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Overlay models for structural analysis under cyclic loading
Computers
Pergamon
00457949(95)00025-9
OVERLAY
MODELS UNDER
FOR STRUCTURAL CYCLIC LOADING
& Slrucrures Vol. 56, No. Z/3, pp. 321-328, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britam. All rights reserved 0045.7949/95 $9.50 + 0.00
ANALYSIS
K. Schiffner Institute of Engineering Mechanics and Control Engineering, University of Siegen, Germany Abstract-The behaviour of structural elements subjected to cyclic loading conditions is described by overlay models and simulated using standard finite element programs. The material model is established by parallel connection of several finite elements, each describing elasto-plastic material behaviour with kinematic or isotropic hardening and sharing the same geometrical position. The death or birth option, respectively, is used to simulate the yielding condition with a higher and lower yielding stress. A method estimating the material parameters is discussed. Results found by using ADINA 6.0 for the analysis of a cylindrical structural element under cyclic axial tension are presented. The effect of locally varying material parameters is discussed.
The present paper builds up overlay models in a general way by overlaying subelements of the P-, K-, and I-type [7]. The material parameters are estimated by an evaluation of experimental data found for test specimen with cylindrical cross section under cyclic loading.
1. INTRODUCTION
The objective of the present paper is to describe elasto-plastic material behaviour under cyclic loading conditions by overlay models. Overlay models consist of a limited number of subelements in parallel connection. Each of the subelements represents one of the following idealized material behaviour as shown in Fig. 1: elastic behaviour (E-subelement), perfectly elasto-plastic behaviour (P-subelement), elasto-plastic behaviour with kinematic hardening (K-subelement) or with isotropic hardening (I-subelement). The linear elastic part is represented by Young’s modulus E. The yield stresses at which the elastic branch of the stress-strain relationships ends is denoted by R, or R:, respectively, in the case of isotropic hardening. The hardening of the K- and I-s&elements is described again by a linear stress-strain relationship with the tangent modulus ET. The unloading path is always linear elastic with the modulus E. The idea of overlaying P-subelements in parallel connection was first introduced in 1918 by Heyn and later realized by Mansing [l] in order to simulate hardening of metallic materials. The Mansing model consists of 10 P-subelements with yield stresses equally spaced in steps of 1,2, 3, . . . , 10. Besseling [2] introduced in his theory of elastic, plastic, and creep deformations a rheological subelement in order to describe creep and the Bauschinger effect. Beginning in 1972 Zienkiewicz and several co-authors [3-61 use a description of the stress-strain relationship for increasing stresses in terms of several straight line segments. This type of representation may be interpreted by overlaying a number of P-subelements which gave the name overlay model. The material parameters of the subelements they used are determined by using experimental stress-strain curves for tension tests only.
2. THE OVERLAY
MODEL
In order to simplify the representation, the stressstrain relationship is reduced to the one-dimensional form cr = u(c) instead of the stress tensor components c,, or the strain tensor components E,,. There are a lot of material models, introduced by several authors, to describe constitutive equations for materials under cyclic loading. Most of the theories are limited to the simulation of special effects of material behaviour, or they are applicable only to certain materials, or the parameters of complex material models are difficult to determine by approximation of experimental data. The K-subelement has the same effect as a parallel connection of a E-subelement and a P-subelement, Fig. 2(a). Given the case that an overlay model contains more than one K-subelement, e.g. nK Ksubelements with the properties (EK, ETK, ReK)i, i = 1, nk, the nK K-subelements can be replaced by 1 K-subelement with the tangential modulus ETK = Fig. 2(b). C;:, (E,,), and (nk - 1) P-subelements, Taking account of these equivalencies, in the following the overlay model consists of nr P-subelements, 1 K-subelement, and n, I-subelements. Figure 3 shows the schematic view of finite element overlay modelling. For each material subelement, the same type of finite elements is used, all sharing the same geometrical space as shown in Fig. 3 for the plane stress state. 321
322
K. Schiffner
(a)
E -subelement
(b)
P -subelement
(c)
(d)
K -subelement
I
Fig. 1. Standard models for the description elasto-plastic; (c) elasto-plastic with kinematic
I -subelement
I
of idealised material behaviour: hardening; and (d) elasto-plastic
np
(a)
P
(a) elastic; (b) perfectly with isotropic hardening.
I K-subelement
-subelements
n,
I -subelement
K -subelement
@) nK K -subelements
+ np P -subelements
1 K-subelement
=
“$_=nK-
Fig. 2. Equivalent subelement E-subelement + P-subelement;
I
models: (a) K-subelement = (b) replacing K-subelements. Fig. 3. Schematic
3. EVALUATION OF OVERLAY MODEL PARAMETERS
f(x) = ;
L2 I=
(CT($)-
and
is
5 a”p(t,) + ap(t,)
k=l
a($))‘]‘-2, (2)
Eb”’- E# >, 0,
+
5 aQl(t,).
k= I
modelling.
The vector of inequality constraints is denoted by g(x) < 0 which are the lower and upper bounds of the design variables, for instance:
I
where approximated stresses a(~,) are formed by superposition of overlay stress components i.e. stresses c?!](c,) of k = 1, rrp P-subelements, the stress c$(t,) of one K-subelement, and the stresses a!](~~) of k = 1, n, I-subelements:
act,) =
overlay
In eqn (3) c$](c,) k = 1, np denotes the stress portion of the P-subelements, d$l(t,) that of the K-subelement, and 6\k1(t,), k = 1, n, that of the Isubelements, respectively. The vector x E R” of the n design variables takes the form:
The parameters of the overlay model are evaluated by the following optimization problem:
where X:= {x E lWJg(x) < 0; h(x) = 0). In eqn (1) f is the objective function defined by
view of finite element
(3)
k = 1, np
or
ErA - Er’ > 0, For h(x)
the
given
k = 1, nr, respectively.
problem
= 0 is formulated:
one
equality
constraints
(5)
Overlay
I
models
for structural
analysis
323
(4
600.
-600.
-600.
STRAIN Fig.
STRAIN
E in % -
E in % -
4. Approximation of uniaxial stress-strain cycles by a graph of continuous straight separation of branches, (b) notations for straight line approximation.
Since the optimisation problem as given in eqn (1) leads to numerous local optima [8], a special strategy is applied to establish the starting vector of optimization: first the experimental data a(~,), i = 1, m, where m is the number of are divided into q single measuring points, branches of loading and deloading states. As an example, the separated branches k = 1,9 are marked in Fig. 4(a) by different labels: l 0 V V A n n 0 +. Then each of these k = 1, q branches with data pairs (Q,, c,), i = 1, mk, is approximated by a graph of r continuous straight lines with r + 1 nodes, Fig. 4(b). The least square method approximation for each of the k = 1, q branches minimizes the function fk
=
+
+---i-t
k=l,q, $[,Z](D(G)- wY]“2>
(7)
lines,
(a)
where 6(ti) is the stress value of the approximated graph of continuous straight lines. For the jth segment, d(t,) is described by linear interpolation
t--c
c?(t)=+5
tj+ I
-4
’
f-
t,,, --t -Oj+l
6,+ I
-
6,
for
L;
(8)
where 4 and t;+ 1 or 6, and 6, + , , respectively, are the co-ordinates of the endpoints of the jth segment. Figure 4(b) exemplarily shows the notation of the fifth branch with r = 4 straight line segments. Every 10th measuring point of that branch is marked by V. The nodes of the straight line segments approximating the fifth branch are labelled by a filled circle l and the inequality constraint equations (5) are evaluated. The co-ordinates t;, c?,of (r + 1) nodes of all straight line segments referring to the same branch are used to describe the function C?(C) of k = 1, q branches by r P-subelements for each branch:
324
K. Schiffner
El’ 11=
1 r, - F, ,
(6 -6,_,)-E”‘...
The yield stress of the P-subelements R;]=iEI”(F,+,-Ccl),
Equation algebraic
(9) is transformed equations
into
system
of linear
(11)
r
is related
j=l,r-I.
r- I
Elf’.
,=I
6, -
a,
6,-c?,
(10)
CT-+1 --,_
and is then solved for each branch. The elastic moduli are evaluated recursively
E[‘l of the r P-subelements
(13)
The yield stresses Ril, j = 1, Y - 1 are plotted as a function of lC, 1which is the maximum strain of each branch at the beginning when the sample is loaded as shown in Fig. 4. In the case that the yield stresses R\‘l are constant, the subelement will be of P-type or K-type, otherwise when Rbl increases with respect to 1Cl1,an I-subelement is assumed. The slope of the function Rpl = Rrile(l F, I) is the starting value for Eki of the optimization problem (1). The starting values for Ep], Ekl, Et” are calculated by taking the mean value of EL” for all branches. The maximum and the minimum values of EL/l are used to formulate the upper and lower bounds of elastic moduli, eqn (5). The applied optimization procedure is based on the method of steepest descent.
2.4
m
2.0
PKI
I
f 1.6
0.8
number of model parameters
Fig. 5. Objective functionsf(x)
(12)
The elastic modulus Err-1of the last P-subelement is interpreted as the sum of the tangent moduli when K- or I-subelements are introduced instead of the first (r - 1) P-subelements E”] = 1
6,--r?,
to
= l/m Z;=, (o(q) - c?(t,)) ’ ]liz of 10 calculated overlay models with different numbers of model parameters.
Overlay models for structural analysis 4. RESULTS
The steel (St 52) samples are tested in a WOLPERT-testing machine which is displacementcontrolled. The sample 1, Fig. 4, is loaded by strain cyciing with steady increasing maxima. The experimental data found for this sample are used to evaluate the model parameters for 10 different overlay models with up to 12 model parameters. In Fig. 5, the objective function _& as given in eqn (2), shows that the approximation of the material behaviour by the overlay models will improve with increasing number of subelements in the
325
overlay model [8]. Since the overlay model which consists of one K-subelement and one I-subelement leads to a reasonable approximation with only six model parameters in comparison to the model with 12 parameters. Therefore this K-I-model is used to simulate the experimental data of three further samples with changed displacement runs in order to check the overlay model. The results are shown in Fig. 6. Figure 6(a) presents the original experimental data for which the overlay model approximation was applied. The simulation results is marked by dashed lines.
b
-600. I -4.0
I -3.0
-2.0
-1.0
0.0
I
I
I
i
1.0
2.0
3.0
4.0
STRAIN
E in % -
600.
(b) t
-3.0
-2.0
-1.0
0.0
1.0
STRAIN Fig. 6(a)-(b).
Continued overleaf.
2.0
3.0
E in % -
4.0
326
K. Schiffner
(cl
400.
200.
0.
-200.
-400.
hudation : measuringpointr
-600.
1 1 I / -2.5 -2.0 -1.5 -1.0 -0.5 0.0
I 0.5
I 1.0
STRAIN
-600. i_ -~ i -2.5 -2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
STRAlN Fig.
6. Simulation
of uniaxial
The K-I-overlay
by a K
1 - overlay
and one I-subelement,
(a) and (b).
stress-strain
K-subelement
model has following
properties
Model
cycles
m
E, = 63,344 N mmw2,
I 2.0
E in
2.5
% d
2.5
2.0
2.5
E m % model
consisting
of one
[7]:
Overlay I& = 140,546 N mm-*,
I 2.5
I
model
parameter
ETK = 2297 N rnrne2, ET, = 337 N mm-‘,
R,,= 157 N mmd2, R,,= 239 N mm-*.
Overlay
models
for structural
1
MODEL (1) activated time
analysis
327
MODEL (2) starting time
500.
I 400.
300.
200.
100.
/
0. 0.
0.3
0.6
0.9
1.2
1.5
STRAIN E in % Fig. 7. Overlay
model consisting
of 2 P-subelement with death and birth option virgin material state of steel samples.
for the simulation
of the
simulated Liider’s band (regions in plastic state) /
\
finite element with Rg= 260. N/mm2 Fig. 8. Simulation
of Liider’s
bands
by material
0
finite elements with Re= 280. N/mm2 models
of different
yield stresses.
328
K. Schiffner
In Ref. [8], the overlay model technique is applied to the three-dimensional stress state. The experimental data found for tubular samples under combined cyclic loading in tension-compression and torsion are simulated by using the finite element program ADINA 6.0. 5. SIMULATION
OF FURTHER
REFERENCES
EFFECTS
Next, the overlay technique is used to simulate two further effects which occur by testing materials: (1) Steel samples in virgin material state show the effect of a higher and a lower yield stress. In order to simulate different yield stresses in the same material sample, an overlay model is established which consists of two P-subelements with death and birth option, Fig. 7. When the first P-subelement is deactivated, the second P-subelement starts. Instead of using a time mark for birth or death of the element, limit strains can be defined also. (2) Liider’s bands are caused by discontinuities of the material with respect to the yield stresses, This effect occurs especially when the material is tested for sufficiently long sample strips with rectangular cross section under tensions. The effect may be simulated easily by using material groups of different yield stresses. As an example in Fig. 8, the stress plot of a finite element analysis is presented where one element with lower yield stress is generated in comparison to all other elements. The plastification starts at this element and extends over the whole strip to the characteristic Liider’s band pattern. 6. CONCLUSIONS
From the results of simulations models it is concluded that:
with
(a) in order to describe elasto-plastic stress-strain behaviour better, overlay models can be introduced without change of the finite element software; (b) the overlay technique can be extended to simulate special material effects.
overlay
1. G. Masing, Zur Heyn’schen Theorie der Verfestigung der Metalle durch verborgen elastische Spannungen. In Wissenschaftliche Veriffentlichungen aus dem SiemensKonzern, Vol. 3, pp. 231-239 (1923). 2. J. F. Besseling, A Theory of elastic, plastic and creep deformations of an initially isotropic material showing anisotropic strain-hardening, creep recovery, and secondary creep. J. Appl. Mech. Trans. ASME 529-536 (1958). 3. 0. C. Zienkiewicz, G. C. Nayak and D. R. J. Owen, Composite and “overlay” models in numerical analysis of elasto-plastic continua. Papers presented at jnt. Svmu. Foundations of Plasticity. Sawczuk, Warsaw (i97i). 4. D. R. J. Owen, A. Prakash and 0. C. Zienkiewicz, Finite element analysis of non-linear composite materials by use of overlay systems. Comput. Struct. 4, 1251-1261 (1974). 5. 0. C. Zienkiewicz, V. Norris and D. J. Naylor, Plasticity and visco-plasticity in soil mechanics with special reference to cyclic loading problems. In Finite Elements in Non-linear Mechanics, Vol. 2. TAPIR Norwegian Institute of Technology, Trondheim (1978). 6. G. N. Pande, D. R. J. Owen and 0. C. Zienkiewicz, Overlay models in time-dependent non-linear materials analysis. Comput. Struct. 7, 435-443 (1977). 1. K. Schiffner and J. He, Optimierte Overlay-modelle zur Beschreibung des Werkstoffverhaltens bei zyklischer Beanspruchung. Forschung im Ingenieurwesen-Engng Res. 58(3), 41-45 (1992). 8. J. He, Overlay-Modelle zur Beschreibung des elastoplastischen und elasto-viskoplastischen Werkstoffverhaltens unter zyklischen Belastungen. Dissertation (1994).
Pergamon
00457949(95)00025-9
OVERLAY
MODELS UNDER
FOR STRUCTURAL CYCLIC LOADING
& Slrucrures Vol. 56, No. Z/3, pp. 321-328, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britam. All rights reserved 0045.7949/95 $9.50 + 0.00
ANALYSIS
K. Schiffner Institute of Engineering Mechanics and Control Engineering, University of Siegen, Germany Abstract-The behaviour of structural elements subjected to cyclic loading conditions is described by overlay models and simulated using standard finite element programs. The material model is established by parallel connection of several finite elements, each describing elasto-plastic material behaviour with kinematic or isotropic hardening and sharing the same geometrical position. The death or birth option, respectively, is used to simulate the yielding condition with a higher and lower yielding stress. A method estimating the material parameters is discussed. Results found by using ADINA 6.0 for the analysis of a cylindrical structural element under cyclic axial tension are presented. The effect of locally varying material parameters is discussed.
The present paper builds up overlay models in a general way by overlaying subelements of the P-, K-, and I-type [7]. The material parameters are estimated by an evaluation of experimental data found for test specimen with cylindrical cross section under cyclic loading.
1. INTRODUCTION
The objective of the present paper is to describe elasto-plastic material behaviour under cyclic loading conditions by overlay models. Overlay models consist of a limited number of subelements in parallel connection. Each of the subelements represents one of the following idealized material behaviour as shown in Fig. 1: elastic behaviour (E-subelement), perfectly elasto-plastic behaviour (P-subelement), elasto-plastic behaviour with kinematic hardening (K-subelement) or with isotropic hardening (I-subelement). The linear elastic part is represented by Young’s modulus E. The yield stresses at which the elastic branch of the stress-strain relationships ends is denoted by R, or R:, respectively, in the case of isotropic hardening. The hardening of the K- and I-s&elements is described again by a linear stress-strain relationship with the tangent modulus ET. The unloading path is always linear elastic with the modulus E. The idea of overlaying P-subelements in parallel connection was first introduced in 1918 by Heyn and later realized by Mansing [l] in order to simulate hardening of metallic materials. The Mansing model consists of 10 P-subelements with yield stresses equally spaced in steps of 1,2, 3, . . . , 10. Besseling [2] introduced in his theory of elastic, plastic, and creep deformations a rheological subelement in order to describe creep and the Bauschinger effect. Beginning in 1972 Zienkiewicz and several co-authors [3-61 use a description of the stress-strain relationship for increasing stresses in terms of several straight line segments. This type of representation may be interpreted by overlaying a number of P-subelements which gave the name overlay model. The material parameters of the subelements they used are determined by using experimental stress-strain curves for tension tests only.
2. THE OVERLAY
MODEL
In order to simplify the representation, the stressstrain relationship is reduced to the one-dimensional form cr = u(c) instead of the stress tensor components c,, or the strain tensor components E,,. There are a lot of material models, introduced by several authors, to describe constitutive equations for materials under cyclic loading. Most of the theories are limited to the simulation of special effects of material behaviour, or they are applicable only to certain materials, or the parameters of complex material models are difficult to determine by approximation of experimental data. The K-subelement has the same effect as a parallel connection of a E-subelement and a P-subelement, Fig. 2(a). Given the case that an overlay model contains more than one K-subelement, e.g. nK Ksubelements with the properties (EK, ETK, ReK)i, i = 1, nk, the nK K-subelements can be replaced by 1 K-subelement with the tangential modulus ETK = Fig. 2(b). C;:, (E,,), and (nk - 1) P-subelements, Taking account of these equivalencies, in the following the overlay model consists of nr P-subelements, 1 K-subelement, and n, I-subelements. Figure 3 shows the schematic view of finite element overlay modelling. For each material subelement, the same type of finite elements is used, all sharing the same geometrical space as shown in Fig. 3 for the plane stress state. 321
322
K. Schiffner
(a)
E -subelement
(b)
P -subelement
(c)
(d)
K -subelement
I
Fig. 1. Standard models for the description elasto-plastic; (c) elasto-plastic with kinematic
I -subelement
I
of idealised material behaviour: hardening; and (d) elasto-plastic
np
(a)
P
(a) elastic; (b) perfectly with isotropic hardening.
I K-subelement
-subelements
n,
I -subelement
K -subelement
@) nK K -subelements
+ np P -subelements
1 K-subelement
=
“$_=nK-
Fig. 2. Equivalent subelement E-subelement + P-subelement;
I
models: (a) K-subelement = (b) replacing K-subelements. Fig. 3. Schematic
3. EVALUATION OF OVERLAY MODEL PARAMETERS
f(x) = ;
L2 I=
(CT($)-
and
is
5 a”p(t,) + ap(t,)
k=l
a($))‘]‘-2, (2)
Eb”’- E# >, 0,
+
5 aQl(t,).
k= I
modelling.
The vector of inequality constraints is denoted by g(x) < 0 which are the lower and upper bounds of the design variables, for instance:
I
where approximated stresses a(~,) are formed by superposition of overlay stress components i.e. stresses c?!](c,) of k = 1, rrp P-subelements, the stress c$(t,) of one K-subelement, and the stresses a!](~~) of k = 1, n, I-subelements:
act,) =
overlay
In eqn (3) c$](c,) k = 1, np denotes the stress portion of the P-subelements, d$l(t,) that of the K-subelement, and 6\k1(t,), k = 1, n, that of the Isubelements, respectively. The vector x E R” of the n design variables takes the form:
The parameters of the overlay model are evaluated by the following optimization problem:
where X:= {x E lWJg(x) < 0; h(x) = 0). In eqn (1) f is the objective function defined by
view of finite element
(3)
k = 1, np
or
ErA - Er’ > 0, For h(x)
the
given
k = 1, nr, respectively.
problem
= 0 is formulated:
one
equality
constraints
(5)
Overlay
I
models
for structural
analysis
323
(4
600.
-600.
-600.
STRAIN Fig.
STRAIN
E in % -
E in % -
4. Approximation of uniaxial stress-strain cycles by a graph of continuous straight separation of branches, (b) notations for straight line approximation.
Since the optimisation problem as given in eqn (1) leads to numerous local optima [8], a special strategy is applied to establish the starting vector of optimization: first the experimental data a(~,), i = 1, m, where m is the number of are divided into q single measuring points, branches of loading and deloading states. As an example, the separated branches k = 1,9 are marked in Fig. 4(a) by different labels: l 0 V V A n n 0 +. Then each of these k = 1, q branches with data pairs (Q,, c,), i = 1, mk, is approximated by a graph of r continuous straight lines with r + 1 nodes, Fig. 4(b). The least square method approximation for each of the k = 1, q branches minimizes the function fk
=
+
+---i-t
k=l,q, $[,Z](D(G)- wY]“2>
(7)
lines,
(a)
where 6(ti) is the stress value of the approximated graph of continuous straight lines. For the jth segment, d(t,) is described by linear interpolation
t--c
c?(t)=+5
tj+ I
-4
’
f-
t,,, --t -Oj+l
6,+ I
-
6,
for
L;
(8)
where 4 and t;+ 1 or 6, and 6, + , , respectively, are the co-ordinates of the endpoints of the jth segment. Figure 4(b) exemplarily shows the notation of the fifth branch with r = 4 straight line segments. Every 10th measuring point of that branch is marked by V. The nodes of the straight line segments approximating the fifth branch are labelled by a filled circle l and the inequality constraint equations (5) are evaluated. The co-ordinates t;, c?,of (r + 1) nodes of all straight line segments referring to the same branch are used to describe the function C?(C) of k = 1, q branches by r P-subelements for each branch:
324
K. Schiffner
El’ 11=
1 r, - F, ,
(6 -6,_,)-E”‘...
The yield stress of the P-subelements R;]=iEI”(F,+,-Ccl),
Equation algebraic
(9) is transformed equations
into
system
of linear
(11)
r
is related
j=l,r-I.
r- I
Elf’.
,=I
6, -
a,
6,-c?,
(10)
CT-+1 --,_
and is then solved for each branch. The elastic moduli are evaluated recursively
E[‘l of the r P-subelements
(13)
The yield stresses Ril, j = 1, Y - 1 are plotted as a function of lC, 1which is the maximum strain of each branch at the beginning when the sample is loaded as shown in Fig. 4. In the case that the yield stresses R\‘l are constant, the subelement will be of P-type or K-type, otherwise when Rbl increases with respect to 1Cl1,an I-subelement is assumed. The slope of the function Rpl = Rrile(l F, I) is the starting value for Eki of the optimization problem (1). The starting values for Ep], Ekl, Et” are calculated by taking the mean value of EL” for all branches. The maximum and the minimum values of EL/l are used to formulate the upper and lower bounds of elastic moduli, eqn (5). The applied optimization procedure is based on the method of steepest descent.
2.4
m
2.0
PKI
I
f 1.6
0.8
number of model parameters
Fig. 5. Objective functionsf(x)
(12)
The elastic modulus Err-1of the last P-subelement is interpreted as the sum of the tangent moduli when K- or I-subelements are introduced instead of the first (r - 1) P-subelements E”] = 1
6,--r?,
to
= l/m Z;=, (o(q) - c?(t,)) ’ ]liz of 10 calculated overlay models with different numbers of model parameters.
Overlay models for structural analysis 4. RESULTS
The steel (St 52) samples are tested in a WOLPERT-testing machine which is displacementcontrolled. The sample 1, Fig. 4, is loaded by strain cyciing with steady increasing maxima. The experimental data found for this sample are used to evaluate the model parameters for 10 different overlay models with up to 12 model parameters. In Fig. 5, the objective function _& as given in eqn (2), shows that the approximation of the material behaviour by the overlay models will improve with increasing number of subelements in the
325
overlay model [8]. Since the overlay model which consists of one K-subelement and one I-subelement leads to a reasonable approximation with only six model parameters in comparison to the model with 12 parameters. Therefore this K-I-model is used to simulate the experimental data of three further samples with changed displacement runs in order to check the overlay model. The results are shown in Fig. 6. Figure 6(a) presents the original experimental data for which the overlay model approximation was applied. The simulation results is marked by dashed lines.
b
-600. I -4.0
I -3.0
-2.0
-1.0
0.0
I
I
I
i
1.0
2.0
3.0
4.0
STRAIN
E in % -
600.
(b) t
-3.0
-2.0
-1.0
0.0
1.0
STRAIN Fig. 6(a)-(b).
Continued overleaf.
2.0
3.0
E in % -
4.0
326
K. Schiffner
(cl
400.
200.
0.
-200.
-400.
hudation : measuringpointr
-600.
1 1 I / -2.5 -2.0 -1.5 -1.0 -0.5 0.0
I 0.5
I 1.0
STRAIN
-600. i_ -~ i -2.5 -2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
STRAlN Fig.
6. Simulation
of uniaxial
The K-I-overlay
by a K
1 - overlay
and one I-subelement,
(a) and (b).
stress-strain
K-subelement
model has following
properties
Model
cycles
m
E, = 63,344 N mmw2,
I 2.0
E in
2.5
% d
2.5
2.0
2.5
E m % model
consisting
of one
[7]:
Overlay I& = 140,546 N mm-*,
I 2.5
I
model
parameter
ETK = 2297 N rnrne2, ET, = 337 N mm-‘,
R,,= 157 N mmd2, R,,= 239 N mm-*.
Overlay
models
for structural
1
MODEL (1) activated time
analysis
327
MODEL (2) starting time
500.
I 400.
300.
200.
100.
/
0. 0.
0.3
0.6
0.9
1.2
1.5
STRAIN E in % Fig. 7. Overlay
model consisting
of 2 P-subelement with death and birth option virgin material state of steel samples.
for the simulation
of the
simulated Liider’s band (regions in plastic state) /
\
finite element with Rg= 260. N/mm2 Fig. 8. Simulation
of Liider’s
bands
by material
0
finite elements with Re= 280. N/mm2 models
of different
yield stresses.
328
K. Schiffner
In Ref. [8], the overlay model technique is applied to the three-dimensional stress state. The experimental data found for tubular samples under combined cyclic loading in tension-compression and torsion are simulated by using the finite element program ADINA 6.0. 5. SIMULATION
OF FURTHER
REFERENCES
EFFECTS
Next, the overlay technique is used to simulate two further effects which occur by testing materials: (1) Steel samples in virgin material state show the effect of a higher and a lower yield stress. In order to simulate different yield stresses in the same material sample, an overlay model is established which consists of two P-subelements with death and birth option, Fig. 7. When the first P-subelement is deactivated, the second P-subelement starts. Instead of using a time mark for birth or death of the element, limit strains can be defined also. (2) Liider’s bands are caused by discontinuities of the material with respect to the yield stresses, This effect occurs especially when the material is tested for sufficiently long sample strips with rectangular cross section under tensions. The effect may be simulated easily by using material groups of different yield stresses. As an example in Fig. 8, the stress plot of a finite element analysis is presented where one element with lower yield stress is generated in comparison to all other elements. The plastification starts at this element and extends over the whole strip to the characteristic Liider’s band pattern. 6. CONCLUSIONS
From the results of simulations models it is concluded that:
with
(a) in order to describe elasto-plastic stress-strain behaviour better, overlay models can be introduced without change of the finite element software; (b) the overlay technique can be extended to simulate special material effects.
overlay
1. G. Masing, Zur Heyn’schen Theorie der Verfestigung der Metalle durch verborgen elastische Spannungen. In Wissenschaftliche Veriffentlichungen aus dem SiemensKonzern, Vol. 3, pp. 231-239 (1923). 2. J. F. Besseling, A Theory of elastic, plastic and creep deformations of an initially isotropic material showing anisotropic strain-hardening, creep recovery, and secondary creep. J. Appl. Mech. Trans. ASME 529-536 (1958). 3. 0. C. Zienkiewicz, G. C. Nayak and D. R. J. Owen, Composite and “overlay” models in numerical analysis of elasto-plastic continua. Papers presented at jnt. Svmu. Foundations of Plasticity. Sawczuk, Warsaw (i97i). 4. D. R. J. Owen, A. Prakash and 0. C. Zienkiewicz, Finite element analysis of non-linear composite materials by use of overlay systems. Comput. Struct. 4, 1251-1261 (1974). 5. 0. C. Zienkiewicz, V. Norris and D. J. Naylor, Plasticity and visco-plasticity in soil mechanics with special reference to cyclic loading problems. In Finite Elements in Non-linear Mechanics, Vol. 2. TAPIR Norwegian Institute of Technology, Trondheim (1978). 6. G. N. Pande, D. R. J. Owen and 0. C. Zienkiewicz, Overlay models in time-dependent non-linear materials analysis. Comput. Struct. 7, 435-443 (1977). 1. K. Schiffner and J. He, Optimierte Overlay-modelle zur Beschreibung des Werkstoffverhaltens bei zyklischer Beanspruchung. Forschung im Ingenieurwesen-Engng Res. 58(3), 41-45 (1992). 8. J. He, Overlay-Modelle zur Beschreibung des elastoplastischen und elasto-viskoplastischen Werkstoffverhaltens unter zyklischen Belastungen. Dissertation (1994).