In the present work, the problem of an infinite elastic perfectly plastic plate under axisymmetrical loading conditions resting on a bilateral Pasternak elastic foundation is considered. The plate is assumed thin, thus making it possible to neglect the shear deformation according to the classical Kirchhoff theory. Yielding is governed by the Johansen's yield criterion with associative flow rule. A uniformly distributed load is applied on a circular area on the top of the plate. As the load is increased, a circular elastic-plastic region spreads out starting from the center of the loaded area, whereas the outer unbounded region behaves elastically. Depending on the size of the loaded area, a further increase of the load may originate two or three different elastic-plastic regions, corresponding to different yield loci. A closed form solution of the governing equations for each region is found for a special value of the ratio between Pasternak soil moduli. The performed analysis allows us to estimate the elastic-plastic behavior of the plate up to the onset of collapse, here defined by the formation of a plastic mechanism within the plate. The corresponding collapse load and the sizes of the elastic-plastic regions are thus found by imposing the boundary and continuity conditions between the different regions. The influence of the soil moduli, plate bending stiffness, and size of the loaded area on the ultimate bearing capacity of the plate is then investigated in detail.

The modal interactions and nonlinear responses of inextensional beams resting on elastic foundations with two-to-one internal resonances are investigated and the primary resonance excitations are considered. The multimode discretization and the method of multiple scales are applied to obtain the modulation equations. The equilibrium and dynamic solutions of the modulation equations are examined by the Newton–Raphson, shooting, and continuation methods. Numerical simulations are performed to investigate the chaotic dynamics of the beam. It is shown that the nonlinear responses may undergo different bifurcations and exhibit rich nonlinear phenomena. Finally, the effects of the foundation models on the nonlinear interactions of the beam are examined.

In this study, the nonlinear equation of motion of the beam on the elastic foundation is obtained via the Newton's second law of motion, and its free vibration nature is investigated. Considering the inextensional condition, the planar model of the beam accounting for the effects of the rotary inertia is derived. Then, the linear vibration and nonlinear vibration of the beam on the elastic foundation are examined. It is shown that the cut-off frequency can be observed in the frequency spectrum of the beam response. The effects of the rotary inertia on the natural frequencies are systematically investigated. Finally, the frequency differences, due to the different foundation models, and the possible modal interaction of the beam are discussed.

This paper presents a Hamiltonian variational formulation to determine the energy minimizing boundary conditions (BCs) of the tensionless contact problem for an Euler–Bernoulli beam resting on either a Pasternak or a Reissner two-parameters foundation. Mathematically, this originates a free-boundary variational problem. It is shown that the BCs setting the contact loci, which are the boundary points of the contact interval, are always given by second order homogeneous forms in the displacement and its derivatives. This stands for the nonlinear nature of the problem and calls for multiple solutions in the displacement, together with the classical result of multiple solutions in the contact loci position. In particular, it is shown that the Pasternak soil possesses an extra solution other than Kerr’s, although it is proved that such solution must be ruled out owing to interpenetration. The homogeneous character of the BCs explains the well-known load scaling invariance of the contact loci position. It is further shown that the Reissner foundation may be given two mechanical interpretations, which lead to different BCs. Comparison with the established literature is drawn and numerical solutions shown which confirm the energy minimizing nature of the assessed BCs.

Jets that emanate from high density porous liners are most widely used for penetration into Earth materials. These shaped-charges differ from those that contain solid copper liners in three major aspects: the porosity, the usually higher initial density, and the very short standoff in which they typically operate. Because penetration depth is commonly very difficult to increase by means of high density solid liners, it is important to understand the benefit of using porous liners in utilization of high density materials. The models published so far to describe the performance of jets formed by porous liners are based on the modification of the virtual origin model to suit this special case, which limits the accuracy of their predictions. Here we present a more general analysis that does not depend on the virtual origin assumption. Our study employs the SCAN semi-analytical code into which a new model for porous jet behavior is incorporated. It is used to explain the benefit of using this type of liner for oil-well penetration, hence penetration into low density targets, as opposed to penetration into hard steel.

In this work an elastoplastic constitutive theory for unsaturated soils is presented. The proposed material model is formulated in the general framework of the theory of porous media and of the flow theory of plasticity. The model is based on an extension of the well-known MRS Lade model whereby the suction and the effective stress tensor are introduced as additional independent and dependent stress components, respectively. Consequently the cap and cone yield conditions of the MRS Lade model both in hardening and softening as well as the internal evolution laws in these regimes are redefined to include the dependency on the suction. The paper illustrates the predictive capability of the extended MRS Lade model for partially saturated soils. Finally, the condition for discontinuous bifurcation in elastoplastic partially saturated porous media as well as the localized failure predictions of the proposed material formulation for different suctions and stress states are also analyzed and discussed.

Explosives are commonly used in terrorist attacks and the craters formed by blast waves can be used as a diagnostic tool. For example, the focus of the explosion and the mass of the explosive used in the attack can be deduced by examining the location, geometry, and dimensions of the crater. However, studies about craters produced by explosions on or above ground level, which would be the case when the explosive charge is situated in a vehicle, are rarely found in the open technical literature. In this paper, a numerical study on craters formed by explosive loads located on the soil surface is presented. The soil parameters used in the numerical model, as well as the analysis procedure, were validated against experimental observations of the crater diameters. Results of numerical tests performed with different amounts of explosive on the soil surface are presented. Moreover, the effect of elevation of the center of energy release of explosive loads located on the soil surface is analyzed and discussed. Simple predictive equations for the crater diameter are presented.

A closed-form solution is presented for the problem of a moving point load on an elastic half-plane with hysteretic damping. The problem has been studied in order to investigate the dynamic amplification of stresses and displacements if the velocity of the load approaches the Rayleigh wave velocity or the shear wave velocity in the elastic medium. This is relevant for the construction of high speed railway lines on relatively soft soils. Hysteretic damping is introduced as pseudo-viscous damping, assuming that the damping in a full cycle of loading and unloading is independent of the frequency.

The response due to a dynamic loading of a poroviscoelastic one-dimensional column is treated analytically. Biot’s theory of poroelasticity is generalized to poroviscoelasticity using the elastic-viscoelastic correspondence principle in the Laplace domain. Damping effects of the solid skeletal structure and the solid material itself are taken into account. The fluid is modeled as in the original Biot’s theory without any viscoelastic effects. The solution of the governing set of two coupled differential equations known from the purely poroelastic case is converted to the poroviscoelastic solution using the developed elastic-viscoelastic correspondence in Laplace domain. The time-dependent response of the column is achieved by the “Convolution Quadrature Method” proposed by Lubich. Some interesting effects of viscoelasticity on the response of the column caused by a stress, pressure, and displacement loading are studied.

Plane-strain contact analysis is presented for compositionally graded materials with power-law strain hardening. The half-space, y ≤ 0 , is modeled as an incompressible, nonlinear elastic material. The effective stress, σ e , and the effective total strain, ε e , are related through a power-law model, σ e = K 0 ε e μ ; 0 < μ ≤ min ( 1 , ( 1 + k ) ). The material property K 0 changes with depth, | y | , as K 0 = A | y | k ; A > 0 , 0 ≤ | k | < 1 . This material description attempts to capture some features of the plane-strain indentation of elastoplastic or steady-state creeping materials that show monotonically increasing or decreasing hardness with depth. The analysis starts with the solution for the normal line load (Flamant’s problem) and continues with the rigid, frictionless, flat-strip problem. Finally, the general solution of normal indentation of graded material by a convex, symmetric, rigid, and frictionless two-dimensional punch is given. Applications of the present results range from surface treatments of engineering structures, protective coatings for corrosion and fretting fatigue, settling of beam type foundations in the context of soil and rock mechanics, to bioengineering as well as structural applications such as contact of railroad tracks.

The directional property of soils, particularly clays, are directly related to the particle orientations. The platy clay particles do possess the tendency to reorient themselves in the most stable condition against applied stress. One may conceive of developing a soil model directly relating particle orientation and applied stress. In this study an angle θ, which is the preferred (therefore stable) orientation of clay particles, is introduced. Using this concept and slip-line theory, the velocity equations of clay in a plane-strain condition are developed. In this approach any change in clay particle orientation is essentially dependent on principal stresses magnitudes as well as their directions. Therefore, the effect of rotation of principal stresses can be included and quantified.

Three-dimensional transient response of both massless and massive multiple, rigid foundations, bonded to an elastic, homogeneous half-space, is investigated to study the effect of dynamic interaction through-soil. The numerical procedure is formulated in terms of the boundary element approach by means of the transient fundamental solutions developed by the authors (1994). This procedure works efficiently for the problem addressed here since the separated foundations are analyzed without discretizing the surface of the half-space outside the contact areas between the half-space and the foundations. It also provides the possibility to study nonlinear problems involved with semi-infinite soils.

A closed-form solution of transient response to suddenly applied loading distributed over a rectangular area on the surface of an elastic homogeneous half-space is developed for special purposes such as analysis of dynamic soil-structure interaction or contact problems. The solution is obtained using Laplace transform with respect to time and Fourier transform with respect to space. Inverse Laplace transform is implemented analytically. As extreme cases of rectangular loading, the solutions for a point force or finite line load can also be obtained. The advantages of this solution over most other solutions by numerical analyses are that the multiple integrations are reduced by one order, the singularity is removed from the integral kernel, and no additional discretization in the vicinity of the region of interest is required.

A viscoelastic model is proposed to describe the dynamic response of the saturated poroelastic materials that obey the Biot theory (1956). The viscoelastic model is defined from the velocity and attenuation of dilatational and distortional waves in poroelastic materials. Its material properties are defined in terms of the elastic moduli, porosity, specific gravity, degree of saturation, and permeability of the soils. The proposed model is tested by comparing its response with the one of poroelastic materials in the case of axial and lateral harmonic loadings of one-dimensional columns. The viscoelastic model is simpler to use than poroelastic materials but yields similar results for a wide range of soils and dynamic loadings.

Results are obtained for the free vibrations of suspended cables for which longitudinal end restraint is supplied by friction. The problem is germane to submarine telecommunication cables. It is shown that all modes of vibration are independent of any characteristic dimensionless parameters, at least insofar as the linear theory applies to flat-sag cables, and the friction capability of the soil at the mudline is low. This sets the problem apart from the traditional, firmly founded suspended cable where, for some of the important in-plane modes, there is variability in the results. The theory is of importance in determining the vortex-induced vibratory response of suspended cable spans to currents of various sorts.

This article describes the measurement and analysis of plate and soil response under low velocity impact. A free-drop impact system was developed to generate the dynamic loading on the plate free surface. The radial strain of the target plate, the longitudinal wave speed and the acceleration of the sand were measured. The measured wave speed data were then used to evaluate the elastic constants of the sand. An analysis based on linear elastodynamics was developed for transient waves on a thin plate resting on an elastic half space. The contact stresses and the normal displacements of the plate were taken as unknown functions. The contact between the plate and the half space were assumed frictionless. The experimental results of the radial strain at the bottom of the target plate and the acceleration of the sand beneath the center of the target plate were compared with the analytical solution. The arrival time, the duration, and the magnitude have good correlation between the analysis and experiment. The overall results appear good and provide an understanding of the transmission of impact load through the plate, the interaction between the plate and the sand, and the propagation of the load into the sand.

This paper desribes the formulation and implementation of a nonreflecting boundary for use with existing finite-element codes to perform nonlinear ground-shock analyses of buried structures. The boundary is based on a first-order doubly asymptotic approximation (DAA 1 ) for disturbances propagating outward from a selected portion of the soil medium surrounding the structure of interest. The resulting set of first-order ordinary differential equations is then combined with the second-order equations of motion for the finite-element model so as to facilitate solution by a staggered solution procedure. This procedure is shown to be computationally stable as long as the time increment is smaller than a limiting value based on the finite-element mass matrix and the DAA-boundary stiffness matrix. Computational results produced by the boundary are compared with exact results for linear canonical problems pertaining to infinite-cylindrical and spherical shells.

An exact analysis is presented here for the three-dimensional dynamics of a long continuous pipeline embedded in an elastic medium. A shell model of the pipe has been used here. It is shown that the dynamic amplification of axial and hoop stresses induced in the pipe due to incident plane longitudinal and shear waves depends crucially on the ratio of rigidities of the surrounding soil and the pipe. Induced stresses are also found to have appreciable frequency dependence for certain combinations of material properties and angles of incidence. Results presented here are also applicable to buried tunnels.

A cumulation of the shear strain energy at the soil surface as a response to underground shear excitations is considered. It is shown that this effect is caused by the vanishing shear modulus at the soil surface. The corresponding mathematical model is presented by a linear hyperbolic equation with variable coefficients. The uniqueness and stability of the solutions are analyzed.