Large strain behavior of two-phase materials with random inclusions

COMPUTATIONAL MATERIALS SCIENCE Computational

Materials

Science 1I ( 1998)221-226

Large strain behavior of two-phase materials with random inclusions H.-P. Ggnser ap*, F.D. Fischer b, E.A. Werner

’

a Christian Doppler Laboratory for Micromechanics of Materials, Institute of Mechanics, Montanuniversitiit Leoben. Franz-Josef-Strafle 18, A-8700 Leoben, Austria b Institute of Mechanics, Christian Doppler Laboratory for Micromechanics of Materials, Montanuniversitiit Leoben, Leoben. Austria ’ Chair A for Mechanics. Technical University of Munich, Boltzmannstrafle 15, D-85748 Garching, Germany

Received23 December 1997;accepted9 February 1998

Abstract The large strain behavior of a two-phase material with random inclusions is investigated using the periodic microfield approach. Two- and three-dimensional representative volume elements (RVES) with different boundary conditions are compared. For low volume fractions of second-phase particles up to about 2% and moderate strains, no marked differences with respect to the stress-strain curves are found between the various models. For higher volume fractions, the differences between the 2D and the 3D representations cannot be neglected anymore. For the simulation of a sheet material, no constraint must be imposed in the thickness direction of a 3D model; otherwise, necking-an essential feature for the large deformation behavior of sheets-cannot be captured. This type of RVE is different from the RVEs normally used for the simulation of the material behavior in bulky specimens. 0 1998 Elsevier Science B.V. All rights reserved. Keywords: Plasticity; Two-phase materials; Large strain behavior; Sheet materials; Periodic microfield approach; Representative volume elements; Finite element method

1. Introduction One of the most common methods for obtaining the macroscopic behavior of multiphase materials by computational techniques is the periodic microfield approach [l-7]. In this approach, a representative volume element (RVE), or ‘unit cell’, is modeled, and the infinitely extending material is simulated by applying periodic boundary conditions in all three spatial directions. By this way, any effects of free

Corresponding author. Tel.: +43-3842-402-870; 3842-46048; e-mail: [email protected]. l

0927-0256/98/$19.00 PII

fax:

+43-

0 1998 Elsevier Science B.V. All rights reserved.

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boundaries are excluded, and the behavior of a piece of material in the interior of a bulky specimen is recovered. However, there remain two aspects for further consideration. The first one is the relation between two-dimensional and three-dimensional models, as has been treated in Refs. [3,8]; the second one is the deformation behavior of thin specimens, like, e.g., sheets. This paper will focus on these two points, assuming a two-phase composite with an elasticplastic matrix and randomly distributed elastic inclusions of cubic shape as a model material. Clearly, the most pronounced difference between 2D and 3D models is the representation of the

222

H.-P. Ginser et d. / Compututionul

underlying geometry. While the 3D model can easily capture the exact geometry of the cell, in a 2D model the cubic inclusions become fibers with square cross-section extending in the thickness direction. Many of the following results can be explained by these restrictions of the 2D model (see also Refs. [3,81). When considering the differences between bulky and flat specimens, a first distinction can be made by investigating the two cases of plane stress and (generalized) plane strain in a 2D model. ’ The generalized plane strain state corresponds roughly to the case of a slice of material in the center regions of a bulky specimen; the deformation in the thickness direction is constrained by the surrounding material. The plane stress state, with no constraint in the thickness direction, then represents a slice of material in a very thin specimen. In a 3D model, the generalized plane strain case may be modeled by periodic or mirror boundary conditions in the thickness direction, and the plane stress case can be modeled simply by unconstrained upper and lower boundary planes.

2. Computational

Materials Science II (1998) 221-226

TIN precipitates, a typical material for sheet forming applications. However, the precipitate volume fractions assumed in some of the simulations are, for computational reasons, much higher than those observed in IF steels. Fig. 1 displays both the three-dimensional and the two-dimensional finite element (FE) discretizations of a typical representative volume element. Both models consist of 25 X 25 second-order, reduced integration, Serendipity-type isoparametric elements in the ’ l’-‘2’ plane. Furthermore, in the 3D model the thickness (‘3’) direction is subdivided into five elements. In any case, periodic boundary conditions are applied in the directions ‘1’ and ‘2’ in order to simulate a specimen infinitely extending in this plane. Four different models are investigated: one 2D and one 3D model for the two cases of a sheet material and of a bulky specimen.

approach

2.1. Finite element model The material investigated consists of an isotropic elastic-J,-plastic matrix with randomly distributed elastic inclusions of cubical shape. The matrix is characterized by a Young’s modulus of E,,, = 2 10 000 [N/mm2], a Poisson’s number V, = 0.3, a yield stress of R, = 150 [N/mm*], and isotropic hardening is described by a power law Us = 532~:~~, where CT,, denotes the matrix equivalent stress, and E,,, denotes the matrix equivalent plastic strain. The respective data for the inclusions are Ei = 600000 [N/mm*], vi = 0.15. This corresponds roughly to an interstitial free (IF) steel with a ferritic matrix and

’ In

the

present study,

(4

the differences between the 2D plane

strain model and the 2D generalized

(b)

plane strain model were

found to be negligibly small. This behavior is explained

by the

Fig.

1. (a)

Three-dimensional

and (b)

two-dimensional

finite

transverse constraint imposed by the (in the 2D model) fiber-like

element mesh of a two-phase material with randomly distributed

arrangement of purely elastic, stiff inclusions.

inclusions (2 vol.% inclusions).

223

H.-P. Giinser et al. / Computational Materials Science 1 I (1998) 221-226

For the simulation of a sheet material in the 3D calculations, no constraint is applied in the thickness direction, which corresponds to the free lower and upper boundary planes of a thin sheet. In the 2D model, this condition is accounted for by using plane stress elements. A 3D representative volume element from the interior of a bulky specimen, on the contrary, is also constrained by periodic boundary conditions in the direction ‘3’. This is the usual RVE known from the literature, and thus will be denoted as 30 volume model. The corresponding 2D models use plane strain or generalized plane strain elements. To be representative for a material with randomly distributed inclusions, the RVEs must contain a sufficient number of precipitates. ’ For this purpose, one inclusion is discretized by one single element. Although the stress fields near the phase boundaries cannot be captured precisely, this rather coarse approximation allows for the required amount of inclusions and for particle-particle interactions while keeping the computational cost reasonably low. Several unit cells with inclusion volume fractions between 2 and 8% are generated and subjected to uniaxial tension in the direction ‘1’. The principal macroscopic in-plane strains are denoted by E, and E2.

2.2. Homogenization

procedure

The true stress-strain curve of the homogenized material (the macroscopic stress-strain curve> is obtained as follows: Given the current edge lengths Ii and the edge length increments dlj of the unit cell, the macroscopic principal strain increments in the directions ‘ 1’ and ‘2’ are obtained as dE,=-,

d/i 4

dE2=F.

(1) 2

* Strictly speaking, the overall particle distribution is only quasi-random because the material is represented by one repeating RVE, even though the particle distribution within the RVE is random.

Neglecting the elastic strains, 3 the macroscopic strain increment in the direction ‘3’ is calculated from the conservation of volume as des = -(dE,

+ dE2),

and the macroscopic fr.

The macroscopic integration:

(2)

equivalent

equivalent

strain increment

strain

E is obtained

is

by

E = Id&.

(4) The macroscopic equivalent Cauchy stress u follows from the increment of the macroscopic equivalent logarithmic strain de, the applied external work increment dW and the volume V = 1,1,1, of the RVE as dW _..

a=i% 3. Results 3. I. Deformation

behavior

To compare the deformation behavior of the 2D and the 3D models, they are subjected to uniaxial tension. The deformation patterns of the 3D sheet model and the 2D plane stress model are depicted in Fig. 2. It can be seen that several inclined grooves form due to the inhomogeneities of the strain field caused by the inclusions. It should be noted that the parallel pattern results from the periodic boundary conditions. Due to the constraints in the thickness direction, the 3D volume model and the 2D plane strain model (both not displayed) do not show such patterns, but deform in a regular manner. The 2D plane stress as well as the 3D sheet and the 3D volume models yield d.s2 = d .s3 = -d&,/2, as it is expected for uniaxial tension. In the 2D plane strain model, the element definition enforces the

3 This simplification is justified because, for a composite consisting of a matrix with a low yield stress and stiff inclusions, the elastic strains and thus the volume dilatation are negligible compared to plastic strains as large as applied here.

224

(b) Fig. 2. Equivalent plastic strain distribution of a (a) 3D sheet RVE and (b) a 2D plane stress RVE (2% inclusions) subjected to uniaxial tension in the

‘I ’

direction. The undeformed shape is drawn as wireframe for comparison.

constraint dcj = 0, and therefore d .s2 = -d E,. Clearly, the 2D plane strain model cannot capture the load case of uniaxial tension, and thus its results are not strictly comparable to those of the other models. 4 3.2. Injluence

of the boundary

conditions

The macroscopic stress-strain curves for inclusion volume fractions of 2, 5, and 8% are depicted in

4 In the present case this holds also true for the generalized plane strain state because, as mentioned above, due to the high stiffness of the inclusions the strain in thickness direction negligibly small.

is

Fig. 3. For the 3D sheet and the 2D plane stress models in the early stages of the deformation the composite shows slightly higher hardening than the matrix due to the reinforcement by the particles. In the later stages the geometric softening caused by the grooves outweighs the reinforcement by the inclusions, finally leading to a drop in the stress-strain curve. This behavior is essentially the same for the 2D and the 3D models; however, the 2D model shows the instability at lower strains because of the somewhat poorer representation of the underlying geometry. Still, the 2D plane stress model may be used as a fairly good approximation for a sheet material, especially at lower precipitate volume fractions.

H.-P. Ginser et al. / Computational

225

Materials Science I I (I 998) 221-226

The 3D volume and the 2D plane strain models do not show any drop in the stress-strain curves, as no necks can develop here. No differences were found between the results of the plane strain and the generalized plane strain models, because even in the generalized plane strain case the upper and lower boundary planes can hardly move due to the large constraint caused by the purely elastic inclusions in the thickness direction. 3.3. Influence of the quasi-random distribution of the

inclusions ICOI 0.0

I 0.2

0.4

0.X

0.6

1.0

1.2

d-1 UXX)

. - matrix -- plwewesn

04

w)

. - 0. planestrain -*

MN) 7(H)

3D.ke.1 3D volume

r-

The differences between several randomly generated RVEs were investigated for the case of 2% inclusions, cf. Fig. 4. No deviations between the results from the different RVEs were found for the ‘bulky’ material, i.e., the 2D plane strain and the 3D volume models. For the sheet material, however, the horizontal tangent of the stress-strain curve is obtained at different strains. This effect is more pronounced for the 3D model than for the 2D model. The reason for this behavior is the influence of the arrangement of the inclusions on the development of the neck: clearly, there exist arrangements which favor the formation of a groove, and such which delay the necking process.

loo0

L -.

/-

p

.. ..-- _.-_.- ..-__.---__---:. . .* \

mmix

--

plantSlrcSS plancrtin

XOIJ

..-. --

-.--. --3Dshee1 matrix 3Dvolume plPnes!Iain p1unesvesli

/.

-!m

3Dsixet 3D vdume

700 I.E” &I b 4ou

\ \ \ \ \

3cuJ

\

200 1. ml ,

0.2

0.4

0.6

0.8

I .o

I.2

et-1 Fig. 3. Macroscopic stress-strain curves for (a) 2%. (b) 5%. (cl 8% inclusions. The stress-strain curve of the pure matrix is also shown.

Fig. 4. Macroscopic stress-strain curves for 2% inclusions and different random RVEs. The stress-strain curve of the pure. matrix is also shown.

226

3.4. Influence

H.-P. Ginser et al. / Computrrtionui Muterids Science I I (I 998) 221-226

of the volume fraction

qf inclusions

The higher the volume fraction of inclusions, the more evident become the differences between the results from the various models. For 2% inclusions and strains up to 0.5, one obtains a quite good agreement between all four models, as, due to the low volume fraction of inclusions, neither the reinforcement by the inclusions plays a big role, nor is necking triggered until rather high strains are reached (Fig. 3al. This picture changes when the volume fraction of inclusions increases (Fig. 3b,cl. On the one hand, the differences between the 2D and the 3D results become more evident. In all cases, the 3D models show a stiffer response than the 2D models. This holds true for the ‘bulky’ as well as for the sheet material, and becomes more pronounced with higher volume fractions, which indicates that the 2D models cannot fully capture the reinforcement effect due to the second-phase particles. On the other hand, the necking behavior of the sheet material is captured quite satisfactorily by the 3D sheet model as well as by the 2D plane stress model. While the differences with respect to the hardening remain, the drop in the stress-strain curve moves in both models to lower strains for higher volume fractions; the ratio of these strains in the 3D model to those in the 2D model is nearly constant.

a 3D model; otherwise, the deformation instability caused by the formation of a neck - an essential feature for the large deformation behavior of sheets - cannot be captured. This type of RVE is different from the RVEs normally used for the simulation of the material behavior in bulky specimens. Sheet necking may as well be simulated by a 2D plane stress model; here, the deformation instability occurs at lower strains than in the 3D model.

Acknowledgements The motivation for this work came from a discussion with V. Tvergaard and A. Needleman, which is most gratefully acknowledged. Furthermore, the authors wish to thank F.G. Rammerstorfer, G. Reisner and T. Antretter for critical discussion.

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The large deformation behavior of a two-phase material with random inclusions was investigated using the periodic microfield approach. 2D and 3D models with different boundary conditions were compared. For low volume fractions of second-phase particles up to about 2% and moderate strains, no marked differences with respect to the stress-strain curves are found between the various models. For higher volume fractions, the differences between the 2D and the 3D representations cannot be neglected anymore. In any case, the 2D models show a softer response than the 3D models, due to the somewhat poorer representation of the underlying geometry. For the simulation of a sheet material, no constraint must be imposed in the thickness direction of

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