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Material model of metallic cellular solids
Pergamoln PII: s0045-7949(%)00310-0
MATERIAL
MODEL
OF METALLIC
Com/mfers & Srrucrures Vol. 62, No. 6. pp. 1049-1057. 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/97 Sl7.00 + 0.00
CELLULAR
SOLIDS
B. HuEkot and L. Faria$ tFaculty of Electrical Engineering and Information Technology, Slovak Technical University, Bratislava, Slovakia IMechanical Engineering Department, Instituto Superio Tkcnico, Lisbon, Portugal (Received I August 1995) Abstract-In this paper we deal with a continuum model of a man-made cellular material. A simple visco-elastic model with softening is used. The basic features of behaviour of cellular material:
linear-elastic region, plateau and locking are presented. The locking function is assumed in terms of volume changes. The stresses are functions of density updating during deformation process. This model has been coded into ABAQUS through user’s subroutine and tested on examples with the strain localization and with the densification (locking). Copyright 0 1996 Elsevier Science Ltd.
INTRODUCTION Cellular materials are currently used in packaging, energy absorbing devices, lightweight structures, etc. The design of these structures requires understanding of the response of these materials to loads. The mechanical properties of cellular materials are related to their microstructural behaviour, i.e. macroscopic stresses are directly related to the forces in the cell walls and mechanisms of cell failure. The behaviour of cellular materials is strongly non-linear and different in tension and compression. Stressstrain curves of cellular materials in uniaxial compression and tension are shown in Refs [l, 21. These curves exhibit three different regimes: linearelastic, plateau and densification. The linear-elastic regimes in compression and tension are very similar and can be described by Hooke’s law. As deformation is increased, cells collapse in a different way. In compression cells collapse by elastic buckling, plastic yielding or brittle crushing, depending on the cell properties [l--3]. This collapse or failure progresses at roughly constant load (giving plateau) until opposing wall in the cell meet and touch. After touching cell walls the deformation stops with increasing stresses (densification or locking), i.e. cellular materials, exhibit deformation until the densification is re.ached. In tension the (collapse of cellular materials does not progress at constant load. Either they elongate, with increasing stiffness of material or yield or simply fracture. Locking or densification was observed in compression [ 1,4]. This problem has been firstly theoretically studied in [5] as an opposite behaviour to an ideal plastic one. Locking functions are presented
in [S, 61, but in reality loads are multiaxial. Therefore we need failure surface of cellular material which reflects multiaxial loading. The mechanism of failure itself may depend on stress state: a foam may be plastic in compression but brittle in tension, for instance. Then the final failure surface is given by the inner envelope of failure surfaces for individual mechanisms [l-3, 71. The failure of cellular material is associated with the initiation of crushing of individual cells. Crushing may occur uniformly or be localized into discrete bands [8-111. The presence of nonhomogenous deformation results from increasing compliance [ 111.Therefore, we distinguish stable and unstable modes of deformation (corresponding to decreasing and increasing compliance). Unstable modes of deformation are source of localization of deformation within the narrow band of cells for both quasi-static and dynamic loads 191. The small influence of strain-rate has been observed on the mode of deformation [12, 131. As we see from experiments [8-10, 12, 131 the real properties of cellular material at a certain point can be affected by the mode of deformation at another point. This coupling leads to use of a length parameter (some characteristic distance) to determine the neighbourhood of each point, which affects to this point. It means that a nonlocal approach [14, 151 should be taken into account for this material model. Modelling of strain localization within the framework of classical continuum theory leads to ill-posed boundary problems, i.e. the governing differential equations lose their ellipticity or their hyperbolicity. The loss of ellipticity of hyperbolicity leads to the extreme mesh sensitivity due to fineness and directions of grid lines. To preserve the
1049
1050
B. Hutko
and L. Faria densification
z ti
00
linear elastic
I
den&cation
plateau
g linear 111 elastic
plateau
Strain I
ED
El
Fig. 1. Stress-strain
curve for cellular
material.
well-posedness of the original problem the regularization can be used. There are different methods of regularization: (1) to enrich standard continuum theory by higher-order gradients [ l&18]; (2) to add rate effects (visco-elasticity, viscoplasticity) [ 19-2 I]; (3) to employ the Cosserat continuum (couple stresses) [22,23]; (4) to use a non-local approach [14, 151. The initiation of strain localization can be analytically computed [24-271 under the condition of homogenous deformation. Then the conditions for the strain localization can be derived. From the above-mentioned review it is clear that the behaviour of cellular material is very complex. Therefore it is necessary to select the main or the most important features of its behaviour. In this paper the simple continuum model for cellular materials is presented in general to reflect the main features of behaviour of cellular material. Then the application to foams is done. The numerical results of a few examples are compared with the experimental ones.
Fig. 3. An approximation
+
EO
of stress-strain
curve.
MATERIAL MODEL FOR CELLULAR SOLIDS
In this paper we deal with an isotropic cellular material. More general behaviour, i.e. transverse isotropic or anisotropic, will be done later, but this is a trial to make very simple model of cellular material. As we mentioned before the cellular material exhibits three regions: linear-elastic, plateau and densification (see Fig. 1). Hooke’s law is a governing equation for the linear elastic region
where
1, p are Lame’s constants, dii is the Kronecker delta, ti, resp. criiis a strain, resp. stress measure, a,, resp. Q are functions of the relative density of cell R = p*/p* and the cell geometry (these functions can be found in Ref. [l]). p* is a density of cellular material, p6 is a density of solid material. The plateau and the densification can be modelled using a constitutive law of viscoelastic material as simple as possible. The simple constitutive law for visco-elastic model (three parameters solid-see Fig. 2) of cellular material has the following form
1
l/E, + l/E, ’
(3) Table
Fig. 2. Visco-elastic
model.
*Elastic: I = 0 MPa *Inelastic: I = 0 MPa v= -1 @ = 1.0, 0.5 El = 0.1 D = 4.0
1. Material
properties
;p = 320 MPa ;p= -lOOMPa ;q = -0.008 s, -0.0073 ;R = 0.1 ;co = 0.8
s
Metallic cellular solids
J
1051
DIS?LACED
ULSI
l.ODOI-02
J
Fig. 4. The initial and final mesh. where R, is a function of the relative density of cell R and the cell geometry (expressions for & can be found in Ref. [l]). This can be generalized for an isotropic solid with small straining as follows f7i + I%‘d= e (D&k + 2/i&j + AC&&k + 2/Z,)
(4)
where 8, 1, ii, ;1, p are material constants. Or introducing relaxation time q into eqn (4), we get
where (6)
and v is a material constant. For finite straining it is necessary to introduce the objective stress and strain measures or to use algorithms to ensure the objectivity [28-301. Employing the additive decomposition of strains the Jaumann stress-rate is used .p . . bij = 6, - qiu,p - wppp where wii is a spin tensor.
Y
-
0.0
0.2
0.4
0.6
0.6
stralnll Fig. 5. Response for different relaxation times.
-0.0073 -0.006
(7)
B. HuEko and L. Faria
1052
6.OOe+6 ,
1
5.008+6
4.008+6 vr
5 3.008+6
Y
-
L L
closed cell opencell
m2.008+6 1.OOe+6 O.OOe+O 0.2
0.0
0.4
0.6
0.6
stralnll Fig. 6. Closed and open cells.
UPDATING
OF RELATIVE
where R. is an initial relative density beginning of current increment.
DENSITY
During deformation the relative density is updating due to volumetric strain-rate [3 11.
tkk= --,
Integrating
of cell at
LOCKING
The locking function can be expressed as
R
(8)
R
eqn (8) we get
R=R,,exp(-l&dt)=Ro(l-Aenk)
(9)
Y = z, - co
(10)
co = 1 - aR
(11)
where co is locking strain, a is a material parameter and Z, is the first strain invariant. If Y < 0 material
2.008+6
r F
al
CQ l.OOe+6
Y
0 L c 0
-
0.4
0.6
stralnll Fig. 7. Density updating.
0.8
constant updating
Metallic cellular solids
1053
V 7,
,’
-
!,,::b *Elastic:
,Y
*Inelastic:
& t?
X
h=O MBa ,p=320MPa h=OMPa,
u=-100
MEa
v=-2.
9 q = -0.006 s
@= 1.0
, R =O.l
E, =O.l
, zso=0.8
D =4.0
-
* velocity jump
V
v= 12,15mZs
Fig. 8.
not lock, if Y = 0 material locks (densifies). From experiments [l, 61 the densification starts before locking strain co is reached. Therefore we are using an approximation of this behaviour, see Fig. 3. The densification process starts at does
a=o00=
1 ckk
=
00
5
to
ho _
case this can
Lkk.
(14)
In the limit case if flkk+ co then &k+ co. Because the volume change is neglected during densification the stress measure is modified as follows
(12)
and then the approximation be described as
dimensional case. For three-dimensional be generalized
of material response can (19
1
fh
(13)
where a0 corresponds to co and D is a material constant. This is the approximation for one-
This is a penalty on a dilatation measure. Practical
calculations
In this subsection we summarize integration of constitutive law of cellular material (see Box 1.). APPLICATION
(i) Calculate the first strain invariant f”‘::+‘, = 6kk. (ii) Check volume changes IF Pn”:‘,< c, THEN *Elastic behaviour integration oj‘eqn (1) to gel (TV EXIT ELSE *Inelastic behaviour
part of total stress
FOR FOAMS
In the previous part we described material model for general isotropic cellular material, i.e. for
(i) Open cells a, = R2
integration oj’eqn(5) and using eqn (7) to get uI
Rz = (; + 0.975) R? &= R.
IF r”‘J’,> LDTHEN *densification of material IF I. < CD‘THEN compute a0 END IF
(ii) Closed cells l We are not taking into account a contribution of internal pressure R, = RW + R(1 - @) Rz = (; i 0.975) (RW + R(l - ‘B))
penalty on the total stress measure eqn (15)
END IF EXIT END IF
R. = JRW
+ R(I - @)
where @ is a volume fraction. J
Box I. Stress integration for cellular material.
Box 2. Material constants for foams.
1054
B. HuEko and L. Faria
3.123B-01
1.697E-01
Min. r’
7.461E-UZ
6.033B01
5.768E-01
5504E-01
5.2398-01
Min.l+
4.97sJs01
Fig. 9. Strain localization before locking.
honeycombs, foams etc. There was no limitation on the cell geometry. In this part we shall define all needed parameters for foams. Foams can be divided
into two groups: open and closed cells [l]. We need to define R,, resp. Rz in eqn (2) and & in eqn (5). These parameters are presented in Box 2.
Metallic
cellular
NUMERICAL RESULTS
The constitutive equations presented above have been implemented into finite element code ABAQUS Version 5.2 [32]. And following numerical experiments haE&en done. One-dimensional compression test
In this example we demonstrate realistic numerical response of cellular material with strain softening due to experimental results. Material properties are presented in Table 1. The finite element model consists of one 3D eight node element. The initial and final mesh are shown in Fig. 4. The response of material to ID compression is shown in Fig. 5 for different relaxation time. For this example the comparison of closed and open cell has been made, see Fig. 6. The influence of density updating on stresses is shown in Fig. 7.
solids
1055
velocity jump on the boundary. Another problem which arises in cellular material is locking of strains. Until now all observations of strain localization with softening have been done with decreasing slope of stress-strain curve. It means that localization of deformation can not propagate to the other places or other directions. For illustration the simple 2D model has been chosen, see Fig. 8. In this example we followed more less study [19,20]. Therefore a homogenous viscoelastic specimen is subjected to a velocity jump u(t) = Heaviside(t)Ao
(16)
on the top and bottom side. Lateral sides are free. In such way elastic waves carry the stress jump onto the symmetry line where they collide. Therefore elastic stresses would be double on this line. So the magnitude of velocity jump is restricted by
Localization of deJrormations in cellular material In this example we deal with modelling of strain localization. As we mentioned in the Introduction there are many methods to model this phenomenon, but most of them need material inhomogenity to start localization for example [ 15,21,26] and to use special crossed triangular elements [21,26], but employing rate-dependence (disco-elasticity, visco-plasticity) as a regularization method we avoid this problem. In this case we need to impose a boundary inhomogenity into our model [l!), 201 only. Therefore we apply a
i
< Au < u,,
where
(18) c is a velocity of elastic wave and ~(6,) is a failure stress. The strain localization started to propagate
Mu._4
- 7.82OE-01
- 7.63OE-01
- 7.44 lE-01
- 7.251E-01 .
6.872E-01
Fig. 10. Strain
(17)
localization
after locking.
B. HuEko and L. Faria
1056
r-l
elastic inelastic locked
imperfection (inhomogenity). For this numerical experiment we chose a simple 1D model, which consists of 40 four node isoparametric elements. The weakest part of structure is the element number 19 (see Fig. 11). As one could have expected the locking propagation started from the weakest part but it continued to the top side in two different ways depending on the overall material properties. One way is the continuous one (see Fig. 11 (A)). Another way is discontinuous, (see Fig. 11 (b), (C)). In this case elastic element is surrounded by collapsed (locked) elements. The similar behaviour has been observed experimentally [12, 131 for 1D model of rings as a wrap-around effect. Now the explanation of this phenomenon is still an open question, because in the case of rings it seems to be the problem of changing geometrical conditions for forces acting on a certain ring. In our case it seems to be the material problem only. This phenomenon has been just observed at dynamical loading.
CONCLUSIONS
Fig. 11. Locking propagation.
from the bottom symmetry line to the top, as one expected, see Fig. 9(a). The distribution of strains has been arranged into horizontal bands until locking was reached see Fig. 9(b). These results are in a good agreement with experimental ones [33] in the sense of quality. After locking the redistribution of strains started due to the volume locking, i.e. the arrangement of strains into horizontal bands disappeared. The locking propagation started from the bottom symmetry line and continues to the top of our specimen. The distributions of volumetric strains after complete locking are presented in Fig. 10. Locking propagation compression
in cellular
material
during
In this example we deal with the locking propagation in the cellular material with an
A simple visco-elastic material model for cellular materials has been presented. The model reflects three different regimes: elastic, plateau and densification. The definition of failure is in the terms of volume changes. The failure of cellular material is associated with crushing of individual cells. Therefore deformations may be occurred uniformly or be localized into narrow bands. Due to material behaviour strains are localized into horizontal narrow bands when the velocity jump is applied on the boundary. After locking material starts to behave like incompressible material with strain hardening. The locking propagation has also been studied, but the explanation in what way will it continue is still an open question for both experimental and numerical studies, but this model does not effect any kind of resistance to shear loading. This is the subject of our future work. For each particular cellular material a simple study can be done to determine the complex failure surface by studying the cell response in the pure shear and in combination with compression and tension.
REFERENCES
L. J. Gibson and M. F. Ashby, Cellular Solids. Structure and Properties. Pergamon Press, Oxford (1988). L. J. Gibson, M. F. Ashby, J. Zhang and T. C. Triantifillou, Failure surfaces for cellular materials under multiaxial loads--I. Modelling. Znt. J. Mech. Sci 31, 635 (1989). T. C. T&ntifillou, J. Zhang, T. L. Shercliff, L. J. Gibson and M. F. Ashbv, Failure surfaces for cellular materials under multiaxial loads-II. Comparison of models with experiment. Znt. J. Mech. Sci. 31, 665 (1989). T. Y. Reddy and R. J. Wall, Axial compression of foam-filled thin-walled circular tubes. Znt. J. Zmpact Engng 7, 151 (1988).
Metallic cellular solids 5. W. Prager, On ideal locking materials. Trans. Sot. Rheology 1, 169 (1957). 6. P. G. Ciarlet, Marhemarical Elasriciry, Volume I: Three-dimensional Elasriciry. North-Holland, Amsterdam (1988). 7. T. C. Triantafillou and L. J. Gibson, Multiaxial failure criteria for cellul.ar materials. Mater. Res. Sot. Symp. 207, 9 (1991).
8. V. P. W. Shim and W. J. Stronge, Laterial crushing in tightly packed arrays of thin-walled metal tubes. Znr. J. Mech. Sci. 28, 709 (1986). 9. W. J. Stronge and V. P. W. Shim, Dynamic crushing of a ductile cellular array. znr. J. Mech. Sci. 29, 381 (1987).
10. J. W. Klintworth and W. J. Stronge, Elasto-plastic yield limits and deformation laws for transversely crushed honeycombs. Znr J. Mech. Sci. 30, 273 (1988). 11. M. C. Shaw and T. Sata, The plastic behaviour of cellular materials. Znr. J. Mech. Sci. 8, 469 (1966). 12. S. R. Reid and T. Y. Reddy, Experimental investigation of inertia effects in one-dimensional metal ring system subjected to end impact-I, fixed-ended systems. Znr. J. Zmpacr Engng 1, 85 (1983). 13. T. Y. Reddy, S. R. Reid and R. Barr, Experimental investigation of inertial effects in one-dimensional metal ring systems subjected to end impact--II, free-ended systems. Znr. J. impact Engng 11,463 (1991). Nonlocal 14. 2. P. Baiant and G. Piiuadier-Cabot. continuum damaze, localization instability and convergence. J. uppl. kiech. 55, 287 (1988). _ 15. Z. P. Baiant and F. B. Lin. Non-local vield limit degradation. Znr J. Numer. kerh. Engng 26, 1805 (1988).
16. H. B. Mulhaus and E. C. Aifantis, A variational principle for gradient plasticity. Znr. J. Solids Srrucr. 28, 845 (1991).
17. D. Lasry and T. Belytschko, Localization limiters in transient problems. Znr. J. Solids Srrucr. 24, 581 (1988). 18. H. L. Schreyer and Z. Chen, One-dimensional softening with localization, J. Appl. Me&. 53, 791 (1986). 19. B. Loret and J. H Prevost, Dynamic strain localization in elasto-(visco)-plastic solids, part 1. General formulation and one-dimensional examples. Compur. Merh. Appl. Mech. Engng 83, 247 (1990).
20. B. Loret and J. H. Prevost, Dynamic strain localization in elasto-(visco)-plastic solids, part 2. Plane strain
1057
examples. Compur. Merh. Appl. Mech. Engng 83, 275 (1990).
21. A. Needleman, Dynamic shear band development in plane strain. J. Appl. Mech. 56, 1 (1989). 22. R. De Borst, Simulation of strain localization: a reappraisal of the Cosserat continuum. Engng Compur. 8, 317 (1991).
23. R. De Borst, M. A. Gutierrez, J. C. J. Schellekens and L. J. Sluys, Regularization and adaptivity in computational modelling of localization phenomena. In Firsr Joint
ASCE-EMD,
ASME-AMD,
SES
Meeting,
Charlottesville (Edited by C. T. Herakovich er al.). Abstracts (1993). 24. L. Annad, Some experimental observations on localized shear bands in plane-strain. Scripra Merall. 18, 423 (1984).
25. Y. K. Lee, Conditions for shear banding and material instability in finite elasto-plastic deformation. Znr. J. Plasticity 5, 197 (1989).
26. M. W. Heinstein and H. T. Y. Yang, Plane strain finite element simulation of shear band formation during metal forming. Znr. J. Numer. Merh. Engng 33, 719 (1992).
27. A. Benallal and C. Comi, Closed form solutions of localization conditions via a geometrical method. In First Joint ASCE-EMD,
ASME-AMD,
SES Meering,
Charlottesville (Edited by C. T. Herakovich er al.). Abstracts (1993). 28. T. J. R. Hughes and J. Winget, Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis. Znr. J. Numer. Merh. EngnglS,
1862 (1980).
29. B. HuEko and J. Murin, Incremental equations for problems with finite elasto-plastic strains. In COMPLAS II (Edited by E. Hinton, D. J. R. Owen and E. Onate), Barcelona,~pp. 1395 (1989). 30. K. Washizu. Variarional Merhods in Elasricirv and Plusriciry. Pergamon Press, Oxford (1982). ’ 31. Y. T. Im and S. Kobayashi, Finite-element analysis of plastic deformation of porous materials. In Metal Forming and Impact Mechanics (Edited by S. R. Reid). Pergamon Press, pp. 103 (1985). 32. ABAQUS, Version 5.2, Hibbit-Karlsson-Sorensen Inc. 33. M. F. Vaz and M. A. Fortes, Characterization of deformation bands in the compression of cellular materials. J. Marer. Sci. Lerr. 12, 1408 (1993).
MATERIAL
MODEL
OF METALLIC
Com/mfers & Srrucrures Vol. 62, No. 6. pp. 1049-1057. 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/97 Sl7.00 + 0.00
CELLULAR
SOLIDS
B. HuEkot and L. Faria$ tFaculty of Electrical Engineering and Information Technology, Slovak Technical University, Bratislava, Slovakia IMechanical Engineering Department, Instituto Superio Tkcnico, Lisbon, Portugal (Received I August 1995) Abstract-In this paper we deal with a continuum model of a man-made cellular material. A simple visco-elastic model with softening is used. The basic features of behaviour of cellular material:
linear-elastic region, plateau and locking are presented. The locking function is assumed in terms of volume changes. The stresses are functions of density updating during deformation process. This model has been coded into ABAQUS through user’s subroutine and tested on examples with the strain localization and with the densification (locking). Copyright 0 1996 Elsevier Science Ltd.
INTRODUCTION Cellular materials are currently used in packaging, energy absorbing devices, lightweight structures, etc. The design of these structures requires understanding of the response of these materials to loads. The mechanical properties of cellular materials are related to their microstructural behaviour, i.e. macroscopic stresses are directly related to the forces in the cell walls and mechanisms of cell failure. The behaviour of cellular materials is strongly non-linear and different in tension and compression. Stressstrain curves of cellular materials in uniaxial compression and tension are shown in Refs [l, 21. These curves exhibit three different regimes: linearelastic, plateau and densification. The linear-elastic regimes in compression and tension are very similar and can be described by Hooke’s law. As deformation is increased, cells collapse in a different way. In compression cells collapse by elastic buckling, plastic yielding or brittle crushing, depending on the cell properties [l--3]. This collapse or failure progresses at roughly constant load (giving plateau) until opposing wall in the cell meet and touch. After touching cell walls the deformation stops with increasing stresses (densification or locking), i.e. cellular materials, exhibit deformation until the densification is re.ached. In tension the (collapse of cellular materials does not progress at constant load. Either they elongate, with increasing stiffness of material or yield or simply fracture. Locking or densification was observed in compression [ 1,4]. This problem has been firstly theoretically studied in [5] as an opposite behaviour to an ideal plastic one. Locking functions are presented
in [S, 61, but in reality loads are multiaxial. Therefore we need failure surface of cellular material which reflects multiaxial loading. The mechanism of failure itself may depend on stress state: a foam may be plastic in compression but brittle in tension, for instance. Then the final failure surface is given by the inner envelope of failure surfaces for individual mechanisms [l-3, 71. The failure of cellular material is associated with the initiation of crushing of individual cells. Crushing may occur uniformly or be localized into discrete bands [8-111. The presence of nonhomogenous deformation results from increasing compliance [ 111.Therefore, we distinguish stable and unstable modes of deformation (corresponding to decreasing and increasing compliance). Unstable modes of deformation are source of localization of deformation within the narrow band of cells for both quasi-static and dynamic loads 191. The small influence of strain-rate has been observed on the mode of deformation [12, 131. As we see from experiments [8-10, 12, 131 the real properties of cellular material at a certain point can be affected by the mode of deformation at another point. This coupling leads to use of a length parameter (some characteristic distance) to determine the neighbourhood of each point, which affects to this point. It means that a nonlocal approach [14, 151 should be taken into account for this material model. Modelling of strain localization within the framework of classical continuum theory leads to ill-posed boundary problems, i.e. the governing differential equations lose their ellipticity or their hyperbolicity. The loss of ellipticity of hyperbolicity leads to the extreme mesh sensitivity due to fineness and directions of grid lines. To preserve the
1049
1050
B. Hutko
and L. Faria densification
z ti
00
linear elastic
I
den&cation
plateau
g linear 111 elastic
plateau
Strain I
ED
El
Fig. 1. Stress-strain
curve for cellular
material.
well-posedness of the original problem the regularization can be used. There are different methods of regularization: (1) to enrich standard continuum theory by higher-order gradients [ l&18]; (2) to add rate effects (visco-elasticity, viscoplasticity) [ 19-2 I]; (3) to employ the Cosserat continuum (couple stresses) [22,23]; (4) to use a non-local approach [14, 151. The initiation of strain localization can be analytically computed [24-271 under the condition of homogenous deformation. Then the conditions for the strain localization can be derived. From the above-mentioned review it is clear that the behaviour of cellular material is very complex. Therefore it is necessary to select the main or the most important features of its behaviour. In this paper the simple continuum model for cellular materials is presented in general to reflect the main features of behaviour of cellular material. Then the application to foams is done. The numerical results of a few examples are compared with the experimental ones.
Fig. 3. An approximation
+
EO
of stress-strain
curve.
MATERIAL MODEL FOR CELLULAR SOLIDS
In this paper we deal with an isotropic cellular material. More general behaviour, i.e. transverse isotropic or anisotropic, will be done later, but this is a trial to make very simple model of cellular material. As we mentioned before the cellular material exhibits three regions: linear-elastic, plateau and densification (see Fig. 1). Hooke’s law is a governing equation for the linear elastic region
where
1, p are Lame’s constants, dii is the Kronecker delta, ti, resp. criiis a strain, resp. stress measure, a,, resp. Q are functions of the relative density of cell R = p*/p* and the cell geometry (these functions can be found in Ref. [l]). p* is a density of cellular material, p6 is a density of solid material. The plateau and the densification can be modelled using a constitutive law of viscoelastic material as simple as possible. The simple constitutive law for visco-elastic model (three parameters solid-see Fig. 2) of cellular material has the following form
1
l/E, + l/E, ’
(3) Table
Fig. 2. Visco-elastic
model.
*Elastic: I = 0 MPa *Inelastic: I = 0 MPa v= -1 @ = 1.0, 0.5 El = 0.1 D = 4.0
1. Material
properties
;p = 320 MPa ;p= -lOOMPa ;q = -0.008 s, -0.0073 ;R = 0.1 ;co = 0.8
s
Metallic cellular solids
J
1051
DIS?LACED
ULSI
l.ODOI-02
J
Fig. 4. The initial and final mesh. where R, is a function of the relative density of cell R and the cell geometry (expressions for & can be found in Ref. [l]). This can be generalized for an isotropic solid with small straining as follows f7i + I%‘d= e (D&k + 2/i&j + AC&&k + 2/Z,)
(4)
where 8, 1, ii, ;1, p are material constants. Or introducing relaxation time q into eqn (4), we get
where (6)
and v is a material constant. For finite straining it is necessary to introduce the objective stress and strain measures or to use algorithms to ensure the objectivity [28-301. Employing the additive decomposition of strains the Jaumann stress-rate is used .p . . bij = 6, - qiu,p - wppp where wii is a spin tensor.
Y
-
0.0
0.2
0.4
0.6
0.6
stralnll Fig. 5. Response for different relaxation times.
-0.0073 -0.006
(7)
B. HuEko and L. Faria
1052
6.OOe+6 ,
1
5.008+6
4.008+6 vr
5 3.008+6
Y
-
L L
closed cell opencell
m2.008+6 1.OOe+6 O.OOe+O 0.2
0.0
0.4
0.6
0.6
stralnll Fig. 6. Closed and open cells.
UPDATING
OF RELATIVE
where R. is an initial relative density beginning of current increment.
DENSITY
During deformation the relative density is updating due to volumetric strain-rate [3 11.
tkk= --,
Integrating
of cell at
LOCKING
The locking function can be expressed as
R
(8)
R
eqn (8) we get
R=R,,exp(-l&dt)=Ro(l-Aenk)
(9)
Y = z, - co
(10)
co = 1 - aR
(11)
where co is locking strain, a is a material parameter and Z, is the first strain invariant. If Y < 0 material
2.008+6
r F
al
CQ l.OOe+6
Y
0 L c 0
-
0.4
0.6
stralnll Fig. 7. Density updating.
0.8
constant updating
Metallic cellular solids
1053
V 7,
,’
-
!,,::b *Elastic:
,Y
*Inelastic:
& t?
X
h=O MBa ,p=320MPa h=OMPa,
u=-100
MEa
v=-2.
9 q = -0.006 s
@= 1.0
, R =O.l
E, =O.l
, zso=0.8
D =4.0
-
* velocity jump
V
v= 12,15mZs
Fig. 8.
not lock, if Y = 0 material locks (densifies). From experiments [l, 61 the densification starts before locking strain co is reached. Therefore we are using an approximation of this behaviour, see Fig. 3. The densification process starts at does
a=o00=
1 ckk
=
00
5
to
ho _
case this can
Lkk.
(14)
In the limit case if flkk+ co then &k+ co. Because the volume change is neglected during densification the stress measure is modified as follows
(12)
and then the approximation be described as
dimensional case. For three-dimensional be generalized
of material response can (19
1
fh
(13)
where a0 corresponds to co and D is a material constant. This is the approximation for one-
This is a penalty on a dilatation measure. Practical
calculations
In this subsection we summarize integration of constitutive law of cellular material (see Box 1.). APPLICATION
(i) Calculate the first strain invariant f”‘::+‘, = 6kk. (ii) Check volume changes IF Pn”:‘,< c, THEN *Elastic behaviour integration oj‘eqn (1) to gel (TV EXIT ELSE *Inelastic behaviour
part of total stress
FOR FOAMS
In the previous part we described material model for general isotropic cellular material, i.e. for
(i) Open cells a, = R2
integration oj’eqn(5) and using eqn (7) to get uI
Rz = (; + 0.975) R? &= R.
IF r”‘J’,> LDTHEN *densification of material IF I. < CD‘THEN compute a0 END IF
(ii) Closed cells l We are not taking into account a contribution of internal pressure R, = RW + R(1 - @) Rz = (; i 0.975) (RW + R(l - ‘B))
penalty on the total stress measure eqn (15)
END IF EXIT END IF
R. = JRW
+ R(I - @)
where @ is a volume fraction. J
Box I. Stress integration for cellular material.
Box 2. Material constants for foams.
1054
B. HuEko and L. Faria
3.123B-01
1.697E-01
Min. r’
7.461E-UZ
6.033B01
5.768E-01
5504E-01
5.2398-01
Min.l+
4.97sJs01
Fig. 9. Strain localization before locking.
honeycombs, foams etc. There was no limitation on the cell geometry. In this part we shall define all needed parameters for foams. Foams can be divided
into two groups: open and closed cells [l]. We need to define R,, resp. Rz in eqn (2) and & in eqn (5). These parameters are presented in Box 2.
Metallic
cellular
NUMERICAL RESULTS
The constitutive equations presented above have been implemented into finite element code ABAQUS Version 5.2 [32]. And following numerical experiments haE&en done. One-dimensional compression test
In this example we demonstrate realistic numerical response of cellular material with strain softening due to experimental results. Material properties are presented in Table 1. The finite element model consists of one 3D eight node element. The initial and final mesh are shown in Fig. 4. The response of material to ID compression is shown in Fig. 5 for different relaxation time. For this example the comparison of closed and open cell has been made, see Fig. 6. The influence of density updating on stresses is shown in Fig. 7.
solids
1055
velocity jump on the boundary. Another problem which arises in cellular material is locking of strains. Until now all observations of strain localization with softening have been done with decreasing slope of stress-strain curve. It means that localization of deformation can not propagate to the other places or other directions. For illustration the simple 2D model has been chosen, see Fig. 8. In this example we followed more less study [19,20]. Therefore a homogenous viscoelastic specimen is subjected to a velocity jump u(t) = Heaviside(t)Ao
(16)
on the top and bottom side. Lateral sides are free. In such way elastic waves carry the stress jump onto the symmetry line where they collide. Therefore elastic stresses would be double on this line. So the magnitude of velocity jump is restricted by
Localization of deJrormations in cellular material In this example we deal with modelling of strain localization. As we mentioned in the Introduction there are many methods to model this phenomenon, but most of them need material inhomogenity to start localization for example [ 15,21,26] and to use special crossed triangular elements [21,26], but employing rate-dependence (disco-elasticity, visco-plasticity) as a regularization method we avoid this problem. In this case we need to impose a boundary inhomogenity into our model [l!), 201 only. Therefore we apply a
i
< Au < u,,
where
(18) c is a velocity of elastic wave and ~(6,) is a failure stress. The strain localization started to propagate
Mu._4
- 7.82OE-01
- 7.63OE-01
- 7.44 lE-01
- 7.251E-01 .
6.872E-01
Fig. 10. Strain
(17)
localization
after locking.
B. HuEko and L. Faria
1056
r-l
elastic inelastic locked
imperfection (inhomogenity). For this numerical experiment we chose a simple 1D model, which consists of 40 four node isoparametric elements. The weakest part of structure is the element number 19 (see Fig. 11). As one could have expected the locking propagation started from the weakest part but it continued to the top side in two different ways depending on the overall material properties. One way is the continuous one (see Fig. 11 (A)). Another way is discontinuous, (see Fig. 11 (b), (C)). In this case elastic element is surrounded by collapsed (locked) elements. The similar behaviour has been observed experimentally [12, 131 for 1D model of rings as a wrap-around effect. Now the explanation of this phenomenon is still an open question, because in the case of rings it seems to be the problem of changing geometrical conditions for forces acting on a certain ring. In our case it seems to be the material problem only. This phenomenon has been just observed at dynamical loading.
CONCLUSIONS
Fig. 11. Locking propagation.
from the bottom symmetry line to the top, as one expected, see Fig. 9(a). The distribution of strains has been arranged into horizontal bands until locking was reached see Fig. 9(b). These results are in a good agreement with experimental ones [33] in the sense of quality. After locking the redistribution of strains started due to the volume locking, i.e. the arrangement of strains into horizontal bands disappeared. The locking propagation started from the bottom symmetry line and continues to the top of our specimen. The distributions of volumetric strains after complete locking are presented in Fig. 10. Locking propagation compression
in cellular
material
during
In this example we deal with the locking propagation in the cellular material with an
A simple visco-elastic material model for cellular materials has been presented. The model reflects three different regimes: elastic, plateau and densification. The definition of failure is in the terms of volume changes. The failure of cellular material is associated with crushing of individual cells. Therefore deformations may be occurred uniformly or be localized into narrow bands. Due to material behaviour strains are localized into horizontal narrow bands when the velocity jump is applied on the boundary. After locking material starts to behave like incompressible material with strain hardening. The locking propagation has also been studied, but the explanation in what way will it continue is still an open question for both experimental and numerical studies, but this model does not effect any kind of resistance to shear loading. This is the subject of our future work. For each particular cellular material a simple study can be done to determine the complex failure surface by studying the cell response in the pure shear and in combination with compression and tension.
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