Furthermore, by applying the orthogonal function expansion technique, the problem can be summarized into the solution of infinite algebraic equation series. Finally, numerical results are obtained for stress concentration factors at the cutout's edge with various opening's ratios and different loading conditions. This new method, at the same time, gives a possibility to the research of cylindrical shells with large non-circular openings or with nozzles.

THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.

Dynamic quasi-continuum model for plate-type nano-materials and analysis of fundamental frequency

A dynamic quasi-continuum model is presented to analyze free vibration of plate-type cubic crystal nano-materials.According to the Hamilton principle,fundamental governing equations in terms of displacement components and angles of rotations are given.As an application of the model,the cylindrical bending deformation of the structure fixed at two ends is analyzed,and a theoretical formula evaluating the fundamental frequency is obtained by using Galerkin's method.Meanwhile,the solution for the classical continuous plate model is also derived,and the size-dependent elastic modulus and Poisson's ratio are taken in computation.The frequencies corresponding to different atomic layers are numerically presented for the plate-type NaC l nano-materials.Furthermore,a molecular dynamics(MD)simulation is conducted with the code LAMMPS.The comparison shows that the present quasi-continuum model is valid,and it may be used as an alternative model,which reflects scale effects in analyzing dynamic behaviors of such plate-type nano-materials.

A theoretical method for solving shock relations coupled with chemical equilibrium and its applications

In this study,a theoretical method is proposed to solve shock relations coupled with chemical equilibrium.Not only shock waves in dissociated flows but also detonation waves in combustive mixtures can be solved.The global iterative solving process is specially designed to mimic the physical and chemical process in reactive shock waves to ensure good stability and fast convergence in the proposed method.Within each global step,the single-variable equations of normal and oblique shock relations are derived and solved with the Newton iteration method to reduce the complexity of the problems,and the minimization of free energy method of NASA(National Aeronautics and Space Administration)is adopted to solve equilibrium compositions.It is demonstrated that the convergent process is stable and very close to the real chemical-kinetic process,and high accuracy is achieved in the solutions of normal and oblique reactive shock waves.Moreover,the proposed theoretical method has also been applied to many problems associated with reactive shocks,including the stability of oblique detonation wave,bow detonation over a sphere,and shock reflection in dissociated air.The great importance of using chemical equilibrium to theoretically predict the theoretical range of the wedge angle for a standing oblique detonation wave(the standing window of the oblique detonation wave),the stand-off distance of bow detonation wave and the transition criterion of shock reflection in dissociated air with high accuracy have been addressed.

Theoretical study of failure in composite pressure vessels subjected to low-velocity impact and internal pressure

A theoretical solution is aimed to be developed in this research for predicting the failure in internally pressurized composite pressure vessels exposed to low-velocity impact.Both in-plane and out-of-plane failure modes are taken into account simultaneously and thus all components of the stress and strain fields are derived.For this purpose,layer-wise theory is employed in a composite cylinder under internal pressure and low-velocity impact.Obtained stress/strain components are fed into appropriate failure criteria for investigating the occurrence of failure.In case of experiencing any in-plane failure mode,the evolution of damage is modeled using progressive damage modeling in the context of continuum damage mechanics.Namely,mechanical properties of failed ply are degraded and stress analysis is performed on the updated status of the model.In the event of delamination occurrence,the solution is terminated.The obtained results are validated with available experimental observations in open literature.It is observed that the sequence of in-plane failure and delamination varies by increasing the impact energy.

Shock relations in gases of heterogeneous thermodynamic properties

Shock relations usually found in literatures are derived theoretically under the assumption of homogeneous thermodynamic properties, i.e., constant ratio of specific heats, γ. However, high temperature effects post a strong shock wave may result in thermodynamic heterogeneities and failure to the original shock relations. In this paper, the shock relations are extended to take account of high-temperature effects. Comparison indicates that the present approach is more feasible than other analytical approaches to reflect the influence of γ heterogeneity on the post-shock parameters.