Out-of-plane shear equilibrium equations

II.1.6.1 Out-of-plane shear equilibrium equations

In Chapter 13 of [Sof04a] one considers the equilibrium in direction $x$ of a small portion of the material (Figure II.1.4) of lengths $d x$ and $d z$ respectively:

\frac{\partial τ_{x z}}{\partial z} + \frac{\partial σ_{x x}}{\partial x} = 0 .

Similarly, the equilibrium of a portion $d x$ of the full laminate is given globally by the expression:

Q_{x z} - \frac{\partial M_{x x}}{\partial x} = 0 .

Figure II.1.4: Out-of-plane XZ shear equilibrium in a laminate (from [Sof04a]).

Then, in Chapter 13 of [Sof04a], developments are done to calculate the relations between $Q_{x z}$ and $τ_{x z}$ . All the developments are based on the local $x$ equilibrium relation.

In this document, a more general presentation of the out-of-plane shear behavior of laminates is done. The $x$ and $y$ components of in-plane local equilibrium are written as follows:

σ_{x x, x} + τ_{x y, y} + τ_{x z, z} = 0,

(II.1.34)

τ_{x y, x} + σ_{y y, y} + τ_{y z, z} = 0 .

(II.1.35)

Correspondingly, a global equilibrium is expressed by the two equations:

M_{x x, x} + M_{x y, y} = Q_{x z},

(II.1.36)

M_{x y, x} + M_{y y, y} = Q_{y z} .

(II.1.37)

Those equations shall be developed and will ultimately allow the calculation of $τ_{x z}$ and $τ_{y z}$ from the global shear $Q_{x z}$ and $Q_{y z}$ .

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