This form of dilation consists of a case where there is linear dilation in two dimensions.
Consider, for example, a square piece of sides that is heated a temperature , so that it is increased in size, but as there is equal expansion for both directions of the piece, it remains square, but has sides .
We can establish that:
And relating to each side we can use:
So that we can analyze the surfaces, we can square the entire expression, obtaining a relationship with its areas:
But the order of magnitude of the linear expansion coefficient (α) é , which when squared becomes of magnitude being immensely smaller than α. How does temperature change (Δθ) hardly exceeds a value of 10³ºC for solid state bodies, we can consider the term α²Δθ² negligible compared to 2αΔθ, which allows us to ignore it during the calculation, like this:
Where, β is the surface expansion coefficient of each material, we have that:
Note that this equation is applicable for any geometric surface, provided that the areas are obtained through the geometric relations for each one, in particular (circular, rectangular, trapezoidal, etc.).
(1) An iron blade has dimensions 10m x 15m at normal temperature. When heated to 500ºC, what is the area of this surface? Given away