4.7 Bending of a Block
203
(increasing κ ∗ ), the elasticity results deviate significantly from the linear relations
given by plate theory. Similar plots show the effects of increasing material nonlinearity (Fig. 4.17c, d). As nonlinearity (c 3 ) increases, strong stress concentrations
develop near the surfaces of the block.
Problems
4.1 A rectangular block of isotropic incompressible tissue has undeformed dimensions a 0 , b 0 , and c 0 in the x, y, and z directions, respectively. The strain-energy
density function per unit undeformed volume is
W = c 1 (I 1 − 3)
2
+ c 2 (I 2 − 3),
where c 1 and c 2 are material constants. With the width of the specimen held
fixed in the y-direction, the tissue is stretched to a new length a in the xdirection. Compute the following:
(a) The deformed thickness (in the z-direction).
(b) The principal Lagrangian strains and Cauchy stresses.
(c) The total force applied to the block in the x and y directions.
4.2 A rectangular bar is composed of isotropic compressible material with the
strain-energy density function
W = c(I 1 + I
−1
3 − 4).
The bar undergoes uniaxial extension λ x = λ. Determine the Cauchy stress
σ x in the axial direction as a function of λ.
4.3 Consider simple shear of a block (see Fig. 4.6). In the undeformed configuration, the block is transversely isotropic relative to the Y −axis, with
incompressible material properties defined by
W = c 1 (I 1 − 3) +
c 3
2c 4
e
c 4 (I 4 −1) 2 − 1
,
where c 1 = c 4 = 1 Pa. For k = 0.5, use aspects of the analysis in Sect. 4.3 to
compute the following:
(a) The principal Cauchy stresses and corresponding directions for c 3 = 0,
1, 10, and 100 Pa. Sketch a differential element oriented in the principal
directions and show the stresses acting on it.
(b) The angle between the principal directions of stress and strain for each
value of c 3 . Explain your results.
203
(increasing κ ∗ ), the elasticity results deviate significantly from the linear relations
given by plate theory. Similar plots show the effects of increasing material nonlinearity (Fig. 4.17c, d). As nonlinearity (c 3 ) increases, strong stress concentrations
develop near the surfaces of the block.
Problems
4.1 A rectangular block of isotropic incompressible tissue has undeformed dimensions a 0 , b 0 , and c 0 in the x, y, and z directions, respectively. The strain-energy
density function per unit undeformed volume is
W = c 1 (I 1 − 3)
2
+ c 2 (I 2 − 3),
where c 1 and c 2 are material constants. With the width of the specimen held
fixed in the y-direction, the tissue is stretched to a new length a in the xdirection. Compute the following:
(a) The deformed thickness (in the z-direction).
(b) The principal Lagrangian strains and Cauchy stresses.
(c) The total force applied to the block in the x and y directions.
4.2 A rectangular bar is composed of isotropic compressible material with the
strain-energy density function
W = c(I 1 + I
−1
3 − 4).
The bar undergoes uniaxial extension λ x = λ. Determine the Cauchy stress
σ x in the axial direction as a function of λ.
4.3 Consider simple shear of a block (see Fig. 4.6). In the undeformed configuration, the block is transversely isotropic relative to the Y −axis, with
incompressible material properties defined by
W = c 1 (I 1 − 3) +
c 3
2c 4
e
c 4 (I 4 −1) 2 − 1
,
where c 1 = c 4 = 1 Pa. For k = 0.5, use aspects of the analysis in Sect. 4.3 to
compute the following:
(a) The principal Cauchy stresses and corresponding directions for c 3 = 0,
1, 10, and 100 Pa. Sketch a differential element oriented in the principal
directions and show the stresses acting on it.
(b) The angle between the principal directions of stress and strain for each
value of c 3 . Explain your results.