Statement of the balance of linear momentum
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The balance of linear momentum can be expressed as:
where is the mass density, is the velocity,
is the Cauchy stress, and is the body force
density.
Recall the general equation for the balance of a physical quantity
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In this case the physical quantity of interest is the momentum density,
i.e., . The source of momentum flux
at the surface is the surface traction, i.e., . The
source of momentum inside the body is the body force, i.e.,
. Therefore, we have
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The surface tractions are related to the Cauchy stress by
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Therefore,
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Let us assume that is an arbitrary fixed control volume. Then,
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Now, from the definition of the tensor product we have (for all vectors
)
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Therefore,
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Using the divergence theorem
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we have
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or,
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Since is arbitrary, we have
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Using the identity
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we get
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or,
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Using the identity
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we get
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From the definition
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we have
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Hence,
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or,
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The material time derivative of is defined as
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Therefore,
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From the balance of mass, we have
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Therefore,
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The material time derivative of is defined as
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Hence,
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