Pressure is Maclaurin series of strain

A statement more general than the Hooke’s law is that pressure in a body can be expressed as a power series (Maclaurin series, to be exact) of (engineering normal) strain with the free coefficient being zero, since pressure disappears with the disappearance of strain. The coefficients of the expansion only depend on the material of the deformed body and on its physical state.

Notation:

1. \(O(f(x))\) is the mathematical Big O. In this law the limit \(e \to 0\) is assumed.

Conditions:

1. The deformations are elastic.

2. The deformations are small, i.e. \(e \ll 1\).

3. This law features the expansion up to the third power of strain, higher terms can be added if needed.

Links:

1. Section 3 on p. 385 of “General Course of Physics” (Obschiy kurs fiziki), vol. 1 by Sivukhin D.V. (1979).

pressure

pressure (or tension) in the deformed body.

Symbol:

p

Latex:

\(p\)

Dimension:

pressure

young_modulus

young_modulus of the body’s material.

Symbol:

E

Latex:

\(E\)

Dimension:

pressure

second_coefficient

Coefficient at the second power of strain in the expansion.

Symbol:

A

Latex:

\(A\)

Dimension:

pressure

third_coefficient

Coefficient at the third power of strain in the expansion.

Symbol:

B

Latex:

\(B\)

Dimension:

pressure

strain

engineering_normal_strain of the body.

Symbol:

e

Latex:

\(e\)

Dimension:

dimensionless

law

p = e^2 * A + e^3 * B + E * e + O(e^4)

Latex:

\[p = e^{2} A + e^{3} B + E e + O\left(e^{4}\right)\]