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Elastoplastic Notched Plate

This validation case belongs to solid mechanics. It aims to validate the following parameters:

  • Plastic material behavior
  • Reduced integration element type

The simulation results of SimScale were compared to the reference values presented in [SSNP123]\(^1\).

Geometry

Find below the notched plate geometry used for this validation case:

notched plate geometry validation simscale
Figure 1: Notched plate geometry used in the present validation case

The plate dimensions are tabulated below:

ABCDEF
x00.0040.0050.00500
y000.0010.0050.0050.001
z0.00010.00010.00010.00010.00010.0001
Table 1: Geometry dimensions in meters

The corresponding nodes marked with an apostrophe (‘) are translated 0.0001 \(m\) along the negative z-direction.

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Nonlinear static

Mesh and Element Types: The meshes used for cases A and C were created in SimScale with the standard algorithm. The mesh used for cases B and D is a hexahedral mesh that was created locally.

CaseMesh TypeNumber of NodesElement Type
(A)2nd order Standard44211Standard
(B)2nd order Hexahedral5508Standard
(C)2nd order Standard44211Reduced integration
(D)2nd order Hexahedral5508Reduced integration
Table 2: Mesh characteristics

Find below the mesh used for cases A and C. It’s a standard mesh with second order tetrahedral cells.

tetrahedral second order mesh notched plate
Figure 2: Second order standard mesh used for cases A and C

Cases B and D are run on a 2nd order hexahedral mesh, which is shown below:

hexahedral second order mesh validation case
Figure 3: Second order hexahedral mesh used for cases B and D

Simulation Setup

Material:

  • Steel (plastic behavior)
    • \(E\) = 200 \(GPa\), \(v\) = 0.3
    • \(\rho\) = 7870 \(kg/m^3\)
    • \(\sigma_y\) = 200 \(MPa\), \(E_T\) = 1000 \(MPa\), as shown below:
young's modulus plastic behavior
Figure 4: Dependency of the Young’s modulus

Boundary Conditions:

  • Constraints
    • \(d_x\) = 0 on face AA’E’EF
    • \(d_y\) = 0 on face AA’B’B
    • \(d_z\) = 0 for all nodes
    • Face EE’D’D: displacement of 0.0001 \(m\) in the positive y-direction

Result Comparison

The results obtained from SimScale for the Cauchy stress \(\sigma_{yy}\) along the FC line are compared to the values of reference from [SSNP123]\(^1\). These values of reference were extracted with WebPlotDigitizer. Figure 5 shows the results for cases A and B, where element technology is set to standard:

notched plate validation results standard elements
Figure 5: Comparison of cases A and B with the reference values from [SSNP123]¹

Similarly, for cases C and D, where reduced integration was used:

notched plate validation results reduced integration elements
Figure 6: Comparison of cases C and D with the reference values from [SSNP123]¹

In both Figures 5 and 6, SimScale’s results show good agreement with the reference values.

Inspecting the Cauchy stress \(\sigma_{yy}\) for case B in the post-processor:

notched plate validation post processing contours
Figure 7: Case B, showing the contours for Cauchy stress σyy, in SimScale’s post-processor
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