**Transactions of the 17****th **_{International Conference on }

**Structural Mechanics in Reactor Technology (****SMiRT 17)**

**Prague, Czech Republic, August 17 –22, 2003**

** Paper # B02-5**

### Modeling Elasto-plastic Behavior of Polycrystalline Grain Structure of Steels at

### Mesoscopic Level

Marko Kovač1)_{, Igor Simonovski}1)_{, Leon Cizelj}1)

### 1)

Jožef Stefan Institute, Reactor Engineering Division, Ljubljana, Slovenia**ABSTRACT**

During severe accidents the pressure boundary of the reactor coolant system can be subjected to the extreme loadings, which might cause failure of the reactor coolant system. Reliable estimation of the extreme deformations of steel can be therefore crucial for determining the consequences of severe accidents. An important drawback of the structural mechanics is idealization of inhomogenous microstructure of materials. Models, which can take inhomogenous microstructure of materials into the account by consideration of grain structure, typically appeared over the last decade. Nevertheless, entirely anisotropic models towards material behavior of a single grain are still not widely used. The proposed model divides the continuum (e.g., polycrystalline aggregate) into a set of sub-continua (grains) by utilizing Voronoi tessellation (random grain structure). Each grain is assumed to be a monocrystal with random orientation of crystal lattice. Strain and stress fields are calculated using finite element method. Proposed model is then used to estimate the minimum size of polycrystalline aggregate above which it can be considered macroscopically homogeneous. The analysis is limited to two-dimensional models. Material parameters for the reactor pressure vessel steel 22 NiMoCr 3 7 with b.c.c. crystal lattice are used in the analysis.

**KEY WORDS: **Polycrystalline material, elasto-plastic material behavior, mesoscale, Voronoi tessellation, finite

elements, crystal plasticity

**INTRODUCTION**

During severe accidents the pressure boundary of reactor coolant system can be subjected to extreme loadings, which might cause failure. Reliable estimation of the extreme deformations of material (steel) can be crucial to determine the course of events and estimate the consequences of severe accidents. Therefore a lot of efforts were made during past decades to determine behavior of materials under large inelastic deformations [1]. Important drawback of structural mechanics, which was not overcome by the most complicated existing models, is the idealization of inhomogenous microstructure of materials [2,3]. Structural mechanics might fail to predict material behavior accurately, when the inhomogeneities start to dominate the response of the structure. For large structures (compared to the size of inhomogeneities), these effects typically become dominant while approaching limit loads. However, for relatively small structures, the effects of inhomogeneities may become noticeable already at the level of normal service loads.

Promising alternatives to structural mechanics were models, which tried to obtain the macroscopic behavior of material from the behavior of material constituents on the microscale (e.g., single atoms with typical size up to 1 nm) or

the mesoscale (crystal grains with typical size of 10–100 µm).

Models, which use stochastic methods to represent grain structure of material and can accommodate the anisotropy of material, were introduced only recently [3,4,5,6]. However, they typically concentrated on few mesoscopical features and simplified or neglected others. A more general mesoscopic model, which describes elastic-plastic behavior of polycrystalline material by accounting for the random grain structure, is therefore proposed. The main idea of proposed mesoscopic model is to divide continuum (e.g., polycrystalline aggregate) into a set of sub-continua (e.g., grains). The overall properties of the polycrystalline aggregate are the outcome of the number of grains in the aggregate and the properties of randomly shaped and oriented grains. The analysis of the macroscopic element is therefore divided into modeling the random grain structure (using Voronoi tessellation and random orientation of crystal lattice) and calculation of strain/stress field. Mesoscopic response of monocrystal grains is modeled with anisotropic elasticity and crystal plasticity. Finite element method, which proved as suitable [3,7], is used to obtain numerical solutions of strain and stress fields.

used in analysis. However, the proposed model can, in general, accommodate a variety of grain structures and material models.

**THEORETICAL BACKGROUND**

Analysis of polycrystalline aggregate is divided into modeling the random grain structure and calculation of strain/stress field.

Basic assumptions of material model are:

• Random polycrystalline structure is represented by a Voronoi tessellation with random orientation of crystal

lattice.

• Each grain is assumed to be a monocrystal with random orientation of crystal lattice. Anisotropic elastic behavior of grains is assumed. A model of plasticity assumes that plastic deformation is caused by crystalline slip on predefined slip planes of crystal lattice. Slip planes and direction are defined by orientation of crystal lattice, which differs from grain to grain (random orientation). Finite element method, which proved as suitable [3], is used to obtain numerical solutions of strain and stress fields.

• Overall properties of the polycrystalline aggregate depend on the properties of randomly shaped and oriented

grains.

**Voronoi tessellation**

The concept of Voronoi tessellation has recently been extensively used in materials science, especially for modeling random microstructures like aggregates of grains in polycrystals, patterns of intergranular cracks, and composites [7,8,9]. A Voronoi tessellation represents a cell structure constructed from a Poisson point process by introducing planar cell walls perpendicular to lines connecting neighboring points. This results in a set of convex polygons/polyhedra (Figure 1) embedding the points and their domains of attraction, which completely fill up the underlying space. All Voronoi tessellations used for the purpose of this paper were generated by the code VorTess [10]. Planar quadrilateral elements were used in the analysis. The reliability of finite element analysis can be improved, if only "meshable" tessellations (with only tolerable distorted elements from ideally square shape to be used) are taken into account [11]. The bias introduced is considered to be small compared to the error caused by the 2-D approximation of grain structure [12].

*x*
*y*
*x'*
*y'*
*x'y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
*x'*
*y'*
* p1*
* p1*
* p2*
* p1*
* p2*
* p2*
Displacement
boundary
condition
Stress
boundary
condition

Figure 1: Voronoi tessellation of 14-grain aggregate with grain boundaries, orientations of crystal lattices, finite element mesh and boundary conditions

**Anisotropic Elasticity and Crystal Plasticity**

grain to grain (random orientation). Rate-independent plasticity with Peirce et al. [16] and Asaro [17] hardening law was used in our research.

Material parameters for the reactor pressure vessel steel 22 NiMoCr 3 7 with body-centered cubic crystal lattice were obtained from literature (e.g., [13,14,18]) and from results of simple tensile test of the reactor pressure vessel steel 22 NiMoCr 3 7 [19]. Further details are available in [20] and references therein.

**Estimation of Representative Volume Element Size**

Large enough and geometrically similar polycrystalline aggregates have equal macroscopic material response. On the macroscale these aggregates appear homogeneous in statistical sense, regardless of mesoscopic material inhomogeneity [21]. However, this is not the case with small size components, where microstructure might play an important role. Estimating minimum polycrystalline aggregates size needed for the homogeneity assumption to become valid is therefore essential to determine effective range of structural mechanics. This minimum polycrystalline aggregates size is usually called representative volume element (RVE) [21]. The homogeneity assumption becomes valid (i.e., polycrystalline aggregate is larger than RVE) when [21]:

## (

_{*}

## )

1* _{≅} −

*ijkl*
*ijkl* *D*

*C* , (1)

where *C**

*ijkl* and *D*ijkl* are macroscopic stiffness and compliance tensors of a polycrystalline aggregate [21].

Equation (1) in general is not valid for polycrystalline aggregates smaller than RVE. The different behavior of both tensors is governed by the size of the aggregate and macroscopic boundary conditions [15]: macroscopic stiffness tensor therefore assumes stress boundary condition, while macroscopic compliance tensor assumes displacement boundary condition. Macroscopic equivalent stresses can be treated as primary variables of polycrystalline aggregate response and they offer reasonably good framework for the interpretation of results. Expressing the RVE size from macroscopic equivalent stresses seems therefore reasonable. Equation (1) can be therefore transformed into [7]:

*d*
*eq*
*s*

*eq* σ

σ ≅

, (2)

where <σ*eq*> represents macroscopic equivalent (Von Mises) stresses and indexes *s* and *d* denote stress and displacement

driven boundary conditions, respectively. Macroscopic equivalent stresses <σ*eq*> are calculated as:

## ∫

=*V* *eq*

*eq* *dV*

*V* σ

σ 1 , (3)

where σ*eq* stands for equivalent stress and *V* for volume of polycrystalline aggregate [6]. A relation between

macroscopic equivalent stresses for both boundary conditions for a polycrystalline aggregate smaller than RVE can be written as [7]:

### (

*RVE*

### )

*d*
*eq*

*s*
*eq*

*i*
*i*
*O*
+

=1

σ σ

, (4)

where *iRVE*represents number of grains in RVE and *i* number of grains size in polycrystalline aggregate smaller than

RVE. A RVE is achieved, when residuum *O* is smaller than 1% [12].

**RESULTS AND DISCUSSION**

Proposed model was used to calculate mesoscopic stress and strain fields in polycrystalline aggregates. An average (macroscopic) response of the polycrystalline aggregate was sought through the appropriate averaging procedure [21]. Macroscopic stresses and strain were than used to estimate RVE size in elasticity and plasticity.

**Mesoscopic Strain/stress Fields**

260 310 360 410 460 510 560 610 660 710 760

### 0%

3% 6% 9% 12% 15% 18% 21% 24% 27% 30%

Figure 2: Equivalent stress [MPa] (left) and equivalent strain [%] (right) for 212-grain polycrystalline aggregate with displacement boundary conditions

The presented polycrystalline aggregate was used, since it represents the characteristic material response. Strain

and stress fields are presented at the biaxial loads *p1*= 1155 MPa and* p2* = 578 MPa (depicted with circle in Figure 3).

This calculation point was selected due to the fact, that macroscopic equivalent stress in this point (<σ*eq*> = 518 MPa) is

higher than yield strength (σ*Y* = 440 MPa), however material is not fully yielded. This enables one to show typical

mesoscopic features.

0 100 200 300 400 500 600

0% 5% 10% 15%

**Macroscopic Equivalent Strain [/]**

**Macr**

**o**

**sco**

**p**

**ic E**

**q**

**u**

**iva**

**len**

**t S**

**tr**

**ess [**

**M**

**P**

**a]**

Figure 3: Macroscopic response of the 212-grains polycrystalline aggregate

Distinctive heterogeneity in the stress and strain fields on the mesoscopic level could be observed in Figure 2. The mesoscopical strains concentrate in the shear bands Figure 2 (right) with the equivalent strains up to 500% higher than the macroscopical (i.e., average) equivalent strain. Shear bands typically develop at grain boundaries (they are less distinctive when passing through the grains) in directions of about 50° from x-axis. The typical distance between shear bands is in the order of 1 grain size. Concentrations of stresses are less distinctive, with local stresses up to 60% higher than macroscopic equivalent stress. Concentrations of local strain and stress due to a grain structure might therefore contribute significantly to localized failure of material and consequently to initialization and growth of microcracks.

**Estimation of RVE Size **

RVE size was estimated for elasticity and plasticity. Analyses were carried out on polycrystalline aggregates
with 14, 23, 53, 110 and 212 grains with sizes from 0.1 mm × 0.07 mm to 0.4 mm × 0.28 mm. 30 different random
orientations of crystal lattices and 2 boundary conditions (stress and displacement boundary conditions) were analyzed
for each polycrystalline aggregate. Analyses were carried out at biaxial loads *p1 *= 200 MPa and *p2 *= 100 MPa for

Equivalent macroscopic strains and stresses are shown in Figure 4 where *d *in the legend refers to displacement

boundary condition, *s *refers to stress boundary condition and *ave *refers to average values (averaged over 30 different

randomly orientated crystal lattices with displacement or stress boundary conditions). The number following
abbreviation denotes number of grains of polycrystalline aggregates with separate values for each boundary condition.
Analytical solution for elasticity for macroscopically homogenous material (<ε*eq*> = 0.0515% and <σ*eq*> = 96.2 MPa) is

95.0 95.2 95.4 95.6 95.8 96.0 96.2 96.4 96.6 96.8 97.0

0.035% 0.040% 0.045% 0.050% 0.055% 0.060%

**Macroscopic Equivalent Strain [/]**

**Ma**
**cr**
**os**
**co**
**pi**
**c **
**Eq**
**ui**
**va**
**le**
**nt**
** S**
**tr**
**es**
**s **
**[MP**
**a]**
d 14
s 14
d 23
s 23
d 53
s 53
d 110
s 110
d 212
s 212
av e d 14
av e s 14
av e d 23
av e s 23
av e d 53
av e s 53
av e d 110
av e s 110
av e d 212
av e s 212
analy tical
440
445
450
455
460
465
470
475

0.0% 0.5% 1.0% 1.5% 2.0%

**Macroscopic Equivalent Strain [/]**

**Ma**
**cr**
**os**
**co**
**pi**
**c **
**Eq**
**ui**
**va**
**le**
**nt**
** S**
**tr**
**es**
**s **
**[M**
**Pa**
**]**
d14 s14
d23 s23
d53 s53
d110 s110
d212 s212
av e d 14 av e s 14
av e d 23 av e s 23
av e d 53 av e s 53
av e d 110 av e s 110
av e d 212 av e s 212

Figure 4: Scatter of macroscopic equivalent strain/stress for elasticity (left) and plasticity (right)

trend *s*

trend *d*

trend *s*

A clear tendency towards decrease of scatter as number of grains in the aggregates increases can be observed. Average values of macroscopic strains and stresses (for both boundary conditions) show a clear trend towards common average solution with increasing number of grains in the aggregate.

RVE size was estimated according to eq. (4). Macroscopic equivalent stresses were taken at macroscopic
equivalent strain <ε*eq*> of 0.0515 % (elasticity) and 1% (plasticity). Figure 5 shows macroscopic equivalent stresses and

scatter depending on number of grains in polycrystalline aggregate for displacement (denoted as *d*) and stress (denoted

80 85 90 95 100 105 110 115

0 100 200 300 400

**Number of grains**

**M**

**ac**

**ros**

**copi**

**c **

**E**

**qui**

**va**

**le**

**nt **

**S**

**tr**

**es**

**s **

**[M**

**P**

**a]**

d s

d extrapolated s extrapolated analytical

410 420 430 440 450 460 470 480 490 500 510 520

0 200 400 600 800 1000

**Number of grains**

**M**

**ac**

**ros**

**copi**

**c **

**E**

**qui**

**va**

**le**

**nt **

**S**

**tr**

**es**

**s **

**[M**

**P**

**a]**

d s

d extrapolated s extrapolated

Figure 5: Convergence of macroscopic equivalent stresses in elasticity (left) and plasticity (right)

corresponds to a polycrystalline aggregate of about 0.45 mm in size. This is comparable with results from literature for

aluminum oxide [12]. The RVE size in plasticity is estimated to 800 grains (with residuum *O* of 1%), which

corresponds to a polycrystalline aggregate of about 0.66 mm in size.

The minimum component size needed for the homogeneity assumption to become valid in plasticity is larger than the one in elasticity. Based on that, the minimum component size is expected to increase even further if material is subjected to larger deformation with the development of damage.

**SUMMARY**

A numerical model, which models elasto-plastic behavior on the mesoscopic level, was proposed. It combines the most important mesoscale features and compatibility with conventional continuum mechanics to model elastic-plastic behavior. Explicit modeling of the random grain structure is used. Grains are regarded as monocrystals and modeled with anisotropic elasticity and crystal plasticity. Proposed model was used to estimate the minimum size of polycrystalline aggregate above which it can be considered macroscopically homogeneous. This can be used as an orientation value to predict the lower bound of domain of structural mechanics (below which it is not able to describe accurately the polycrystalline material behavior). The estimation is based on comparison of macroscopic quantities of polycrystalline aggregates. The analysis is limited to two-dimensional models. Material parameters for the reactor pressure vessel steel 22 NiMoCr 3 7 with bainitic microstructure with b.c.c. crystals are used in analysis. The minimum size of polycrystalline aggregate needed for the homogeneity assumption to become valid was estimated to be below 0.7 mm and is larger in plasticity than the one in elasticity. This suggests that the minimum component size is expected to increase if material is subjected to damage. Future work will include broadening proposed model with simulation of initialization and growth of microcracks and its effect on estimations of minimum polycrystalline aggregate size for the homogeneity assumption validity.

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