Basic Equations

i j≡∂A_i

∂rj and

∂²B

∂r∂s

i jkl≡ ∂²B_{i j}

∂rk∂sl.

The superscript^T indicates the transpose matrix and⁻¹ its inverse. The identity matrix is written as 1, and its components asδi j.

Vectors and tensors in Fourier space are denoted by a tilde on top of the symbol, e.g. ˜u_i.

3.2 Basic Equations

This section will present the basic relations of thermoelasticity in an outline form. The summary of the following subsection is based upon the following books on elasticity [Lan91,Wal70,Wal72, Wal02,Lov72,PG00]. More detailed information can be found therein.

3.2.1 Strains, Stresses and Elastic Constants

Assume a system in an initial equilibrium state where the particles are located at positions R.

Under the influence of an applied stress the initial configuration changes to the final position r.

For each particle an instantaneousdisplacement vectoru(R,t)is defined having the components ui(R,t) =ri(t)−Ri. (3.1) Forhomogeneous strain,i.e.constant strain throughout the material, the transformation fromRto ris linear and can be written as

ri(R) =αi jRj, (3.2)

where the elements of the transformation gradient tensor αi j ≡∂ri/∂Rj are constants. Equiva-lently the transformation can be characterized by the displacement gradientsu_{i j}≡∂u_i/∂R_j, which are related to the transformation gradients by

α_{i j}=δ_{i j}+u_{i j}. (3.3)

Thus for homogeneous strain the u_{i j} are also constants. The volumeV(r) after deformation is related to the original reference volumeV(R)by

det[αi j] = V(r)

V(R). (3.4)

Elastic theory describes the deformation of any configuration from a reference configuration in terms of a strain tensor. Pairs of particles in the initial (reference) configuration and in the final configuration are separated by∆Rand∆rrespectively. For constant strainsεi jover the region∆R, the relative positions of all particles in configuration{r}are completely specified by the reference configuration{R}and the strainsε_{i j}

|∆r|²− |∆R|²=2ε_{i j}∆R_i∆R_j. (3.5) Here the so-calledLagrangian strain tensor¹at reference positionRis defined as

(ε)_{i j}≡ε_{i j}=1

This tensor obviously is symmetric (ε_ij=ε_ji). Its’ significance is based on the fact that it is not limited to small deformations. Even for a finite strain it completely determines the deformation of the body. But in most cases the deformationsu_i are small and the non-linear term which is of second order, can be neglected:

In this work, only the case of small deformations is considered. So it will be totally safe to work with the linearized Lagrangian strain tensor (3.7). Inserting (3.5) in (3.6) it follows that the Lagrangian strains are related to the transformation gradients by the relationship

ε_{i j}=1

2 α_kiα_{k j}−δ_{i j}

. (3.8)

In contrast to a characterization of the deformation in terms ofα_{i j} oru_{i j}, the Lagrangian strains contain no information about the rotations of the solid body.

Inversion of the relation (3.8) results in the power series αi j=δi j+εi j−1

2εkiεk j+... (3.9)

From Eq. (3.4) the volume ratio of the body in the starting (V0) and the final configuration (V) is V

V₀ =det[α_{i j}] =1+ε_ii+... (3.10)

σ

11

σ

12

σ

13

y z

x

σ

σ σ

σ

31

σ

33 32

23 22

σ

21

Figure 3.1: Components of the Cauchy stress tensorσ_{i j}acting on a body element.

which gives the connection between the traceε_iiof the strain tensor and the relative volume change.

On deformation of an elastic body the particles are moved from their equilibrium position. The internal forces which are calledstressestry to bring the body back to its equilibrium situation. The forces f_i acting on the surface of a crystal can be written as divergences of the applied Cauchy stress tensorσ_ik[Lan91,Cha00],

f_i=∂σ_ik

∂R_k. (3.11)

Positive stresses are directed outward from the crystal surfaces as depicted in figure3.1. In equi-librium the forces due to internal strains have to compensate each other in every partial volume of the deformed body,i.e. f_i=0. So the equilibriumcondition for mechanical stabilityis

∂σ_ik

∂Rk =0. (3.12)

The configurational dependency of the state functions is given by the relative positions of all particles only. Thus for essentially constant strains within the range of the interaction the internal energyU and the free energyF depend on the actual configuration{r}only through{R}andε according to equation (3.5): U=U({r},S) =U({R},ε,S) andF =F({r},T) =F({R},ε,T).

The thermodynamics of a deformation of a solid body can be described by the differentials of the state functions²

dU=dU({R},ε,S) =T dS+δW=T dS+V0τ_ik^Sdε_ik (3.13) or equivalently by the Helmholtz free energy differential

dF=dF({r},T) =dF({R},ε,T) =−SdT+V₀τ_ik^Tdε_ik, (3.14) sinceF=U−TSwhich is the result of the Legendre transformation fromStoT . The variableV₀ denotes the volume of the (non-deformed) reference system,Sis the entropy, andδW=−V₀τ_ikdε_ik is the work due to the change of the strain tensor. τ_ik^S and τ_ik^T are the adiabatic or isothermal

1The components of the strain tensor are functions of the coordinates and thus not independent of each other, because the six different componentsε_{i j}are expressed by derivatives of just three independent functions, the components of the displacement vectoru(cf. sub-section (3.2.4)).

2For hydrostatic compression the stress tensor isτ_{i j}=−pδ_{i j}, wherepis the pressure normal to the body interfaces.

Soτ_ikdu_ik=−pdu_ii=−pdVand thus equation (3.13) reduces to the usual formdU=T dS−pdV.

stress tensor respectively. It is also assumed that at every instant of the deformation process the interior of the body is in equilibrium according to the external conditions. From equation (3.14) we therefore get for the (isothermal)thermodynamic stress tensor(tension) evaluated at the initial configuration{R}

where the derivative is evaluated at constant temperature keeping all other strain components ε_kl6=εi j fixed.^3,4Analogous relations also hold for the adiabatic situation which are not explicitly written down.

For small homogeneous deformations, as it is certainly the case for particles oscillating about the initial equilibrium configuration{R}, the Helmholtz free energy can be expanded according to

F({R},ε,T) =F({R},0,T) +V₀τ_{i j}^Tε_{i j}+1

2V₀C_{i jkl}^T ε_{i j}ε_kl+O(ε³) (3.16) Here the fourth rank tensorC_{i jkl}^T is thetensor of (isothermal) elastic constants.

The isothermal or adiabatic elastic constants can also be obtained from the second derivative of the strain energyW which refers either to the free energy or to the internal energy

C_{i jkl} = ∂τ_{i j} This scheme can be extended to define higher order elastic constants like

D_{i jklmn}= 1

Due to the symmetry of the strain tensor and due to the permutability of partial derivatives, the elasticity tensor is symmetric under interchange of the first two indices, the last two indices and the first and last index pair,i.e.

C_ijkl=C_jikl, C_ijkl=C_ijlk and C_ijkl=C_klij. (3.19) Instead of maximal 81 independent values, there exists a maximum of 21 independent elastic constants. This number can be drastically reduced taking into account the point group symmetry of the crystal equilibrium configuration (cf. Section3.3). In general, the number of independent elastic constants decreases as the point group symmetry of the crystal increases. Cubic crystals have three independent elastic constants. Hexagonally ordered two-dimensional crystalline solids and isotropic solids have two independent elastic constants.

3Note that the expressionτ_{i j}^T=1/V₀ ∂F/∂ε_{i j}

Thas to be understood asdF=τ_{i j}du_{i j}, where all elements withi6=j of the symmetric tensordu_{i j}occur twice. Thus the differentiation of the free energyFreturns values forτ_{i j}which have a value twice as high fori6= jcompared to the evaluation from expressionτ_{i j}^T=C_{i jkl}^T ε_kl. [Lan91]

4When the stress components are evaluated at the final configuration{r}starting from an arbitrary initial configuration {R}, then [Wal70]

Because of the symmetry requirements of equation (3.19) there exist only six independent pairs of subscriptsij. These can be numbered from 1 to 6 according to the following scheme, which was first introduced by Voigt [Voi10]:

i j = 11 22 33 32 or 23 31 or 13 21 or 12

α = 1 2 3 4 5 6 (3.20)

The elastic constants inVoigt’s notationtake the formC_αβ ≡C_{i jkl}whereα andβ take the values 1 to 6. From the third equality of equation (3.19) we haveC_αβ =C_βα.

If the strain tensor depends on the locationR, the free energy is a functional of the strain tensor F[ε(R)] = 1

V₀ Z

dRF({R},ε(R),T). (3.21)

Further important thermodynamic functions, useful in this context are the pressureP, the (isother-mal)bulk modulus B^T, and thecompressibilityκ^T being defined as [Wal72]

P = −

The two constantsB^T andκ^T are a measure of the substances’ resistance to volume changes.

3.2.2 Fourier Space

Sometimes it is helpful to work in Fourier space instead of the position space, because in Fourier space the differential equations become algebraic equations. The atomic displacements for aN particle system transform as and the Lagrangian strains in Fourier space read⁵

ε˜_{i j}(k) =i

2[k_iu˜_j(k) +k_ju˜_i(k)]. (3.25)

5Fourier transform of the strain tensor written explicitly:

ε˜_{i j}(k) =

In the third step the first and the third summand vanish, because the displacement vector is zero on the box edges.

In general, spatial partial derivatives in position space ∂/∂Ri change to factors of ki in Fourier space. The equilibrium condition for mechanical stability of equation (3.12) is

k_jσ˜_{i j}=0. (3.26)

So the elastic free energy∆F per particle for an elastically homogeneous and isothermal system of constant particle numberNin absence of external stress (σ=0) transforms as:⁶

∆F = 1

For small deformations the stresses depend linearly on the strains [Wal70,Wal72]

τ_{i j}=B_{i jkl}ε_kl, (3.30)

with thestress-strain coefficients(δ_ikis the Kronecker delta symbol) B_{i jkl}=1

2 σ_ilδ_jk+σ_jlδ_ik+σ_ikδ_jl+σ_jkδ_il−2σ_{i j}δ_kl

+C_{i jkl}. (3.31)

This is the generalized version of the well knownHooke’s law. The stressesσ in equation (3.31) are those existing prior to the imposition of the strain,i.e.the stresses in the initial state. For sys-tems under no external stress (σ=0) the stress-strain coefficients reduce to the elastic constants,

6Side calculation to get from equation (3.28) to (3.29):

∆F = 1

In the last step the symmetry properties of the tensor of elasticity are exploited and use is made of the possibility that indices occurring twice can be renamed arbitrarily. The following representation of theδ-function is used:

δ_k,k⁰= 1 N

∑

N

α=1e^i(k−k0)·r^α.

i.e. B_{i jkl}=C_{i jkl}. The adiabatic and isothermal stress-strain coefficients, which sometimes are also calledBirch moduliobey the relations

B^S_{i jkl}= ∂τi j

∂ε_kl

S and B^T_{i jkl}=

∂τi j

∂ε_kl

T. (3.32)

The superscriptsS andT will be omitted in the following, because all relations hold for the adi-abatic and for the isothermal case. From the expression (3.31) it is clear that the stress-strain coefficients B_{i jkl} are symmetric in the first and the in last pair of indices. So, they can also be written in Voigt notation, but do not satisfy all Voigt symmetry relations, becauseB_αβ 6=B_βα. Thecompliance tensor S_αβ is defined as the inverse of the elastic tensorB_αβ,i.e.

S_αβB_βγ=δαγ. (3.33)

By multiplication of Hooke’s law byS_αβ one obtains in Voigt notation

S_γατ_α ^(3.30)= S_γαB_αβε_β^(3.33)= δ_γβε_β =ε_γ (3.34)

⇒S_αβ = ∂εα

∂τ_β, (3.35)

which obviously is the inverse of (3.32). Equation (3.34) is analogous to the paramagnetic case, where a fieldH induces a magnetizationM=ξHand the compliance tensorS plays the role of the magnetic susceptibilityξ =∂M/∂H.

For isotropic pressure Pthe stress tensor is diagonal and can be written as⁷ τ_{i j} =−Pδ_{i j}. This reduces the relation (3.31) to

B_{i jkl}=−P δ_jlδ_ik+δ_ilδ_jk−δ_{i j}δ_kl

+C_{i jkl}. (3.36)

3.2.4 Compatibility Relations for the Strains

The components of the Lagrangian strain tensor(ε)_{i j}≡εi j are defined by equation (3.7) as partial derivatives of the displacement field with respect to some given reference lattice. Different strain tensor components cannot be arbitrary functions of the coordinates x, y, and z. They obey the followingcompatibility relations

O×(O×ε)^T =0, (3.37)

which were first derived by Saint Venant in 1890. These relations express the physical fact, that all material of a body before and after the deformation is continuous and connected. Inside of the body there do not occur dislocations or penetrations of body material due to deformation.

The compatibility relations can be obtained as mathematical identities after execution of the

sec-7The sign difference between the stress tensor components and the pressure is due to the sign convention: Positive stress on a body is exerted outward whereas positive pressure acts in the opposite direction.

ond derivatives of the equation (3.7) being interpreted as a kinematic equation.⁸ They read in full index notation

∂²ε_{i j}

∂Rk∂Rl+ ∂²ε_kl

∂Ri∂Rj− ∂²ε_ik

∂Rj∂Rl − ∂²ε_jl

∂Ri∂Rk =0. (3.38)

In three dimensions the expression (3.38) represents altogether 81 equations which contain 6 inde-pendent compatibility relations. All other equations are fulfilled identically or are just repetitions of one of the compatibility relations. For simply connected regions the compatibility relations are sufficient and necessary for the existence of a unique displacement field. Rigorous proofs of these relations are given in [Lov72,Sok56].

In Fourier space they read in vector respectively in full index notation

k×ε˜×k = 0 (3.39)

∋ mpi ∋

n jlkpε˜i jkl = 0 (3.40)

where the Levi-Civita symbol⁹ ∋

i jkis used.

Im Dokument Computer-Simulationen zu Strukturen und Phasenumwandlungen in Modell-Kolloiden (Seite 27-34)