16.3 Elastic–Plastic Inflation of a Spherical Vessel
225
Fig. 16.4 Epure of residual
stresses
a
b
Fig. 16.5 Spherical vessel (a) and its element (b)
16.3 Elastic–Plastic Inflation of a Spherical Vessel
A spherical vessel with an external radius b and internal radius a is under the action
of even internal pressure p (Fig. 16.5). Due to central symmetry, the principal axes
of stress and strain tensors will be the direction of the central radius r and any two
directions perpendicular to it on the sphere r = const. Let us designate two latter
directions using the indexes ϕ and ψ (Fig. 16.5b). Then radial stresses, strains, and
displacements will be designated as σ r , ε r , and u = u(r), respectively, and tangential stresses, strains, and displacements as σ ϕ = σ ψ , ε ϕ = ε ψ , and v = w = 0.
All components of stresses, strains, and displacements other than zero are
functions of the variable r only. Based on formulas (1.24) and (1.25) in any point of
the sphere wall, we have
225
Fig. 16.4 Epure of residual
stresses
a
b
Fig. 16.5 Spherical vessel (a) and its element (b)
16.3 Elastic–Plastic Inflation of a Spherical Vessel
A spherical vessel with an external radius b and internal radius a is under the action
of even internal pressure p (Fig. 16.5). Due to central symmetry, the principal axes
of stress and strain tensors will be the direction of the central radius r and any two
directions perpendicular to it on the sphere r = const. Let us designate two latter
directions using the indexes ϕ and ψ (Fig. 16.5b). Then radial stresses, strains, and
displacements will be designated as σ r , ε r , and u = u(r), respectively, and tangential stresses, strains, and displacements as σ ϕ = σ ψ , ε ϕ = ε ψ , and v = w = 0.
All components of stresses, strains, and displacements other than zero are
functions of the variable r only. Based on formulas (1.24) and (1.25) in any point of
the sphere wall, we have