# 物理代写|固体物理代写Solid-state physics代考|PHYSICS3544

## 物理代写|固体物理代写Solid-state physics代考|The constitutive equation

So far we have prepared the formal environment to describe force actions and corresponding deformations. The next point is to look for the mathematical relationship linking the strain and stress tensors, which is usually referred to as the constitutive equation of elasticity.

In this regard, it must be preliminarily observed that elasticity theory is unable to provide this relationship which, instead, must be assumed ‘a priori’ of the problem we aim at investigating. Accordingly, any result of continuum elasticity will specifically depend on the adopted constitutive equation. This is the level at which the atomistic theory plays a major role since it provides the needed fundamental knowledge. In fact, once assigned the most appropriate model for the lattice many-body potential energy $U=U(\mathbf{R})$ governing the ion displacements (see sections 1.3.4 and 3.1), the constitutive stress-strain relationship is there contained, even if not always immediately apparent.

Consistently with the hypothesis of small deformations, we guess that they are linearly dependent on their causing action ${ }^7$ and formally write
$$T_{i j}=\sum_{k h} C_{i j k h} \epsilon_{k h}$$

which can be simply looked at as the most general form of the Hooke law. By this linear elastic constitutive equation we introduce the fourth rank elastic tensor $C_{i j k h}$. In general, among its $3^4=81$ components only 21 are independent as determined by the symmetric character of the strain and stress tensor which imposes
$$C_{i j k h}=C_{j i k h} \quad C_{i j k h}=C_{i j h k}$$
Another symmetry is imposed by the guessed constitutive equation (5.18) which in fact represents the macroscopic counterpart of the harmonic crystal model developed in chapter 3. Therefore, by analogy with equation (3.4) we can surely state that there exists a formal dependence of the elastic energy density $u=u\left(\epsilon_{i j}\right)$ on the strain tensor which, within the adopted constitutive model, is cast in the harmonic form
$$u=\frac{1}{2} \sum_{i j k h} C_{i j k h} \epsilon_{i j} \epsilon_{k h}$$

## 物理代写|固体物理代写Solid-state physics代考|Elasticity of homogeneous and isotropic media

We are now going to investigate the elasticity of a homogeneous and isotropic solid, that is a system where (i) the elastic constants are just the same everywhere and (ii) its elastic response is just the same along any direction. More formally, homogeneity and isotropicity are found whenever the elastic tensor is invariant upon translations and rotations. Although these features define a somewhat idealised situation, this case study is paradigmatically important and leads to very general results which, in fact, can be widely applied in practice.

By choosing a frame of reference where the stress tensor is diagonal (see section 5.1.3) and considering a uniaxial traction along the $x_1$ axis, it is empirically found that the system response is twofold: (i) it stretches along the $x_1$ direction and (ii) it shrinks in the $\left(x_2, x_3\right)$ plane. Since the only non-zero stress component is $T_{11}^$, we can formalise the observed phenomenology by defining the following strain tensor $$\epsilon_{11}^=+\frac{1}{E} T_{11}^* \quad \epsilon_{22}^=-\frac{\nu}{E} T_{11}^ \quad \epsilon_{33}^=-\frac{\nu}{E} T_{11}^ \quad \epsilon_{12}^=\epsilon_{23}^=\epsilon_{31}^*=0,$$
where the two $E$ and $\nu$ constants are introduced in the proposed combination for further convenience. It is important to remark that just two constants are in fact needed to fully accomplish with this elastic problem since we must only describe the observed material stretching along $x_1$ and its corresponding shrinking in a normal plane. We summarise the physical situation by saying that a homogeneous and isotropic linear elastic medium has only two independent elastic moduli, namely $E$ and $\nu$ which are known as the Young modulus and the Poisson ratio, respectively. In table 5.2 we report their value for some elemental crystalline system.

## 物理代写|固体物理代写Solid-state physics代考|The constitutive equation

$$T_{i j}=\sum_{k h} C_{i j k h} \epsilon_{k h}$$

$$C_{i j k h}=C_{j i k h} \quad C_{i j k h}=C_{i j h k}$$

$$u=\frac{1}{2} \sum_{i j k h} C_{i j k h} \epsilon_{i j} \epsilon_{k h}$$

## 物理代写|固体物理代写Solid-state physics代考|Elasticity of homogeneous and isotropic media

(ii) 它的弹性响应在任何方向上都是相同的。更正式地说，只要弹性张量在平移和旋转时不变，就会发现同 质性和各向同性。尽管这些特征定义了一个有点理想化的情况，但这个案例研究在范式上很重要，并产生 了非常普遍的结果，事实上，可以在实践中广泛应用。

$$\epsilon_{11}^{\overline{=}}+\frac{1}{E} T_{11}^* \quad \epsilon_{22}^{=}-\frac{\nu}{E} T_{11} \epsilon_{33}^{\overline{=}}-\frac{\nu}{E} T_{11} \quad \epsilon_{12}^{=} \epsilon_{23}^{=} \epsilon_{31}^*=0$$

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