NONLINEAR FINITE ELEMENT ANALYSIS AND ADINA Proceedings of the 4th ADINA Conference Massachusetts Institute of Technology 15-17 June 1983 Guest Editor K. J. BATHE Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. PERGAMON PRESS OXFORD NEW YORK TORONTO SYDNEY PARIS FRANKFURT U.K. Pergamon Press Ltd., Headington Hill Hall, Oxford 0X3 OBW, England U.S.A. Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A. CANADA Pergamon Press Canada Ltd., Suite 104, 150 Consumers Road, Willowdale, Ontario M2J 1P9, Canada AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., P.O. Box 544, Potts Point, N.S.W. 2011, Australia FRANCE Pergamon Press SARL, 24 rue des Ecoles, 75240 Paris, Cedex 05, France FEDERAL REPUBLIC Pergamon Press GmBH, Hammerweg 6, OF GERMANY D-6242 Kronberg-Taunus, Federal Republic of Germany Copyright © 1983 Pergamon Press Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or trans- mitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopy- ing, recording or otherwise, without permission in writing from the publishers. ISBN 0 08 030566 0 Published as a special issue of the journal Computers & Structures, Vol. 17, Numbers 5-6 and supplied to subscribers as part of their normal subscription. Also available to non-subscribers. Computers ά Structures Vol. 17, No. 5 6, p. ν (1983). Pergamon Press Ltd. Printed in Great Britain. PREFACE The thoughts I presented in the preface of the proceedings of the 3rd ADINA Conference (see Computers and Structures, Vol. 13, No. 5-6, 1981) are also quite applicable to this 4th Conference Nonlinear Finite Element Analysis and ADINA. However, there are some thoughts that, at this opportunity, I would like to mention again and discuss further. It is well-accepted that finite element methods will be employed to an increasing extent in engineering practice, and that the appropriate application of these analysis methods requires a considerable amount of education and experience. The effective use of finite element computer programs is based on a thorough understanding of the basic principles of mechanics and materials and a knowledge of the finite element procedures and assumptions used when operating on these principles. Much of the basic knowledge to properly employ finite element methods must be taught at the Universities in the undergraduate and graduate education, and we must expect a considerable evolution—some of which has taken place already—of the University curricula in this respect. This educational goal is clearly not fulfilled by simple use of some finite element computer programs in the courses taught, but there is a deep intellectual challenge to change some of the traditional teaching of mechanics, for both analysis and design. However, another important area of professional education is the Continuing Education Study so important to all of us in engineering practice. With the methods of analysis and design evolving quite rapidly, we are continuously faced with many new developments to learn and absorb. Also, considering the currently available very impressive capabilities for analysis using finite element methods, the art of good engineering analysis is a very exciting and rewarding field with much room for learning and engineering ingenuity. The objective of the ADINA Conferences is to contribute to this continuing education process. In these conferences a number of valuable papers are presented on the usage of the ADINA system that provide the focal point for in-depth discussions of current state-of-the-art modeling and solution of complex problems. The reason for concentrating on the use of one finite element program system is to give a common basis of interest and understanding to the conference participants. However, many experiences discussed are quite general and should be of interest to most finite element researchers and practitioners. This volume contains the papers presented at the 4th ADINA Conference. I am once again personally very pleased that these proceedings are published in Computers and Structures and would like to express my appreciation to Professor H. Liebowitz for this help in my research and educational goals. I am also very thankful to the authors of the papers for their efforts and cooperation, and to Ms. C. Simonsen of the Center for Advanced Engineering Study of M.I.T. and Ms. T. Nolan, my secretary, for their help in organizing this conference. Finally, I would like to thank the ADINA users group whose lasting support is making this complete endeavor possible. Massachusetts Institute of Technology K. J. BATHE Cambridge, MA 02139 U.S.A. V Computers ά Structures Vol. 17, No. 5 6, pp. 625 633, 1983 0045-7949/83 $3.00+ .00 Printed in Great Britain. Pergamon Press Ltd. THE USE OF ADINA FOR ANALYSIS OF MINES WITH EXPLOSIVE FILLS FREDERICK H. GREGORY and AARON D. GUPTA U.S. Army Ballistic Research Laboratory, U.S. Army Armament Research and Development Command, Aberdeen Proving Ground, MD 21005, U.S.A. Abstract—The structural response of a U.S. M-15 and a Soviet TM-46 land mine with explosive fills subjected to an externally applied pressure wave has been analyzed with the ADINA finite element code. The finite element model of the two-dimensional axisymmetric configurations predicted response out to 2 msec of real time. Failure of the mine case was predicted, based on a comparison of the value of the three-dimensional second invariant of plastic strain with that of the one-dimensional value obtained from tensile tests. 1. INTRODUCTION thick and has three steps. This cover connects to a This paper describes the response of antitank mines of central circular plate formed by spot-welding of a two different configurations to a transient blast load. thick plate to the thin cover section. The intermediate The rationale for this analysis is the need to develop a wall is formed from 0.94 mm thick steel sheet to which remote, expeditious means of clearing a path through a hollow cylindrical piece 0.56 mm thick is attached to an enemy mine field. A technique of delivering rela- form the centrally located top fuze well. The fuze well tively large transient pressure to the surface of the contains a 40 g tetryl booster charge for activation. earth by means of explosives is under development. The lower part of the mine body is formed by a deep The object of this study is to determine the extent of drawing operation which results in very in- structural damage to mine bodies from a given level of homogeneous material properties. The central cavity blast wave amplitude and shape. The principle dam- in the main body of the mine is filled with a charge of age mechanism is to be a serious distortion or rupture 5.7 kg TNT explosive. The cavity between the top and of the mine body rather than fuze initiation or pressure intermediate walls is unfilled. However compression plate removal since the activation mechanisms could of air in this region can contribute to alteration of the be changed easily from one type of mine to another response behavior of the mine and subsequent un- and a failure could not be guaranteed based on a crimping of the joint. particular mode of actuation. The normal method of activation of the fuze is by The mines investigated represent typical antitank means of force applied to the pressure cap depressing mines, both foreign and U.S. manufacture, which con- the fuze and releasing the striker to strike the booster sist basically of round thin metal bodies filled with charge in the fuze well. This activates the tetryl booster explosives. These types of antitank mine constitute a which in turn detonates the primary TNT charge. The large part of the inventory of U.S. and foreign mines. secondary fuze well on the TM-46 mine gives it an The components most distinctive are the fuze mech- anti-disturbance capability. anisms. There are a variety of radically different fuzes for these mines, different both in mechanical designs 2.2 M-15 antitank mine description and method of activation. Therefore the numerical The M-15 mine has a cylindrical body similar to models adapted for the two mines are representative of the TM-46 mine. However there is no intermediate a large class of both foreign and U.S. mines. wall or unfilled space in the U.S. mine. The mine has The paper has four major areas as follows: (a) prob- a nominal diameter of 32.13 cm, height of 9.88 cm, lem definition, (b) determination of material proper- and weighs 14.3 kg. The center of the top of the mine ties and selection of failure criteria, (c) finite element has a depressed area which houses the pressure plate model description and calculations, and (d) dynamic assembly. Isometric and side views of the mine are response prediction of the structural assembly. shown in Fig. 2. The mine is made essentially of two pieces of WD-1010 steel which are joined at the lower periph- 2. PROBLEM DEFINITION ery by a 360 crimp. The upper part of the mine body 2.1 TM-46 antitank mine description is formed by a deep drawing operation which results The TM-46 land mine has a cylindrical steel body in very inhomogeneous materials properties as is the with a primary fuze well in the center of the top and one on the bottom, presumably for antilift or booby trapping purposes. In addition, it has a secondary fuze well in the sidewall underneath the carrying handle. A sectional drawing of the mine is shown in Fig. 1. The mine has a moninal diameter of 29.7 cm, height of 7.3 cm, and weighs 8.7 kg with a main charge of 5.7 kg III TNT. The mine body is made of three pieces of sheet steel which are joined at the upper periphery by a 360 crimp. The top of the mine body is only 0.635 mm Fig. 1. Soviet antitank mine. 625 626 F. H. GREGORY and A. D. GUPTA ARMING PLUG IN PRESSURE Fig. 2. U.S. M-15 antitank mine. case with the Soviet TM-46 mine. The central cavity Also, inclusion of these unsymmetrically located in the lower half of Fig. 2 is filled with 10 kg of structures would have necessitated the use of a three- composition Β explosive. dimensional (3-D) finite element model resulting in The fuze is activated by means of force applied to significant increase in computing time and costs. The the pressure plate (1250-2000 newtons) which in turn dimples at the base of both mines were eliminated for is transferred to the belleville springs. At a certain the same reasons. Because of these simplifications, deflection, the belleville springs snap through, driving the 2-D axisymmetric models were adequate for the firing pin into the detonator. The explosion of the dynamic response evaluation. detonator activates the tetryl booster which in turn detonates the primary composition Β charge. There 2.4 Base support and surface loading are two auxiliary fuze wells on the M-15 mine to During field emplacement, the mines may be placed allow anti-disturbance capability similar to the Soviet on the surface and covered with grass or other mine. materials for concealment. In other cases, the mines may be shallow buried. In either case, the mines will 2.3 Guidelines for the numerical model experience transient pressure loading on the top In keeping with the philosophy of identifying a surface due to detonation of a counter-mine explosive general failure mechanism independent of some in the vicinity. The base and side boundary condi- specific design feature, all pressure caps or plates, tions were treated in two different ways in the M-15 fuzes and springs were omitted from the finite ele- mine study. It is expected that typical field boundary ment model of both mines. This was done in accord- support conditions would be bracketed by the two ance with the previously stated guideline of not extreme conditions simulated. In one case, the base identifying failure of the fuze components. The mod- was supported on non-linear springs simulating soil. els shown do not include secondary fuzes and filling In this case, the mine was simulated as being buried holes. However the secondary tetryl booster charge is in soil up to its top surface by allowing downward included in the Soviet mine to facilitate assessment of acceleration/movement of the mine based on dy- the influence of trapped air in the unfilled space below namic properties of the soil medium as described in the top wall. [1]· The auxiliary fuze wells were not considered in the The other support condition used for the M-15 current investigation since they make the mine bodies and TM-46 mines was a rigid support which closely highly susceptible to damage due to stress concen- modeled the experimental conditions described in [2]. trations near the junction between the body and the A roller support condition was used allowing lateral, fuze. Thus, the simplified model is conservative in but no vertical, motion. The indirect loading of the terms of blast load required for mine deactivation. mine through shock waves passing through the soil The use of ADINA for analysis of mines with explosive fills 627 TOP SIDE WALL Time (ms) (a) LOCATION OF SPECIMENS, M-15 MINE Fig. 3. Shock loading function for antitank mines. medium was not modeled. In this rigid support condition, the input shock load is applied to the top and sides of the mine; whereas, in the spring support condition, only the top of the mine was loaded directly. For structural loading the pressure pulse used in this paper simulated peak pressure and impulse mea- sured from experiments conducted with mine clear- SIDE WALL—' ance types of explosives in [2]. The peak pressure was 13.8 MPa and the impulse delivered was 6.5 kPa-sec. (b) LOCATION OF SPECIMENS, TM-46 MINE A decaying exponential function was fitted to these parameters resulting in the following equation />(*) = 13.76 e"211'.7 (1) A curve of this function varying in time is shown in Fig. 3. 3. MATERIAL PROPERTIES AND FAILURE CRITERIA J 60i,12R 2.03 i.25 Material properties were required for the steel jackets, the explosive filler materials, the trapped air, and the soil in which the mine is emplaced. Mechan- (c) PREPARATION OF SPECIMEN DIMENSIONS (cm) ical properties were measured for the steel jackets by employing uniaxial tensile tests. The data for the Fig. 4. Details of tensile specimen sampling and prepara- tion. explosive and soil were taken from available publica- tions. Failure criteria used for the steel jackets and the filler materials were similar to the formulations in [3]· 3.1 Steel casing The M-15 jacket is made of a medium strength steel alloy with a density of 7.80 g/cm 3 and a thickness of 0.94 mm. The TM-46 jacket is made of a low Pressure plate well tensile tests carbon soft magnetic steel equivalent to mild steel. 500 The lower part of the casing was deep drawn, but it retained an equiaxed grain microstructure with iso- Bilinear tropic properties. Two tensile specimens were cut approximation Test 2 from each of the significant surfaces of the mine Q_ body. Locations of these specimens are shown in Fig. Test I 4(a, b). The specimens were machined with a large radius on the test section as shown in Fig. 4(c). An extensometer and a biaxial strain gage were attached at the location of the minimum width and the speci- mens were tested in an Instron Testing Machine. Typical stress-strain curves for the U.S. and the Soviet mine body are shown in Figs. 5 and 6 re- spectively. Evidence of work hardening and residual 6.0 9.0 stress was significant in the Soviet mine due to the True strain (%) forming operation. Fig. 5. Stress-strain curves for the pressure plate well Bilinear approximations to the stress-strain curves specimens for the M-15 mine. 628 F. H. GREGORY and A. D. GUPTA equation of state[1,6] is given as Γ(Γ + 1)(Λμ2 +V + <V) K~ 2^Jf (4) where κ = the loading bulk modulus; Γ = the Grüneisen coefficient; A, B, C = the coefficients ap- pearing in the Grüneisen equation of state in terms of ? 100 μ; Α' = Α(Γ+ l) + 2£; B' = Β(Γ + 2) + 3C; C = C(T + 3); μ = e,(\ - t ); e = (V- K)/K , vol- r v 0 0 ume strain taken positive in compression; and V = 1 /pQ = specific volume at normal conditions. 0 The values for the material constants of the ex- 5 10 plosives used are shown in Table 1. ENGINEERING STRAIN (X) Note that when e = 0, μ = 0, κ = A and V = V. v 0 0 Fig. 6. Stress-strain curves for the top cover plate speci- Also, in the Grüneisen EOS, at e = 0, we take the v mens for the TM-46 mine. pressure and internal energy to be zero. Because no data were available to relate the un- loading bulk modulus to the volumetric strain, the obtained by averaging the data for the individual same values of the bulk modulus for unloading as for specimens are shown superimposed in Figs. 5 and 6. loading were used for all explosives. The loading The ADINA [4, 5] finite element code used in this shear modulus, G was obtained from the loading analysis has a bilinear, elastic-plastic, von Mises yield h bulk modulus, κ,, by use of the relationship, condition, kinematic hardening, axisymmetric 2-D element for the steel jacket. The criterion selected to predict failure of the steel 3φ -2v) G,= (5) casing material was described in [3] as the value of the 2(1+v) * second invariant of plastic deviatoric strain at failure, Ijjie1*), defined as Figures 7-9 show the graphical relationships of the three explosives represented by eqns (4) and (5). (2) Table 2 gives the values of the two moduli as they were used in the ADINA program. ADINA uses where the strains indicated are to be the strains at linear interpolation between discrete points. failure. In the uniaxial tension test where the load is The tensile volumetric strain at failure for the applied in the Z-direction, we have, composition B-3 explosive is given in [6] as — 0.1%. This criterion was used in calculations for all ex- = 3/4(££ )2. (3) z plosives in this investigation. The technique used in the ADINA code to apply this failure is by the artifice 3.2 Characterization of explosives- of superimposing on the applied load-induced strains, There are two types of explosives employed in the an in-situ gravitational pressure sufficient to cause a TM-46 mine, i.e. TNT as the main charge and tetryl hydrostatic compression equal in magnitude to the as the fuze well charge. For the U.S. M-15 mine, tensile failure. Then, when the total strain becomes composition B-3 explosive consisting of 60% RDX negative, a tension cut-off plane is assumed to form and 40% TNT is used in cast form as the main charge. normal to the principal strain. The normal and shear After surveying the available material properties of stiffness across this plane are reduced by a factor explosives and the various 2-D axisymmetric materi- determined by an input value. One or two additional als models in the ADINA code, it was decided that planes orthogonal to existing tension cut-off plane(s) the curve description material model (see [4] Section are allowed to form if the strain criterion is met. The XII, pp. 17-22) was the appropriate model to use. planes become inactive if compression again develops This model requires tables of loading and unloading in the direction normal to it. bulk moduli and shear moduli versus volume strain. The pseudo-hydrostatic pre-strain is applied by A relationship between the volume strain and the positioning the vertical coordinate (Z-coordinate) at bulk modulus obtained from the Mie-Grüneisen the proper negative value. The hydrostatic pressure Table 1. Material constants for expl osives and soil Type p Γ A Β C V o 3 (g/cm ) (Gpa) (Gpa) (Gpa) Comp B-3 1 .68 .947 13.5 9.5 100.6 .29 TNT 1.614 .737 10.367 9.101 138.33 .3 Tetryl 1 .70 1.6 10.498 17.8 20.6 .3 Wet Tuff 2.0 1 .5 21.77 32.5 18.33 — The use of ADINA for analysis of mines with explosive fills 629 Κ - BULK MODULUS BULK MODULUS / 3 -I s I . ο ο G - SHEAR MODULUS I « 2.0 0.0 0.02 0.04 0.06 0.08 0.10 £v - VOLUMETRIC STRAIN Fig. 8. Bulk and shear moduli vs volume strain for TNT explosive. SHEAR MODULUS Κ-BULK MODULUS ,^20.0 S» 3 Î5 [I FAILURE IN TEN If AT € v =-0 Γ 0 2 4 6 0.0 0.02 0.04 0.06 0.08 0.10 VOLUME STRAIN (%) £v - VOLUMETRIC STRAIN Fig. 7. Bulk and shear moduli vs volume strain for com- Fig. 9. Bulk and shear moduli vs volume strain for tetryl position B-3 explosive of the M-15 mine. explosive. Table 2. ADINA Input values for bulk and shear moduli for filler materials COMPOSITION B-3 EXPLOSIVE Point No. ε,.V κ I n K u G ι (%) (GPa) (GPa) (GPa) 1 0 13. 52 13.52 6.60 2 1.0 14. 00 14.00 6.84 3 2.5 14.91 14.91 7.28 4 3.75 15.83 15.83 7.73 5 5.0 16 92 16.92 8.26 6 10.0 23 36 23.36 11.41 TNT EXPLOSIVE 1 0 21 72 21 .72 10.62 2 1.0 23 03 23.03 11.24 3 3.0 25 65 25.65 12.55 4 5.0 28 68 28.68 14.01 5 9.0 35 85 35.85 17.51 6 11.0 40 20 40.20 19.65 TETRYL FILLER 1 0 10 5 10.5 4.03 2 1 .0 11 15 11.15 4.27 3 3.0 12 59 12.59 4.83 4 5.0 14 .24 14.24 5.46 5 8.0 17.2 17.2 6.60 6 10.0 19 .56 19.56 7.50 630 F. H. GREGORY and A. D. GUPTA applied at an element integration point is given for an transfer some load to the middle plate as the volume element, j, by of the cavity is decreased. An attempt was made to model the air with 2-D axisymmetric fluid elements composed of an inviscid linear compressible material. (6) A constant bulk modulus was used in lieu of a pressure dependent bulk modulus due to lack of where p is the density of the overburden; is the available data for air. However, the primary difficulty e shape function for node / of element j; and Z is the with this model was that there was nothing in the i} vertical coordinate for node / in element / The model to prevent the upper plate from penetrating the position of the system vertical coordinate can be middle plate as the deformation progressed. obtained from the equation, Since the air was judged to apply only a minimal restraint on the motion of the upper plate and due to the need to prevent the two plates from passing through one another, a different model has been gPe adopted. The model consists of axial truss elements where κ is the initial bulk loading modulus; ej is the connecting the two circular plates. The material 0 volumetric failure strain, negative in tension; and g is model for the trusses is nonlinear and develops only the acceleration due to gravity. a small force up until the axial strain in trusses approaches —1. At this strain, a large stiffness is 3.3 Soil simulation specified to simulate contact between the two plates. For the structural response calculations of the Constraints are applied to the upper end of the shallow buried M-15 mine, only the top of the mine trusses to insure that its radial coordinate is the same was exposed to blast pressure while the remainder as the radial coordinate of its lower end. Also, the was assumed to be embedded in soil. An implicit axial coordinate of upper end is constrained to modeling technique was employed whereby nodal tie translate with the upper plate. elements were used to model the base support as nonlinear springs. No simulation of the soil was 4. FINITE ELEMENT MODEL DESCRIPTION AND necessary for the rigid support calculations. CALCULATIONS Three different types of nodal tie elements were The two mines were modeled as axisymmetric 2-D available in the Ballistic Research Laboratory version structures using the ADINA finite element code. The of the ADINA code. The particular type chosen is the steel components were modeled with six-node ele- boundary type element defined by one node only and ments including mid-side nodes on the plate surface. is capable of three translational and three rotational The explosive components were modeled with four- degrees of freedom. In the M-15 mine, the elements node QUAD elements except where they interfaced along the base of the mine were used to transmit a the steel jacket, in which case a mid-side node was vertical force (Fz), while those along the side exerted included on the interface edge. a horizontal force (FY). The time step used for the calculations was deter- Due to the large variety of soils in which mines mined from the Courant stability condition would be emplaced, it is possible only to select a soil simulation model which would be representative of At = (7) some subclass of soils. Thus, a typical load deflection curve [7] was selected to define the nodal tie element properties. The average load-deflection for slowly varying loads in the elastic loading range from [7] is where At is the minimum Courant stability time cnt 0.0815 MPa/cm. To account for the dynamic re- step; Δ/ is the distance between the two closest nodes sponse of soil at the base of the mine, a nonlinear in the system; Ε is the Young's modulus for the quadratic component was added to the force stiffest material; ρ is the density of the material; and deflection property. η is the number of time steps which we wish to For the support along the vertical sides of the mine, represent the shock wave in passing through the a linear spring force was used due to consideration of distance Δ/. The value of Ar was approximately 200 crti small lateral movement. The linear nodal tie element nanoseconds, for both the TM-46 and M-15 mines. stiffness values along the vertical side are propor- A value of η of four was used, so that the time step tional to the height of the particular element onto for the central difference explicit time integration which the nodal tie boundary element is attached. method was 50 nanoseconds. Similarly, the nonlinear stiffness values for springs at the base in the ADINA input are adjusted by a factor 4.1 M-15 mine calculations proportional to the annular sector of π radians and As indicated previously in Section 2.4, two a radial extent appropriate for the particular nodal tie different boundary conditions were used in modeling element. For soil modeling of the TM^6 mine in the the M-15 mine. The primary difference between the ADINA code, an explicit technique using two layers two calculations was in the base support condition. of compressible soil elements surrounding the mine One used a nonlinear spring support and the other will be employed. used a rigid vertical base support. The mesh configuration for both M-15 models is shown in Fig. 3.4 Simulation of void in ΓΜ-46 mine 10. The TM-46 mine has a cavity between the upper Eigenfrequencies were generated and the associ- pressure plate and the middle plate covering the ated mode shapes were plotted via the ADINA primary charge. This cavity has air in it which would post-processor, PLOT3D[8]. The natural frequencies The use of ADINA for analysis of mines with explosive fills 631 very inefficient load transfer from the top cover to the main mine body. On the other hand, any viable failure mechanism for the mine must inevitably in- volve a failure of the main mine body. Since the ADINA 1981 code does not have a contact element to sense when the top cover plate and middle plate impact, we have used nonlinear truss elements to approximate the interaction of the two plates. This approach was described in Section 3.4. Eigenfrequencies and mode shapes were also ob- tained for the TM-46 mine model. The eigen- frequencies and associated periods for the lower Fig. 10. Finite element mesh for the M-15 mine. modes are given in Table 4. All calculations described herein used the total are important for estimating the rate of response of Lagrangian formulation with a lumped mass matrix a structure. For similar loadings, the higher the with the exception of the nodal tie and truss elements. natural frequencies of a structure, the faster the The formulations used for these were material non- structure will respond. In addition, rapid responses linearity only and updated Lagrangian analysis pro- cause higher strain rates to be effected. This is cedure, respectively. significant for strain rate sensitive materials such as mild steel which both of the subject mines embody. However, strain rate sensitivity was not modeled in 5. DYNAMIC RESPONSE PREDICTIONS these calculations. The mode shapes associated with Several modifications to the ADINA program were the lower eigenfrequencies often give a good indi- made to assist us in interpreting the response predic- cation of the deformed shape which will result from tions. These are described fully in [1]. A summary of the application of typical loads. This was especially these modifications will be given here. Due to the very evident in the deformation of the TM-46 mine. The large amount of stress-strain data available from the lower eigenfrequencies and periods for the M-15 ADINA results, some means of selectively extracting mine are given in Table 3. significant parts of the results was desired. Since the component which involved the most credible failure 4.2 TM-46 mine calculations mechanisms was the steel jacket, we focused our atten- The ADINA calculations for the TM-46 mine tion on it. The modifications were made in two have not been completed. However, some of the different areas. First, routines were written to monitor salient features of the model have been developed the extreme (maximum/minimum) stresses and strains from progress made in studies of the mine thus far. in the steel components. Information on the location, A drawing of the current mesh configuration is time, and value of these extreme stresses and strains shown in Fig. 11. were saved and printed at intervals during the calcu- From our experience with the M-15 mine, we lation. Second, routines were written to calculate and expected significantly different materials properties in monitor the second invariant of plastic strain. The the outer steel jacket of the TM-46 mine. Mea- value of this quantity was compared to an input value surements showed that this was, indeed, the case. in order to predict failure of the steel jacket. Tables of Several different sets of materials properties were the maximum value of this quantity were stored and used to model the various steel components of the printed at preselected intervals. mine. In addition to the above modifications to ADINA, It was evident from the first that two particular one further modification was necessary to successfully difficulties would be encountered in modeling the obtain the solution to such long response times using TM-46 mine. First, the difference in stiffness between the explicit time integration scheme. In the standard the steel plates and the air filled region leads to ADINA program, (1977 version) whenever plasticity numerical problems. The collapse of the air filled occurs in the kinematic hardening model for a solid region leads to the impact of the upper plate on the element, a linearized correction is applied to bring the middle plate. This phenomenon needs to be modeled stress tensor back to the von Mises yield surface. rather carefully. Second, the thin stepped top cover Because the linearized correction leaves the stress at a shown in the upper right part of Fig. 11 leads to a position in stress space along a tangent to the convex Table 3. Eigenfrequencies and periods for the M-15 mine Spring Supported Mine Rigidly Supported Mine Frequency Period Frequency Period (cps) (sec) (cps) (sec) 36* 2.744 χ 10"? 6426 1.556 χ 10"u 3636 2.750 χ 10"u 7899 1.266 χ 10"u 6710 1.490 χ ]0~k 9685 1.032 χ ^0~^ 8531 1 .172 χ 10"14 12186 8.205 χ 10"5 *Rigid body mode.