# How are the thermal expansion of a solid and the stress tensor related?

• A
My idea is this: tensor stress is directly related to the internal pressure of a solid. That is to the force that the neighboring atoms exert each other in relation to a unit of surface.

When I heat a solid we can have the phenomenon of thermal expansion: this is connected to the fact that a deformation of the solid takes place which leads to an increase in the interatomic distances and therefore to a decrease in the frequencies of the phonons and a lowering of the energy of the system. For this deformation occurs. Now an increase in interatomic distances implies a decrease in internal pressure. The decrease in internal pressure is due only to the variation of the positions of the atoms in the crystal lattice with the expansion.

Given that the stress tensor is also calculated as a variation of the potential energy, it seems to me strongly linked to the internal pressure of the solid. In fact, even the potential energy depends only on the atomic positions in the lattice. I think this is the meaning of what is written in the article https://arxiv.org/pdf/2006.12867.pdf (equation (4)). Obviously, to find the correct parameters of the deformed cell I will have to apply an iterative procedure that leads me to equality between the internal pressure of the solid and the external one ("target pressure" -sentence extracted from the article-). This condition is necessary for the solid to be in equilibrium.

Also I don't understand why it is written that "The second term on the right-hand side of equation 4 corresponds to the partial derivative of the free energy with respect to volume in the quasiharmonic approximation".

From what I know the partial derivative of the free energy with respect to the volume at constant temperature is equal to the partial derivative of the kinetic energy with respect to the volume at constant entropy. This relation contains the Gruneisen parameters that allow me to describe the dependence of frequencies on volume and therefore on temperature, since in thermal expansion a change in volume corresponds to a change in temperature.

Why did the QSCAILD method "propose a way to include pressure from kinetic energy within this static pattern"? I understood that this sentence derives from the fact that the first term can be traced back to a kinetic term through the virial theorem while the second to the derivative of Helmoltz free energy with respect to volume in the quasi-harmonic approximation. Even this term if I am not mistaken is always equivalent to a kinetic term. So I wonder: maybe equation 4 in the article is just the expression for internal energy?

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Astronuc
Staff Emeritus
tensor stress is directly related to the internal pressure of a solid. That is to the force that the neighboring atoms exert each other in relation to a unit of surface.
Usually, stress is due to a mechanical load or force, and the load/force is distributed over a cross-sectional area, which is stress, and there can be tensile/compressive and shear forces involved. Depending on the stress level, deformation of a solid will occur (displacement of atoms from their positions prior to the load application). There is a relationship between the stress and displacement (or strain) in the elastic range for which the relationship between stress and strain are expressed by a set of equations with the elastic modulus and Poisson's ratio. See constitutive model.

https://en.wikipedia.org/wiki/Linear_elasticity
http://web.mit.edu/16.20/homepage/3_Constitutive/Constitutive_files/module_3_with_solutions.pdf

Note that for metals, the elastic modulus decreases with increasing temperature. There is also a shear modulus that also decreases with increasing temperature. Plastic (permanent) deformation and creep are more complicated.

With thermal expansion (with increasing temperature), no stress is produced, and displacement or thermal strain is expressed by a thermal expansion coefficient and a differential temperature.

In describing the deformation of a solid subject to load with increasing temperature, one has to account for the thermal expansion and deformation/strain appropriately.

• cito93
Usually, stress is due to a mechanical load or force, and the load/force is distributed over a cross-sectional area, which is stress, and there can be tensile/compressive and shear forces involved. Depending on the stress level, deformation of a solid will occur (displacement of atoms from their positions prior to the load application). There is a relationship between the stress and displacement (or strain) in the elastic range for which the relationship between stress and strain are expressed by a set of equations with the elastic modulus and Poisson's ratio. See constitutive model.

https://en.wikipedia.org/wiki/Linear_elasticity
http://web.mit.edu/16.20/homepage/3_Constitutive/Constitutive_files/module_3_with_solutions.pdf

Note that for metals, the elastic modulus decreases with increasing temperature. There is also a shear modulus that also decreases with increasing temperature.
It is amorphous substances that can be described by 2 moduli alone - elastic modulus and either Poisson ratio or shear modulus.
Solids generally have 21 independent moduli connecting the 6 components of stress and 6 of strain.
With thermal expansion (with increasing temperature), no stress is produced, and displacement or thermal strain is expressed by a thermal expansion coefficient and a differential temperature.

In describing the deformation of a solid subject to load with increasing temperature, one has to account for the thermal expansion and deformation/strain appropriately.
True - thermal expansion of a homogenous solid should produce strain but no stress.
The "internal pressure" actually is connected to compressibility. How does elastic modulus change on thermal expansion?

DrDu