Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 150

Combining these two equations, it follows that 2 c L @p c L @f

@ x ð

x 1 Þ¼

@ Z ð

x 1 Þþ g ð x 1 Þ

and; integration of these two equations yields

Z x

1
2 f

1
2 c L

p ðxÞ¼ ðxÞþ g ðzÞ d z þ constant and

0

Z Z

1
2 f

1
2 c L

q ðZÞ¼ ðZÞ g ðzÞ d z constant

0

Substitution of these results for the functions p (

x

) and q ( ) into u 1 ð

x 1 ; t Þ¼ p ð x 1

þ c L t Þþ q ð x 1 c L t Þ

yields

u 1 ð x 1 ; t Þ¼ p ð x 1 þ c L t Þþ q ð x 1 c L t Þ

1
2 ½

¼ f ð x 1 þ c L t Þþ f ð x 1 c L t Þ

Z x 1 þc L t Z x 1 c L t

1
2 c L

þ g ðzÞ d z g ðzÞ d z

0 0

Z x 1 þc L t

1
2 ½

1
2 c L

¼ f ð x 1 þ c L t Þþ f ð x 1 c L t Þ þ g ðzÞ d z;

x 1 c L t

where the constant terms in this expression that are inherited from the functions p ( x )

and q ( A similar result

holds for the S wave; one has only to change the c L to c S and the subscript on x 1 from
1 to 2 to obtain the S result. The difference between the two results is that the shear
or S wave is a propagating shearing motion, as opposed to a propagating compres-
sion/tension motion, and that it travels at a different wave speed. If the same results
were obtained for an isotropic elastic material, the wave speeds would be

) that are zero due to the initial condition u 1 ð x 1 ; 0 Þ¼ f ð x 1 Þ .

c L ¼ l þ

2 m 2 m

r

and c S ¼ for ð L Þ for ð S Þ:

r

Problems

6.3.1. Verify that ( 6.21 ) may be determined directly from ( 6.20 ).
6.3.2. Record the form of (6.22) for transversely isotropic materials.
6.3.3. Calculate ( 6.24 ) from ( 6.23 ).
6.3.4. Record the explicit matrix form for the constitutive relation for Hooke's
law (5.6H).
6.3.5. Record the explicit matrix form for the constitutive relation for Hooke's
law in a homogeneous orthotropic material.
6.3.6. Record the explicit matrix form for the constitutive relation for a trans-
versely isotropic, homogeneous viscoelastic material.
6.3.7. Verify the Navier equation ( 6.29 ) for isotropic linear elasticity. Accomplish
this by substituting (2.49) into ( 6.24 ) and then place the modified ( 6.24 ) for