MultiPhysicsVault/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_049.md

	Echoes the remaining data after input data errors have been diagnosed.
EXPLIT	Section 10.6.5 (DYNPAK)Carries out explicit time integration.
FEAM	Section 9.6.2 (MINDLAY)Organising routine for the elasto-plastic analysis of layered Mindlin plates.
FEMP	Section 9.5.2 (MINDLIN)Organising routine for the elasto-plastic analysis of nonlayered Mindlin plates.
FIXITY	Section 10.6.6 (DYNPAK)Boundary conditions are inserted.
FLOWMP	Section 9.5.5 (MINDLIN, MINDLAY)Determines $\partial F/\partial \sigma_{f}$ (i.e. yield function derivatives) for elastoplastic layered and nonlayered Mindlin plates.
FLOWPL	Section 7.8.4.2 (PLANET, MIXDYN)Determines the vector $d_{D}$ for elasto-plastic analysis.
FLOWVP	Section 8.9 (VISCOUNT, DYNPAK)Determines the viscoplastic strain rate for each Gauss point according to (8.7).
FRONT	Section 6.4.12 (PLANET, VISCOUNT, MINDLIN, MINDLAY)Performs element assembly and equation solution by the frontal method. Contains a facility for efficient resolution of equations.
FUNCTA	Section 10.6.8 (DYNPAK, MIXDYN)Interpolates acceleration ordinate at $\Delta t$ intervals.
FUNCTS	Section 10.6.9 (DYNPAK, MIXDYN)Evaluates factor for Heaviside and Harmonic time function at $\Delta t$ apart.
GAUSSQ	Section 6.4.2 (PLANET, VISCOUNT, MINDLIN, MINDLAY, DYNPAK, MIXDYN)Evaluates the sampling point positions and weighing factors for numerical integration by Gauss quadrature.
GEOMST	Section 11.5.8 (MIXDYN)Evaluates the stress stiffness matrix.
GRADMP	Section 9.5.6 (MINDLIN)Evaluates the total displacement and rotation derivatives ( $\partial w/\partial x$ , $\partial w/\partial y$ , $\partial \theta_{x}/\partial x$ , $\partial \theta_{x}/\partial y$ , $\partial \theta_{y}/\partial x$ , $\partial \theta_{y}/\partial y$ ).
GSTIFF	Section 11.5.9 (MIXDYN)Evaluates the global stiffness matrix in compacted profile form.
IMPEXP	Section 11.5.10 (MIXDYN)Sets the constants of integration and evaluates partial effective load vector.

INCREM	Section 6.5.3 (PLANET, VISCOUNT, MINDLIN, MINDLAY)Controls the incrementing of the applied loads for two-dimensional applications.
INPUT	Section 6.5.1 (PLANET, VISCOUNT, MINDLIN, MINDLAY)Data input subroutine for two-dimensional applications.
INPUTD	Section 10.6.10 (DYNPAK, MIXDYN)Data input subroutine. Reads the mesh data, properties etc
INTIME	Section 10.6.11 (DYNPAK, MIXDYN)Reads the data necessary for time integration.
INVAR	Section 7.8.3 (PLANET, VISCOUNT, DYNPAK, MIXDYN)Evaluates the effective stress level at a given point for monitoring plastic yielding.
INVERT	Section 8.7.3 (VISCOUNT)This subroutine determines the inverse of any arbitrary square matrix.
INVMP	Section 9.5.7 (MINDLIN)Evaluates the Mindlin plate stress resultant invariants for nonlayered plates.
ITRATE	Section 11.5.11 (MIXDYN)Evaluates the total effective load and iterates until convergence is reached.
JACOBD	Section 10.6.13 (DYNPAK, MIXDYN)Evaluates the deformation Jacobian matrix.
JACOB2	Section 6.4.4 (PLANET, VISCOUNT, MINDLIN, MINDLAY, DYNPAK, MIXDYN)Evaluates the Jacobian matrix, its inverse and the Cartesian derivatives of the element shape functions for two-dimensional applications.
LAYMPA	Section 9.6.5 (MINDLAY)Evaluates the matrix of flexural rigidities and the matrix of shear rigidities for the layered elastoplastic Mindlin plate.
LINEAR	Section 7.8.6 (PLANET, MIXDYN)Determines the stresses from given displacements assuming linear elastic behaviour.
LINGNL	Section 10.6.14 (DYNPAK, MIXDYN)Evaluates the linear stresses for small and large deformation analysis.
LINKIN	Section 11.5.12 (MIXDYN)This routine links with the profile solver.
LOADPB	Section 6.4.6 (MINDLIN, MINDLAY)Evaluates the consistent nodal forces for plate bending problems.

LOADPL	Section 10.6.15 (DYNPAK, MIXDYN)Generates the load vector.
LOADPS	Section 6.4.5 (PLANET, VISCOUNT)Evaluates the consistent nodal forces due to gravity and distributed edge loads for two-dimensional problems.
LUMASS	Section 10.6.16 (DYNPAK, MIXDYN)Generates the consistent mass matrix for implicit elements and special lumped mass matrix for explicit elements.
MDMPA	Section 9.6.6 (MINDLAY)Evaluates the constitutive matrices for use in layered Mindlin plate analysis.
MINDPB	Section 9.5.8 (MINDLIN, MINDLAY)Reads additional input data for elasto-plastic, layered and nonlayered Mindlin plates.
MIXDYN	Section 11.5.2 (MIXDYN)Organises implicit/explicit transient dynamic program.
MODPB	Section 6.4.10 (MINDLIN)Evaluates the D matrix for plate bending applications.
MODPS	Section 6.4.9 (PLANET, VISCOUNT, DYNPAK, MIXDYN)Evaluates the D matrix for plane and axisymmetric situations.
MULTPY	Section 11.5.13 (MIXDYN)Multiplies square matrix to a vector or vector to a vector.
NODEXY	Section 6.4.1 (PLANET, VISCOUNT, MINDLIN, MINDLAY)Interpolates the coordinates of midside nodes for elements with straight sides. This routine is modified in MINDLIN and MINDLAY where a hierarchical formulation is adopted for the ninth node. (See Section 9.5).
NODXYR	Section 10.6.18 (DYNPAK, MIXDYN)Evaluates the midside node of elements. In case of axisymmetric problems if (R, Θ) coordinates are read r, z coordinates are evaluated within it.
OUTDYN	Section 10.6.19 (DYNPAK, MIXDYN)Writes the output on output file and stress and displacement histories of required Gauss points and nodes respectively on specified tapes.
OUTMP	Section 9.5.10 (MINDLIN)Outputs displacements, reactions and Gauss point stress resultants for elasto-plastic nonlayered Mindlin plates.
OUTMPA	Section 9.6.7 (MINDLAY)Outputs displacements, reactions and Gauss point layer stresses for elasto-plastic layered Mindlin plates.

OUTPUT	Section 7.8.8 (PLANET, VISCOUNT)Outputs the results for two-dimensional problems at specified intervals.
PLAST	Section 7.8.9 (PLANET)The main or master segment for two-dimensional elastoplastic applications.
PREVOS	Section 10.6.20 (DYNPAK, MIXDYN)Reads the initial force and stresses.
REDBAK	Section 11.5.14 (MIXDYN)Solves equations after matrix decomposition, using forward and backward substitution.
RESEPL	Section 11.5.15 (MIXDYN)Evaluates the internal force for different yield criteria in the implicit explicit program.
RESMP	Section 9.5.11 (MINDLIN)Evaluates the internal nodal forces

$$ \boldsymbol {p} = \int_ {\Omega} \boldsymbol {B} _ {f} ^ {T} \sigma_ {f} d \Omega + \int_ {\Omega} \boldsymbol {B} _ {s} ^ {T} \sigma_ {s} d \Omega $$

for the stress resultants \sigma_{f} and \sigma_{s} for elasto-plastic, non-layered Mindlin plates.

RESMPA	Section 9.6.8 (MINDLAY)
	Evaluates the residual force vector for layered elasto-plastic Mindlin plates.
RESIDU	Section 7.8.7 (PLANET)
	Evaluates the nodal forces which are statically equivalent to the stress field satisfying elasto-plastic conditions.

RESVPL	Section 10.6.21 (DYNPAK)
	Evaluates the internal forces for different yield criteria in the explicit transient dynamic program.

SFR2	Section 6.4.3 (PLANET/ VISCOUNT, MINDLIN, MINDLAY, DYNPAK, MIXDYN)
	Evaluates the element shape functions and their local derivatives for 4, 8 and 9 node isoparametric quadrilateral elements. SFR2 is modified in MINDLIN and MINDLAY to allow for a hierarchical representation for the 9th central node.

STEADY

Section 8.12 (VISCOUNT)Monitors convergence to steady state conditions for two-dimensional elasto-viscoplastic problems.

STEPVP	Section 8.8 (VISCOUNT)
	Evaluates quantities, such as stresses and viscoplastic strains, at the end of each time step of a viscoplastic solution.

STIFFP

Section 7.8.5 (PLANET)

	Evaluates the stiffness matrix for each element for elastoplastic problems employing either D or $D_{ep}$ as appropriate.
STIFMP	Section 9.5.13 (MINDLIN)
	Evaluates the stiffness matrices for nonlayered elastoplastic Mindlin plate elements.
STIFVP	Section 8.7.1 (VISCOUNT)
	Evaluates the stiffness matrix for each element in turn for two-dimensional elastoplastic applications.
STIMPA	Section 9.6.9 (MINDLAY)
	Evaluates the stiffness matrices for layered elastoplastic Mindlin plate elements.
STRESS	Section 8.10 (VISCOUNT)
	Evaluates the increment in stress occurring during a timestep of a viscoplastic analysis according to (8.20).
STRMP	Section 9.5.14 (MINDLIN)
	Evaluates stress resultants $[M_x, My, M_{xy}, Q_x, Q_y]^T$ for elastoplastic nonlayered Mindlin plates.
STRMPA	Section 9.6.10 (MINDLAY)
	Evaluates the stresses $[\sigma_x, \sigma_y, \tau_{xy}, \tau_{xz}, \tau_{yz}]^T$ for elastoplastic layered Mindlin plates at each layer and each Gauss point.
SUBMP	Section 9.5.15 (MINDLIN, MINDLAY)
	Carries out matrix multiplications in elastoplastic layered and nonlayered Mindlin plates.
TANGVP	Section 8.7.2 (VISCOUNT)
	Evaluates the $D^n$ matrix for viscoplastic analysis by implicit time stepping schemes.
VISCO	Section 8.13 (VISCOUNT)
	The main or master segment for two-dimensional elastopiscoplastic applications.
VZERO	Section 9.5.16 (MINDLIN, MINDLAY)
	Zeroes a vector in elastoplastic layered and nonlayered Mindlin plates.
YIELDF	Section 7.8.4.1 (PLANET, VISCOUNT, MIXDYN, DYN-PAK)
	Determines the flow vector a for plastic and viscoplastic applications. (Amended in Section 10.6.22 for dynamic transient problems).
ZERO	Section 7.8.2 (PLANET, VISCOUNT)
	Sets to zero the contents of several arrays employed in the programs. (Modified for viscoplastic applications in Section 8.11).
ZEROMP	Section 9.5.16 (MINDLIN, MINDLAY)
	Zeroes various arrays in elastoplastic layered and nonlayered Mindlin plate programs.

12.3 Alternative material models

The plastic behaviour of most solids is adequately described by the four yield criteria presented in Chapter 7; namely the Tresca, Von Mises, Mohr-Coulomb and Drucker-Prager yield surfaces. However, for some engineering materials, notably concrete, rocks and soils, some modifications must be made to the above criteria or new yield surfaces postulated if an accurate prediction of the material response is required.

For soils, the Mohr–Coulomb and Drucker–Prager criteria suffer from two deficiencies. Firstly, the assumption of an associated flow rule leads to excessive dilatency and secondly it is seen from Fig. 7.4 that both models imply that the material can support an unlimited hydrostatic compression. These deficiencies can be removed by use of the so-called critical state model, which assumes that the yield surface comprises two distinct parts. ^{(1-3)} The surface is shown plotted in terms of deviatoric \sigma_{d} and hydrostatic stress, \sigma_{s} , in Fig. 12.1. In the subcritical region yielding is stable due to strain hardening of the material whilst the supercritical region exhibits strain softening so that this portion of the yield surface forms a failure criterion.

text_image

supercritical region subcritical region C critical state line F SUPER = 0 non-associated flow rule F SUB = 0 (elliptical section) associated flow rule B 1 Scs A A' 0 σc 2σc σ, D

Fig. 12.1 Critical state model for the behaviour of soil, [\sigma_d = |\sigma_1 - \sigma_3|, \sigma_s = \frac{1}{2} (\sigma_1 + \sigma_3)] .

A nonassociative flow rule is adopted in the supercritical region and the conical yield surface implied in Fig. 12.1 may be circular or hexagonal in form corresponding to a Mohr–Coulomb behaviour. In the subcritical region, the two most common shapes for the so-called cap is a log spiral or an ellipse and an associated flow rule is assumed to be obeyed. The yield surface can be expressed in the form

F _ {\mathrm{SUPER}} = \sigma_ {d} - 2 \sin \phi \sigma_ {s} - 2 c \cos \phi = 0

F _ {\mathrm{SUB}} = \frac {\sigma_ {d} ^ {2} - S _ {c s} ^ {2} \sigma_ {s} (2 \sigma_ {c} - \sigma_ {s})}{\sigma_ {d} + S _ {c s} \sigma_ {c}} = 0, \tag {12.1}

in which S_{cs} is the slope of the critical state line.

In the tensile zone, various options are open for modelling the limited tensile strength of the soil. The curved line BA' can be employed or, more simply the vertical intercept OB (implying zero tensile strength) may be assumed. Complete details of the critical state model for soils can be found in Refs. 1–3 including its application to the numerical solution of practical problems.

The Mohr–Coulomb and Drucker–Prager criteria exhibit the same deficiencies for modelling concrete behaviour as occur in the case of soils. In particular they overestimate the tensile strength of the material and also allow the material to support an unlimited hydrostatic compression. Many models have been proposed to more accurately predict the behaviour of concrete; a review of which can be found in Ref. 4.

The most common method of predicting the tensile behaviour of concrete (and rocks) is by use of the no-tension model (or limited tension model). ^{(5)} In this, the tensile principal stresses are monitored throughout the structure and as soon as the value at any point exceeds the specified limiting tensile strength of the concrete, the material is assumed to crack in a plane normal to the principal direction. The tensile stress must then be reduced to zero by evaluating its nodal force equivalent and regarding these as residual forces to be applied and redistributed in an iterative process. Should the crack close on load reversal a frictional behaviour between the surfaces of the crack can be modelled. It is worth recording that the numerical stability of such solution processes is relatively poor since on initiation of tensile cracking the existing stress must be eliminated by redistribution, whereas for elasto-plastic problems, yielding merely necessitates that the existing stress level be maintained.

An example of this type of analysis is illustrated in Fig. 12.2 where a cylindrical prestressed concrete reactor vessel is shown. The geometry of the vessel, together with the location of the prestressing system is indicated and the finite element mesh employed in solution is also shown. The concrete is assumed to behave as a limited tension material and the steel components as a Von Mises elasto-plastic solid. The effects of prestressing are included as an initial stress system and the vessel is incrementally loaded by a progressively increasing internal pressure. Figure 12.3 shows the vertical deflection of the centre point of the end slab with increasing load and good agreement is observed with both the experimental results and numerical analysis of Ref. 6. The zones of tensile cracking are shown in Fig. 12.4 for various applied pressure values and again good agreement with the results of Ref. 6 is evident.

text_image

circumferential prestressing 40 in 20 in 9 in 5 in 15 in longitudinal prestressing cable cros section through vessel longitudinal prestressing load internal pressure pressure equivalent to prestressing forces parabolic isoparametric element mesh and loading system

Fig. 12.2 Finite element idealisation of a prestressed concrete reactor vessel by quadratic isoparametric elements.

line

displacement (in)	internal pressure (psi)	Series
0.00	0	31 parabolic elements, ν=0
0.01	~450	31 parabolic elements, ν=0.15, hoop pressure=620 psi
0.02	~550	31 parabolic elements, ν=0
0.03	~600	31 parabolic elements, ν=0.15, hoop pressure=510 psi
0.04	~650	31 parabolic elements, ν=0.15, hoop pressure=510 psi
0.05	~680	31 parabolic elements, ν=0
0.06	~700	31 parabolic elements, ν=0.15, hoop pressure=620 psi
0.04	~650	experimental ref (6)
0.03	~600	experimental ref (6)
0.02	~550	experimental ref (6)
0.01	~500	experimental ref (6)
0.00	0	experimental ref (6)
0.04	~650	experimental ref (6)
0.05	~680	experimental ref (6)
0.06	~700	experimental ref (6)
0.04	~650	31 parabolic elements, ν=0.15
0.03	~600	31 parabolic elements, ν=0.15
0.02	~550	31 parabolic elements, ν=0.15
0.01	~500	31 parabolic elements, ν=0.15
0.00	0	31 parabolic elements, ν=0
0.04	~650	experimental ref (6)
0.05	~680	experimental ref (6)
0.06	~700	experimental ref (6)
0.04	~650	31 parabolic elements, ν=0.15
0.03	~620	31 parabolic elements, ν=0.15
0.02	~580	31 parabolic elements, ν=0.15
0.01	~550	31 parabolic elements, ν=0.15
0.00	0	31 parabolic elements, ν=0
0.04	~650	experimental ref (6)
0.05	~680	experimental ref (6)
0.06	~700	experimental ref (6)

Fig. 12.3 Load/deflection curves for the vessel of Fig. 12.2 failing in slab flexural mode.

trace of circumferential cracks

zones of radial cracking

Fig. 12.4 Zones of tensile cracking for the vessel of Fig. 12.2 failing in slab flexural mode.

For predicting the compressive behaviour of concrete as well as the tensile response many failure surfaces have been proposed and a typical model is illustrated in Fig. 12.5. In addition to a brittle behaviour in tension, the model allows a viscoplastic range of behaviour before material failure. For further details the reader is directed to Ref. 4.

A final approach to concrete behaviour which is worthy of mention is afforded by the so-called endochronic theory pioneered by Valanis ^{(7,8)} and generalised to concrete structures by Bazant. ^{(9,10)} To account for the strain history dependence of materials (in addition to their strain rate dependence) the concept of intrinsic time z is introduced which is related to the Newtonian time scale, t according to

d z ^ {2} = \alpha^ {2} (d \zeta^ {2} + \beta^ {2} d t ^ {2}), \tag {12.2}

where d\zeta is effectively a measure of the deformation path length, \beta is a

material parameter and \alpha depends on \dot{\zeta} . Bazant has generalised the endochronic model to account for inelastic dilatancy, hydrostatic and shear compaction and fracture behaviour. ^{(10)}

text_image

initial viscoplastic loading surface elastic region plastic yield surface elasto—viscoplastic region brittle failure surface -σ₂ -σ₃ -σ₁

Fig. 12.5 Typical yield and failure surfaces for concrete.

12.4 Further applications

12.4.1 Flow problems

In this class of problem we are concerned with the continuing viscous flow of materials under steady state conditions. Typical examples include the extrusion of material through a die and flow of lubricating muds in oil drilling applications. In each case the problem is characterized by the fact that the elastic strains are negligible in comparison to the plastic components. For this reason, the viscoplastic numerical process described in Chapter 8 is unsuitable, since the increment of stress occurring during a timestep was based on the elastic strain increment according to (8.15). Thus an alternative formulation is clearly necessary and in fact a considerable simplification is achieved if the elastic components of strain are neglected in solution. ^{(11)}

The plastic strain rate, \dot{\epsilon}_{vp} , which is now assumed to be the total strain rate, \dot{\epsilon} , is given from (8.7) to be

\dot {\epsilon} = \dot {\epsilon} _ {v p} = \gamma \langle \Phi (F) \rangle a, \tag {12.3}

and we recall that a is the flow vector defined by (7.42), \Phi is an appropriate flow function (given for example by (8.8) or (8.9)) and \gamma is a fluidity parameter. For the particular case of a Von Mises yield surface we have from (7.11) that

F (\sigma , \kappa) = \sqrt {3} (J _ {2} ^ {\prime}) ^ {1 / 2} - \sigma_ {Y} (\kappa), \tag {12.4}

where J_{2}^{\prime} is the second deviatoric stress invariant and \sigma_{Y} is the uniaxial yield stress of the material which may be a function of the strain hardening