Use the Thermal Expansion subnode to add an internal thermal strain caused by changes in temperature. It is possible to model bending due to a temperature gradient in the thickness direction of the shell.

Thermal expansion can be modeled for the Linear Elastic Material, Linear Elastic Material, Layered, Hyperelastic Material, Layered, or Piezoelectric Material, Layered. The thermal expansion can be applied to arbitrary layers in a multilayered shell when the Composite Materials Module analysis is available.

This section is present when this node is added under a Linear Elastic Material, Layered or Layered Hyperelastic Elastic Material node. In this section, select the layers in which thermal expansion needs to be modeled.

For a multilayered shell, it is often easiest to add one Thermal Expansion node per layer, if the temperature input is manual.

If the same layer is selected in two Thermal Expansion nodes being active on the same boundary, the second definition will override the previous.

The Volume reference temperature Tref is the temperature at which there are no thermal strains. As a default, the value is obtained from a Common model input. You can also select User defined to enter a value or expression for the temperature locally.

The Temperature T is by default obtained from a Common model input. You can also select an existing temperature variable from a heat transfer interface (for example, Temperature (htsh/sol1)), if any temperature variables exist, or manually enter a value or expression by selecting User defined. This is the midsurface temperature of the shell, controlling the membrane part of the thermal expansion. For layered shells, it is the mid-layer temperature for each layer.

If needed, you can add a through-thickness temperature gradient in the Thermal Bending section.

Select an Input type to specify how the thermal strain is entered. For the default Secant coefficient of thermal expansion the thermal strain is given by

where the secant coefficient of thermal expansion α can be temperature dependent.

When Input type is Tangent coefficient of thermal expansion, the thermal strain is given by

where αt is the tangential coefficient of thermal expansion.

When Input type is Thermal strain, enter the thermal strain dL as function of temperature explicitly.

In all three cases, the default is to take values From material. When entering data as User defined, select Isotropic, Diagonal, or Symmetric to enter one or more components for a general coefficient of the thermal expansion tensor or the thermal strain tensor. When Diagonal or Symmetric input is used, the axis orientations are given by the coordinate system selection in the parent node.

From the list, select Temperature difference in thickness direction or Temperature gradient in thickness direction.

When Temperature difference in thickness direction is selected, enter the temperature difference ΔTz. This is the temperature difference between the top and bottom surfaces.

When Temperature gradient in thickness direction is selected, enter the temperature gradient T’ in the direction from the bottom surface to the top surface.

When Temperature difference in thickness direction is selected, enter the temperature difference ΔTz between the top surface of the topmost of the selected layers and bottom surface of the bottommost of the selected layers.

When Temperature gradient in thickness direction is selected, enter the temperature gradient T’ in the direction from the bottom surface to the top surface.

Physics tab with Linear Elastic Material, Linear Elastic Material, Layered, Hyperelastic Material, Layered, Piezoelectric Material, Layered, or Section Stiffness node selected in the model tree: