Elastic-plastic Wave Propagation Research Articles

The propagation of spherical waves in rate-sensitive elastic-plastic materials

An integral equation method for the analysis of elastic-plastic wave propagation is presented. The elastic-plastic solution is thereby found as the superposition of the corresponding elastic result with waves produced by dynamically induced plastic strains. The solutions are represented in the form of integrals with elastodynamic Green's functions as integration kernels. The spherically symmetric problem of a dynamically loaded spherical cavity is considered and the corresponding Green's functions for this geometry are derived in closed form. Time convolution is carried out analytically over a prescribed time step and the spatial integration is performed by Gaussian quadrature. If the wave travels within each time step just the distance of one spatial element the evaluation of the integrals leads to a tridiagonal system of algebraic equations. Numerical results are compared to some known analytical solutions, proving the accuracy of the method. Computations are carried out for rate sensitive power law hardening-thermal softening materials.

A dynamic model for a Timoshenko beam in an elastic-plastic state

In this paper, equations which describe the propagation of elastic-plastic waves of combined stress resultants in a Timoshenko beam under symmetrical bending and tension or compression are derived. It is assumed that the beam material exhibits a weak work-hardening and strain-rate independent behaviour. The normality rule is postulate for the plastic strain rates of an infinitesimal volume element. The wave propagation is described by a hyperbolic system of differential equations, the state variables of which are the stress resultants and the strain rates of an infinitesimal beam element. It is shown that the actual distribution of shear stress in the plastic range need not be taken into account by means of a plastic shear correction factor.

A Riemann solver and a second-order Godunov method for elastic-plastic wave propagation in solids

By use of three basic paths of elastic-plastic loading, a Riemann solver is established for the one-dimensional combined longitudinal and torsional stress wave problem of the thin-walled elastic-plastic tube. Based on this, a second-order Godunov method is developed for the numerical computation of elastic-plastic waves. Finally the numerical method is applied to some basic problems with known exact solutions and the comparison of results is made.

One-Dimensional Theory of Elastic-Plastic-Viscoplastic Stress Wave Propagation.

In the present paper, a one-dimensional theory of elastic-plastic-viscoplastic stress wave propagation is evolved. The concepts of understress and overstress are introduced to the constitutive equation used. The strain dependence and the strain-rate dependence of the wave propagation speed of the stress wave are given. A few forms of the wave propagation speed of the elastic-plastic-viscoplastic stress wave are newly derived. It is also shown that the theory of elastic-plastic-viscoplastic stress wave propagation given here contains a theory of elastic-plastic stress wave propagation and a theory of elastic-viscoplastic stress wave propagation. Moreover, a precursor and an unloading wave are also mentioned.

Plastic wave front propagation in cylindrical bars

Making use of the dynamic, elastic-plastic finite element program, the elastic-plastic wave propagation in a cylindrical bar with a circular cross section is analysed for linear strain hardeningt Special attention is given to the influence of strain rate on. the axial stress distribution for variou. impact loading conditions. Comparison is made between numerical two-dimensional and analytical one-dimensional results for structural low alloy steel (C-Mo-Cr) having strain rate dependens dynamic mechanical properties. The implications of practical importance for experimental dynamic plasticity studies by means of Hopkinson Split Pressure Bar (HSPB) technique are discussed in detail.

Mechanical behaviour of austenitic steel and aluminium with respect to elastic-plastic wave propagation

Wave propagation phenomena influence the material behaviour of structures under dynamic loading. Parameters of constitutive equations must be determined experimentally with respect to these effects. Impact tension tests were performed using specimens of austenitic steel X 6 CrNi 18 11. The specimen geometry as well as the induced impact energy were varied. The validity of the determined material law parameters was examined by combination with numerical calculations. Using the same procedure data received in impact torsion test for aluminium 99,98 were confirmed. A confident description of material behaviour over wide ranges of strain rate and strain was obtained by coupling experiment and calculation. Wellenausbreitungsphänomene beeinflussen das Werkstoffverhalten bei dynamisch beanspruchten Bauteilen. Die Parameter konstitutiver Gleichungen müssen unter Berücksichtigung dieser Effekte experimentell ermittelt werden. An Proben aus dem austenitischen Stahl X 6 CrNi 18 11 wurden Schlag-Zug-Versuche durchgeführt, wobei die Probengeometrie sowie die induzierte Schlagenergie variiert wurden. Durch Kombination mit numerischen Berechnungen wurde die Gültigkeit der ermittelten Stoffgleichungsparameter überprüft. Mit der gleichen Vorgehensweise wurden im Schlag-Torsionsversuch ermittelte Daten für Aluminium 99,98 untermauert. Durch die Rückkopplung zwischen Versuch und Rechnung gelang eine gesicherte Beschreibung des Werkstoffverhaltens über weite Bereiche der Dehngeschwindigkeit und der Dehnung.

An arbitrary lagrangian-eulerian finite element method for path-dependent materials

The conservation laws, the constitutive equations, and the equation of state for path-dependent materials are formulated for an arbitrary Lagrangian-Eulerian finite element method. Both the geometrical and material nonlinearities are included in this setting. Computer implementations are presented and an elastic-plastic wave propagation problem is used to examine some features of the proposed method.

Elastic-plastic wave propagation in cylindrical bars

Etude de la distribution de la contrainte uniaxiale pour differentes conditions de charge. Comparaison entre les resultats numeriques a deux dimensions et analytiques a une dimension d'un acier ayant des proprietes mecaniques dynamiques independantes de la vitesse de deformation

Propagation of elastic-plastic compression-shear waves in materials having delayed yield

Propagation of elastic-plastic compression-shear waves in materials having delayed yield

The problem of resonance for longitudinal small amplitude excitation of an elastic-plastic bar

Resonance of a finite elastic-plastic bar excited by longitudinal stress impulses of amplitudes smaller than the yield stress is studied. The shape of the impulses is rectangular and the stress-strain curve is assumed to be bilinear. Solution of the problem is based on the classical assumptions of elastic-plastic wave propagation and is being handled by means of the position-time diagram. Frequency changes of the excitation forces are accepted to obtain simple equations governing the build up of maximum stress, strain and particle velocity within the sample.

Dynamic Response of Axisymmetric Solids Subjected to Impact and Spin

A numerical method is presented to obtain solutions for axisymmetric solids subjected to impact and spin. The method is based on a finite element formulation wherein the equations of motion are integrated directly rather than through the traditional stiffness matrix approach. The formulation is given for axisymmetric triangular elements which can experience large strains and displacements in the radial, axial, and angular directions. The effects of material strength and compressibility are included to account for elastic-plastic flow and wave propagation. The addition of angular displacements allows the effect of twisting to be included in the analysis. The radial and axial equations of motion are applied in the usual manner, whereas the angular (spin) equations of motion are obtained by conserving angular momentum; this allows the angular velocities of the nodes to be altered by a change in radial position and/or shear stress induced tangential forces acting on the concentrated masses. The numerical method can also be used to obtain steady-state solutions for spinning solids.

Mechanisms of elastic-plastic wave propagation in rods and plates made of a strain-rate-sensitive material

Mechanisms of elastic-plastic wave propagation in rods and plates made of a strain-rate-sensitive material

Estimation of yield stress in polymers at high strain-rates using G.I. Taylor's impact technique

M easurement of the flow stress of polymers at strain-rates of 10 3−10 4 Sec −1 using G.I. T aylor's (1948) projectile impact method is discussed. Taylor's own analysis, and other related ones developed since then, are shown to be unsuitable for application to polymeric materials, and a method based on one-dimensional elastic-plastic wave propagation analysis is presented which is more satisfactory. The assumptions made in this analysis are justified by the experimental data reported here for polycarbonate; the yield stress and strain for this material in compression at a mean strain-rate of 7 × 10 3 sec −1 are estimated, respectively, as 180±10 MPa and 6.7±0.3%.

High Velocity Impact Calculations in Three Dimensions

A three-dimensional analysis is presented for high velocity impact problems. A Lagrangian finite-element technique is formulated for tetrahedron elements subjected to large strains and displacements. The effects of material strength and compressibility are included to account for elastic-plastic flow and wave propagation. The strains and strain rates in each element are determined from the displacements and velocities of the nodes. The strains, strain rates, and material properties are used to determine the elastic stresses, plastic deviator stresses, hydrostatic pressure, and artificial viscosity. The stresses are equated to concentrated forces acting on concentrated masses at the nodes, and the nodal equations of motion are numerically integrated. Illustrative examples are also included.

Finite element analysis of elastic-plastic wave propagation effects

The feasibility of dynamic analysis by means of the finite element method, in conjunction with direct time step integration procedures, has been demonstrated and documented in the literature. However, there appears to be some uncertainty concerning the validity of such analyses in cases involving wave propagation effects. It is the purpose of this paper to demonstrate that under appropriate conditions, nonlinear finite element analysis techniques, utilizing a variable time step integration procedure, may be applied to obtain a reliable description of the response of a body undergoing wave propagation effects. The method employs Taylor series expansions to obtain predictor-corrector expressions, truncated to difference form, of the solution to a system of first order nonlinear equations. The corresponding second order system of equations of motion are employed to construct the corrector expression. The incorporation of this procedure within the tangent modulus method of nonlinear analysis is described and numerical results are presented for a number of axisymmetric structures subjected to elastic-plastic wave propagation effects, including those cases involving reflections from boundaries.

Analytical investigation of the unloading wave: PMM vol. 40, n≗ 2, 1976, pp. 327–336

In the general case, the determination of the unloading wave shape [1] in the theory of elastic plastic wave propagation is reduced to the solution of a functional equation of complex structure. A characteristics method [2] is proposed for the approximate construction of the unloading wave, in particular loading cases formulas are obtained to determine its initial slope [3] and the next derivatives at the initial point [4–6]. An investigation of the general properties of an unloading wave is given in [7]. It is shown that as the load tends to zero asymptotically, the unloading wave at the end of a semi-infinite bar has an asymptote with a slope determined by the elastic wave velocity. An investigation of the functional equation is given in this paper and a method of solution of this equation in the form of a power series is proposed. This approach to the problem permits obtaining both known and some new results. In the general loading case, formulas are obtained to determine the initial slope of the unloading wave and a method of determining the next derivatives at the initial point is indicated. Conditions are found for linear hardening for which the unloading wave is a straight line. The existence of an asymptote different from those mentioned in [7] is proved; it is shown how to continue the solution to adjacent sections by means of some known section, and an unloading wave in a material with delayed yielding is investigated.

Finite-amplitude elastic plastic wave propagation in laminated composites: Hegemier, G. A. J App Physics, Vol 45, No 10 (Oct 1974) p 4248

Finite-amplitude elastic plastic wave propagation in laminated composites: Hegemier, G. A. J App Physics, Vol 45, No 10 (Oct 1974) p 4248

Comparison of Shock-Induced Two- and Three- Dimensional Incipient Turbulent Seperation

2 Garnet, H. and Armen, H., Evaluation of Numerical Time Integration Methods as Applied to Elastic-Plastic Dynamic Problems Involving Wave Propagation, Rept. RE-475, March 1974, Research Dept., Grumman Aircraft Corp., Bethpage, N.Y. 3 Stoodley, G. R. and Ball, D. J., Mathematical Background of Two Numerical Integration Techniques for Ordinary Differential Equations, Memo. RM-192, Oct. 1961, Research Dept, Grumman Aircraft Corp., Bethpage, N.Y. 4 Mantus, M., Lerner, E., and Elkins, W., Landing of the Lunar Excursion Module (Method of Analysis), Rept. LED-5206A, Revised April 10, 1967, Grumman Aircraft Corp., Bethpage, N.Y. 5 Lerner, E. and Mantus, M., Dynamics of Unsymmetric Landing, Rept. ADR 02-10-10-62-1, Jan. 1963, Grumman Aircraft Corp, Bethpage, N.Y. 6 Donnell, L. H., Longitudinal Wave Transmission and Impact, Transactions of the ASME, Vol, 52, 1930, pp. 153-167. 7 DeJuhasz, K. J., Graphical Analysis of Impact of Bars Stressed Above the Elastic Range, Journal of the Franklin Institute, Vol. 248, No. 2, Aug. 1949, pp. 113-142. 8 Davids, N. and Kumar, S., A Contour Method for One Dimensional Pulse Propagation in Elastic-Plastic Materials, Proceedings of the Third U.S. National Congress of Applied Mechanics, Brown University, Providence, R.I., 1958, pp. 502-512. 9 Garnet, H. and Armen, H., One Dimensional Elastic-Plastic Wave Propagation and Boundary Reflections, Memo, Grumman Research Department, Bethpage, N.Y. (in preparation).

On the reduction of the hodograph equations for one-dimensional elastic-plastic wave propagation

On the reduction of the hodograph equations for one-dimensional elastic-plastic wave propagation

Finite-amplitude elastic-plastic wave propagation in laminated composites

An approximate nonlinear theory is developed to describe finite-amplitude wave propagation parallel to the alternating layers of a laminated composite with elastic-plastic constituents. Model construction is based upon an asymptotic method in which dominant signal wavelengths are assumed large compared to typical composite microdimensions. The final equations evolve as a one-dimensional binary mixture theory.