Surfaces Of Strong Discontinuity Research Articles

Numerical Analysis of Unsteady Vibrations of a Plate Resting on an Elastic Isotropic Half-Space

The paper is devoted to the numerical solution of the problem of vibrations of an infinite elastic plate resting on an elastic isotropic half-space using the analytical method based on the ray method with its numerical realization via the Maplesoft package. Unsteady oscillations are caused by the action of instantaneous loads on the plate, resulting in the appearance of two plane wave surfaces of strong discontinuity in the elastic half-space, behind the fronts of which, up to the contact boundary, the solution is constructed using ray series. The unknown functions entering the coefficients of the ray series and the equation of plate motion are determined from the boundary conditions of the contact interaction between the plate and the half-space. Previously, the approximate solution of this problem was obtained analytically without using mathematical packages, and the dynamic deflection of the plate involving only the first three terms of the ray series was written down. In this work, a two-layer medium with different properties was investigated using an algorithm developed to solve contact dynamic problems related to the occurrence and propagation of strong and weak discontinuity surfaces.

On the flat strong discontinuities in incompressible polymeric liquids

We studied the discontinuous stationary solutions for the rheological mesoscopic modified model of Pokrovskii-Vinogradov, which describes the dynamics of liquid polymers. The Rankine-Hugoniot conditions for the model were introduced. We justified the existence of stationary solutions with flat surface of strong discontinuity for the case of constant velocity direction across the discontinuity and for the case with change of direction (rotating discontinuity). The stability of such solutions was also considered. For linearized equations of the model we posed the eigenvalue problem for partial solutions with unlimited grow in time. It was shown that such solutions exists in anisotropic case wich means the stationary solutions with flat discontinuity are unstable within the given model.

A boundary element method formulation for quasi-brittle material fracture analysis using the continuum strong discontinuity approach

The implicit formulation of the boundary element method is applied to bi-dimensional problems of material failure involving, sequentially, inelastic dissipation with softening in continuous media, bifurcation and transition between weak and strong discontinuities. The bifurcation condition is defined by the singularity of the localization tensor. Weak discontinuities are related to strain localization bands of finite width, which become increasingly narrow until to collapse in a surface with discontinuous displacement field, called strong discontinuity surface. To associate such steps to the fracture process in quasi-brittle materials, an isotropic damage constitutive model is used to represent the behaviour in all of them, considering the adaptations that come from the strong discontinuity analysis for the post-bifurcation steps. The crack propagation across the domain is done by an automatic cells generation algorithm and, in this context, the fracture process zone in the crack tip became totally represented.

One-Dimensional Interaction of a Cylindrical Unloading Wave with a Moving Elastic–Plastic Boundary

The one-dimensional dynamic problem of the theory of large elastic–plastic deformations is considered for the interaction of an unloading wave with an elastic–plastic boundary. It is shown that before the occurrence of the unloading wave, the increasing pressure gradient leads to quasistatic deformation of the elasti©viscoplastic material filling the round tube, which is retained in the tube due to friction on its wall, resulting in the formation of near-wall viscoplastic flow and an elastic core. The unloading wave is initiated at the moment of the onset of slippage of the material along the inner wall of the tube. Calculations were conducted using the ray method of constructing approximate solutions behind strong discontinuity surfaces, and ray expansions of the solutions behind the cylindrical surfaces of discontinuities were obtained.

Incompressible Polymer Fluid Flow Past a Flat Wedge

A problem of an incompressible polymer fluid flow past an infinite flat wedge is considered. The flow moves parallel to the plane of symmetry of the wedge and normal to the wedge rib. It is demonstrated that two surfaces of strong discontinuities are needed for the no-slip condition to be satisfied on the wedge surface. Steady solutions of the problem are studied, and the flow is shown to be asymmetric with respect to the plane of symmetry of the wedge.

An algorithm based on incompatible modes for the global tracking of strong discontinuities in shear localization analyses

Numerical methods for predicting localized shear failure in elasto-plastic solids have experienced considerable advancements in the last decades. Among these approaches, the so-called “Embedded Strong Discontinuity (ESD)” method is often successfully used to accurately simulate the post-localization response with negligible dependence on the finite element discretization. However, it was observed that the employed discontinuity tracking strategy plays a crucial role in the successful localization analysis. In this contribution, we propose a novel strategy for the global tracking of discontinuity surfaces. It is based on exploiting information obtained from the enhanced parameters employed in Enhanced Assumed Strain (EAS) formulations. It is well known, that enhanced strain element formulations are able to better capture localized shear deformations as compared to standard finite elements. This can be explained as a consequence of the improved performance in bending. We observed, that the approximation of the strain jumps delimiting the shear band is connected with a deformation field characterized by opposite bending curvatures across these two discontinuities. Hence, in view of the relations existing between the kinematics of strong and weak discontinuities, we formulate a proper scalar function of the enhanced parameters to identify potential strong discontinuity surfaces, which are evaluated in each step of the analysis with negligible computational cost. This proposed approach has a global character, as it is based upon evaluating discontinuity surfaces defined in the complete analysis domain that are, by construction, continuous across elements. We demonstrate that the tracking algorithm correctly identifies the potential strong discontinuity surface already in early loading stages, even before a localization condition is fulfilled. In those elements which are crossed by the potential failure surface and which also satisfy the localization condition, the kinematics of embedded strong discontinuities is activated to capture the shear failure surface. The performance of the new tracking algorithm is demonstrated by means of several numerical shear localization analyses using associative and non-associative Drucker–Prager elastoplastic models to simulate 2-D and 3-D benchmarkanalyses.

Analysis of the Viscoelastic Sphere Impact against a Viscoelastic Uflyand-Mindlin Plate considering the Extension of Its Middle Surface

In the present paper, the problem on impact of a viscoelastic sphere against a viscoelastic plate is considered with due account for the extension of plate’s middle surface and local bearing of sphere and plate’s materials via the Hertz theory. The standard linear solid models with conventional derivatives and with fractional-order derivatives are used as viscoelastic models, respectively, outside and within the contact domain. As a result of impact, transient waves (surfaces of strong discontinuity) are generated in the plate, behind the wave fronts of which up to the boundaries of the contact domain the solution is constructed in terms of one-term ray expansions due to short-time duration of the impact process. The motion of the contact zone occurs under the action of extension forces acting in the plate’s middle surface, transverse force, and the Hertzian contact force. The suggested approach allows one to find the time-dependence of the impactor’s indentation into the target and the Hertzian contact force.

Impact response of a Timoshenko-type viscoelastic beam considering the extension of its middle surface.

In the present paper, the problem of low-velocity impact of an elastic sphere against a viscoelastic Timoshenko-type beam is studied considering the extension of its middle surface. The viscoelastic features of the beam out of the contact domain are governed by the standard linear solid model with derivatives of integer order, while within the contact domain the fractional derivative standard linear solid model is utilized, in so doing rheological constants of the material in both models are the same. However the presence of the additional parameter, i.e. fractional parameter which could vary from zero to unit, allows one to vary beam’s viscosity, since the structure of the beam’s material within this zone may be damaged, resulting in the decrease of the beam material viscosity in the contact zone. Consideration for transient waves (surfaces of strong discontinuity) propagating in the target out of the contact zone via the theory of discontinuities and determination of the desired values behind the surfaces of discontinuities upto the contact domain with the help of ray series, as well as the utilization of the Hertz theory in the contact zone allow one to obtain a set of two integro-differential equations, which govern the desired values, namely: the local bearing of the target and impactor’s materials and the displacement of the beam within the contact domain.

Transient wave velocities in pre-stressed thin-walled beams of open profile with Cosserat-type micro-structure

In the present paper, a new wave theory of thin-walled beams of open profile with Cosserat-type micro-structure is suggested based on the approach proposed by Rossikhin and Shitikova [1] for pre-stressed spatially curved thin-walled beams of open profile. The aim of the authors is to create the theory of propagation of transient waves (surfaces of strong discontinuity) in thin-walled beams of open profile, which should be quite different from Timoshenko-like theories, resulting in the data comparable with those corresponding to transient wave propagation in the three-dimensional Cosserat continuum.

Impact response of a viscoelastic beam considering the changes of its microstructure in the contact domain

The problem on low-velocity impact of a long thin elastic rod with a flat end upon an infinite viscoelastic Timoshenko-type beam, the dynamic behaviour of which is described by a set of equations taking the rotary inertia, transverse shear deformation and extension of the beam’s middle surface into account, is considered. The viscoelastic features of the beam are governed by the standard linear solid model with derivatives of integer order. At the moment of impact, shock waves (surfaces of strong discontinuity) are generated both in the impactor and target, the influence of which on the contact domain is considered via the theory of discontinuities. The contact zone moves like a rigid whole under the action of the contact force and longitudinal and transverse forces applied to the boundary of the contact region, which are obtained on the basis of one-term ray expansions. During the impact process, decrosslinking within the domain of the contact of the beam with the rod occurs, resulting in more free displacements of molecules with respect to each other, and finally in the decrease of the beam material viscosity in the contact zone. This circumstance allows one to describe the behaviour of the beam material within the contact domain by the standard linear solid model involving fractional derivatives, since variation in the fractional parameter (the order of the fractional derivative) enables one to control the viscosity of the beam material. The contact force has been determined analytically via the Laplace transform technique.

ON WEAK DISCONTINUITIES AND JUMP EQUATIONS ON WAVE SURFACES IN MICROPOLAR THERMOELASTIC CONTINUA

The present study is devoted to problem of propagating surfaces of weak and strong discontinuities of translational displacements, microrotations and temperature in micropolar (MP) thermoelastic (TE) continua. Problems of propagation of weak discontinuities in type-I MPTE continua are discussed. Geometrical and kinematical compatibility conditions due to Hadamard and Thomas are used to study possible wave surfaces of weak discontinuities. Weak discontinuities are discriminated according to spatial orientations of the discontinuities polarization vectors (DPVs). It is shown that the surfaces of weak discontinuities can propagate exist without weak discontinuities of the temperature field. Second part of the paper is concerned the discussions of the propagating surfaces of strong discontinuities of field variables in type-II MPTE continua. Constitutive relations for hyperbolic thermoelastic type-II micropolar continuum is derived by the field theory. The special form of the first variation of the action integral is used in order to obtained 4-covariant jump conditions on wave surfaces. Three-dimensional form of the jump conditions on the surface of a strong discontinuity of thermoelastic field are derived from 4-covariant form.

Impact Response of a Fractionally Damped Spherical Shell

AbstractIn the present paper, the problem on normal low-velocity impact of a solid upon an isotropic spherical shell is studied without considering the changes in the geometrical dimensions of the contact domain. At the moment of impact, shock waves (surfaces of strong discontinuity) are generated in the target, which then propagate along the shell during the process of impact. Behind the wave fronts up to the boundary of the contact domain, the solution is constructed with the help of the theory of discontinuities and one-term ray expansions. The ray method is used outside the contact spot, but the Laplace transform method is applied within the contact region. As a result, the exact solution of the contact force is determined as a function of time. This model is intended to be used in simulating crash scenarios in frontal impacts, and to provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks.

Dynamics Response of a Fractionally Damped Spherical Shell Impacted by a Body of Finite Dimensions

In the present paper, the problem on normal low-velocity impact of a solid upon an isotropic spherical shell is studied without considering the changes in the geometrical dimensions of the contact domain. At the moment of impact, shock waves (surfaces of strong discontinuity) are generated in the target, which then propagate along the shell during the process of impact. Behind the wave fronts up to the boundary of the contact domain, the solution is constructed with the help of the theory of discontinuities and one-term ray expansions. The ray method is used outside the contact spot, but the Laplace transform method is applied within the contact region. It is assumed that the viscoelastic features of the shell are exhibited only in the contact domain, while the remaining part retains its elastic properties. In this case, the contact spot is assumed to be a plane disk with constant radius, and the viscoelastic features of the shell are described by the fractional derivative standard linear solid model. In the case under consideration, the governing differential equations are solved analytically by the Laplace transform technique. As a result, the exact solution of the contact force is determined as a function of time.

Transverse impact of a ball on a sphere with allowance for waves in the target

The transverse impact of a solid projectile on an elastic spherical shell with a pivoting contour support has been studied. Inside the contact zone, the projectile-target interaction is described by a solution of the standard system of equations. Outside the contact zone, the points of the shell are displaced and the shell is deformed due to propagation of a nonstationary wave front. A solution in this region is constructed using ray series with variable coefficients representing jumps of the time derivatives of the unknown functions on the wave surface of strong discontinuity. These coefficients are determined to within arbitrary constants using momentless equations of motion of the shell points. The constants are determined by matching two solutions at the contact zone boundary. Using the obtained analytical expressions and plotted dependences for the contact force and dynamic inflection, it is possible to judge on the influence of the shell structure design on the dynamic characteristics of impact interaction.

Elastoplastic model of cylindrical projectile-plate interaction

Elastoplastic models of the transverse impact of a cylindrical solid projectile on an isotropic elastic Uflyand-Mindlin plate have been studied using numerical simulation methods. Displacements of the plate points outside the contact region are caused by a transverse shear wave propagating at a finite velocity. This wave is formed at the moment of impact and represents a surface of strong discontinuity. Behind the transverse wave front, the unknown dynamic quantities are represented in the form of single-term ray expansions. The local crumpling of the target material proceeds in a quasi-static manner and its dependence on the contact force is described using an elastoplastic model. The dependence of the contact force on the elastoplastic properties of a target plate at the site of impact and on the contact disk inertia has been considered.

Transient Response of Thin Bodies Subjected to Impact: Wave Approach

In this paper, we examine the problem of the response of rods, beams, plates and shells to low velocity impact. Thin bodies may possess elastic, viscoelastic, thermoelastic, isotropic, anisotropic features, as well as other features. In this paper, the emphasis is on wave theories of shock interaction. These theories are based on the fact that at the moment of impact transient waves (surfaces of strong discontinuity) are generated within the contact domain, which further propagate along the thin bodies and thereby influence the process of the shock interaction. The desired functions behind the strong discontinuity surfaces are found in terms of one-term, two-term or multiple-term ray expansions, the coefficients of which are determined with an accuracy of arbitrary functions from a set of equations describing the dynamic behavior of the thin body. On the contact domain boundary, the ray expansions for the desired functions go over into the truncated power series with respect to time and are matched further with the desired functions within the contact region that are represented by the truncated power series with respect to time with uncertain coefficients. As a result of such a procedure, it has been possible to determine all characteristics of shock interaction and, among these, to find the time dependence of the contact force and the displacements of the contact region. The proposed procedure is illustrated by numerous examples, which are combined into sections depending on the characteristics of the contact force and the dimensions of the thin bodies. The results are compared with those obtained by other approaches and procedures.

Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part II

Asymptotic stability by Lyapunov of the steady-state solution to the linear initial-boundary value problem which is formulated in the first part of this paper [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Well-posedness of a modified initial-boundary value problem on stability of shock waves in a viscous gas. Part I, J. Math. Anal. Appl. 331 (1) (2007) 408–423 (this issue)] proved under two conditions. The main of these conditions is that zeros of Lopatinsky determinant excluding η = 0 , s = 0 lie in the left half-plane. The problem arises while providing grounds for replacing of thin transitional zones of strong gradients of basic flow parameters for viscous heat-conducting gas with surfaces of strong discontinuity.

Dynamic Stability of a Circular Pre‐Stressed Elastic Orthotropic Plate Subjected to Shock Excitation

The problem on low‐velocity impact of an elastic body upon a pre‐stressed circular orthotropic plate possessing cylindrical anisotropy is considered. The dynamic behavior of the plate is described by equations taking the rotary inertia and transverse shear deformations into account. Longitudinal compressing forces are uniformly distributed along the plate’s median plane. Contact interaction is modeled by a linear spring, and a force arising in it is the linear approximation of Herts’z contact force. During the shock interaction of the impactor with the plate, the waves which are the surfaces of strong discontinuity are generated in the plate and begin to propagate. Behind the fronts of these waves, the solution is constructed in terms of ray series, which coefficients are the different order discontinuities in partial time‐derivatives of the desired functions, and a variable is the time elapsed after the wave arrival at the plate’s point under consideration. The ray series coefficients are determined from recurrent equations within accuracy of arbitrary constants, which are then determined from the conditions of dynamic contact interaction of the impactor with the target. The found arbitrary constants allow one to construct the solution both within and out of the contact region. The analysis of the solution obtained enables one to find out the new effect and to make the inference that under a certain critical magnitude of the compression force the orthotropic plate goes over into the critical state, what is characterized by ‘locking’ the shear wave within the contact region, resulting in plate damage within this zone as soon as the compression force exceeds its critical value.

The response of an elastic half‐space under a momentary shear line impulse

AbstractThe response of an ideal elastic half‐space to a line‐concentrated impulsive vector shear force applied momentarily is obtained by an analytical–numerical computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. The shear force is concentrated along an infinite line, drawn on the surface of the half‐space, while being normal to that line as well as to the axis of symmetry of the half‐space. An exact loading model is introduced and built into the computational method for this shear force. With this model, a compatibility exists among the prescribed applied force, the geometric decay of the shear stress component at the precursor shear wave, and the boundary conditions of the half‐space; in this sense, the source configuration is exact. For the transient boundary‐value problem described above, a wave characteristics formulation is presented, where its differential equations are extended to allow for strong discontinuities which occur in the material motion of the half‐space. A numerical integration of these extended differential equations is then carried out in a three‐dimensional spatiotemporal wavegrid formed by the Cartesian bicharacteristic curves of the wave characteristics formulation. This work is devoted to the construction of the computational method and to the concepts involved therein, whereas the interpretation of the resultant transient deformation of the half‐space is presented in a subsequent paper. Copyright © 2003 John Wiley & Sons, Ltd.

Source signature and elastic waves in a half‐space under a sustainable line‐concentrated impulsive normal force

AbstractFirst, the response of an ideal elastic half‐space to a line‐concentrated impulsive normal load applied to its surface is obtained by a computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. Then, the geometry is determined of the obtained waves and the source signature—the latter is the imprint of the spatiotemporal configuration of the excitation source in the resultant response. Behind the dilatational precursor wave, there exists a pencil of three plane waves extending from the vertex at the impingement point of the precursor wave on the stress‐free surface of the half‐space to three points located on the other two boundaries of the solution domain. These four wave‐arresting points (end points) of the three plane waves constitute the source signature. One wave is an inhibitor front in the behaviour of the normal stress components and the particle velocity, while in the behaviour of the shear stress component, it is a surface‐axis wave. The second is a surface wave in the behaviour of the horizontal components of the dependent variables, while the third is an inhibitor wave in the behaviour of the shear stress component. An inhibitor wave is so named, since beyond it, the material motion is dying or becomes uniform. A surface‐axis wave is so named, since upon its arrival, like a surface wave, the dependent variable in question features an extreme value, but unlike a surface wave, it exists in the entire depth of the solution domain. It is evident from this work that Saint‐Venant's principle for wave propagation problems cannot be formulated; therefore, the above results are a consequence of the particular model proposed here for the line‐concentrated normal load. Copyright © 2002 John Wiley & Sons, Ltd.