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LinearElasticity1#

A cantilever beam undergoing bending deformation.

Note that this simulation is also performed in examples/Elastic/Elas1.py.

  • $uy$
  • LinearElasticity1
 14 import matplotlib.pyplot as plt
 15 import numpy as np
 16
 17 from EasyFEA import Display, ElemType, Models, Simulations
 18 from EasyFEA.FEM import Field, BiLinearForm, FeArray, Sym_Grad, Trace
 19 from EasyFEA.Geoms import Domain
 20
 21 if __name__ == "__main__":
 22     Display.Clear()
 23
 24     # ----------------------------------------------
 25     # Configuration
 26     # ----------------------------------------------
 27
 28     dim = 2
 29
 30     # geom
 31     L = 120  # mm
 32     h = 13
 33
 34     # load
 35     F = -800  # N
 36
 37     # model
 38     E = 210000  # MPa
 39     v = 0.3
 40
 41     elastic = Models.Elastic.Isotropic(dim, E, v, planeStress=True, thickness=h)
 42     lmbda = elastic.get_lambda()
 43     mu = elastic.get_mu()
 44
 45     # ----------------------------------------------
 46     # Mesh
 47     # ----------------------------------------------
 48
 49     contour = Domain((0, 0), (L, h), h / 3)
 50
 51     if dim == 2:
 52         mesh = contour.Mesh_2D([], ElemType.QUAD9, isOrganised=True)
 53     else:
 54         mesh = contour.Mesh_Extrude(
 55             [], [0, 0, h], [3], ElemType.HEXA27, isOrganised=True
 56         )
 57
 58     nodesX0 = mesh.Nodes_Conditions(lambda x, y, z: x == 0)
 59     nodesXL = mesh.Nodes_Conditions(lambda x, y, z: x == L)
 60
 61     # ----------------------------------------------
 62     # Formulations
 63     # ----------------------------------------------
 64
 65     field = Field(mesh.groupElem, dim)
 66
 67     def S(u: Field) -> FeArray:
 68         Eps = Sym_Grad(u)
 69         return 2 * mu * Eps + lmbda * Trace(Eps) * np.eye(dim)
 70
 71     @BiLinearForm
 72     def ComputeK(u: Field, v: Field):
 73         Sig = S(u)
 74         Eps = Sym_Grad(v)
 75         return Sig.ddot(Eps)
 76
 77     weakForms = Models.WeakForms(field, ComputeK)
 78
 79     # ----------------------------------------------
 80     # Simulations
 81     # ----------------------------------------------
 82
 83     simu = Simulations.WeakForms(mesh, weakForms)
 84
 85     simu.add_dirichlet(nodesX0, [0] * dim, simu.Get_unknowns())
 86     simu.add_surfLoad(nodesXL, [F / h**2], ["y"])
 87
 88     simu.Solve()
 89
 90     # ----------------------------------------------
 91     # Results
 92     # ----------------------------------------------
 93
 94     Display.Plot_Result(simu, "uy", 10, plotMesh=True)
 95
 96     def von_mises_stress(u):
 97         Stress = S(u)
 98         if dim == 2:
 99             xx = Stress[..., 0, 0]
100             yy = Stress[..., 1, 1]
101             xy = Stress[..., 0, 1]
102             return np.sqrt(xx**2 + yy**2 - xx * yy + 3 * xy**2)
103         else:
104             Stress_d = Stress - 1 / 3 * Trace(Stress) * np.eye(3)
105             return np.sqrt(3 / 2 * Stress_d.ddot(Stress_d))
106
107     Svm_e = field.Evaluate_e(von_mises_stress, simu.u)
108
109     Display.Plot_Result(simu, Svm_e, plotMesh=True, ncolors=11)
110
111     plt.show()

Total running time of the script: (0 minutes 0.433 seconds)

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