Thermal Stresses in a Layered Plate

In this example, thermal stresses in a layered plate are analyzed. The plate consists of three layers: a coating, a substrate, and a carrier. The coating is deposited onto the substrate at a temperature of 800°C. At this temperature both the coating and the substrate are stress-free. During the first stage of the analysis, the temperature of the plate is lowered to 150°C, which induces thermal stresses in the coating/substrate assembly. At this temperature the coating/substrate assembly is epoxied to a stress-free carrier layer. During the second stage of the analysis, the temperature in the entire assembly is lowered to 20°C, and the thermal stresses are examined.

Model Definition

The plate is considered to be thick and therefore in a state of plane strain. It is modeled using the 2D Solid Mechanics interface. The geometry of the plate is shown in Figure 1. The bottom layer of the geometry is the carrier, the middle layer is the substrate, and the top layer is the coating.

Figure 1: The plate geometry.

Material Properties

The three layers are modeled as isotropic and linear elastic. Their coefficients of thermal expansion are constant. The material properties of the layers are shown in Table 1, Table 2, and Table 3.

Table 1: Material properties of the carrier.
Material property	Value
E	215 GPa
ν	0.3
ρ	1000 kg/m3
α	6.6·10-6 K-1

Table 2: Material properties of the substrate.
Material property	Value
E	130 GPa
ν	0.28
ρ	1000 kg/m3
α	3·10-6 K-1

Table 3: Material properties of the coating.
Material property	Value
E	70 GPa
ν	0.17
ρ	1000 kg/m3
α	5·10-7 K-1

Activation of the Carrier

The carrier is only present at the second stage of the analysis. The activation of this layer is readily performed using the subnode under . Note that the carrier will be activated in a stress-free state, even though its strain reference temperature (800°C) is different from the temperature at activation.

Loading and Boundary Conditions

Loading on the plate consists of an applied homogeneous temperature field. First, the temperature of the coating and substrate is reduced from the initial temperature 800°C to 150°C. During this temperature change, the carrier is not yet present. At 150°C, the carrier is activated using the subfeature, and the temperature of the whole assembly is reduced to 20°C.

The plate is constrained using the feature.

Results and Discussion

Figure 2 shows the normal stress in the x direction after the first stage of the analysis. The substrate material has a higher coefficient of thermal expansion than the coating material. This means that the substrate shrinks more than the coating, causing tensile stresses in the substrate area next to the coating and compressive stresses in the coating.

Figure 2: Normal stress in the x direction for the first stage of the analysis.

Note that during the first stage of the analysis, the carrier is inactive.

Figure 3 shows the normal stress in the x direction after the second stage of the analysis, where the temperature is lowered to 20 °C. The stress levels in the substrate have increased slightly near to the coating, as have the compressive stresses in the coating compared to after the first stage.

Figure 3: Residual thermal stress at room temperature.

The coefficient of thermal expansion is higher in the carrier than in the substrate. As the temperature is decreased, the carrier experiences tensile stresses, while the substrate near the carrier experiences compressive stresses.

MEMS_Module/Actuators/layered_plate

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Tdeposition	800[degC]	1073.2 K	Coating deposition temperature
Tepoxying	150[degC]	423.15 K	Temperature when the coating/substrate is epoxied to the carrier
Troom	20[degC]	293.15 K	Room temperature
Temp	1[K]	1 K	Temperature parameter

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.02.

In the text field, type 0.014.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	0.002
Layer 2	0.01

Click

Click the

button in the toolbar.

Solid Mechanics (solid)

In the window, under click .

Click in the window and then press Ctrl+A to select all domains.

Linear Elastic Material 1

In the window, under click .

Thermal Expansion 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the Tref list, choose . In the associated text field, type Tdeposition.

From the T list, choose . In the associated text field, type Temp.

The carrier is only active during the second stage of the analysis. Use an node for conditional activation of the domain.

Linear Elastic Material 1

In the window, click .

Activation 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Select Domain 1 only.

Locate the section. In the text field, type Temp<Tepoxying.

Rigid Motion Suppression 1

In the toolbar, click

and choose .

Select Domain 2 only.

Materials

Carrier

In the window, under right-click and choose .

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

Locate the section. From the list, choose .

Click

Select Domain 1 only.

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


Property	Variable	Value	Unit	Property group
Young’s modulus	E	2.15e11	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.3	1	Young’s modulus and Poisson’s ratio
Density	rho	1000	kg/m³	Basic
Coefficient of thermal expansion	alpha_iso ; alphaii = alpha_iso, alphaij = 0	6e-6	1/K	Basic

Substrate

Right-click and choose .

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

Select Domain 2 only.

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


Property	Variable	Value	Unit	Property group
Young’s modulus	E	1.3e11	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.28	1	Young’s modulus and Poisson’s ratio
Density	rho	1000	kg/m³	Basic
Coefficient of thermal expansion	alpha_iso ; alphaii = alpha_iso, alphaij = 0	3e-6	1/K	Basic

Coating

Right-click and choose .

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

Select Domain 3 only.

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


Property	Variable	Value	Unit	Property group
Young’s modulus	E	7e10	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.17	1	Young’s modulus and Poisson’s ratio
Density	rho	1000	kg/m³	Basic
Coefficient of thermal expansion	alpha_iso ; alphaii = alpha_iso, alphaij = 0	5e-7	1/K	Basic