Controlled Thermal Expansion: Derivation of calculations
Derivation of calculations
The 'Controlled Thermal Expansion' model predicts some of the properties exhibited by a novel type of lattice structure which uses two materials in a specific geometry to control the thermal expansion coefficient. Most of the work done on this type of structure has been to do with 2-D lattices, however this model deals with 3-D volumetric lattices.
These lattices are still in the developmental stage and so this model is merely a tool to investigate the theory and predict how this type of lattice may behave, they are not currently commercially available.
The attraction of these lattices is that through optimization of the geometry (assuming suitable lattice materials have been chosen), the coefficient of thermal expansion can be controlled (it can be made zero or even negative). This is while remaining structurally robust, as they are stretch dominated lattices.
Figure 1. Schematic of the lattice structure
Physical properties
Density
The mass per unit volume of the unit cell is calculated using the geometry of the lattice and is given by:
Where the size of the unit cell, L, the diameter of the outer and inner lattices, D1 and D2, and the offset angle of the outer lattice, θ, are user specified. The source of this is equation (B.6) from the appendix of the paper: "Concepts for structurally robust materials that combine low thermal expansion with high stiffness" by Steeves et al published in the "Journal of the Mechanics and Physics of Solids" in 2007.
Mechanical properties
Hydrostatic stiffness
In order to calculate the elastic mechanical properties of the lattice it is necessary to find the hydrostatic stiffness. From appendix of the paper mentioned above the hydrostatic stiffness is given by:
Where the displacement at each vertex, u, is given by:
If the forces F1 and F2 are given by:
And if it can be assumed that:
Then the hydrostatic stiffness is given by:
Where A1, A2, l1 and l2 have the same meanings as defined in the calculation for density.
After the hydrostatic stiffness has been found, then the regular elastic constants can be calculated.
Bulk modulus
Using the assumption that the hydrostatic stiffness is simply three times the bulk modulus then:
Young's modulus
From the bulk modulus it is possible to calculate the Young's modulus if a value for the Poisson's ratio is found. Currently the model uses an assumed value of 0.3 for the Poisson's ratio and so the Young's modulus is given by:
Shear modulus
In the same way, the shear modulus can be calculated from the standard relation of elastic constants for an isotropic material:
Thermal properties
Coefficient of thermal expansion
If the temperature change is homogenous then the thermal expansion of the lattice is assumed to be isotropic. The coefficient of thermal expansion only depends upon the materials used and the offset angle of the outer lattice:
Optimization
As well as calculating some mechanical and thermal properties, the model will also calculate the optimum offset angle required for the lattice to have a user specified coefficient of thermal expansion, assuming that the desired value can be achieved with the specified geometry and materials. The angle is found by making tan(θ) the subject of the equation used to find the coefficient of thermal expansion. To do this the following quadratic must be solved:
So therefore an expression for tan(θ) is:
To then get the value for theta, θ, the positive value of tan(θ) is used such that:
Copyright © 2020, Granta Design, Cambridge, UK