tetragonal symmetry.pdf

8/14/2019 tetragonal symmetry.pdf

http://slidepdf.com/reader/full/tetragonal-symmetrypdf 1/20

Modelling Simul. Mater. Sci. Eng. 7 (1999) 909–928. Printed in the UK PII: S0965-0393(99)07765-7

The relation between single crystal elasticity and the effective

elastic behaviour of polycrystalline materials: theory,

measurement and computation

J M J den Toonder†, J A W van Dommelen‡ and F P T Baaijens‡

† Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands‡ Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands

E-mail: [email protected], [email protected] [email protected]

Received 8 April 1999, accepted for publication 1 September 1999

Abstract. Due to continuing miniaturization, characteristic dimensions of electroniccomponentsare now becoming of the same order of magnitude as the characteristic microstructural scales of the constituent materials, such as grain sizes. In this situation, it is necessary to take into accountthe influence of microstructure when studying the mechanical behaviour. In this paper, we focuson the relation between the (anisotropic) properties of individual grains and the effective elasticbehaviour of polycrystalline materials. For large volumes of materials, the conventional averagingtheory may be applied. This is illustrated with experiments on various barium titanates. For smallvolumesof material, we examine therelationshipby means of micromechanical computationsusinga finite-element model, allowing the simulation of a real microstructure, based on a microscopicimage of the grainstructure. Various cubic and tetragonalmaterialsare studied. The computationalresults clearly show the influence of the specific microstructural properties on the effective elasticbehaviour.

1. Introduction

The classical way to model the mechanical response of a structure to a certain load is to

use continuum mechanics theory. This theory assumes that the constituent materials are

homogeneous, and in most cases isotropic, without explicitly considering the microstructure of

the materials. This approach is justified as long as the structure has characteristic dimensions

much larger than the microstructural features of the materials such as grain sizes. This will

indeed be the case for most engineering structures, such as steel bridges or concrete buildings.

However, the continuum approach breaks down if the characteristic dimensions of the structure

are of the same order of magnitude as the characteristic microstructural scales. Due to the

distinct trend of miniaturization in electronic component technology, we have now indeedapproached the point where this is the case. An example is the trend in development of the

multilayer ceramic capacitor (MLCC).

The MLCC, of which a schematic illustration is represented in figure 1, consists of stacks

of ceramic dielectric layers interspersed with thin metal electrodes. The ceramic materials are

commonly based on polycrystalline barium titanate (Hennings [1]). Typical thicknesses of

ceramic and metal layers in present-day MLCCs are 10 and 2 µm, respectively. The electric

capacity per volume of an MLCC is enhanced by reduction of dielectric layer thickness.

0965-0393/99/060909+20$30.00 © 1999 IOP Publishing Ltd 909



910 J M J den Toonder et al

Figure 1. Left: schematicrepresentation of a multilayer ceramic capacitor(MLCC).Right: ceramiclayers in an MLCC. The metal electrodes are plates containing holes, so in an intersection theyappear as interrupted lines.

Currently, there is a development to reduce the ceramic layer thicknesses to less than 3 µm

and internal electrodethicknessesto less than 1 µm, whereas thenumber of stacksmay increaseto up to 500 layers.

The consequence of this tendency of miniaturization is that the individual ceramic layers

can no longer be considered as continuum materials. Since thelayer thickness is reduced nearly

to grain sizes, the layers will consist of only a few grains in the out-of-plane direction. Figure 1

shows a practical example of this; in this case, the ceramic layer thickness even virtually

equals the average grain size. The structural integrity of these layers will be dominated by the

properties of the individual grains.

It is clear that the continuum approach will fail to provide an accurate description of

the mechanical behaviour of the MLCC and also of other micro-electronic components, and

the microstructure of the material must be accounted for explicitly. The relatively new field

of micromechanical modelling of materials, or, in short, micromechanics, does exactly this.

In micromechanics, materials are modelled on a microstructural scale that is larger than the

atomic scale, but smaller than the continuum level. In polycrystals, the microstructural features

considered act typically on the scale of the grains. The ultimate goal of micromechanics is

to find the relation between the microstructural properties of a material and its macroscopic

behaviour.

In the past few years there has been an increased interest in micromechanics. This has

originated from the demand to be able to design a material with certain favourable properties

by manipulating its microstructure, and it has been stimulated by the increased miniaturization

of components, as discussed above. Examples of recent micromechanical studies are the

modelling of brittle intergranular failure in microstructures (Grah et al [2]) and computation

of effective elastic constants of polycrystalline thin films (Mullen et al [3]).

The aim of the present paper is to make a contribution to the field of micromechanics by

studying the influence of the elastic properties of the separate grains on the effective elastic

behaviour of polycrystalline materials. This was also the focus of Mullen et al [3], but ourapproach has several new aspects. First, we couple analytical, experimental and computational

results. Second, our calculations are based on a real microstructure instead of an artificial

one. Finally, we also consider the tetragonal crystal structure in addition to the cubic crystal

structure. In particular, we consider barium titanate. This material has a cubic crystal structure

above its Curie temperature of about T c = 130 ◦C and a tetragonal structure below T c.

The plan of the paper is as follows. We begin by reviewing the existing theory that

links the microstructural properties to the macroscopic behaviour, which is valid only for



Single crystal elasticity and effective elastic behaviour 911

large volumes of material. This theory is subsequently illustrated with measurements on

various barium titanates over a range of temperatures. The last part of the paper is devoted to

micromechanical computational modelling of thin sheets of polycrystalline materials, as are

present in the MLCCs. The computational results clearly show the influence of the specific

microstructural properties on the effective elastic behaviour.

2. Macroscopic effective constants of large volumes of polycrystalline materials: theory

A sufficiently large volume of untextured polycrystalline ceramic material can be

macroscopically described as an isotropic elastic material that is characterized by a set of two

independent material parameters. However, microscopically, the behaviour of the individual

grains will in general be anisotropic. In this section, we will discuss the elastic behaviour

of individual cubic and tetragonal crystal structures. Both structures exhibit orthotropic

behaviour. Furthermore, we will give a short review of the theory that links the microscopic,

anisotropic properties to the macroscopic, isotropic elastic behaviour.

2.1. Linear isotropic elasticity

Elastic materialsare characterized by a directrelationbetween thelocal stressstateand thelocal

strain state. In the case of linear elasticity this relation can be represented by the generalized

formulation of Hooke’s law, which states that the stress is proportional to the strain

σ = 4C : E (1)

with σ the Cauchy stress tensor and

E = 1

2( ∇ u + ( ∇ u)T) (2)

the infinitesimal strain tensor, where u is the displacement in the material. Equation (1) is

commonly written in a matrix notation. Due to symmetry of the stress tensor and the strain

tensor only six state variables need to be evaluated. Thus, the stress–strain relation is written

as (Nye [4], Ting [5])

σ ∼

= Cε∼

(3)

with

σ ∼

T= [ σ 11 σ 22 σ 33 σ 23 σ 31 σ 12 ] (4)

and

ε∼

T= [ ε11 ε22 ε33 2ε23 2ε31 2ε12 ]. (5)

For general anisotropic elasticitythe elasticity matrixC willbe symmetricand therefore contain

21 independent components.

It is well known that the behaviour of isotropic linear elastic materials can be described

with a set of two independent parameters, for example Young’s modulus E and Poisson’s ratioν. In terms of these parameters, the elasticity matrix C for isotropic materials reads

C = E

(1 + ν)(1 − 2ν)

1 − ν ν ν

ν 1 − ν ν

ν ν 1 − ν12

(1 − 2ν)12

(1 − 2ν)12

(1 − 2ν)

. (6)




Alternatively, isotropic linear elastic behaviour can also be characterized with the shear

modulus G in combination with the bulk modulus κ . The sets of material parameters (E, ν)

and (G, κ) are related as

G =

E

2(1 + ν) κ =

E

3(1 − 2ν) . (7)

2.2. Crystal elasticity

Single-crystal elasticity will in general not be isotropic. The number of independent material

parameters depends on the level of symmetry of the crystal structure. For the cubic structure,

the elasticity matrix can be written in terms of three independent material parameters (Nye [4],

Ting [5]):

C =

c11 c12 c12

c12 c11 c12

c12 c12 c11

c44

c44c44

. (8)

If the relation 2c44 = c11 − c12 is satisfied, the material will be isotropic and the elasticity

matrix can be written in the form of equation (6).

Tetragonal crystal structures have lower symmetry properties. As a result, for a complete

description of the constitutive behaviour six independent parameters are needed:

C =

c11 c12 c13

c12 c11 c13

c13 c13 c33

c44

c44

c66

. (9)

In this study, we are particularly interested in barium titanate (BaTiO3

) at various

temperatures. At temperatures above approximately 130 ◦C, which is the Curie temperature

for this material, the barium titanate crystals have a cubic structure. At lower temperatures

the crystal structure is tetragonal. In addition to barium titanate, we consider two more

ceramic materials, namely indium (In) and zircon (ZrSiO4). In table 1 the various independent

components of the elasticity matrix are given for the materials investigated (Hellwege [6],

Berlincourt and Jaffe [7]).

2.3. Effective elastic constants

Generally a ceramic material will consist of a number of grains, each with a different and

unique orientation with respect to a reference frame. All individual grains will microscopically

show anisotropic material behaviour that is dependent on the crystal structure and orientation.

However, when the number of grains is sufficiently large and the orientations are randomlydistributed, the effective macroscopic behaviourwill be isotropic and the constitutive properties

can be characterized by an effective Young’s modulus and an effective Poisson’s ratio.

The macroscopic effective elastic constants are found by averaging the anisotropic elastic

properties of the individual crystals over all possible crystal orientations. To this end a rotation

tensor between a reference frame and a rotated vector basis attached to the crystal structure is

defined:

e∼

= R · e

∼ with R · R

T= I. (10)




Table 1. Elastic constants (GPa).

BaTiO3 BaTiO3 In ZrSiO4

cubic tetragonal tetragonal tetragonal

c11 173 275.1 45.2 424

c12 82 179.0 40.0 70

c44 108 54.3 6.52 113

c33 — 164.9 44.9 490

c13 — 151.6 41.2 150

c66 — 113.1 12.0 48.5

The components of the fourth-order elasticity tensor with respect to the rotated frame are

obtained by

4Cijkl =

4Cmnop RlpRkoRj nRim . (11)

The macroscopic effective elastic constants are obtained by averaging this tensor over all

possible rotation tensors (Hearmon [8]):

4Cijkl =

all possible R

4CmnopRlpRko Rjn Rim. (12)

This average is called the Voigt average and for orthotropic materials (see Cook and Young

[16]) it results in the following macroscopic effective elastic constants:

EV = (A − B + 3C)(A + 2B)

2A + 3B + CGV =

A − B + 3C

5 νV =

A + 4B − 2C

4A + 6B + 2C(13)

with

A = c11 + c22 + c33

3 B =

c23 + c13 + c12

3 C =

c44 + c55 + c66

3 . (14)

Another approach is to average the inverse of the elasticity tensor, i.e. the fourth-order

compliance tensor 4S, over all possible orientations

4S ijkl =

all possible R

4S mnopRlpRko Rj nRim (15)

which results in the so-called Reuss effective constants:

ER = 5

3A + 2B + C GR =

5

4A − 4B + 3C νR = −

2A + 8B − C

6A + 4B + 2C (16)

with

A=

s11 + s22 + s33

3 B

= s23 + s13 + s12

3 C

= s44 + s55 + s66

3 . (17)

The Voigt averaging method assumes the strains to be continuous, whereas the stresses are

allowed to be discontinuous. As a resultthe forcesbetween thegrains will notbe in equilibrium.

This method will give upper bounds for the actual effective elastic constants (E,G). When the

Reuss averaging method is applied, the stresses are assumed to be continuous and the strainscan be discontinuous. Consequently, the deformed grains do not fit together and lower bounds

for the effective elastic constants are found.

The Voigt–Reuss–Hill approach combines the upper and lower bounds by assuming the

average of the Voigt and the Reuss elastic constants to be a good approximation for the actual

macroscopic effective elastic constants:

EVRH = EV + ER

2 GVRH =

GV + GR

2 νVRH =

EVRH

2GVRH

− 1. (18)




Table 2. Macroscopic effective elastic constants (GPa), (—), according to the conventionalaveraging procedures.



EV 199.8 162.2 16.8 305.6

GV 83.0 59.8 5.87 119.4

νV 0.204 0.355 0.434 0.279

ER 173.2 130.0 10.6 256.9

GR 69.7 47.6 3.65 98.1

νR 0.243 0.367 0.456 0.310

EVRH 186.5 146.1 13.7 281.3

GVRH 76.3 53.7 4.76 108.8

νVRH 0.221 0.360 0.445 0.293

The macroscopic effective elastic constants of the materials that will be the subject of

investigation in section 4 are represented in table 2. The values are obtained by substitution of

the constants in table 1 into the foregoing expressions.

3. Macroscopic effective elastic constants of barium titanate: measurements

As an illustration of the theory summarized in the previous section, in this section we present

measurements of the macroscopic effective elastic constants of a number of barium titanates

as a function of temperature. The barium titanates are being used, or are considered for use, as

dielectrics in multilayer ceramic capacitors (MLCCs). Owing to the phase transformation

that barium titanate undergoes at the Curie temperature (T c = 130 ◦C), the relationship

between macroscopic effective behaviour and elastic properties of the individual grains is

clearly demonstrated within the temperature range of our measurements, which is 20–400 ◦C.

3.1. The experimental technique and set-up

We determined the elastic constants with the pulse excitation method. The experimental set-up

is sketched in figure 2. A disc-shaped barium titanate specimen is placed on four ceramic balls

on a sample holder. During a measurement, a graphite projectile hits the specimen, causing

it to vibrate with a certain natural frequency. As is well known, the value of the frequency

depends on the specimen geometry, the mass of the specimen, the elastic constants of the

material and the mode of vibration. The latter can be influenced by the way in which the

specimen is supported and by the exact position at which the projectile hits the specimen. In

our experiment, the frequency is captured via a waveguide by a microphone and the signal

is subsequently analysed by a commercially available apparatus (Grindosonic). The whole

set-up is placed in an oven, so that measurements can be made up to 400 ◦C.

We measured two natural frequencies of the specimens, namely that of the fundamental

flexural mode f f (one nodal circle) and that of the fundamental torsional mode f t (two nodaldiameters). Since we know the geometry, the mass and the vibrational modes, we can compute

the Young’s modulus and the Poisson’s ratio of the materials from the two resonant frequencies

measured. For the details, we refer to Glandus [9]. The measurements were conducted over

the temperature range 20–400 ◦C. The test specimens used were circular discs with diameters

ranging from 44.80 to 45.60 mm, and thicknesses between 2.04 and 2.14 mm. The materials

used were five barium titanates; all these materials are BaTiO3 based, each having specific

chemical substitutions (or ‘dopants’), the function of which is to tune the dielectric properties




Figure 2. The experimental set-up used formeasuring the elastic material constants bythe pulse excitation method.

of the material. The materials are used as dielectrics in multilayer ceramic capacitors. The

grain size of all barium titanates is approximately 1 µm.

3.2. Results

A typical plot of the resonant frequencies measured with the pulse-excitation method (torsionalf t and flexural f f ) is shown in figure 3.

We computed the elastic constants, as explained above, from the measured frequencies.

All results are summarized in figure 4, whichshows the effective macroscopic Young’s modulus

and Poisson’s ratio for all materials. It is clear that both quantities are strongly temperature

dependent. The most striking feature for all materials is the sudden change in behaviour for

Figure 3. A typical plot of the resonant frequenciesmeasured with the pulse-excitation method (torsionalf t and flexular f f ).




Figure 4. The temperature dependence of the macroscopic effective Young’s modulus E (left) andPoisson’s ratio ν (right) measured for various barium titanates (curves). The markers indicate thepresent theoretical estimates for pure polycrystalline barium titanate (table 2, VRH averages).

both quantities around the Curie temperature T c = 130 ◦C of barium titanate. It is obvious

from figure 4 that the transition in crystal structure which occurs at T c has a direct influence

on the elastic properties.

Figure 4 also shows that the elastic properties are virtually constant above the transition

temperature. Below the sharp transition, however, the properties still change with temperature.

The Young’s modulus, for example, increases with increasing temperature below 120 ◦C.

This is probably caused by the fact that the dopants present in the materials result in regions

with a cubic crystal structure even below 120 ◦C (i.e. the dopants smear out the global Curie

temperature towards lower temperatures). As a final observation from figure 4, we note that

the Poisson’s ratio shows some fluctuations; it appears that the value ν is sensitive to small

fluctuations in the resonant frequencies measured, which are caused by measurement noise.

The qualitative tendency we found for E is consistent with results reported by Duffy et al

[10] for another MLCC barium titanate.

3.3. Comparison with theory

The theoretical Voigt–Reuss–Hill estimates of table 2 for cubic and tetragonal barium titanate

are plotted in figure 4 along with the experimental results. The theoretical values are

qualitatively consistent with the measurements, i.e. they also exhibit a lower Young’s modulus

at temperatures below T c (tetragonal structure) than above T c (cubic structure). The theoretical

values for Poisson’s ratio match the measurements well. Hence, we may conclude that the

general behaviour of the measurements can be explained primarily by the change in the crystal

structure of individual grains as a function of temperature.

However, a quantitative comparison of the theoretical and experimental results showsthat there is some discrepancy, especially for the Young’s modulus at room temperature. The

explanation for this may be that the theoretical values concern pure barium titanate, whereas

the measurements were conducted with doped BaTiO3. This would also explain why in the

experiments the Young’s modulus increases even at low temperatures, i.e. below T c: the doped

polycrystalline barium titanates at these lower temperatures actually consist of a mixture of

tetragonal and cubic crystals, with the contribution of the cubic structures increasing as the

temperature goes up.




4. Effective elastic behaviour of small volumes of various polycrystals: simulations

In the previous sections we saw how the macroscopic effective elastic behaviour of large

volumes of polycrystalline materials can be theoretically related to the anisotropic elastic

properties of the individual grains. This relationship was also illustrated with measurementsof cubic and tetragonal barium titanate. However, when the volume of material is small, so that

it consists of a limited number of grains, the overall elastic behaviour may be highly dependent

on the specific elastic properties of the individual grains. This means that their influence on the

effective elastic properties of the material may not be averaged out, so that the theory discussed

in section 2 is not applicable.

Considering the trend towards miniaturization in many applications (e.g. microelectronic

devices), this situation already occurs in practice. For that reason, we present in this section

numerical simulations of the effective elastic behaviour of a limited aggregate of grains with

various (anisotropic) elastic properties. The results are related to the theory we discussed

above. The finite-element simulations we present here may be considered a first step towards a

more comprehensive modelling of the general relation between the microstructure of a material

and its overall effective mechanical properties.

4.1. Mesh generation

We used a finite-element model (FEM) to compute the elastic behaviour of a microstructure.

The first step in such modelling is the generation of a computational mesh. As the starting

point for the generation of the mesh we took a digitized image of the actual microstructure of

a polycrystalline material. This image is depicted in figure 5, and it represents a microscopic

enlargement of the grain structure of aluminium oxide (Al2O3). The image contains 56 grains

with a typical grain size of 60 µm and consists of 649 × 489 pixels with a grey-value in the

range 0–255.

The LEICA QWin image processing package [11] was used to process the image. Grain

boundaries were detected, and the image was converted to a binary image in which grain

boundaries were represented by zero-valued pixels.

In order to extract from this image a limited set of globally numbered points and curves

which can serve as an input for a mesh generator, a series of operations was performed in

MATLAB [12]. Figure 6 shows the result, which consists of a set of globally numbered points

representing the corners of the grains; a set of globally numbered curves representing parts

of grain boundaries, each of which is described by the numbers of the sequential points it

connects; and, finally, a set of surfaces, each of which represents a grain. The numbers and

directions of the curves describing the grain boundary are known for all surfaces.

Finally, a finite-element mesh was generated with use of the SEPRAN mesh generator

[13]. Because of the irregular shapes of most grains, the surfaces were meshed with triangular

elements. All points defined in the input for the mesh generator appeared as nodal points of

the finite-element mesh. The result was a complete mesh topology description of the total

grain structure. For each individual grain the set of elements was available and could be givenspecific anisotropic material properties. In figure 7 the finite-element mesh for the aluminium

oxide grain structure, consisting of 3996 elements, is represented.

4.2. Numerical procedure

We carried out various FEM calculations on the mesh in figure 7, using the FEM package

MARC [14]. Details of the element formulation are given in appendix B. We considered




Figure 5. Digitized microscopic image of the microstructure of Al2O3.

Figure 6. Input for mesh generator.

four cases, corresponding to the microstructure consisting of the four (anisotropic) materials

represented in table 1. In each calculation, all grains, represented by distinct element sets,

were given identical anisotropic properties. However, the orientation of the elastic tensor was




Figure 7. Finite-element mesh with 3996 triangular elements.

Figure 8. Boundary conditions used in the FEMcalculations.

varied randomly for all grains within one calculation. Thus, although all grains had identical

properties with respect to their own local coordinate system, for each grain the orientation of

the local coordinate system was randomly chosen with respect to the reference frame e∼

. The

interfaces between adjoining grains (the grain boundaries) were not explicitly modelled, but

were merely assumed to be borders of grains over which the orientation of the anisotropic

elastic properties changes.

From the outcome of the finite-element calculations, we determined the effective elastic

constants that would yield the same average normal stresses along the edges as observed in the

finite-element calculation if the material were considered to be isotropic. In order to obtain ameasure for the average stresses and strains, the boundary conditions (which are represented

in figure 8) were chosen such that the edges of the domain remained straight. At the upper,

lower and left-hand sides the displacements in the normal direction were suppressed, while

at the right-hand side the displacements in the normal direction were set to a certain positive

value. The average stress component σ 11 was found by averaging the stress component σ 11

along the left-hand edge. The σ 22 stress component was obtained by averaging the σ 22 value

along the lower edge of the domain.




For each of the four materials we considered, we performed a number of calculations (N )

with different sets of random orientations and calculated a set of effective material parameters

(EFEM, νFEM) for each calculation. Then, we computed the average effective elastic constants

of these N calculations as follows:

EFEM = 1

N

N i=1

EFEMi νFEM =

1

N

N i=1

νFEMi. (19)

If N is large enough, these quantities should converge to the macroscopic effective elastic

constants, which are measured for large volumes of polycrystalline materials. We put N = 30;

recall that we had 56 grains in our microstructure.

The numerical analyses were carried out both for plane strain and plane stress situations.

The thickness t was set at approximately the average grain size

t =

X · Y

n(20)

where X and Y are the dimensions of the total structure, which consists of n grains, in

respectively the e1 and e2 directions. A macroscopic strain of ε11 = 0.018 was imposed.

In a plane strain calculation the strain in the out-of-plane direction, ε33, isforced tobe zero.

Effectively, this implies that the structure extends infinitely in this direction. The boundary

conditions at the upper and lower sides ensure the average strain ε22 to be zero also, while a

certain positive average ε11 is imposed.

For isotropic materials the stress components σ 11 and σ 22 follow from equation (3):

σ 11 = E

(1 + ν)(1 − 2ν)((1 − ν)ε11 + νε22 + νε33)

σ 22 = E

(1 + ν)(1 − 2ν)(νε11 + (1 − ν)ε22 + νε33). (21)

After substitution of the plane strain condition and theboundary conditions, the effective elastic

constants that for an isotropic elastic material would yield a homogeneous stress state withσ 11 = σ 11 and σ 22 = σ 22 are found to be

EεFEM =

(σ 11 + 2σ 22)(σ 11 − σ 22)

ε11(σ 11 + σ 22)νε

FEM = σ 22

σ 11 + σ 22

. (22)

In plane stress calculations the stress in the out-of-plane direction, σ 33, is forcedto bezero,

which is a valid approximation for infinitely thin geometries. Again, the applied boundary

conditions ensure that the average strain ε22 is zero, while for ε11 a certain positive value is

imposed.

From the inverse of equation (3) it can be deduced that for linear elastic isotropic materials

the subsequent relations hold:

ε11 =

σ 11 − νσ 22 − νσ 33

E ε22 =

−νσ 11 + σ 22 − νσ 33

E . (23)

Aftersubstitution of the boundary conditionsand the planestress relation σ 33 = 0, the following

expressions for the effective elastic constants that would result in the observed average stress

and strain states can be found:

Eσ FEM =

σ 211 − σ 222

σ 11ε11

νσ FEM =

σ 22

σ 11

. (24)




Table 3. Plane strain FEM average effective elastic constants (GPa), (—).



Eε

FEM

186.6 162.8 15.7 279.9

std(EεFEM) 7.1 12.0 1.2 14.4

νεFEM 0.223 0.355 0.438 0.292

std(νεFEM) 0.012 0.009 0.005 0.012

0

50

100

150

200

250

300

E f f e c t i v e Y o u n g ’ s m o d u l u s [ G P a ]

VoigtReussVoigt−Reuss−HillFEM

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

E f f e c t i v e P o i s s o n ’ s r a t i o [ − ]


0

50

100

150

200

250

300



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

E f f e c t

i v e P o i s s o n ’ s r a t i o [ − ]


Figure 9. Plane strain effective constants: barium titanate, cubic (top); barium titanate,tetragonal (bottom).

4.3. Results

First we will discuss the results for the plain strain calculations. A total of 30 different sets of

56 random grain orientations were generated. For each previously described material, FEM

effective elastic constants were determined using equation (22) for each set of orientations andwere subsequently averaged with equation (19). In table 3 the resulting mean elastic constants

are summarized. The results are given in figures 9 and 10. In these figures each ‘◦’ marks

the FEM effective elastic constant resulting from one distinct set of orientations. The mean

and standard deviation of all 30 calculations are given for each material, as are the analytically

determined macroscopic effective elastic constants from section 2.

For the barium titanate material with a cubic structure and for zircon, the average FEM

effective constants virtually equal the Voigt–Reuss–Hill (VRH) value, which was assumed to




0

2

4

6

8

10

12

14

16

18

20

E f f e c t i v e Y o u n g ’ s m o d u l u

s [ G P a ]


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

E f f e c t i v e P o i s s o n ’ s r a t i o

[ − ]

VoigtReussVoigt−Reuss−Hill

FEM

0

50

100

150

200

250

300

350

400

E f f e c t i v e Y o u n g ’ s m o d u l u s

[ G P a ]


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

E f f e c t i v e P o i s s o n ’ s r a t i o [

− ]


Figure 10. Plane strain effective constants: indium (top); zircon (bottom).

Table 4. Plane stress FEM average effective elastic constants (GPa), (—).



Eσ FEM 174.0 145.1 11.8 263.8

std(Eσ FEM) 7.0 9.2 1.0 13.2

νσ FEM 0.221 0.322 0.351 0.320

std(νσ FEM) 0.025 0.030 0.056 0.030

be a good approximation for the macroscopic effective constants of a large polycrystalline

aggregate. However, both in the case of tetragonal barium titanate and that of indium, the

FEM average effective Young’s modulus substantially exceeds the VRH value. While for the

indium the results are well within the upper and lower bounds, for tetragonal barium titanate

the averages approximately equal the Voigt bound.The plane stress analyses were performed for the same sets of orientations and material

properties as for the plain strain calculations. Also, the same triangular mesh was used. In

table 4 the average plane stress effective constants are represented. The results are shown in

figures 11 and 12.

For cubic barium titanate, the mean effective Young’s modulus approximately equals the

lower bound, while the mean effective Poisson’s ratio is equal to the VRH value. The variation

in νσ FEM is considerably larger than for the plane strain situation. This applies for all four




0

50

100

150

200

250

300



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5



0

50

100

150

200

250

300


[ G P a ]


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


[ − ]


Figure 11. Plane stress effective constants: barium titanate, cubic (top); barium titanate,tetragonal (bottom).

materials. For the other three materials, the average νσ FEM is not within the range between

the upper and lower bounds. The average effective Young’s modulus for tetragonal barium

titanate matches the VRH value, while for the remaining two materials the average E σ FEM is

considerably lower, but within the Voigt and Reuss bounds.

4.4. Discussion

For the different calculations with distinct sets of orientations, the effective elastic behaviour

of the simulated microstructures shows substantial scatter around a mean value. The effective

properties are determined by the exact orientations, and hence the specific properties of

the microstructures determine the mechanical behaviour on the scale considered. The most

important message from the computations, therefore, is that for relatively small polycrystalline

aggregates the microstructure must be explicitly taken into account in mechanical studies, andthe use of effective isotropic elastic properties is not correct. This confirms the conclusions

drawn by Mullen et al [3], who carried out similar computations.

It was expected that the average of the effective elastic behaviour of many small

microstructures would lead to the theoretical average behaviour explained in section 2, i.e.

that the mean value of the scattered point in the previous figures would correspond to the

Voigt–Reuss–Hill average. For some of our results, this is clearly not the case. Several

possible explanations for this unexpected behaviour can be given.




0

2

4

6

8

10

12

14

16

18

20

E f f e c t i v e Y o u n g ’ s m o d u l u

s [ G P a ]


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


[ − ]


0

50

100

150

200

250

300

350

400


[ G P a ]


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

E f f e c t i v e P o i s s o n ’ s r a t i o [

− ]


Figure 12. Plane stress effective constants: indium (top); zircon (bottom).

First, as we show in appendix A, the tetragonal barium titanate and the zircon material

properties may show nearly incompressible material behaviour for certain stress states. For

plane strain calculations, this may result in an overestimation of the elastic stiffness due to

mesh-locking (Hughes [15,p 208]). This may lead to the observed deviations.

Furthermore, a plane strain calculation forces the local strains in the e3 direction to be

zero everywhere, while for the calculation of the effective constants only global restrictions

are required, i.e. we have

ε33 = 0 instead of ε33 = 0

ε23 = 0 instead of ε23 = 0 (25)

ε31 = 0 instead of ε31 = 0.

However, this may lead to an overestimation of the effective constants for polycrystalline

aggregates for all anisotropic materials.

Another possible explanation for the average effective Young’s modulus being relativelyhigh for some plane strain calculations andrelatively low for most plane stresscalculations may

be the two-dimensional nature of the analysis. The analytically derived approximations for the

macroscopic effective elastic constants are based on a three-dimensional configuration of many

randomly oriented grains. However, the numerical approximations for the effective elastic

constants are based on the analysis of a two-dimensional configuration of (three-dimensionally

oriented) grains. Furthermore, boundary effects may be present and may influence the observed

macroscopic behaviour.




Table 5. Influence of mesh refinement. Computed effective elastic constants for one set of orientations in the case of tetragonal barium titanate.

Number of

elements EεFEM (GPa) νε

FEM (—)

1 066 156.4 0.359

1 759 154.3 0.360

3 996 152.4 0.362

6 025 151.1 0.363

11 303 152.3 0.362

33 268 151.4 0.362

0

50

100

150

200

250

300

E f f e c t i v e Y o u n g ’ s m

o d u l u s [ G P a ]


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5



Figure 13. Planestrain effective constantswith quadrilateral elements: barium titanate, tetragonal.

Other errors might be influenced by the nature of the computational mesh and the element

formulation, which should always be considered when performing finite-element modelling.

Details of the element formulations used are given in appendix B.

The calculations discussed above were carried out with a mesh consisting of 3996 second-

order triangular elements. In order to investigate the effects of mesh dependence, we repeated

the plane strain analysis for one set of orientations with several mesh refinements in the case

of tetragonal barium titanate (see table 5). Further mesh refinement appears to have little

influence on the observed results.

For barium titanate the plane strain calculations were also repeated for all sets of grain

orientations with a quadrilateral mesh consisting of 3833 linear elements, since the second-

order triangular elements may be sensitive to errors due to element distortion. The results

are represented in figure 13. Compared to the triangular mesh (figure 9), deviations are small.

Application of alternative integration procedures as an assumed strainformulation or a constant

dilatation method [14], which may prevent potential element locking, did not have significant

influence on the observed effective behaviour either.It still remains unclear which of the above explanations apply. In the case of the plane

strain computations, the incompressible nature of the deviating materials seems to be the most

likely candidate. Mullen et al [3] appear to always find the Voigt–Reuss–Hill average for their

averaged effective FEM Young’s modulus. The explanation may be that all of their materials

have a cubic crystal structure and do not exhibit incompressibility. Furthermore, their analysis

was restricted to plane strain modelling. In our case as well, this is the combination that leads

to the analytical averages.




5. Conclusions

In this paper we examined the relation between the (anisotropic) properties of the individual

grains and the effective macroscopic elastic behaviour of polycrystalline materials. For large

volumes of material, the conventional analytical theory of averaging the crystal properties canbe used to establish this relationship. We illustrated this with measurements of barium titanates

at various temperatures.

However, for relatively small volumes of materials with dimensions of the order of the

grain size, such as thin polycrystalline sheets, this analytical theory is no longer appropriate.

Therefore, we carried out micromechanical computations using a finite-element model to study

such structures. Our numerical model allows the simulation of the elastic properties of a real

microstructure, based on a microscopic image of the grain structure.

For various cubic and tetragonal materials the effective elastic properties were determined.

The results indeed turned out to be dependent on the precise orientations of the individual

grains. Hence, our main conclusion is that for relatively small polycrystalline aggregates, the

microstructure must be explicitly taken into account in mechanical studies, and the use of

average isotropic elastic properties is not correct.The computational results we presented here may be considered a first step towards a more

comprehensive modelling of the general relation between the microstructure of a material and

its overall effective mechanical properties. Ongoing work is directed towards simulation of

fracture, and the modelling of internal stresses between the grains that appear during sintering

of the microstructure.

Acknowledgments

The pulse excitation set-up was designed by M H M Rongen and J P van den Brink

(Philips Research Laboratories). S Oostra (Philips Central Development Passive Components)

prepared the barium titanate specimens. Figure 1 was provided by H Nabben (Philips Central

Development Passive Components), and figure 5 by R Apetz (Philips Research Laboratories).

Appendix A. Orthotropic properties of the materials studied

For certain stress states, the tetragonal barium titanate and the zircon material may show nearly

incompressible behaviour. This can be demonstrated by their orthotropic Poisson’s ratios.

Table A1. Orthotropic properties (GPa), (—).



E11 120.1 124.2 6.7 375.9

E33

120.1 64.7 5.0 400.0

G23 108 54.3 6.52 113

G12 108 113.1 12.0 48.5

ν12 0.32 0.29 0.30 0.064

ν23 0.32 0.65 0.64 0.29

ν31 0.32 0.33 0.48 0.30

1 − ν12 − ν13 0.36 0.06 0.06 0.65

1 − ν31 − ν32 0.36 0.34 0.04 0.40




The cubic and tetragonal elasticity matrices are special cases of orthotropic material

behaviour since these structures contain three orthogonal directions of symmetry. For these

materials, the inverse relation of equation (3) can be written as (Cook and Young [16])

ε∼=

Sσ ∼ (26)with

S =

1

E11

−ν21

E22

−ν31

E33

−ν12

E11

1

E22

−ν32

E33

−ν13

E11

−ν23

E22

1

E331

G231

G311

G12

. (27)

When this notation is applied, the material can be characterized by the following set of

parameters

E11, E22, E33

ν12, ν21, ν13, ν31, ν23, ν32

G23, G31, G12

(28)

which, because of the symmetry condition for the matrix S , satisfy the relations

ν12E22 = ν21E11

ν13E33 = ν31E11 (29)

ν23E33 = ν32E22.

The volume strain of a material is defined as

εv = tr(E ). (30)

For a linear elastic orthotropic material the volume strain follows from equation (27):

εv = σ 11

E11

(1 − ν12 − ν13) + σ 22

E22

(1 − ν21 − ν23) + σ 33

E33

(1 − ν31 − ν32). (31)

When εv ≈ 0, the material behaviour will be nearly incompressible. This relation is satisfied

when

1 − ν12 − ν13 ≈ 0

1 − ν21 − ν23 ≈ 0

1 − ν31 − ν32 ≈ 0.

(32)

Nearly incompressible material behaviour may give rise to problems when applied in planestrain finite-element calculations.

The elastic properties of the cubic and tetragonal materials previously discussed (table 1)

can also be given in terms of the orthotropic elastic constants given in table A1. Moreover,

the expressions (32) are also evaluated. Since for cubic and tetragonal materials ν21 = ν12

and ν23 = ν13, only two expressions need to be evaluated. As can be observed, for certain

stress states, tetragonal barium titanate and indium may show nearly incompressible behaviour,

which can be troublesome in plane strain finite-element calculations, as we present in section 4.




Appendix B. Element formulation

We used the FEM package MARC to carry out the calculations. The following element

formulations were used. Details of the elements may be found in [14].

For the plane strain calculations, MARC element 125 was used. This is a second-orderisoparametric two-dimensional plane strain triangular element with nodes at the three corners

and the three midsides. A disadvantage of this element is that distortion during solution may

cause poor results. Therefore, as described in section 4.4, we also used element 11 for the plane

strain calculations. This is a linear four-node isoparametric plane strain quadrilateral element.

Although it is less sensitive to element distortions than element 125, shear behaviour may be

poorly represented. Theshear behaviour may be improved by using an alternative interpolation

function, such as the assumed strain procedure, which we also used. For nearly incompressible

behaviour, use of element 11 may lead to element locking, which can be eliminated with use

of the constant dilatation method, which we also applied to our problem (see section 4.4).

In the case of the plane stress calculations, we applied element 124, which is a quadratic

six-node plane stress triangle. Distortion of the element could lead to poor results, and a

quadrilateral element would be preferable in that case. Since we concentrate on the planestrain calculations in this paper, results with the quadrilateral plane stress element are not

reported here.

References

[1] HenningsD 1987Bariumtitanateceramic materialsfor dielectricuse Int. J. High Technology Ceramics 3 91–111

[2] Grah M, Alzebdeh K, ShengP Y, Vaudin M D, Bowman K J andOstoja-Starzewski M 1996 Brittle-intergranular

failure in 2D microstructures: experiments and computer simulations Acta Mater. 44 4002–18

[3] Mullen R L, Ballarini R and Heuer A H 1997 Monte Carlo simulation of effective elastic constants of

polycrystalline thin films Acta Mater. 45 2247–55

[4] Nye J F 1985 Physical Properties of Crystals (Oxford: Clarendon)

[5] Ting T C T 1996 Anisotropic Elasticity: Theory and Applications (Oxford: Oxford University Press)

[6] Hellwege K-H 1979 Numerical Data and Functional Relationships in Science and Technology

(Landolt–B¨ ornstein New Series, group III, vol 11) (Berlin: Springer)[7] Berlincourt D and Jaffe H 1958 Elastic and piezoelectric coefficients of single-crystal barium titanate Phys. Rev.

111 143–8

[8] Hearmon R F S 1969 The elastic constants of polycrystalline aggregates Physics of the Solid State

ed S Balakrishne, M Krishnamurthi and B R Rao (New York: Academic) pp 401–14

[9] Glandus J C 1981 Rupture fragile et resistance aux chocs thermiques de ceramiques a usages mechaniques PhD

Thesis University of Limoges, France

[10] Duffy W, Cheng B L, Gabbay M and Fantozzi G 1995 Anelastic behaviour of barium-titanate-based ceramic

materials Metall. Mater. Trans. A 26 1735–9

[11] 1996 LEICA QWin Reference Guide & User Guide Leica Imaging Systems Ltd, Cambridge, UK

[12] 1994 MATLAB The Language of Technical Computing: Computation, Visualization, Programming: Version 5

(Natick, MA: MathWorks)

[13] 1993 SEPRAN Users Manual (Delft: Ingenieursbureau SEPRA)

[14] 1994MARCAnalysis ResearchCorporationVolumeA: UserInformation, VolumeB: Element Library, VolumeC:

Program Input (Palo Alto, CA: MARC)

[15] Hughes T J R 1987 The Finite Element Method (Englewood Cliffs, NJ: Prentice-Hall)[16] Cook R D and Young W C 1985 Advanced Mechanics of Materials (London: Macmillan)

tetragonal symmetry.pdf

Documents

Transcript of tetragonal symmetry.pdf