2/15/2016 1 1Lecture #4 – Fall 2015 1D. Mohr
151-0735: Dynamic behavior of materials and structures
by Dirk Mohr
ETH Zurich, Department of Mechanical and Process Engineering,
Chair of Computational Modeling of Materials in Manufacturing
Lecture #4:
Integration Algorithms for Rate-independent Plasticity (1D)
© 2015
2/15/2016 2 2Lecture #4 – Fall 2015 2D. Mohr
151-0735: Dynamic behavior of materials and structures
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
3.00E+02
3.50E+02
4.00E+02
0.00E+00 5.00E-02 1.00E-01 1.50E-01 2.00E-01 2.50E-01
Recall: Important difference
E E
①Elastic loading
②Elasto-plastic
loading
③Elastic
unloading
pe
①Elastic loading
②Elastic
unloading
e
ELASTO-PLASTIC NON-LINEAR ELASTIC(e.g. metals, concrete, thermoplastics ) (e.g. rubbers, foams)
2/15/2016 3 3Lecture #4 – Fall 2015 3D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent perfect plasticity
p
• Simplified rheological model:
The strain is split into an elastic and a plastic part
i.e. the elastic strain is
INITIAL CONFIGURATION
DEFORMED (CURRENT) CONFIGURATION
pe
pe
linear springfrictional device
2/15/2016 4 4Lecture #4 – Fall 2015 4D. Mohr
151-0735: Dynamic behavior of materials and structures
i. Constitutive equation for stress
)( pE
ii. Yield functionkkf ],[
iii. Flow rule][sign p
iv. Loading/unloading conditions
0f0 if
0f0 if
0f0 if
0fand
0fand
Rate-independent perfect plasticity - Summary
Material model parameters: (1) Young’s modulus E, and (2) flow stress k.
2/15/2016 5 5Lecture #4 – Fall 2015 5D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent perfect plasticity - Application
p
Ek /
Ek /
time
time
time
k
Total strain
Plastic strain
Stress
k
①
①
②
③
③
④
④
⑤
① ② ③ ④ ⑤
2/15/2016 6 6Lecture #4 – Fall 2015 6D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent isotropic hardening plasticity
The magnitude of the stress increases due to strain hardeningwhen the material is deformed in the elasto-plastic range. Forisotropic hardening materials, it is described through an evolutionequation for the flow stress k.
E E
①Elastic loading
②Elasto-plastic
loading
③Elastic
unloading
④Elastic
re-loading
⑤Elasto-plastic
loading
2/15/2016 7 7Lecture #4 – Fall 2015 7D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent isotropic hardening plasticity
to measure the amount of plastic flow (slip). This measure is oftencalled equivalent plastic strain. Unlike the plastic strain, themagnitude of the equivalent plastic strain can only increase!
][ pkk
Firstly, we introduce a scalar valued non-negative function
dtp
It is then assumed that the flow stress is a monotonically increasingsmooth differentiable function of the equivalent plastic strain
This equation describes the isotropic hardening law.
2/15/2016 8 8Lecture #4 – Fall 2015 8D. Mohr
151-0735: Dynamic behavior of materials and structures
Rate-independent isotropic hardening plasticity
Frequently used parametric forms of the function are the Swift and Voce laws:
n
pS Ak )( 0
][ pkk
]exp[10 pV Qkk
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
3.00E+02
3.50E+02
4.00E+02
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
3.00E+02
3.50E+02
4.00E+02
0.00E+00
5.00E+01
1.00E+02
1.50E+02
2.00E+02
2.50E+02
3.00E+02
3.50E+02
4.00E+02
SV kkk )1(
Swift Voce Swift-Voce
Qkkd
dk
p
0 ,0
Hardening saturation
pp
p
k k k
2/15/2016 9 9Lecture #4 – Fall 2015 9D. Mohr
151-0735: Dynamic behavior of materials and structures
0
50
100
150
200
250
300
350
400
0 0.05 0.1 0.15 0.2
Rate-independent isotropic hardening plasticity
In engineering practice, the isotropic hardening function is often represented by a piece-wise linear function
][ p
][ MPak
PEEQ k
0.000 199.1
0.020 246.3
0.050 283.9
0.100 321.0
0.200 365.6
2/15/2016 10 10Lecture #4 – Fall 2015 10D. Mohr
151-0735: Dynamic behavior of materials and structures
i. Constitutive equation for stress
)( pE
ii. Yield function][],[ pp kf
iii. Flow rule][sign p
iv. Loading/unloading conditions
0f0 if
0f0 if
0f0 if
0fand
0fand
Isotropic hardening plasticity - Summary
v. Isotropic hardening law
][ pkk with dtp
2/15/2016 11 11Lecture #4 – Fall 2015 11D. Mohr
151-0735: Dynamic behavior of materials and structures
)( pE
][],[ pp kf
][sign p
• First-order ordinary differential equation
0f0 if
0f0 if
0f0 if
0fand
0fand
Differential equation to be solved
• Multiplier to satisfy the constraints
with dtp
Ett pp ][][sign
• Initial condition
0]0[ tp
• Prescribed loading
][t
and
)( pE ,
2/15/2016 12 12Lecture #4 – Fall 2015 12D. Mohr
151-0735: Dynamic behavior of materials and structures
Numerical solution of Differential Equations
Without the loading and unloading conditions, the plasticity problemreduces to solving an ordinary first-order differential equation forthe plastic strain, considering time as the only independent variable:
Ett pp ][][sign
0]0[ tp
][ ygdt
dy• D.E.
• I.C.0]0[ yty
Such equations are solved numerically using integration algorithms.Instead of the calculating the exact analytical solution, we limit ourattention to calculating the approximated solution
][ nn tyy
at equally-spaced instants tn, n=1,…,N with the time step Dt,
nn ttt D 1
2/15/2016 13 13Lecture #4 – Fall 2015 13D. Mohr
151-0735: Dynamic behavior of materials and structures
Numerical solution of Differential Equations
A first popular method is the so-called forward (explicit) Euleralgorithm:
][1 nnn ytgyy D
][ 001 ytgyy D
][ 112 ytgyy D
0y
Starting with the initial condition, the approximations can beprogressively calculated.
2/15/2016 14 14Lecture #4 – Fall 2015 14D. Mohr
151-0735: Dynamic behavior of materials and structures
Numerical solution of Differential Equations
Recall that and thus the forward (explicit) Euler algorithm may also be written as
nnn ytyy D1
001 ytyy D
112 ytyy D
0y
][' ygy
In other words, the time derivative at time tn is given by the approximation
t
yyty nn
nD
1][
y
tn tn+1
exact derivative
approximation
2/15/2016 15 15Lecture #4 – Fall 2015 15D. Mohr
151-0735: Dynamic behavior of materials and structures
Numerical solution of Differential Equations
A second popular method is the so-called backward (implicit) Euleralgorithm:
1ny
1y
2y
0y
Starting with the initial condition, the approximations can beprogressively calculated. However, at each time step ti, an oftenimplicit equation needs to be solved.
][ 101 ytgyy Dwhich is obtained from solving the implicit equation
][ 212 ytgyy Dwhich is obtained from solving the implicit equation
][ 11 D nnn ytgyywhich is obtained from solving the implicit equation
2/15/2016 16 16Lecture #4 – Fall 2015 16D. Mohr
151-0735: Dynamic behavior of materials and structures
Numerical solution of Differential Equations
According to the backward (implicit) Euler algorithm, the time derivative at time tn is given by the approximation
t
yyty nn
nD
1][
][1 nnn ytgyy D
][ ygy
y
tntn-1
exact derivative
approximation
2/15/2016 17 17Lecture #4 – Fall 2015 17D. Mohr
151-0735: Dynamic behavior of materials and structures
IllustrationExample: y
dt
dy
1]0[ ty
The approximate solution with forward (explicit) Euler algorithm for a time step of Dt=1 reads (we have g[y]=y and y0=1):
]exp[ty
(differential equation)
(initial condition)
(exact solution)
2111][ 001 D ytgyy
4212][ 112 D ytgyy
10 y
8414][ 223 D ytgyy
16818][ 334 D ytgyy
3216116][ 445 D ytgyy 0
20
40
60
80
100
120
140
160
0 1 2 3 4 5
1Dt
1.0Dt
01.0Dtexact
Observe from the graph that the method converges for Dt
2/15/2016 18 18Lecture #4 – Fall 2015 18D. Mohr
151-0735: Dynamic behavior of materials and structures
0
20
40
60
80
100
120
140
160
180
200
0 1 2 3 4 5
Comparison implicit vs. explicit
The approximate solutions with forward (explicit) Euler and backward (implicit) Euler algorithms for a time step of Dt=0.1
exact
implicit
explicit
2/15/2016 19 19Lecture #4 – Fall 2015 19D. Mohr
151-0735: Dynamic behavior of materials and structures
… back to the plasticity problem
Ett pp ][][sign
Et
ttg
p
nn
p
n
n
p
n
p
n
11
11
sign
][
D
D
D
Differential equation:
plus “discrete” evolution constraints:
0D
01 nf
0)( 1 D nf
Initial condition:
00 p
Dependent variables:
)( 111
p
nnn E
State variable:
D
p
n
p
n 1
][ 11
p
nn kk
111 nnn kf
1sign D n
p
n
0D then 01 nfif
0Dthen01 nfif
00 p
2/15/2016 20 20Lecture #4 – Fall 2015 20D. Mohr
151-0735: Dynamic behavior of materials and structures
Ett pp ][][sign
Et
ttg
p
nn
p
n
n
p
n
p
n
11
11
sign
][
D
D
D
Differential equation:
plus “discrete” evolution constraints:
0D
01 nf
0)( 1 D nf
Initial condition:
00 p
Dependent variables:
)( 111
p
nnn E
State variable:
D
p
n
p
n 1
][ 11
p
nn kk
111 nnn kf
1sign D n
p
n
0D then 01 nfif
0Dthen01 nfif
00 p
Main unknown:
Plastic multiplier D
… which makes our problem more complexthan solving an ordinary first orderdifferential equation!
… back to the plasticity problem
2/15/2016 21 21Lecture #4 – Fall 2015 21D. Mohr
151-0735: Dynamic behavior of materials and structures
Return Mapping Algorithm
We solve the plasticity problem assuming a strain-driven process, i.e. for a given increment in the applied total strain,
nn D 1
we determine numerical approximations of the corresponding stress and state variables at time tn+1 based on their values at time tn.
Applied total strain increment
D
RETURN MAPPING ALGORITHM
State variables at time tn
p
n
p
n ,
State variables at time tn+1
p
n
p
n 11 ,
OUTPUT:
)( 111
p
nnn E Stress at time tn+1
2/15/2016 22 22Lecture #4 – Fall 2015 22D. Mohr
151-0735: Dynamic behavior of materials and structures
Return mapping procedure
When computing the solution at time tn+1, we first compute the trial elastic state by assuming that the material response is purely elastic (no plastic evolution) when applying D :
)()()( 11 DD EEE n
p
nn
p
nn
trial
n
p
n
trialp
n
,
1
p
n
trialp
n
,
1
][11
p
n
trial
n
trial
n kf
01
trial
nf then elastic loading stepif
then plastic loading stepif 01
trial
nf
2/15/2016 23 23Lecture #4 – Fall 2015 23D. Mohr
151-0735: Dynamic behavior of materials and structures
Return mapping procedure
)(1 D En
trial
n
nn
nn
trial
n 1
trial
n 1trial
n 1
trial
n 1
01
trial
nf 01
trial
nf 01
trial
nf 01
trial
nf
n
trial
n 1
01
trial
nf
elastic step
elastic step
plastic step
plastic step
elastic step
2/15/2016 24 24Lecture #4 – Fall 2015 24D. Mohr
151-0735: Dynamic behavior of materials and structures
Return mapping procedure
Elastic loading step: 0D
011
trial
nn ff
State variables at time tn+1
p
n
p
n 1
OUTPUT:
)( 111
p
nnn E
Stress at time tn+1
p
n
p
n 1
Applied total strain increment
D
Calculate Trial State
State variables at time tn
p
n
p
n ,
01
trial
nf
trial
n
trial
n f 11 ,
0D
2/15/2016 25 25Lecture #4 – Fall 2015 25D. Mohr
151-0735: Dynamic behavior of materials and structures
Return mapping procedure
Plastic loading step: 0D01
trial
nf
In a plastic loading step, the plastic multiplier D>0must be determined such that the yield condition at time tn+1 is full filled.
111 nnn kf
p
trial
n
p
n
p
n
p
nn
p
nnn EEE D 111111 )()(
pD
11 sign DD n
p
n
p
np
Firstly, we express the absolute value of the stress n+1 as a function of the unknown plastic multiplier:
(1)
while
And hence
][sign)()( 11111 D n
trial
n
p
nnn EE
2/15/2016 26 26Lecture #4 – Fall 2015 26D. Mohr
151-0735: Dynamic behavior of materials and structures
Return mapping procedure
1 D
p
n
p
n ] [1 D
p
nn kk
0][1111 DD p
n
trial
nnnn kEkf
Then, using the results (2) and (3) in (1), we obtain the so-called discrete consistency condition:
Secondly, we express the flow stress kn+1 as a function of the unknown plastic multiplier:
(3)
D Etrial
nn 11 (2)
)(][sign
][sign)(][sign
][sign
11
111
111
D
D
E
E
trial
nn
n
trial
nn
nnn
observe that][sign][sign 11
trial
nn
2/15/2016 27 27Lecture #4 – Fall 2015 27D. Mohr
151-0735: Dynamic behavior of materials and structures
Return mapping procedure
0][11 DD p
n
trial
nn kEf
n
trial
n 1
n 1n
D
DeD
nk
1nk
)( DE
1n
][ Dp
nk
2/15/2016 28 28Lecture #4 – Fall 2015 28D. Mohr
151-0735: Dynamic behavior of materials and structures
Solving the discrete consistency condition
0)(
)(
)(
1
01
011
D
DD
DD
EHf
EHHk
HkEf
trial
n
p
n
trial
n
p
n
trial
nn
Example #1: Linear hardening law
pp Hkk 0][ with constant hardening modulus H
The discrete consistency condition then reads
EH
f trial
n
D 1
from which we determine the plastic multiplier
trial
nf 1
D
xEHf trial
n )(y 1
2/15/2016 29 29Lecture #4 – Fall 2015 29D. Mohr
151-0735: Dynamic behavior of materials and structures
Solving the discrete consistency condition
Example #2: General non-linear concave hardening law ][ pk
The discrete consistency condition then reads
Which corresponds to seeking the root of the convex function
0][
][
][
1
1
11
DD
DD
DD
n
p
n
trial
n
n
p
nn
trial
n
p
n
trial
nn
kkEf
kkEk
kEf
n
p
n
trial
n kxkExf ][y[x] 1
p
k
nk
p
n
D
x
trial
nf 1
D
2/15/2016 30 30Lecture #4 – Fall 2015 30D. Mohr
151-0735: Dynamic behavior of materials and structures
Solving the discrete consistency condition
Seeking the root of a C1-continuous function is a standard problem in applied mathematics. For example, it can be found using a Newton-Raphson scheme:
x
00 x]['
][
0
001
xy
xyxx
]['
][
1
112
xy
xyxx
]['
][1
n
nnn
xy
xyxx
1x0x2x
TOLxy n ][ 1iterate until then 1D nx
2/15/2016 31 31Lecture #4 – Fall 2015 31D. Mohr
151-0735: Dynamic behavior of materials and structures
Elasto-plastic Tangent Modulus
The derivative d/d is called elasto-plastic tangent modulus. During plastic tensile loading , we have
0)(
d
d
dkddEdkddf
p
and thus
)0 ,0 ,0( pddd
d
d
dkE
Ed
p
d
d
dkE
d
dkE
ddEd
p
p
)(
p
p
d
dkE
d
dkE
d
d
2/15/2016 32 32Lecture #4 – Fall 2015 32D. Mohr
151-0735: Dynamic behavior of materials and structures
Elasto-plastic Tangent Modulus
Formally, we note the incremental stress-strain response as
)( dEd ep
with
p
p
d
dkE
d
dkE
epE
E if 0
if 0
Eep
2/15/2016 33 33Lecture #4 – Fall 2015 33D. Mohr
151-0735: Dynamic behavior of materials and structures
Summary: Return Mapping Algorithm
State variables at time tn+1
p
n
p
n 1
OUTPUT:
)( 111
p
nnn E
Stress at time tn+1
p
n
p
n 1
Applied total strain increment
D
Calculate Trial State
State variables at time tn
p
n
p
n ,
0][1 DD p
n
trial
n kE
01
trial
nf
Solve:
0D
01
trial
nf
State variables at time tn+1
][sign)( 11
trial
n
p
n
p
n D
D
p
n
p
n 1
)( 111
p
nnn E
Stress at time tn+1
trial
n
trial
n f 11 ,
0D
OUTPUT:
2/15/2016 34 34Lecture #4 – Fall 2015 34D. Mohr
151-0735: Dynamic behavior of materials and structures
Reading Materials for Lecture #4
• J.C. Simo and T.J.R. Hughes, “Computational Inelasticity” (first chapter): http://link.springer.com/book/10.1007%2Fb98904
• M.E. Gurtin, E. Fried, L. Anand, “The Mechanics and Thermodynamics of Continua”, Cambridge University Press, 2010.