150
3 Continuum Mechanics and Nonlinear Elasticity
3.10 A flat membrane undergoes homogeneous in-plane deformation. Before
deformation, two line segments ae 1 and c e 2 are drawn on the membrane,
where e 1 and e 2 are orthogonal unit vectors. After deformation, these line
segments become b(e 1 + 2e 2 ) and d(e 2 − 2e 1 ), respectively. Determine the
deformation gradient tensor in the plane of the membrane and write it in
dyadic form in terms of e 1 and e 2 .
3.11 Relative to Cartesian coordinates (x, y, z), the Cauchy stress tensor at a point
is given by
σ = [σ ij ] =
⎡
⎣
20 30 −10
30 −60 25
−10 25 70
⎤
⎦ .
(a) Determine the stress components σ rr and σ θz at this point in cylindrical
coordinates.
(b) Determine the Cauchy stress vector across a plane defined by the relation
S = 2x + 2y + z − C = 0,
where C is a constant. Hint: A vector normal to a surface with equation
S = 0 is given by n = ∇S.
(c) The stress vector can be written in the form T = σ nn n + σ ns s, where
n and s are unit vectors normal and tangent to the plane S (n not
summed). Determine the stress components σ nn and σ ns . Hint: Consider
the magnitude of T.
3.12 Consider a circular tube with an internal pressure. In cylindrical polar
coordinates, the differential equation of radial equilibrium is
∂σ rr
∂r
+
σ rr − σ θθ
r
= 0,
where the σ ij (r) are Cauchy stresses and r is the deformed radial coordinate.
Derive this equation in the following ways:
(a) Sum forces on a differential element (Fig. 3.26).
(b) Use the equation ∇ · σ = 0.
3.13 Consider a deformed circular tube with inner radius a and outer radius b.
Relative to Cartesian coordinates (x, y, z) the stress distribution in the tube is
σ = x
2 ye x e x + (x − y)e y e y − y(e x e y + e y e x ).
What external loads are required to hold the tube in equilibrium with the
given stress distribution? Write the loading conditions in terms of cylindrical
coordinates (r, θ, z).
3 Continuum Mechanics and Nonlinear Elasticity
3.10 A flat membrane undergoes homogeneous in-plane deformation. Before
deformation, two line segments ae 1 and c e 2 are drawn on the membrane,
where e 1 and e 2 are orthogonal unit vectors. After deformation, these line
segments become b(e 1 + 2e 2 ) and d(e 2 − 2e 1 ), respectively. Determine the
deformation gradient tensor in the plane of the membrane and write it in
dyadic form in terms of e 1 and e 2 .
3.11 Relative to Cartesian coordinates (x, y, z), the Cauchy stress tensor at a point
is given by
σ = [σ ij ] =
⎡
⎣
20 30 −10
30 −60 25
−10 25 70
⎤
⎦ .
(a) Determine the stress components σ rr and σ θz at this point in cylindrical
coordinates.
(b) Determine the Cauchy stress vector across a plane defined by the relation
S = 2x + 2y + z − C = 0,
where C is a constant. Hint: A vector normal to a surface with equation
S = 0 is given by n = ∇S.
(c) The stress vector can be written in the form T = σ nn n + σ ns s, where
n and s are unit vectors normal and tangent to the plane S (n not
summed). Determine the stress components σ nn and σ ns . Hint: Consider
the magnitude of T.
3.12 Consider a circular tube with an internal pressure. In cylindrical polar
coordinates, the differential equation of radial equilibrium is
∂σ rr
∂r
+
σ rr − σ θθ
r
= 0,
where the σ ij (r) are Cauchy stresses and r is the deformed radial coordinate.
Derive this equation in the following ways:
(a) Sum forces on a differential element (Fig. 3.26).
(b) Use the equation ∇ · σ = 0.
3.13 Consider a deformed circular tube with inner radius a and outer radius b.
Relative to Cartesian coordinates (x, y, z) the stress distribution in the tube is
σ = x
2 ye x e x + (x − y)e y e y − y(e x e y + e y e x ).
What external loads are required to hold the tube in equilibrium with the
given stress distribution? Write the loading conditions in terms of cylindrical
coordinates (r, θ, z).