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| t × 10⁻³ secs | Crown deflection (in) | | ------------- | --------------------- | | 0 | 0.00 | | 2 | 0.06 | | 4 | 0.05 | | 6 | 0.05 | | 8 | 0.07 | | 10 | 0.08 |

Fig. 11.5(b) Spherical shell results. Cases (iii) and (vi). Figure 11.5(b) illustrates the elasto-plastic small deformation response. The time periods are elongated with the inclusion of plasticity effects. In Fig. 11.5(c) the results for the problem with the stiff element are presented with explicit, implicit and mixed explicit–implicit analysis (cases (vii)–(ix)). The execution times and results are compared. The relative computer times are: (i) all elements considered as explicit - 120.0 sec (ii) stiff elements as implicit and rest explicit - 80.8 sec (iii) all elements considered as implicit - 16.4 sec 

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| Method | Time (t × 10⁻³ secs) | Crown deflection (in) | |--------|----------------------|------------------------| | Explicit | 0.25 × 10⁶ sec | 0.025 | | Implicit-explicit | 0.40 × 10⁶ sec | 0.040 | | Implicit | 0.60 × 10⁶ sec | 0.060 |

Fig. 11.5(c) Spherical shell results. Cases (vii)-(ix). This shows that by representing the stiff elements implicitly computer time can be saved. The analysis in which all elements are treated implicitly gives the lowest execution time for this small problem. However, with increasing problem size (and band width) the solution time for an implicit solution increases very rapidly because of the large core requirement and the increased number of computer operations. Finally it should be noted that Hughes has recently shown how the implicit-explicit schemes may be used in a more general context where there are, for example, nonsymmetric stiffness matrices involved or an implicit-explicit dynamic relaxation solution is required. $^{(13)}$ # 11.7 Problems 11.1 Repeat Problems 10.1–10.4 using program MIXDYN. Use fully explicit, fully implicit and mixed implicit/explicit meshes. # 11.8 References 1. BELYTSCHKO, T. and MULLEN, R., Mesh partitions of explicit-implicit time integration, In: Formulations and Computational Algorithms in Finite Element Analysis, Ed. K. J. Bathe et al., MIT Press (1977). 2. BELYTSCHKO, T. and MULLEN, R., Stability of explicit-implicit mesh partitions in time integration, Int. J. Num. Meth. Engng. 12, 1575–1586 (1978). 3. BELYTSCHKO, T., YEN, H. J. and MULLEN, R., Mixed methods for time integration, Proc. Int. Conf. on Finite Elements in Non-linear Mechanics (FENO-MECH 78), Univ. Stuttgart, Germany (Sept. 1978). 4. HUGHES, T. J. R. and LIU, W. K., Implicit-explicit finite elements in transient analysis: stability theory, J. Appl. Mech. 45, 371–374 (1978). 5. HUGHES, T. J. R. and LIU, W. K., Implicit-explicit finite elements in transient analysis: implementation and numerical examples, J. Appl. Mech. 45, 375-378 (1978). 6. HUGHES, T. J. R., PISTER, K. S. and TAYLOR, R. L., Implicit-explicit finite elements in nonlinear transient analysis, Comp. Meth. Appl. Mech. Engng. 17/18, 159-182 (1979). 7. PARK, K. C., FELIPPA, C. A. and DERUNTZ, H. A., Stabilization of staggered solution procedures for fluid-structure interaction analysis, In: Computational Methods for Fluid-Structure Interaction Problems, Ed. T. Belytschko and T. L. Geers, ASME Applied Mechanics Symposia Series, AMD, 26, 94–124 (1977). 8. PARK, K. C., Partitioned transient analysis procedures for coupled field problems, to be published J. Appl. Mech. (1980). 9. BATHE, K. J. and WILSON, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey (1976). 10. PAUL, D. K. and HINTON, E., User guide and report on program MIXDYN for implicit-explicit transient dynamic analysis, Research Report, University College of Swansea (1980). 11. NAGARAJAN, S. and POPOV, E. P., Elastic-plastic dynamic analysis of axisymmetric solids, Computers and Structures, 4, 1117-1134 (1974). 12. BATHE, K. J. and OZDEMIR, H., Elastic-plastic large deformation static and dynamic analysis, Computers and Structures, 6, 81–90 (1976). 13. HUGHES, T. J. R., Implicit-explicit finite element techniques for symmetric and nonsymmetric systems, Proc. Int. Conf. Numerical Methods for Nonlinear Problems, Swansea, 127–139, Pineridge Press, Swansea, U.K. (1980). # Chapter 12 # Alternative formulations and further applications # 12.1 Introduction Throughout this text we have considered several specific elasto-plastic material problems and, apart from Chapter 3, treatment has been limited to the use of elasto-plastic quasi-static incremental theory or an elasto-viscoplastic formulation. These theories and the application areas of solids and plates form, undoubtedly, the area of most interest and importance in nonlinear material analysis and it is for this reason that they have been chosen for study in this text. However, other topics and applications of possibly equal importance have had to be omitted for reasons of space and it is the aim of this chapter to indicate to the reader some areas for future studies. The developments which will be discussed can be categorised into the following classes: \- Further applications. The elasto-plastic and elasto-viscoplastic theories described earlier in this text can be extended to cover some alternative structural forms. Of prime importance in this area is the analysis of both thick and thin three-dimensional shell structures and the main changes necessary to the corresponding linear elastic finite element process relate to expressing the yield criterion in terms of the appropriate stress resultants. \- Alternative material models. The behaviour of some engineering materials may not be adequately described by the yield criteria presented in Chapter 7. This is particularly true of soils, rocks and concrete, since these materials, for example, have a limited tensile strength which is not accurately reflected in either the Mohr–Coulomb or Drucker–Prager failure laws. For such materials appropriate failure criteria must be developed. Additionally for soils the assumption of associated plasticity leads to excessive dilatency necessitating alternative formulations for accurate material modelling. ● Further problem classes. Many physical situations exist which are governed by nonlinear equation systems which are not suitable for solution by the techniques described so far in the text. One such example is the time dependent deformations which take place during a metal forming process. In this application the elastic strains are negligible compared with the plastic components and therefore the stress increments can no longer be expressed by use of (8.15). For dynamic situations, coupled media problems frequently have to be solved. This may involve a fluid/structure interaction problem of the seismic analysis of water retaining structures or the impulsive loading of a nuclear containment vessel together with the coolant fluid. All the above problems may be complicated by further nonlinear behaviour due to gross geometrical deformations. \- Improved numerical techniques. Since nonlinear solution processes are necessarily expensive with regard to computational time, any savings which can be made in this area are of prime importance. Developments in this area include improved nonlinear equation solution techniques and self-adaptive schemes for optimisation of the finite element mesh and load incrementation. A further enhancement is the use of sub-structuring techniques to separate elastic and elasto-plastic regions leading ultimately to coupled boundary integral/finite element solutions. In this chapter we explore the above developments (and others) in more detail and provide the reader with references for future study. Many of the subroutines presented earlier in the text can be employed (possibly in a modified form) in the development of computer codes for these further applications. Therefore the role of each subroutine presented is summarised and its location in the text also listed. # 12.2 List of subroutines In this section we record details of each subroutine that has been presented in this text. This library of subroutines can be employed to develop computer codes for the further applications discussed later in this chapter. The section of the chapter in which the subroutine is presented is recorded and the codes in which it is used are also indicated, employing the following program names: One-dimensional applications

QUITER	Solution of quasiharmonic problems by direct iteration (Chapter 3).
QUNEWT	Solution of quasiharmonic problems by the Newton-Raphson process (Chapter 3).
NONLAS	Solution of nonlinear elastic problems (Chapter 3),
ELPLAS	Solution of elasto-plastic problems (Chapter 3).
UNVIS	Solution of elasto-viscoplastic problems (Chapter 4).
TIMOSH	Solution of elasto-plastic nonlayered Timoshenko beams (Chapter 5).
TIMLAY	Solution of elasto-plastic layered Timoshenko beams (Chapter 5).

Two-dimensional applications

PLANET	Elasto-plastic analysis of plane stress, plane strain and axisymmetric solids (Chapter 7).
VISCOUNT	Elasto-viscoplastic analysis of plane stress, plane strain and axisymmetric solids (Chapter 8).
MINDLIN	Elasto-plastic analysis of nonlayered Mindlin plates (Chapter 9).
MINDLAY	Elasto-plastic analysis of layered Mindlin plates (Chapter 9).
DYNPAK	Elasto-plastic transient dynamic analysis of two dimensional solids (Chapter 10).
MIXDYN	Implicit-explicit elasto-viscoplastic transient dynamic analysis of two dimensional solids (Chapter 11).

12.2.1 Subroutines for one-dimensional applications

ASSEMB	Section 3.4.2 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Assembles the element contributions to form the global stiffness matrix and global load vector. (Simple equation solver).
ASTIF1	Section 3.10.1 (QUNEWT)Formulates the stiffness matrix for each element according to (2.25) and (2.29) for the solution of one dimensional quasi-harmonic problems by the Newton Raphson method.
BAKSUB	Section 3.4.4 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Performs the backsubstitution phase of the Gaussian reduction process. (Simple equation solver).
BEAM	Section 5.4.5 (TIMOSH)The master routine for elasto-plastic nonlayered Timoshenko beam program TIMOSH.
BEML	Section 5.5.5 (TIMLAY)The master routine for elasto-plastic layered Timoshenko beam program TIMLAY.
CONUND	Section 3.10.3 (QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Monitors convergence of the nonlinear solution process based on the residual forces according to (3.27).
CONVP	Section 4.9 (UNVIS)Monitors convergence to steady state conditions according to (4.41) for one-dimensional elasto-viscoplastic problems.
DATA	Section 3.2 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Data input subroutine for one-dimensional applications.

GREDUC	Section 3.4.3 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Undertakes equation elimination by Gaussian reduction. (Simple equation solver).
INCLOD	Section 3.7 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Controls the incrementing of the applied loads for one-dimensional applications (modified for viscoplastic problems in Section 4.10).
INCVP	Section 4.8 (UNVIS)Evaluates quantities at the end of the time step and the equilibrium correction terms for one-dimensional elastoviscoplastic problems.
INITAL	Section 3.6 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Initialises to zero some arrays used by other subroutines for one-dimensional applications.
MONITR	Section 3.9.2 (QUITER)Monitors convergence of the direct iteration process for one-dimensional quasiharmonic problems.
NONAL	Section 3.3 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Controls the nonlinear solution process according to the value of NALGO specified, for one-dimensional applications.
REFOR1	Section 3.10.2 (QUNEWT)Evaluates the ‘equivalent nodal forces’ according to (3.26) for one-dimensional quasiharmonic problems. (Newton Raphson solution).
REFOR2	Section 3.11.2 (NONLAS)Evaluates the equivalent nodal forces according to (3.32) for one-dimensional nonlinear elastic problems.
REFOR3	Section 3.12.2 (ELPLAS)Evaluates the equivalent nodal forces for one-dimensional elastoplastic problems.
REFORB	Section 5.4.5 (TIMOSH)Evaluates the residual forces for a nonlayered elastoplastic Timoshenko beam.
RFORBL	Section 5.5.5 (TIMLAY)Evaluates the residual forces for a layered elastoplastic Timoshenko beam.
RESOLV	Section 3.4.5 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Undertakes reduction of the R.H.S. terms for equation resolution (Simple equation solver).

RESULT	Section 3.5 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Outputs the results for one-dimensional applications.
STIFF1	Section 3.9.1 (QUITER)Formulates the stiffness matrix for each element according to (2.25) for the solution of one-dimensional quasiharmonic problems by direct iteration.
STIFBL	Section 5.5.5 (TIMLAY)Evaluates the elasto-plastic stiffness matrix for each element for the solution of layered Timoshenko beams.
STIFFB	Section 5.4.5 (TIMOSH)Formulates the elasto-plastic stiffness matrix for each element for the solution of nonlayered Timoshenko beams.
STIFF2	Section 3.11.1 (NONLAS)Formulates the stiffness matrix for each element according to (2.33) for nonlinear elastic one-dimensional problems.
STIFF3	Section 3.12.1 (ELPLAS)Formulates the stiffness matrix for each element according to either (2.38) or (2.43) for one-dimensional elasto-plastic problems.
STUNVP	Section 4.7 (UNVIS)Formulates the stiffness matrix for each element in turn for one-dimensional elasto-viscoplastic applications.
UNDIM	Section 3.8 (QUITER, QUNEWT, NONLAS, ELPLAS)The main or master segment for one-dimensional nonlinear problems. See Fig. 3.1 for the small changes in the different applications.
UNVISC	Section 4.11 (UNVIS)The main or master segment for one-dimensional visco-plastic problems.

12.2.2 Subroutines for two-dimensional applications

ADDBAN	Section 11.5.3 (MIXDYN)
	Generates the global matrix from the element stiffness matrices.
ADDRES	Section 11.5.4 (MIXDYN)
	Addresses the diagonal term of a matrix.
ALGOR	Section 6.5.2 (PLANET, VISCOUNT, MINDLIN, MIND-LAY)
	Controls the nonlinear solution process according to the value of NALGO specified, for two-dimensional applications.
BLARGE	Section 10.6.3 (DYNPAK, MIXDYN)
	Evaluates the strain matrix B for small and large deformation.
BMATPB	Section 6.4.8 (MINDLIN)
	Evaluates the strain matrix, B, for plate bending problems.

BMATPS	Section 6.4.7 (PLANET, VISCOUNT)
	Evaluates the strain matrix, B, for plane and axisymmetric situations.
CHECK1	Section 6.4.13 (PLANET, VISCOUNT, MINDLIN, MINDLAY)
	Scrutinises the problem control parameters for possible errors (two-dimensional applications).
CHECK2	Section 6.4.15 (PLANET, VISCOUNT, MINDLIN, MINDLAY)
	Checks the geometric data, boundary conditions and material properties for possible errors (two-dimensional applications).
COLMHT	Section 11.5.5 (MIXDYN)
	Evaluates the height of column above the diagonal of a matrix from the known addresses of diagonal terms.
CONTOL	Section 10.6.4 (DYNPAK, MIXDYN)
	Reads control data for dynamic dimensioning and also checks the dimension limits.
CONVER	Section 6.5.4 (PLANET)
	Monitors convergence of the nonlinear solution iteration process for two-dimensional applications.
CONVMP	Section 9.5.3 (MINDLIN, MINDLAY)
	Checks for convergence of solution of elasto-plastic layered and nonlayered Mindlin plates.
DBE	Section 6.4.11 (PLANET, VISCOUNT)
	Forms the matrix product DB.
DECOMP	Section 11.5.6 (MIXDYN)
	Decomposes positive definite matrix into LDL $^T$ .
DEPMPA	Section 9.6.4 (MINDLAY)
	Sets up the layered discretisation for the layered elasto-plastic Mindlin plate.
DIMEN	Section 7.8.1 (PLANET, VISCOUNT)
	Presets the value of variables associated with dynamic dimensioning.
DIMMP	Section 9.5.4 (MINDLIN, MINDLAY)
	Sets up dynamic dimensions in programs MINDLIN and MINDLAY for the elasto-plastic analysis of layered and nonlayered plates.
DINTOB	Section 11.5.7 (MIXDYN)
	Multiplies the modulus and strain matrices to give DB.
DYNPAK	Section 10.6.2 (DYNPAK)
	Organises the explicit viscoplastic transient dynamic analysis.
ECHO	Section 6.4.14 (PLANET, VISCOUNT, MINDLIN, MINDLAY)