Notch Stress

Three-dimensional linear-elastic analysis.

• create a hexahedron mesh

• define a linear-elastic solid body

• project the linear-elastic stress tensor to the mesh-points

• plot the max. principal stress component

• evaluate the fatigue life

This example requires external packages.

pip install pypardiso

A linear-elastic notched plate is subjected to uniaxial tension. The cell-based mean of the stress tensor is projected to the mesh-points and its maximum principal value is plotted.

import pypardiso

import felupe as fem

meshes = []

radius = fem.mesh.Point(a=-2.5).revolve(n=9, phi=90).translate(5, axis=1)
radius = fem.mesh.flip(radius)
middle = fem.mesh.Line(a=-7.5, b=0, n=9).expand(n=0)
meshes.append(middle.fill_between(radius, n=6))

left = fem.mesh.Line(-7.5, 5, n=11).expand(n=0).rotate(90, axis=2, center=[-7.5, 0])
right = (
    fem.mesh.Line(a=-2.5, b=5, n=11)
    .expand(n=0)
    .rotate(90, axis=2, center=[-2.5, 0])
    .translate(5, axis=1)
)
meshes.append(right.fill_between(left, n=6))
meshes.append(fem.Rectangle(a=(-50, 0), b=(-7.5, 12.5), n=(36, 11)))

mesh = fem.MeshContainer(meshes, merge=True).stack()
mesh = fem.MeshContainer([mesh, mesh.mirror(axis=0)], merge=True, decimals=6).stack()
mesh = fem.MeshContainer([mesh, mesh.mirror(axis=1)], merge=True, decimals=6).stack()
mesh = mesh.expand(n=3, z=2.5)

region = fem.RegionHexahedron(mesh)
field = fem.FieldContainer([fem.Field(region, dim=3)])

boundaries = fem.dof.uniaxial(
    field, clamped=True, sym=False, move=0.02, return_loadcase=False
)
solid = fem.SolidBody(umat=fem.LinearElastic(E=2.1e5, nu=0.30), field=field)
step = fem.Step(items=[solid], boundaries=boundaries)
job = fem.Job(steps=[step]).evaluate(parallel=True, solver=pypardiso.spsolve)

solid.plot(
    "Principal Values of Stress",
    show_edges=False,
    view="xy",
    project=fem.topoints,
    show_undeformed=False,
).show()

Static Scene

Interactive Scene

The number of maximum endurable cycles between zero and the applied displacement is evaluated with a SN-curve as denoted in Eq. (1). The range of the maximum principal value of the stress tensor is used to evaluate the fatigue life. For simplicity, the stress is evaluated for the total solid body. To consider only stresses on points which lie on the surface of the solid body, the cells on faces cells_faces() must be determined first.

(1)\[\frac{N}{N_D} = \left( \frac{S}{S_D} \right)^{-k}\]

S_D = 100  # MPa
N_D = 2e6  # cycles
k = 5  # slope

S = fem.topoints(fem.math.eigvalsh(solid.evaluate.stress())[-1], region)
N = N_D * (abs(S) / S_D) ** -k

view = solid.view(point_data={"Endurable Cycles": N})
plotter = view.plot(
    "Endurable Cycles",
    show_undeformed=False,
    show_edges=False,
    log_scale=True,
    flip_scalars=True,
    clim=[N.min(), 2e6],
    above_color="lightgrey",
)
plotter.camera.zoom(6)
plotter.show()

Static Scene

Interactive Scene