Thin Layer Interfaces

This model demonstrates alternative implementations for describing a thin layer, and the impact of the choice on the continuity of the displacement and stress fields. It is shown how a perfect interface can be obtained by asymptotically changing the material parameters.

Model Definition

Figure 1 shows the undeformed geometry composed by two domains forming a square of side L and the contacting surface where the thin layer interface will be placed. The full domain is a square of side 1 m and the interface is built by joining two circular arcs of radius 0.707.

Figure 1: Model geometry.

A nearly incompressible Saint Venant–Kirchhoff hyperelastic material is used for both domains. The thin layer of thickness d << L contains a compressible Saint Venant–Kirchhoff hyperelastic material. The domain material properties are shown in Table 1.

Table 1: domain material properties.
variable	value
Bulk modulus	50 N/mm2
Shear modulus	10 N/mm2

thin layer approximations

This study uses three approximations:

•

Solid

•

Membrane

•

Spring

In the solid approximation, a slit is introduced and consequently a discontinuity of the displacement field is allowed. The slit is filled with a 3D thin material whose deformation gradient, F, is approximated as

(1)

Here, ue is the extension of the layer, ua is the average displacement of the layer mid-plane, and N is the normal.

In the membrane approximation, no slit is introduced; the continuity of the displacement is assured and only a jump in the stress is permitted. The deformation gradient is approximated as follows:

(2)

The material properties for both the solid and membrane approximations are given in terms of the bulk modulus kb and the shear modulus μb.

If a spring material is used, a slit is introduced as in the solid case and the two sides of the interface are connected by springs. The deformation gradient is approximated as

(3)

and the resulting geometric nonlinear spring force fs per unit area is defined as

(4)

where α is the spring stiffness constant.

Boundary conditions

A prescribed displacement boundary condition is applied on the lateral faces in the normal direction up to a stretch of 50%. The upper and lower faces are constrained with a roller.

Results and Discussion

In the case of a perfect interface, the displacement and stresses are continuous, as shown in Figure 2.

Figure 2: First Piola-Kirchhoff stress (xX component) for the perfect interface case after a 50% stretch. Continuity of displacements and stresses along the interface are enforced.

If a thin layer of material is inserted between the domains, the perfect continuity of the stress and displacements is no longer ensured. For example, when using a solid approximation, both the stress and displacement fields are discontinuous, as shown in Figure 3.

Figure 3: First Piola–Kirchhoff stress distribution with a thin layer using a solid approximation.

The displacement can be enforced to be continuous using the membrane approximation. Moreover, if the ratio between the shear modulus of the thin layer and the bulk material goes to zero, the continuity of the stress is also restored.

Figure 4: First Piola–Kirchhoff stress distribution with a thin layer using the membrane approximation.

Figure 5: Stress at the center point. Comparison between thin layer (membrane approximation) and perfect interface.

If a spring material is used, the displacements are no more continuous but the stresses are. If the stiffness of the spring increases, the displacement jump disappears resulting in a perfect interface.

Figure 6: First Piola–Kirchhoff stress distribution with a thin layer using a spring material.

Figure 7: Stress at the center point. Comparison between thin layer (spring material) and perfect interface.

Reference

1. A. Javili, “Variational formulation of generalized interfaces for finite deformation elasticity,” Math. Mech. Solids, vol. 23, 2018.

Nonlinear_Structural_Materials_Module/Hyperelasticity/thin_layer_interfaces

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Name	Expression	Value	Description
L	1[m]	1 m	Length
R	1[m]/sqrt(2)	0.70711 m	Radius
d	1[m]	1 m	Out-of-Plane dimension

Parameters: Bulk Material

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Name	Expression	Value	Description
RhoBulk	1[kg/m^3]	1 kg/m³	Density
KBulk	50 [N/mm^2]	5E7 N/m²	Bulk Modulus
MuBulk	10 [N/mm^2]	1E7 N/m²	Lame Parameter

Parameters: Thin Layer

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Name	Expression	Value	Description
th	L/10	0.1 m	Thin layer thickness
RhoBnd	1[kg/m^3]	1 kg/m³	Density
KRatio	0.2	0.2	Scale factor for bulk modulus
KBnd	KRatio*KBulk	1E7 N/m²	Bulk modulus
MuRatio	100	100	Scale factor for Lame parameter
MuBnd	MuRatio*MuBulk	1E9 N/m²	Lame parameter
AlphaRatio	1	1	Scale factor for stiffness
AlphaBnd	AlphaRatio*(KBulk/th)	5E8 N/m³	Spring stiffness

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Name	Expression	Value	Description
stretch	0	0	Stretch

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Name	Expression	Value	Description
nel	20	20	Number of elements

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Property	Variable	Value	Unit	Property group
Bulk modulus	K	KBulk	N/m²	Bulk modulus and shear modulus
Shear modulus	G	MuBulk	N/m²	Bulk modulus and shear modulus
Density	rho	RhoBulk	kg/m³	Basic

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stretch (Stretch)

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Property	Variable	Value	Unit	Property group
Bulk modulus	K	KBnd	N/m²	Bulk modulus and shear modulus
Shear modulus	G	MuBnd	N/m²	Bulk modulus and shear modulus
Density	rho	RhoBnd	kg/m³	Basic

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stretch (Stretch)	range(0.1,0.1,0.5)

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MuRatio (Scale factor for Lame parameter)	range(1,-0.25,0.25) 0.01

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MuRatio (Scale factor for Lame parameter)	10^{range(-3,1,4)}

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Membrane Approximation

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Table Annotation 1

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x-coordinate	y-coordinate	Annotation
0	3/2*L	$\mu_{ratio}$ = 10000
2*L	3/2*L	$\mu_{ratio}$ = 1000
4*L	3/2*L	$\mu_{ratio}$ = 100
0	-3/2*L	$\mu_{ratio}$ = 1
2*L	-3/2*L	$\mu_{ratio}$ = 0.1
4*L	-3/2*L	Perfect interface

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Legends
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Expression	Unit	Description
0	1
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1e4	1

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Expression	Unit	Description
aveop1(solid.PxX)	MPa	Average 1
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Legends
Perfect Interface

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Add a thin layer using a spring approximation to mimic a cohesive interface model.

Solid Mechanics (solid)

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Parameter name	Parameter value list	Parameter unit
stretch (Stretch)	range(0.1,0.1,0.5)

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In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
AlphaRatio (Scale factor for stiffness)	10^{range(-5,1,3)}

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Spring Approximation

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Table Annotation 1

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In the table, enter the following settings:


x-coordinate	y-coordinate	Annotation
0	3/2*L	$\alpha_{ratio}$ = 0.001
2*L	3/2*L	$\alpha_{ratio}$ = 0.1
4*L	3/2*L	$\alpha_{ratio}$ = 1
0	-3/2*L	$\alpha_{ratio}$ = 10
2*L	-3/2*L	$\alpha_{ratio}$ = 1000
4*L	-3/2*L	Perfect interface