Nonlinear Finite Elements for Continua and Structures Part 8 potx

Elastic plastic materials are further subdivided into rate-independent materials, where the stress is independent of the strain rate, i.e., the rate of loading has no effect on the stres

and ${\bf C}$ are related ( ), it can also be shown that the components ofthe nominal stress tensor is given by

\begin{equation}

{\bf P} = {\partial W\over \partial {\bf F}^T}, \qquad

P_{ij} = {\partial W\over \partial F_{ji}}

(Exercise: Show this.)

\noindent {\bf Modified Mooney-Rivlin Material}\par

In 1951, Rivlin and Saunders [ ] published their experimental results on thelarge elastic deformations of vulcanized rubber - an incompressible

homogeneous isotropic elastic solid, in the Journal of Phil Trans A.,

Vol 243, pp 251-288 This material model with a few refinements is

still the most commonly used model for rubber materials It is assumed

in the model that behavior of the material is initially isotropic and

path-independent, i.e., a stored energy function exists The stored

energy function is written

where $\bar{c}_{ij}$ are constants

They performed a number of experiments on different

types of rubbers and discovered that Eq ( ) may be reduced to

\begin{equation}

\Psi = c(I_1 - 3) + f(I_2 - 3)

\end{equation}

where $c$ is a constant and

$f$ is a function of $I_2 - 3$ For a Mooney-Rivlin material, $W$ can bereduced

further to

\begin{equation}

\Psi = \Psi(I_1, I_2) = c_1 (I_1 - 3) + c_2 (I_2 - 3)

\end{equation}

An example of the set of $c_1$ and $c_2$ is: $c_1 = 18.35 {\rm psi}$ and

$ c_2 = 1.468 {\rm psi}.$ Equation ( ) is also an example of a

Neo-Hookean material, and the components of the second Piola-Kirchhoff stress can be obtained by

differentiating Eq ( ) with respect to the components of the right

Cauchy Green deformation tensor

tensor; however, the deformation is constrained such that

which represents a constraint on the deformation One way in which the

constraint ( ) can be enforced is through the use of a constrained

potential, or stored energy, function [Ref] Alternatively, a penalty

function formulation (Hughes, 1987) can be used In this case,

the modified strain energy function

and the constitutive equation become:

\begin{eqnarray}

\bar{\Psi} &=& \Psi + p_0\,{\rm ln}I_3 + {1\over 2} \lambda({\rm ln}I_3)^2 \\{\bf S} &=& 2{\partial \Psi\over \partial {\bf C}} + 2(p_0 + \lambda({\rm

ln}I_3)){\bf C}^{-1}

\end{eqnarray}

respectively The penalty parameter $\lambda$ must be large enough so thatthe

compressibility error is negligible (i.e., $I_3$ is approximately equal

to $1$), yet not so large that numerical ill-conditioning occurs

Numerical experiments reveal that $\lambda = 10^3\times {\rm max}(C_1, C_2)$ to $\lambda = 10^7\times {\rm max}(C_1,

C_2)$ is adequate for floating-point

word length of 64 bits The constant $p_o$ is chosen so that the components

of ${\bf S}$ are all zero in the initial configuration, i.e,

\subsection {Plasticity in One Dimension}

Materials for which permanent strains are developed upon unloading arecalled plastic materials Many materials (such as metals) exhibit elastic

(often linear) behavior up tp a well defined stress levlel called the

yeild strength Onec loaded beyond the initial yield strength, plastic

strains are developed Elastic plastic materials are further subdivided

into rate-independent materials, where the stress is independent of the

strain rate, i.e., the rate of loading has no effect on the stresses and

rate-dependent materials , in which the stress depends on the strain

rate; such materials are often called strain rate-sensitive.

The major ingredients of the theory of

plasticity are

\begin{enumerate}

\item A decomposition of each increment of strain into an elastic,

reversible component $d\varepsilon^e$ and an irreversible plastic part

$d\varepsilon^p$

\item A yield function $f$ which governs the onset and continuance ofplastic deformation

\item A flow rule which governs the plastic flow, i.e., determines the

plastic strain increments

\item A hardening relation which governs the evolution of the yield function

\end{enumerate}

There are two classes of elastic-plastic laws:

\begin{itemize}

\item Associative models, where the yield function and the potential

function are identical

\item Nonassociative models where the yield function and flow potential aredifferent

deformation, and can not

be written as a single valued function of the strain as in ( ) and ( )

The stress is path-dependent and dependes on the history of the

deformation We cannot therefore write an explicit relation for the stress

in terms

of strain, but only as a relation between rates of stress and strain

The constitutive relations for rate-independent and rate-dependent

plasticity in one-dimension are given in the following sections

\subsubsection{\bf Rate-Independent Plasticity in One-Dimension}

A typical stress-strain curve for a metal under uniaxial stress is shown

in Figure~\ref{fig:stress-strain} Upon initial loading, the material

behaves elastically (usually assumed linear) until the initial yield stress

is attained The elastic regime is followed by an elastic-plastic

regime where permanent irreversible plastic strains are induced upon further loading

Reversal of the stress is called unlaoding In unloading, the

stress-strain response is typically

assumed to be governed by the elastic modulus and the strains

which remain after complete unloading are called

the plastic strains The increments in strain are

assumed to be additively decomposed into elastic and plastic parts Thus

we write

\begin{equation}

d\varepsilon = d\varepsilon^e +d\varepsilon^p

\end{equation}

Dividing both sides of this equation by a differential time increment

$dt$ gives the rate form

\begin{equation}

\dot{\varepsilon} = \dot{\varepsilon}^e + \dot{\varepsilon}^p

\end{equation}

The stress increment (rate) is related to the increment (rate) of

elastic strain Thus

stress-strain curve In rate form, the relation is written as

is scaled by an arbitrary factor, the constitutive relation remains

unchanged and therefore the material response is {\em rate-independent}even though it is expressed in terms of a strain rate In the sequel,

the rate form of the constitutive relations will be used as the notation

because the incremental form can get cumbersome especially for large strainformulations

$\bullet$ kinematic hardening

The increase of stress after initial yield is called work or strain

hardening The hardening behavior of the material is generally a

function of the prior history of plastic deformation

In metal plasticity, the history of plastic deformation is often

plastic strain $\bar{\varepsilon}$, is

an example of an internal variable used to characterize the inelastic

response of the material An alternative, internal variable used in the

representation of hardening is the plastic work which is given by (Hill,

1958)

\begin{equation}

W^P = \int \sigma\dot{\varepsilon}^p dt

\end{equation}

The hardening behavior is often expressed through the

dependence of the yield stress, $Y$, on the accumulated plastic strain, i.e.,

$Y = Y(\bar{\varepsilon})$ More general constitutive relations use

additional internal variables

A typical hardening curve is shown in Figure ( ) The slope of this

curve is the plastic modulus, $H$, i.e.,

where the result $\partial \bar{\sigma}/\partial \sigma = {\rm

sign}(\sigma)$ has been used For plasticity in one-dimension

The plastic switch parameter $\alpha$ is introduced with $\alpha=1$

For plasticity, the conditions are:

\begin{equation}

\dot{\lambda}\dot{f}=0,\quad \dot{\lambda}\ge 0, \quad \dot{f}\le 0

\end{equation}

Thus for plastic loading, $\dot{\lambda}\ge 0$ and the consistency

condition $\dot{f}=0$ is satisfied For purely elastic loading or

unloading, $\dot{f}\ne 0$ and it follows that $\dot{\lambda}=0$

The constitutive relations for rate-independent plasticity in 1D are

summarized in Box 9.1

\subsubsection{Rate-Dependent Plasticity in One Dimension}

In rate dependent plasticity, the plastic response of the material

depends on the rate of loading The elastic response is given as before(in rate form) as

rate-dependent plasticity the stress can exceed the yield stress

The plastic strain rate is given by

For example, the overstress model introduced by Perzyna (19xx) is given by

\begin{equation}

\phi = Y\bigl({\bar{\sigma}\over Y}-1\bigr)^n

\end{equation}

where $n$ is called the rate-sensivity

exponent Using ( ) and ( ) the expression for the stress rate is given by

\begin{equation}

\dot{\sigma}=E\biggl(\dot{\varepsilon} - {\phi(\sigma, \bar{\varepsilon})

\over \eta}{\rm sgn}(\sigma)\biggr)

\end{equation}

which is a differential equation for the evolution of the stress

Comparing this expression to ( ), it can be seen that ( ) is

inhomogeneous in the rates and therefore the material response is

{\em rate-dependent} Models of this type are often used to model thestrain-rate dependence observed in materials More elaborate models withadditional internal variables and perhaps different response in differentstrain-rate regimes have been developed (see for examplethe unified creepplasticity model [Ref]) Nevertheless, the simple overstress model ( )has been very successful in capturing the strain rate dependence of

metals over a large range of strain rates [Refs]

An alternative form of rate-dependent plasticity that has been used withconsiderable success by Needleman ( ) and others is given by

\begin{equation}

\dot{\bar{\varepsilon}}^p = \dot{\varepsilon_0}\biggl({\bar{\sigma}\overY}\biggr)^{1/m}

\end{equation}

without any explicit yield surface For plastic straining at the rate

$\dot{\varepsilon_0}$,

the response $\bar{\sigma}=Y$ is obtained This response is called thereference response and can be obtained by performing a unixial stress

test with a plastic strain rate $\dot{\varepsilon}_0$ In the case of

small elastic strain rates, the test can be run at total strain rate of

$\dot{\epsilon}_0$ without significant error (Check!) For rates which exceed

$\dot{\varepsilon}_0$ the stress

is elevated above the reference stress while for lower rates the stress

falls below this value A case of particular interest is the

near rate-independent limit when the rate-sensitivity exponent $m\to 0$

It can be

seen from ( ) that, for $\bar{\sigma}<Y$,

the effective plastic strain rate is negligible while for a finite plastic

strain rate the effective stress is approximately equal to the reference

stress, $Y$ In this way, the model exhibits an effective yield limit

together with near elastic unloading and rate-independent response

The constitutive relations for rate-dependent plasticity in 1D are

summarized in Box 9.2

T Belytschko & B Moran, Solution Methods, December 16, 1998

CHAPTER 6

SOLUTION METHODS AND STABILITY

Very Rough Draft-use equations at your own peril

by Ted Belytschko and Brian Moran

The first part of the chapter describes time integration, the procedures usedfor integrating the discrete momemtum equation and other time dependentequations in the system, such as the constitutive equation We begin with thesimplest of methods, the central difference method for explicit time integration.Next the family of Newmark β-methods, which encompass both explicit andimplicit methods, are described Explicit and implicit methods are compared andtheir relative advantages described As part of implicit methods, the solution ofequilibrium equations is also examined

A critical step in the solution of implicit systems and equilibrium problems

is the linearization of the governing equations Linearization procedures for theequations of motion, and as a special case, the equilibrium equations aredescribed

6.2 EXPLICIT METHODS

In this Section the major features of explicit and implicit time integrationmethods for the discretized momentum equation and solution methods for thediscrete equilibrium equations are described The methods are described in thecontext of Lagrangian meshes, but can be extended to Eulerian and ALE mesheswith some techniques described in Chapter 7 The description of the solutionprocedures of equilibrium problems is combined with the description of implicitprocedures for dynamic problems, because, as we show later, the methodologiesare almost identical; the solution of a static problem by an implicit method onlyrequires that the inertial term be dropped

To illustrate the major features of explicit and implicit methods for timeintegration, the solution of the equations of motion is first considered for rate-independent materials In this class of equations, we can avoid some of thecomplications that arise in the treatment of rate-dependent materials but stillillustrate the most important properties of explicit and implicit methods We willfirst describe explicit and implicit methods using only a single time integrationformula: the central difference method for explicit time integration and the

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T Belytschko & B Moran, Solution Methods, December 16, 1998

Newmark β-methods for implicit integration In Section X, other time integration

formulas are considered

6.2.1 Central Difference Method The central difference method is

among the most popular of the explicit methods in computational mechanics andphysics It has already been discussed in Chapter 2, where it was chosen todemonstrate some nonlinear solutions in one dimension The central differencemethod is developed from central difference formulas for the velocity andacceleration We consider here its application to Lagrangian meshes with rate-independent materials Geometric and material nonlinearites are included, and infact have little effect on the time integration algorithm

For the purpose of developing this and other time integrators we will use thefollowing notation Let the time of the simulation 0≤t ≤t E be subdivided intotime intervals, or time steps, ∆t n , n=1 to n TS where n TS is the number of time

steps and t E is the end-time of the simulation; ∆t n is also called the nth time increment The variables at any time step are indicated by a superscript; thus t nis

the time at time step n, t0=0 is the beginning of the simulation and dn≡d t( )n is

the matrix of nodal displacements at time step n Time increment n is given by

∆t n=t n−t n−1

∆t n+1

2 =1

where the second equation gives the midpoint time step

The central difference formula for the velocity is

(6.2.3b)For the case of equal time steps the above reduces to

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T Belytschko & B Moran, Solution Methods, December 16, 1998

We now consider the time integration of the undamped equations of

motion for rate-independent materials, Eq (4.x.x.), which at time step n are given

X, it is time independent for a Lagrangian mesh Methods for Eulerian meshesare discussed in Chapter 7 The internal and external nodal forces are functions ofthe nodal displacements and the time The external loads are usually prescribed asfunctions of time; they may also be functions of the nodal displacements becausethey may depend on the configuration of the structure, as when pressures areapplied to the surfaces which undergo large deformations The dependence of theinternal nodal forces on displacements is quite obvious: the nodal displacementsdetermine the strains, which in turn determine the stresses and hence the nodalinternal forces by Eq (4.4.11) Internal nodal forces are generally not directlydependent on time, but there are situations of engineering relevance when this isthe case; for example, when the temperature is prescribed as a function of time,the stresses and hence the internal nodal forces depend directly on time

The equations for updating the nodal velocities and displacements areobtained as follows Substituting Eq (6.2.4a) into (6.2.3b) gives

At any time step n, the displacements d n will be known The nodal

forces fn can be determined by using in sequence the strain-displacementequations, the constitutive equation and the relation for the nodal internal forces.Thus the entire right hand side of (6.2.5) can be evaluated, which gives υn+1 ,

and the displacements dn+1

at time step n+1 can be determined by (6.2.2b) The

entire update can be accomplished without solving any system equations provided

that the mass matrix M is diagonal This is the salient characteristic of an explicit

method:

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T Belytschko & B Moran, Solution Methods, December 16, 1998

in an explicit method, the time integration of the discrete momentum equations for

a finite element model does not require the solution of any equations.

In numerical analysis, integration methods are classified according to thestructure of the time difference equation The difference equations for first andsecond derivatives are written in the general forms

for the function at time step n only involves the derivatives at previous time steps.

Difference equations which are explicit according to this classification generallylead to solution schemes which require no solution of equations In most casesthere is no benefit in using explicit schemes which involve the solution ofequations, so the use of such explicit schemes is rare There are a few exceptions.For example, if the consistent mass is used with the central difference method,even though the difference equation is classified as explicit, system equations stillneed to be solved in the update

6.2.2 Implementation A flow chart for explicit time integration of a finite

element model with rate-independent materials is shown in Box 6.1 Thisflowchart generalizes the flowchart given in Chapter 2 by considering nonzeroinitial conditions, a variable time step and including elements which require morethan one-point quadrature The primary dependent variables in this flowchart arethe velocities and the Cauchy stresses Initial conditions must be given for thevelocitites, the Cauchy stresses, and all state variables of the materials in themodel The displacements are initially considered to vanish

Flowchart inorrect, half missing on time steps, not n order

Box 6.1 Flowchart for Explicit Time Integration

1 Initial conditions and initialization:

set v0,σ0, and other material state variables;

4 compute kinetic energy and check energy balance, see Section ??

5 update nodal velocities: vn+ 1

2 =vn+1

2∆t nan

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T Belytschko & B Moran, Solution Methods, December 16, 1998

6 enforce velocity boundary conditions:

8 update counter and time: n←n+ 1, t←t+∆t

9 update nodal velocities: vn+1=vn+ 1

1 compute external nodal forces f ext ,n which are global

2 loop over elements e

i GATHER element nodal displacements and velocities

END quadrature point loop

iv compute external nodal forces on element, f e

ext , n

v f e

n =fe ext ,n−fe int, n

vi compute ∆t crit e , if∆t crit e <∆t crit then∆t crit= ∆t crit e

vii SCATTER fe n to global fn

3 END loop over elements

In this algorithm, the accelerations are first integrated to obtain thevelocities The integration of the velocities is broken into two half-steps so thatthe velocities are available at an integer step in the computation of the energybalance The displacements are computed in each time step by integrating thevelocities

The main part of the procedure is the calculation of the nodal forces from

the nodal displacements at a given time step, which is performed in getf In this

subroutine, the equations governing a continuum are used along with thegather/scatter procedures:

6-5