České vysoké učení technické v Praze Fakulta jaderná a fyzikálně inženýrská

Katedra matematiky

Mathematical Modelling of Mobility of

Martensitic Microstructures

Diplomová práce

Ondřej Glatz

Vedoucí práce: Ing. Hanuš Seiner, Ph.D. (ÚT AV ČR a KMAT FJFI)
Konzultant: Ing. Michal Landa, CSc. (ÚT AV ČR)
Akademický rok: 2009/2010

Prohlašuji, že jsem předloženou práci vypracoval samostatně a že jsem uvedl veškerou použitou literaturu.

V Praze dne 5. 1. 2010

Ondřej Glatz I would like to thank my supervisor Ing. Hanuš Seiner, PhD., and my consultant Ing. Michal Landa, CSc. for their willingness, patience, and support.

Contents

Contents  i
List of Symbols  v
Introduction  1
1 Introduction to Martensitic Phase Transition  3
1.1 Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Deformation, Deformation Gradient . . . . . . . . . . . . . . . . . . . 3
1.1.2 Local Properties of Deformation . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Kinematic Compatibility of Gradients . . . . . . . . . . . . . . . . . . 6
1.2 Continuum Mechanics and Crystal Structure . . . . . . . . . . . . . . . . . . 7
1.2.1 Crystal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Crystal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Cauchy–Born Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Reversible Martensitic Transition . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Mathematical Description of Transition . . . . . . . . . . . . . . . . . 11
1.3.2 Variants of Martensite . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Twins, Twinning Systems . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Martensitic Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Transformation Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Austenite–martensite Interface . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Martensitic Transition in CuAlNi . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5.1 Symmetries, Variants of Martensite . . . . . . . . . . . . . . . . . . . . 22
1.5.2 Twinning Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.3 Austenite–martensite Interface . . . . . . . . . . . . . . . . . . . . . . 25
2 Analysis and Modelling of Incompatible Microstructures  27
2.1 Analysis of the Lambda and X Microstructures . . . . . . . . . . . . . . . . . 27
2.1.1 The Lambda Microstructure . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.2 The X Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Modified Model of the Martensitic Microstructure . . . . . . . . . . . . . . . 38
2.2.1 Helmholtz Free Energy Estimation . . . . . . . . . . . . . . . . . . . . 38
2.2.2 Geometry Parametrisation and Optimisation . . . . . . . . . . . . . . 41
2.3 Mobility of the Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.1 Spontaneous Isothermal Motion of non-Equilibrium Microstructures and Rate-Dependent Dissipation 43
2.3.2 Mobility Modelling of the Lambda and X Microstructures . . . . . . . 45
3 Implementation and Results  49
3.1 Elastic Strain Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.1 Model Implementation in COMSOL Multiphysics . . . . . . . . . . . . 49
3.1.2 Mesh Settings, Convergence . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Modelling of the Stable Microstructures . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Optimisation Settings, Convergence . . . . . . . . . . . . . . . . . . . 51
3.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Modelling of the Microstructure Mobility . . . . . . . . . . . . . . . . . . . . 53
3.3.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.2 Microstructure Mobility Investigation . . . . . . . . . . . . . . . . . . 54
3.3.3 Microstructure Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 58
Conclusions  63
Bibliography  65
A Fotografie mikrostruktur  71
List of Figures  77

List of Tables  79

List of Symbols

Symbol  Description
a, b, c, α, β, γ  scalar values
u, v  vector values
A, B  matrices
ℝ, ℤ, ℕ  set of real, integral, and natural numbers
u·v, u × v, u ⊗ v standard dot product, cross product, and dyadic product of vectors u and v
I  identity matrix in ℝ^{3} (identity)
u_{k}, a_{ij}, A_{ij}  vector components and matrix elements
(a_{ij}), (A_{ij})  matrix with elements a_{ij} and A_{ij}
det A, rank A, tr A determinant, rank, and trace of matrix A
ker A  kernel of matrix A
A^{−1}, A^{T}  inverse and transpose of matrix A
A^{−T}, adj A  inverse transpose and adjoint matrix
∂Ω, Ω^{C}  boundary and complement of set Ω
\frac{∂y_{j}}{∂x_{i}}, ∂_{x_{i}}y_{j}  derivative of y_{j} with respect of x_{i}
∇y  gradient of function y (see definition 1.1.1)
L(e_{1}, e_{2}, e_{3})  crystal lattice generated by vectors e_{1}, e_{2}, and e_{3}
G(e_{i}), L(e_{i})  space group and Laue group of lattice L(e_{1}, e_{2}, e_{3})
L_{A}, L_{M}  point group of austenite and martensite
{A:B}  twinning system comprising gradients A and B
ε_{ij}, C_{ijkl}  small strain tensor (Cauchy’s strain tensor) and stiffness tensor

Introduction

Následující práce si klade za cíl aplikovat známou teorii fázových transformací v monokrystalech slitin s tvarovou pamětí na dva konkrétní vzorky těchto slitin vyšetřované v ÚT AV ČR. Úkolem je identifikovat jednotlivé fáze podílející se struktuře pozorované v daných vzorcích a předpovědět i další obtížně zjistitelné parametry této struktury.

Pro případ, že nebude klasická teorie použitelná, má být navrženo její zobecnění v jehož rámci již bude možno dané struktury vyšetřit. Výsledky této analýzy pak poslouží ke konstrukci modelu pohyblivých fázových rozhraní ve zkoumaných materiálech.

V úvodní kapitole tedy představíme zmíněnou klasickou teorii, ve velkém zde čerpáme z bakalářské práce zpracované právě na toto téma. V dalších kapitolách pak navazujeme samotnou analýzou.

Hlubší porozumění mechanismům fázové transformace ve slitinách s tvarovou pamětí najde jistě uplatnění v technické praxi, kde se již delší dobu tato třída inteligentních materiálů s úspěchem používá.

Chapter 1

Introduction to Martensitic Phase

Transition

This chapter summarises the mathematical description of the reversible martensitic transition theory developed particularly by Ball and James [1]. It extends some parts of author’s bachelor thesis [2].

1.1 Continuum Mechanics

Most of the theoretical framework used in this thesis is formulated in the terms of continuum mechanics. Thus it is necessary to introduce some basics used in the context of the martensitic transition.

1.1.1 Deformation, Deformation Gradient

The essential terms we are using are the deformation and the deformation gradient. Definition follows:

Definition 1.1.1. A homeomorphism y: x → y(x) on Ω ⊂ ℝ^{n} into ℝ^{n} is called deformation, if y_{i|G_{k}^{∘}} ∈ C^{(2)}(G_{k}^{∘}), where G_{k}^{∘} are open connected sets (regions) in ℝ^{n} and ⋃_{k=1}^{r} G_{k} = Ω, r ∈ ℕ.

PSfrag replacements x y y(x)

Figure 1.1: Deformation of a body [3]

Further we require that det F(x) > 0 ∀ x ∈ G_{k}^{∘}. The matrix F(x) = ∇y(x) is defined as

F(x) = ∇y(x) = (\begin{matrix}\frac{∂y_{1}}{∂x_{1}} & \frac{∂y_{1}}{∂x_{2}} & … & \frac{∂y_{1}}{∂x_{n}} \\ \frac{∂y_{2}}{∂x_{1}} & \frac{∂y_{2}}{∂x_{2}} & … & \frac{∂y_{2}}{∂x_{n}} \\ ⋮ & ⋮ & ... & ⋮ \\ \frac{∂y_{n}}{∂x_{1}} & \frac{∂y_{n}}{∂x_{2}} & … & \frac{∂y_{n}}{∂x_{n}} \end{matrix} )  for short: F_{ij} = \frac{∂y_{i}}{∂x_{j}} .  (1.1)

The matrix F(x) is called the deformation gradient.

Let Ω be a subset of ℝ^{n} representing a body before the deformation. Then, if x ∈ Ω is a point of the body in reference configuration, the y(x) is the corresponding point of the deformed body (see Fig. 1.1). Instead of the deformation y the displacement field

u(x) = y(x) − x  (1.2)

is often used.

1.1.2 Local Properties of Deformation

If deformation gradient F(x) is continuous in a point x, it characterises the deformation on the infinitesimal neighbourhood of this point, and

dy = F(x) dx  (1.3)

holds.

Since the value of det F(x) is equal to the Jacobian of the deformation, it describes a volume change of the neighbourhood of the point x. We can write

dṼ = (det F) dV ,  (1.4)

where dṼ stands for an infinitesimal volume of the deformed body, and dV denotes the respective volume in the reference configuration (for clarity the arguments are omitted, but keep in mind that in general all used quantities are functions of position in the reference configuration).

Thereafter, the text is restricted to deformations in ℝ^{3}. For the volume of an infinitesimal parallelepiped

dṼ = (dy_{1} × dy_{2})·dy_{3}  (1.5)

(here, vectors dy_{i} correspond to edges of given volume) we can derive formula (1.4) using (1.3).

An infinitesimal parallelogram of area dS̃, with normal ñ can be written as

dS̃ = ñ dS̃ = dy_{1} × dy_{2} .  (1.6)

Analogously to (1.4), we obtain the formula for transformation of surface elements by deformation y:

dS̃ = (det F)F^{−T}dS .  (1.7)

The deformation is called homogeneous, if the deformation gradient is independent on x (i.e. F(x) is constant on body Ω). Then, the mapping y is an affine transformation:

y(x) = Fx + x_{0}  F ∈ ℝ^{n,n}, x_{0} ∈ ℝ^{n} .  (1.8)

Let the point x be a point of continuity of the deformation gradient. Then, we can consider the deformation to be homogeneous on the infinitesimal neighbourhood of x. As mentioned above, such deformation is an affine transform of the neighbourhood of the point x. Every affine transform can be decomposed into pure stretch, a following rotation, and a final translation by vector x_{0} from (1.8). This unique decomposition is obtained by application of theorem 1.1.1 to the deformation gradient F.

Theorem 1.1.1 (Polar decomposition). Let F be a regular, real matrix; det F > 0. Then, there exist a unique symmetric positive definite matrix U and an orthogonal matrix R, det R = 1 (i.e. R is a proper rotation), such that

F = RU ,  (1.9)

where U = \sqrt{F^{T}F} and R = FU^{−1}.

The matrix U, often called the Bain matrix (tensor), represents just metric properties of the deformation on the neighbourhood of x. Rotation R and vector x_{0} (in (1.8)) carry the information on the transformation of the neighbourhood as a whole.

PSfrag replacements n m

y = Fx + c y = Gx + c

Figure 1.2: Compatibility of gradients [3]

1.1.3 Kinematic Compatibility of Gradients

Consider a deformation y defined on region Ω which is homogeneous on regions G ⊂ Ω and G^{C} = Ω \ \overline{G}, where ∂G is a part of a sufficiently smooth hypersurface. Let us examine the relation between the gradients F and G corresponding to the deformations y_{|G} and y_{|G^{C}} respectively.

Since the deformation is continuous, the surface ∂G has to transform equally “from both sides” (see Fig. 1.2). Let z ∈ ∂G, n be a normal to ∂G in the point z. If T is a tangent plane to ∂G at the point z (hence n is a normal of T ), for all vectors t from the tangent space of T the relation Ft = Gt has to hold. This can be expressed by the Hadamard jump condition

∃ a ∈ ℝ^{3} : F − G = a ⊗ n ,  (1.10)

where ⊗ denotes dyadic product defined below.

Definition 1.1.2. Binary operation ⊗ : ℝ^{n}× ℝ^{n} → ℝ^{n,n} defined by formula

u ⊗ v = uv^{T} = (\begin{matrix}u_{1}v_{1} & u_{1}v_{2} & … & u_{1}v_{n} \\ u_{2}v_{1} & u_{2}v_{2} & … & u_{2}v_{n} \\ ⋮ & ⋮ & ... & ⋮ \\ u_{n}v_{1} & u_{n}v_{2} & … & u_{n}v_{n} \end{matrix})  ∀ u, v ∈ ℝ^{n} ,  (1.11)

is called the dyadic product of vectors u and v.

One can see that

rank (a ⊗ b) ≤ 1 .  (1.12)

Therefore, we can write the Hadamard condition equivalently as

rank(F − G) ≤ 1 .  (1.13)

According to the findings above, ker(F − G) is a tangent space of ∂G at every point in ∂G (supposing F ≠ G). Since F and G are constant, the tangent space remain same at all points in ∂G. Hence ∂G has to be a part of plane.

If constant matrices F and G are prescribed, a deformation y homogeneous on some regions G, G^{C} ∈ Ω with gradients F and G exists if and only if the Hadamard condition (1.13) or (1.10) is fulfilled for some vectors a, n ∈ ℝ^{3}. Then, the surface, which separates G and G^{C}, is a plane with normal n (after the deformation, the normal changes, using (1.7), to (det F)F^{−T}n). Gradients F and G satisfying the Hadamard condition are called compatible.

We need to extend the idea of compatibility to symmetric positive definite matrices U_{a} and U_{b}, which come from the polar decomposition (see 1.1.1) of the corresponding deformation gradients. The question is if there exist rotations R_{1} and R_{2} such that matrices F = R_{1}U_{a} and G = R_{2}U_{b} are compatible gradients. When

R_{1}U_{a} − R_{2}U_{b} = a ⊗ n  (1.14)

holds for some rotations, we call the matrices U_{a} and U_{b} compatible.

This redefinition of the compatibility for symmetric positive definite matrices is more general. Thus, there is no simple criterion (like the Hadamard jump condition) for this new compatibility. Some more complex criteria are mentioned in section 1.3.3.

1.2 Continuum Mechanics and Crystal Structure

Since this work deals with monocrystalline material, we need to introduce basic properties of crystalline solids into our description in the terms of the continuum mechanics. Especially, the effects of a crystal lattice symmetries are important.

1.2.1 Crystal Lattice

For simplicity, in this section we are considering an ideal crystal, which is of infinite size and has a periodic structure without defects. The characteristic property of such ideal crystal is a repetition of some basic motif in the directions of crystallographic axes. The motif can be asymmetric comprising number of atoms.

PSfrag replacements e_{1}

e_{2}
f_{1}
f_{2}

Figure 1.3: Crystal lattice and lattice vectors [3]

Consider three linearly independent vectors (e_{1}, e_{2}, e_{3}) defining the elementary translations in the crystallographic axes directions. Then the set

L(e_{1}, e_{2}, e_{3}) = {μ_{1}e_{1} + μ_{2}e_{2} + μ_{3}e_{3} | μ_{i} ∈ ℤ} .  (1.15)

form a three-dimensional grid. By placing a selected motif at the all grid points the ideal crystal is obtained. This grid is called the crystal lattice. Equivalently: The crystal lattice is a set of points, which have identical and identically oriented neighbourhood. The generating vectors are called lattice vectors or primitive basis.

The lattice direction is understood as a direction defined by two distinct lattice points. Likewise, the lattice plane is understood as a plane defined by three affine independent lattice points.

1.2.2 Crystal Symmetries

It is clear that two different triplets of vectors can generate identical lattices. See example in Fig. 1.3 (in 2D), where the same lattice is generated by vectors (e_{1}, e_{2}) as well as by vectors (f_{1}, f_{2}). Let us explore the relation between such triplets. The following definition comes in useful for this.

Definition 1.2.1 (primitive unit cell). The primitive unit cell of a crystal lattice is a parallelopiped defined by respective lattice vectors.

Let (e_{1}, e_{2}, e_{3}) and (f_{1}, f_{2}, f_{3}) be triplets such that

L(e_{1}, e_{2}, e_{3}) = L(f_{1}, f_{2}, f_{3}) .  (1.16)

Then each vector in the first triplet must be a integral linear combination of vectors from the second triplet:

e_{i} = μ_{ij}f_{j}  μ_{ij} ∈ ℤ  (1.17)

This ensures that the set L(e_{1}, e_{2}, e_{3}) is a subset of L(f_{1}, f_{2}, f_{3}). To achieve equality in (1.16) one more condition has to be added. The necessary and sufficient condition for the equality in (1.16) is (1.17) along with

|det(μ_{ij})| = 1 ,  (1.18)

where (μ_{ij}) denotes a matrix in ℤ^{3,3} with coefficients μ_{ij}. This tells us that primitive unit cells of both lattices have the same volume. Obviously, the matrices (μ_{ij}) form a multiplicative group. This group is denoted S.

Another useful groups are the symmetry groups of the lattice. First, we must define what is understood by the lattice symmetry.

Definition 1.2.2 (lattice symmetry). Lattice symmetry is an affine map on ℝ^{3} into ℝ^{3}, which maps the lattice back to itself.

For example, the elementary translations by lattice vectors are possible symmetries for all lattices. But it is essential to distinguish between the lattice symmetries and symmetries of atoms spatial arrangement. Because of possible asymmetry of the motif, the atoms spatial arrangement adopts, in general, only the translational lattice symmetries.

From the text above, it follows that a regular linear map H is a symmetry of lattice L(e_{1}, e_{2}, e_{3}) if and only if

∃ (μ_{ij}) ∈ S : He_{i} = μ_{ij}e_{j} .  (1.19)

The set of all lattice symmetries as well as the set of all regular linear maps forms a group. Thus we can define the group of all linear lattice symmetries (or the group of matrices representing these symmetries):

G(e_{i}) = G(e_{1}, e_{2}, e_{3}) = {H ∈ ℝ^{3,3} | ∃ (μ_{ij}) ∈ S : He_{i} = μ_{ij}e_{j}} .  (1.20)

A more important group than the space group G(e_{i}) is its finite subgroup containing only proper rotations:

L(e_{i}) = L(e_{1}, e_{2}, e_{3}) = {Q ∈ G(e_{i}) | Q ∈ SO(3)} .  (1.21)

This group is called the Laue group of crystal lattice. One can show that there exist only 11 distinct Laue groups. These are further divided into 7 known crystal systems (for details see [4]).

Since it is possible that no primitive unit cell has the same symmetries as a whole lattice, it is useful to define a conventional unit cell. The conventional unit cell respects lattice