Block Pressing on Arch

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Block Pressing on Arch

This conceptual example shows how to calculate critical points in models with contact. The model consists of a block modeled with the Solid Mechanics interface pressing on an arch modeled with the Shell interface and also exemplifies how to model the contact between a shell and a solid. During loading, the arch exhibits a snap-through behavior. The definition of the problem is based on a benchmark example from Ref. 1.

Model Definition

The model geometry consists of an arch and a block as shown in Figure 1. Since the arch is modeled with the Shell interface, a 3D geometry is used. However, a 2D plane strain behavior is intended, and consequently symmetry conditions are applied to all boundaries and edges in the y direction to suppress any out-of-plane deformation.

Figure 1: Model geometry

Only contact without friction is considered and the augmented Lagrangian contact method is used.

A boundary load is applied the top surface of the block. Its magnitude is controlled by the monotonically increasing deflection of the arch, which makes it possible to track the entire load path, even though the force does not increase monotonically. The ends of the arch are fixed and the displacement of the block is constrained in the x direction.

Results and Discussion

Figure 2 depicts the deformed shape and the von Mises stress distribution at the last step of the simulation. The snap-through of the arch is clearly observed. The arch is represented by a shell dataset that shows both its top and bottom surface.

Figure 2: Deformation and von Mises stress at the final step.

The load versus deflection curve is shown in Figure 3. The load is in the figure represented by a dimensionless load factor. Two limit points can be observed, the first occurs for a load factor equal to 18 and a deflection of 36 mm. At this point the arch becomes unstable and a snap-through occurs. When the deflection of the arch reaches 80 mm, the load factor has decreased to 14. At this point the second limit point is reached, and the arch finds a new stable configuration. After this point the load factor increases with increasing deflection.

Figure 3: Load versus deflection curve.

The progressive deformation of the block and the arch, including the snap-through of the arch, is shown in Figure 4 for six values of the continuation parameter. Figure 5 shows the contact pressure exerted by the block on the arch during the snap-through.

Figure 4: Deformation of the model for six different parameter values.

Figure 5: Contact pressure acting on the arch.

Notes About the COMSOL Implementation

When a Shell interface is used in a contact simulation, it is recommended that the destination boundary always belongs to the shell. Moreover, the contact definition should be made in the Shell interface. In this example, the block modeled with a Solid Mechanics interface is thus, in the node, considered as external to the current physics.

Contact problems are often unstable in their initial configuration. To help the solver find an initial solution, a is added to the otherwise unconstrained block during the first parameter step.

Modeling the post-critical behavior of a system is not possible by incrementally increasing the boundary load. The unstable behavior is even more pronounced when contact is present. To be able to find all limit points and to track the full load versus deflection curve, a displacement controlled load scheme is used by adding a . Here, the magnitude of the boundary load is controlled through the monotonically increasing deflection of the arch. Alternatively, the vertical displacement could be prescribed on the top surface of the block, but this is a less general technique that fails for some cases. Also, a prescribed displacement would not give an evenly distributed load.

1. P. Wriggers, Computational Contact Mechanics, Springer-Verlag, 2006

Structural_Mechanics_Module/Verification_Examples/block_on_arch

Modeling Instructions

From the menu, choose .

In the window, click

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1

In the window, click

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In the tree, select .

3

4

In the tree, select .

5

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In the text field, type u.

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In the tree, select .

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Global Definitions

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In the window, under click .

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In the window for , locate the section.

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Browse to the model’s Application Libraries folder and double-click the file block_on_arch_parameters.txt.

Work Plane 1 (wp1)

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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Work Plane 1 (wp1)>Circle 1 (c1)

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In the toolbar, click

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In the window for , locate the section.

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From the list, choose .

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Locate the section. In the text field, type R_arch.

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In the text field, type seg_arch.

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Locate the section. In the text field, type -R_arch.

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Locate the section. In the text field, type 90-seg_arch/2.

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9

button in the toolbar.

Work Plane 1 (wp1)>Delete Entities 1 (del1)

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In the window, right-click and choose .

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On the object , select Boundaries 2 and 3 only.

Work Plane 1 (wp1)>Partition Edges 1 (pare1)

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In the toolbar, click

and choose .

2

On the object , select Boundary 1 only.

Work Plane 1 (wp1)>Circle 2 (c2)

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type R_block.

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In the text field, type seg_block.

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Locate the section. In the text field, type R_block.

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Locate the section. In the text field, type -90-seg_block/2.

7

.

8

button in the toolbar.

Work Plane 1 (wp1)>Rectangle 1 (r1)

1

In the toolbar, click

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In the window for , locate the section.

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In the text field, type R_block.

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In the text field, type height_block.

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Locate the section. In the text field, type -R_block/2.

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.

Work Plane 1 (wp1)>Intersection 1 (int1)

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In the toolbar, click

and choose .

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Select the objects and only.

Work Plane 1 (wp1)

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In the window, click .

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In the window for , locate the section.

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Clear the check box.

Extrude 1 (ext1)

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In the toolbar, click

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In the window for , locate the section.

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In the table, enter the following settings:


Distances (m)
d

4

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5

button in the toolbar.

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In the toolbar, click

and choose .

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In the window for , type Arch in the text field.

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Locate the section. From the list, choose .

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Select the object only.

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Locate the section. From the list, choose .

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1

Right-click and choose .

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In the window for , type Block in the text field.

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Locate the section. In the list, select .

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Select the object only.

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Locate the section. From the list, choose .

Form Union (fin)

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In the window, under click .

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In the window for , locate the section.

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From the list, choose .

4

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5

button in the toolbar.

Material 1 (mat1)

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In the window, under right-click and choose .

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In the window for , locate the section.

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In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	10[GPa]	Pa	Basic
Poisson’s ratio	nu	0.2	1	Basic
Density	rho	1	kg/m³	Basic

Material 2 (mat2)

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

From the list, choose .

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Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	70[GPa]	Pa	Basic
Poisson’s ratio	nu	0.3	1	Basic
Density	rho	1	kg/m³	Basic

Average 1 (aveop1)

1

In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

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Select Point 11 only.

Average 2 (aveop2)

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Right-click and choose .

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In the window for , locate the section.

3

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Select Point 3 only.

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In the window, right-click and choose .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Unit	Description
disp_block	aveop1(-w)	m	Block displacement
disp_arch	aveop2(-w)	m	Arch displacement

Contact Pair 1 (p1)

1

In the toolbar, click

and choose .

2

Select Boundaries 4 and 8 only.

3

button in the toolbar.

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In the window for , locate the section.

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From the list, choose .

The destination boundary should be on a boundary modeled with the Shell interface.

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In the window, under click .

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In the window for , locate the section.

3

From the list, choose .

Thickness and Offset 1

1

In the window, under click .

2

In the window for , locate the section.

3

In the d text field, type d.

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From the list, choose .

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In the zreloffset text field, type -1.

Prescribed Displacement/Rotation 1

1

In the toolbar, click

and choose .

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Select Edges 1 and 7 only.

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In the window for , locate the section.

4

Select the check box.

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Select the check box.

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Locate the section. From the list, choose .

1

In the toolbar, click

and choose .

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Select Edges 2, 3, 5, and 6 only.

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In the toolbar, click

and choose .

2

In the window for , locate the section.

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Under , click

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In the dialog box, select in the list.

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6

In the window for , locate the section.

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From the list, choose .

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Locate the section. Select the check box.

The source boundary is in the Solid Mechanics interface.

Solid Mechanics (solid)

In the window, under click .

Prescribed Displacement 1

1

In the toolbar, click

and choose .

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Select Edges 13 and 19 only.

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In the window for , locate the section.

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Select the check box.

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In the toolbar, click

and choose .

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Select Boundaries 5 and 6 only.

Boundary Load 1

1

In the toolbar, click

and choose .

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Select Boundary 7 only.

3

In the window for , locate the section.

4

Specify the FA vector as


0	x
0	y
load*F_ref	z

The dependent variable load will be created in the next step using a global equation.

5

button in the toolbar.

6

In the dialog box, in the tree, select the check box for the node .

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Global Equations 1

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

In the table, enter the following settings:


Name	f(u,ut,utt,t) (1)	Initial value (u_0) (1)	Initial value (u_t0) (1/s)	Description
load	disp_block-para*max_disp	0	0

4

Locate the section. Click

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5

In the dialog box, type displacement in the text field.

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7

In the tree, select .

8

Add a small spring stiffness to the block to stabilize the model during the initial step.

Spring Foundation 1

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. In the kV text field, type 1e3*(para<0.01).

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

1

Right-click and choose .

2

Select Edges 2 and 5 only.

3

In the window for , locate the section.

4

In the text field, type n_elem_arch.

1

In the toolbar, click

and choose .

2

Select Boundary 5 only.

1

Right-click and choose .

2

Select Edges 10 and 17 only.

3

In the window for , locate the section.

4

In the text field, type n_elem_block.

1

In the window, right-click and choose .

2

Select Edges 9 and 20 only.

1

In the toolbar, click

.

2

In the window, right-click and choose .

3

button in the toolbar.

Step 1: Stationary

1

In the window, under click .

2

In the window for , click to expand the section.

3

Select the check box.

4

5

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
para (Load parameter)	range(0,0.05,1)

Solution 1 (sol1)

1

In the toolbar, click

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2

In the window, expand the node, then click .

3

In the window for , locate the section.

4

In the text field, type 0.0005.

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In the window, expand the node, then click .

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In the window for , locate the section.

7

From the list, choose .

8

In the window, expand the node, then click .

9

In the window for , locate the section.

10

Under , click

.

11

In the dialog box, select in the list.

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13

In the window, under right-click and choose .

Structural mechanics interfaces should be solved in a single segregated step.

14

In the toolbar, click

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1

In the window, expand the node.

2

Right-click and choose .

3

In the window for , locate the section.

4

From the list, choose .

5

From the list, choose .

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Locate the section. In the text field, type solid.mises.

7

Click to expand the section. From the list, choose .

8

In the toolbar, click

.

9

button in the toolbar.

10

button in the toolbar.

Contact Forces (shell)

1

In the window, click .

2

In the window for , locate the section.

3

From the list, choose .

Contact 1, Pressure

1

In the window, expand the node, then click .

2

In the window for , locate the section.

3

Select the check box.

4

In the associated text field, type 5e-10.

1

In the window, expand the node, then click .

2

Select Boundary 1 only.

3

In the window for , locate the section.

4

From the list, choose .

5

In the toolbar, click

.

1

In the toolbar, click

and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Right-click and choose .

Load vs. Deflection

1

In the toolbar, click

and choose .

2

In the window for , type Load vs. Deflection in the text field.

1

Right-click and choose .

2

In the window for , locate the section.

3

In the table, enter the following settings:


Expression	Unit	Description
disp_block	mm	Block displacement
disp_arch	mm	Arch displacement

4

Locate the section. From the list, choose .

5

In the text field, type load.

6

Click to expand the section. Find the subsection. From the list, choose .

Load vs. Deflection

1

In the window, click .

2

In the window for , locate the section.

3

Select the check box.

4

Locate the section. From the list, choose .

5

Locate the section. Select the check box.

6

In the associated text field, type Deflection (mm).

7

In the toolbar, click

.

1

In the toolbar, click

and choose .

2

In the window for , type Deformation in the text field.

3

Locate the section. From the list, choose .

4

In the text field, type range(1,4,21).

5

Click to expand the section. From the list, choose .

1

Right-click and choose .

2

Select Edges 2 and 5 only.

3

In the window for , locate the section.

4

In the text field, type z.

5

Locate the section. From the list, choose .

6

In the text field, type x.

7

Click to expand the section. In the text field, type 2.

1

Right-click and choose .

2

In the window for , locate the section.

3

toggle button.

4

Select Edges 9, 10, 14, 17, and 20 only.

5

Locate the section. Find the subsection. From the list, choose .

6

From the list, choose .

1

In the window, click .

2

In the window for , click to expand the section.

3

Select the check box.

4

Find the subsection. In the text field, type para = .

5

In the toolbar, click

.

button in the toolbar.