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:

The plate dimensions are tabulated below:

	A	B	C	D	E	F
x	0	0.004	0.005	0.005	0	0
y	0	0	0.001	0.005	0.005	0.001
z	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001

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.

Case	Mesh Type	Number of Nodes	Element Type
(A)	2nd order Standard	44211	Standard
(B)	2nd order Hexahedral	5508	Standard
(C)	2nd order Standard	44211	Reduced integration
(D)	2nd order Hexahedral	5508	Reduced integration

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

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

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:

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:

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

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: