Appendix 3A: Transition macro ↔ micro
The description of the constitutive equation and the determination of the elastic constants are developed in this article in the model of the elastic continuum (macroscopic view). On the other hand, the description in the elastic lattice is used to describe a point defect (microscopic view).
In this section the transition between the two models according to the current state of knowledge will be illuminated. On the one hand, the transition contains a normalization factor, which is determined as follows:
[math]\displaystyle{ k = k_1 \cdot k_2 = {4\mu}^\prime c^2 \cdot \left( {2\mu}^\prime + {4\lambda}^\prime \right)\cdot e^* }[/math]
[math]\displaystyle{ [k] = m^2 \qquad \qquad e^* = N^2 s^2 }[/math]
On the other hand – and maybe more importantly – some approximations and assumptions are made that simplify the transition.
There are two parts to the transition: On the one hand, there is the well-studied transition from the continuum limit to the undisturbed elastic lattice ([math]\displaystyle{ k_1 }[/math]).
The second part deals with the transition to a point defect, i.e. to the elastic lattice with perturbation ([math]\displaystyle{ k_2 }[/math]).
Transition from the macroscopic model to the undisturbed lattice
The question arises as to how the microscopic potentials used in the linear lattice are related to the macroscopic elasticity constants.
This transition succeeds in the case of an undisturbed elastic lattice and leads to the Born-Huang condition, which allows to identify – depending on the lattice structure – the individual spring constants with macroscopic quantities.
The procedure is relatively lengthy and can be read e.g. in [1], [2]. However, for a lattice that is isotropic on average (e.g. averaged over several unit cells), the identification becomes easy and microscopic Lamé coefficients (marked with a tilde) can be defined as follows:
[math]\displaystyle{ \displaystyle \overset{\sim}{\mu}=\frac{\mu}{\rho_0} \qquad \qquad \overset{\sim}{\lambda}=\frac{\lambda}{\rho_0} }[/math]
Furthermore, in the isotropic elastic medium it holds that (see [3]):
[math]\displaystyle{ \displaystyle \rho_0=\frac{\mu}{c^2}=\frac{1}{4\mu^\prime c^2}=\frac{1}{k_1} }[/math]
And therefore the definition of the microscopic compression and longitudinal moduli becomes:
[math]\displaystyle{ \displaystyle \overset{\sim}{K}=\frac{K}{\rho_0} = k_1 K = 4\mu^\prime c^2 K }[/math]
[math]\displaystyle{ \displaystyle \overset{\sim}{M}=\frac{M}{\rho_0} = k_1 M = 4\mu^\prime c^2 M }[/math]
The microscopic quantities are marked with a tilde.
Transition from the macroscopic model to the defect in the lattice
Through the transition, the factor [math]\displaystyle{ k_2=\left( 2\mu^\prime + 4\lambda^\prime \right) }[/math] remains. This factor is determined from the normalization of the defect strength by considering a single, small, isotropic defect. We remember the constitutive equation, in the form with separated homogeneous and trace-free stress terms:
[math]\displaystyle{ \mathbf{ε}=2\mu^\prime \cdot \mathbf{σ_\perp} + \left( 2\mu^\prime + 4\lambda^\prime \right) \cdot \mathbf{σ_\parallel} }[/math]
[math]\displaystyle{ \displaystyle \mathbf{σ_\parallel} = \frac{1}{4}\cdot \mathrm{tr}\left(\mathbf{σ} \right) \cdot \mathbf{g} \qquad \qquad \mathrm{tr}\left(\mathbf{σ_\perp} \right) = 0 \qquad \qquad \mathbf{σ}=\mathbf{σ_\parallel} + \mathbf{σ_\perp} }[/math]
In the case of a small, isotropic defect, the trace-free part of the stress tensor is zero [math]\displaystyle{ (\mathbf{σ_\perp}=\mathbf{0}) }[/math], and the stress tensor becomes equivalent to its homogeneous part:
[math]\displaystyle{ \mathbf{σ}=\mathbf{σ_\parallel} \qquad \qquad \mathbf{ε}=\left( 2\mu^\prime + 4\lambda^\prime \right) \cdot \mathbf{σ} }[/math]
On the other side of the equation, the deformation tensor also assumes diagonal shape with the same eigenvalue [math]\displaystyle{ \varepsilon }[/math]:
[math]\displaystyle{ \varepsilon_{ij}= \delta_{ij} \cdot \varepsilon \cdot \delta(\mathbf{x_0}) }[/math]
On the other hand, the deformation tensor is defined as:
[math]\displaystyle{ \displaystyle \varepsilon_{ij}= \frac{1}{2}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) }[/math]
And used in the present case:
[math]\displaystyle{ \displaystyle \varepsilon_{ij}= \delta_{ij} \cdot \varepsilon \cdot \delta(\mathbf{x_0}) =\delta_{ij} \frac{\partial u_i}{\partial x_i} =\delta_{ij} \varepsilon \frac{\partial n_i}{\partial x_i} }[/math]
In the last expression using a normalized displacement field, so that [math]\displaystyle{ \varepsilon }[/math] is exactly the relative defect strength.
The expression for the force density is thus significantly simplified:
[math]\displaystyle{ \displaystyle \mathbf{F}=-\mathbf{\nabla σ}=\frac{\mathbf{\nabla ε}}{(2\mu^\prime + 4\lambda^\prime)} =\frac{-\varepsilon}{(2\mu^\prime + 4\lambda^\prime)} \delta_{ij} \frac{\partial}{\partial x_j} \left(\frac{\partial n_i}{\partial x_i}\right) =\frac{-\varepsilon}{(2\mu^\prime + 4\lambda^\prime)} \cdot \Delta \mathbf{n} =\frac{-\varepsilon}{k_2} \Delta \mathbf{n} }[/math]
On the other hand, for the force density of a small defect in the elastic lattice it applies (Chapter 2):
[math]\displaystyle{ \mathbf{F}_{Defekt,rel.}=-\xi \Delta_d \mathbf{u_\alpha} }[/math]
The normalization condition is that the defect strength remains the same in both descriptions:
[math]\displaystyle{ \varepsilon:=\xi }[/math]
This can be achieved by using the given conversion factor [math]\displaystyle{ k_2=\left(2\mu^\prime+4\lambda^\prime\right) }[/math] when changing from the macroscopic to the microscopic description.
Conclusion
The transition from the macroscopic to the microscopic model is shown according to the current state of knowledge of the author. The results may not yet be fully documented, but the following important points should be pointed out:
- Approximations: In the microscopic model small, isotropic defects are treated. This means a significant simplification of the macroscopic model, where the defect shape is arbitrary.
- Preliminary factor: The elasticity constants agree in the microscopic and macroscopic model up to a constant factor. Mathematically, this factor can be chosen freely in the microscopic model, since the solutions of the system of differential equations can only be determined up to a constant factor.
The treatment shown here should plausibilize by means of physical considerations that the constant factor is the term [math]\displaystyle{ k=k_1 \cdot k_2=4\mu^\prime c^2\ \cdot \left(2\mu^\prime+4\lambda^\prime\right) }[/math].
References
- ↑ Dynamical theory of crystal lattices, Max Born, Kun Huang, Oxford, Clarendon Press, 1954
- ↑ Theory of lattice dynamics in the harmonic approximation, A. A. Maradudin, E. W. Montroll, G. H. Weiss, I. P. Ipatova, New York & London: Academic Press, 1971.
- ↑ Kontinuumsmechanik, M. Sigrist, ETH Zürich, 2005 [1]
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