The Theory of Linear Viscoelasticity by D. R. Bland

Author:D. R. Bland [D. R. Bland]
Language: eng
Format: epub
Publisher: INscribe Digital
Published: 2016-04-04T04:00:00+00:00

or

where Cj is a constant vector, possibly complex, Z is a complex constant and nk is the unit vector in the direction of propagation of the wave. Substitute from equation (25) into equation (11):

whenever

i.e.

This is a vector equation and therefore either the two vectors are equal in magnitude and direction or the two vectors both have zero coefficients. In the former case

and

where A is a complex constant; and in the latter case

and

The displacement uj in eq. (26) satisfies uj,i – ui,j = 0 and ui,i ≠ 0. The displacement uj in eq. (27) satisfies uj,j = 0 and uj,i – ui,j ≠ 0. The results of this paragraph can be summarized as

“The only plane sinusoidal waves possible in a linear viscoelastic material under no sinusoidal body force are a wave of dilatation and no rotation and a wave of rotation and no dilatation. In the former the motion is along the direction of propagation, and in the latter the motion is along any direction normal to the direction of propagation.”

This result is identical to that for an elastic medium.

The stress associated with either wave can be found by substitution from eq. (25) into eq. (10):

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