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When a continuous medium is in mechanical equilibrium, the equations of motion (2.109) reduce to the form

(2.119)

It is useful express the equilibrium equations (2.119) in terms of three coordinate systems that will be used later in this book. In terms of a set of Cartesian coordinates (x1,x2,x3) the equilibrium equations (2.119) may be expressed as the following three independent equilibrium equations (see, for example, [3])

(2.120)

(2.121)

(2.122)

where the stress tensor is symmetric such that

(2.123)

When using cylindrical polar coordinates (r,θ,z) such that

(2.124)

the equilibrium equations (2.120)–(2.122) are written as

(2.125)

(2.126)

(2.127)

where the stress tensor is symmetric such that

(2.128)

When using spherical polar coordinates (r,θ,ϕ) such that

(2.129)

the equilibrium equations (2.120)–(2.122) are written as

(2.130)

(2.131)

(2.132)

where the stress tensor is symmetric such that

(2.133)