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Axisymmetric formulation for elastic structures of revolution

In this numerical tour, we will deal with axisymmetric problems of elastic solids. We will consider a solid of revolution around a fixed axis $$(Oz)$$, the loading, boundary conditions and material properties being also invariant with respect to a rotation along the symmetry axis. The solid cross-section in a plane $$\theta=\text{cst}$$ will be represented by a two-dimensional domain $$\omega$$ for which the first spatial variable (x[0] in FEniCS) will represent the radial coordinate $$r$$ whereas the second spatial variable will denote the axial variable $$z$$.

Problem position

We will investigate here the case of a hollow hemisphere of inner (resp. outer) radius $$R_i$$ (resp. $$R_e$$). Due to the revolution symmetry, the 2D cross-section corresponds to a quarter of a hollow cylinder.


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from __future__ import print_function from dolfin import * from mshr import * import matplotlib.pyplot as plt %matplotlib notebook  Re = 11. Ri = 9. rect = Rectangle(Point(0., 0.), Point(Re, Re)) domain = Circle(Point(0., 0.), Re, 100) - Circle(Point(0., 0.), Ri, 100) domain = domain - Rectangle(Point(0., -Re), Point(-Re, Re)) \                 - Rectangle(Point(0., 0.), Point(Re, -Re))  mesh = generate_mesh(domain, 40) plot(mesh)  class Bottom(SubDomain):     def inside(self, x, on_boundary):         return near(x[1], 0) and on_boundary class Left(SubDomain):     def inside(self, x, on_boundary):         return near(x[0], 0) and on_boundary class Outer(SubDomain):     def inside(self, x, on_boundary):         return near(sqrt(x[0]**2+x[1]**2), Re, 1e-1) and on_boundary  facets = MeshFunction("size_t", mesh, 1) facets.set_all(0) Bottom().mark(facets, 1) Left().mark(facets, 2) Outer().mark(facets, 3) ds = Measure("ds", subdomain_data=facets)

Definition of axisymmetric strains

For axisymmetric conditions, the unkown displacement field is of the form:

\boldsymbol{u} = u_r(r,z)\boldsymbol{e}_r + u_z(r,z)\boldsymbol{e}_z

As a result, we will work with a standard VectorFunctionSpace of dimension 2. The associated strain components are however given by:

\boldsymbol{\varepsilon} = \begin{bmatrix} \partial_r u_r & 0 & (\partial_z u_r + \partial_r u_z)/2 \\ 0 & u_r/r & 0 \\ (\partial_z u_r + \partial_r u_z)/2 & 0 & \partial_z u_z\end{bmatrix}_{(\boldsymbol{e}_r,\boldsymbol{e}_\theta,\boldsymbol{e}_z)}

The previous relation involves explicitly the radial variable $$r$$, which can be obtained from the SpatialCoordinate x[0], the strain-displacement relation is then defined explicitly in the eps function.

Note: we could also express the strain components in the form of a vector of size 4 in alternative of the 3D tensor representation implemented below.

Jeremy BLEYER committed Jun 12, 2018  1093                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               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Jeremy BLEYER committed May 28, 2018  1094     1095     1096     1097     1098     1099     1100     1101     1102     1103     1104     1105     1106     1107     1108     1109     1110     1111     1112     1113     1114     1115     1116     1117     1118     1119     1120     1121     1122     1123     1124     1125     1126     1127     1128     1129     1130     1131     1132     1133     1134     1135     1136     1137     1138     1139     1140     1141     1142     1143     1144     1145     1146     1147     1148     1149     1150                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
x = SpatialCoordinate(mesh)  def eps(v):     return sym(as_tensor([[v[0].dx(0), 0, v[0].dx(1)],                           [0, v[0]/x[0], 0],                           [v[1].dx(0), 0, v[1].dx(1)]]))  E = Constant(1e5) nu = Constant(0.3) mu = E/2/(1+nu) lmbda = E*nu/(1+nu)/(1-2*nu) def sigma(v):     return lmbda*tr(eps(v))*Identity(3) + 2.0*mu*eps(v)

Resolution

The rest of the formulation is similar to the 2D elastic case with a small difference in the integration measure. Indeed, the virtual work principle reads as:

\text{Find } \boldsymbol{u}\in V \text{ s.t. } \int_{\Omega} \boldsymbol{\sigma}(\boldsymbol{u}):\boldsymbol{\varepsilon}(\boldsymbol{v}) d\Omega = \int_{\partial \Omega_T} \boldsymbol{T}\cdot\boldsymbol{v} dS \quad \forall\boldsymbol{v} \in V

where $$\boldsymbol{T}$$ is the imposed traction on some part $$\partial \Omega_T$$ of the domain boundary.

In axisymmetric conditions, the full 3D domain $$\Omega$$ can be decomposed as $$\Omega = \omega \times [0;2\pi]$$ where the interval represents the $$\theta$$ variable. The integration measures therefore reduce to $$d\Omega = d\omega\cdot(rd\theta)$$ and $$dS = ds\cdot(rd\theta)$$ where $$dS$$ is the surface integration measure on the 3D domain $$\Omega$$ and $$ds$$ its counterpart on the cross-section boundary $$\partial \omega$$. Exploiting the invariance of all fields with respect to $$\theta$$, the previous virtual work principle is reformulated on the cross-section only as follows:

\text{Find } \boldsymbol{u}\in V \text{ s.t. } \int_{\omega} \boldsymbol{\sigma}(\boldsymbol{u}):\boldsymbol{\varepsilon}(\boldsymbol{v}) rd\omega = \int_{\partial \omega_T} \boldsymbol{T}\cdot\boldsymbol{v} rds \quad \forall\boldsymbol{v} \in V

where the $$2\pi$$ constants arising from the integration on $$\theta$$ have been cancelled on both sides. As a result, the bilinear and linear form are similar to the plane 2D case with the exception of the additional $$r$$ term in the integration measures.

The final formulation is therefore pretty straightforward. Since a uniform pressure loading is applied on the outer boundary, we will also need the exterior normal vector to define the work of external forces form.


Jeremy BLEYER committed Jun 12, 2018  1151                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               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n = FacetNormal(mesh) p = Constant(10.)  V = VectorFunctionSpace(mesh, 'CG', degree=2) du = TrialFunction(V) u_ = TestFunction(V) a = inner(sigma(du), eps(u_))*x[0]*dx l = inner(-p*n, u_)*x[0]*ds(3)  u = Function(V, name="Displacement")

First, smooth contact conditions are assumed on both $$r=0$$ (Left) and $$z=0$$ (Bottom) boundaries. For this specific case, the solution corresponds to the classical hollow sphere under external pressure with a purely radial displacement:

u_r(r) = -\dfrac{R_e^3}{R_e^3-R_i^3}\left((1 − 2\nu)r + (1 + \nu)\dfrac{R_i^3}{2r^2}\right)\dfrac{p}{E}, \quad u_z=0

Jeremy BLEYER committed Jun 12, 2018  1178                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               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bcs = [DirichletBC(V.sub(1), Constant(0), facets, 1),        DirichletBC(V.sub(0), Constant(0), facets, 2)] solve(a == l, u, bcs) print("Inwards radial displacement at (r=Re, theta=0): \ {:1.7f} (FE) {:1.7f} (Exact)".format(-u(Re, 0.)[0], float(Re**3/(Re**3-Ri**3)*((1-2*nu)*Re+(1+nu)*Ri**3/2/Re**2)*p/E))) print("Inwards radial displacement at (r=Ri, theta=0): \ {:1.7f} (FE) {:1.7f} (Exact)".format(-u(Ri, 0.)[0], float(Re**3/(Re**3-Ri**3)*((1-2*nu)*Ri+(1+nu)*Ri/2)*p/E)))
Inwards radial displacement at (r=Re, theta=0): 0.0018375 (FE) 0.0018387 (Exact) Inwards radial displacement at (r=Ri, theta=0): 0.0020879 (FE) 0.0020894 (Exact)

Jeremy BLEYER committed Jun 12, 2018  1203                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               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plt.figure() plot(100*u, mode="displacement") plt.show()