Thermal Expansion
As the temperature changes, most materials react by a change of volume. For a constrained structure, the stresses that evolve even with moderate temperature changes can be considerable. The volume change can be represented a thermal strain εth, which produces stress-free deformations. For a linear elastic material, the constitutive law is
In the computations, the thermal expansion appears as a load, even though it formally is a part of the constitutive relations.
You can include thermal expansion in a model either by adding a Thermal Expansion (for Materials) subnode to the chosen material, or by using the Thermal Expansion.
Temperature Dependence of the Thermal Expansion Coefficient
When performing an analysis over a larger range of temperatures, you often need to consider the temperature dependence in the values of the thermal expansion coefficient itself.
As long as you are using materials from the COMSOL Material Library, everything is handled internally. When you want to enter data from your own measurements or from the literature, you do, however, need to be aware of some details in the definitions used.
Tangent or Secant Data
Thermal expansion coefficients can appear in two forms: tangent and secant.
The tangent form states that the increment in length is
(2-29)
where αt is the tangential thermal expansion coefficient. This form, which is the thermodynamic definition, is conceptually simple, because αt is uniquely defined at each temperature. It is, however, less convenient to use in practice because an integration is required for determining the actual change in length for a finite temperature difference.
The secant formulation, which is the default in COMSOL Multiphysics, is often used in engineering:
In the secant formulation, the actual values of α will however depend on the choice of reference temperature, Tref, at which the material has the reference length L0:
Converting from Tangent Form to Secant Form
Equation 2-29 can be integrated, giving
(2-30)
Define
Thus,
giving the secant thermal expansion coefficient as
For most materials and temperature ranges , which makes it possible to approximate with the simpler expression
(2-31)
If you have access to tangent data, you can choose between two different methods for using them in COMSOL Multiphysics:
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In most of the physics interfaces, you can enter tangent data directly by selecting Tangent coefficient of thermal expansion in the settings for Thermal Expansion. When using this option, a numerical integration of Equation 2-30 will be performed each time the thermal strain is used. This will have a negative impact on the performance, when compared to using a secant coefficient of thermal expansion.
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Precompute the expression in Equation 2-29 externally for the intended range of temperatures. This can for example be done in a spreadsheet program. Enter the computed result as a function, which is then used as any other secant temperature dependent thermal expansion coefficient.
When using Tangent coefficient of thermal expansion, the integrate operator is used. It is called with the two integration limits being the reference temperature <phys>.Tref and the current temperature <phys>.T, where <phys> is the tag of the physics interface. If you define the expression for the coefficient of thermal expansion yourself, you must ensure that it depends on a ‘free’ variable, and not use the same temperature variable as you use to prescribe the current temperature <phys>.T.
Thermal Expansion Coefficient Dependence on Reference Temperature
Let αm(T) be the temperature-dependent function that represents the measured values of the secant thermal expansion coefficient. The change in length of a sample at a given temperature T with respect to the sample’s original length at a temperature Tm is called dilation.
Note that by definition, the dilation at T = Tm is zero, so Tm denotes the strain-free state of the material as far as the measured values of αm(T) is concerned. Denote the length of the sample at a temperature T as L(T) and the strain-free length as L0 = L(Tm). The dilation can be then expressed as L(T) − L(Tm). Using the definition of the secant coefficient of thermal expansion, L(T) can be written as:
(2-32)
When using the measured data, it is possible that the strain-free state occurs at a temperature Tref which differs from Tm. The dilation at any temperature T would then be defined as L(T) − L(Tref), where L(Tref) can be written as.
(2-33)
As a result of this shift in the strain-free temperature, it is necessary to redefine the thermal expansion coefficient so that L(T) and L(Tref) can be related using Equation 2-32 but with Tm replaced by Tref.
(2-34)
Here αr(T) is the redefined thermal expansion coefficient, based on Tref. It can be derived from the relations above. Using Equation 2-32 and Equation 2-34 there are two ways of writing the current length L(T), so that
(2-35)
Equation 2-33 makes it is possible to eliminate L(Tref) and L(Tm) from Equation 2-35:
(2-36)
It is now possible find αr(T), expressed in known quantities. After some algebra, the final expression is
(2-37)
In order to arrive at this form of αr(T), the numerator has been rewritten, using
(2-38)
Representation in COMSOL Multiphysics
Most materials listed in the material libraries and databases available with COMSOL Multiphysics and its add-on products contain a function for the measured temperature-dependent thermal expansion coefficient curve. You can find this from the Materials branch, as shown in Figure 2-19. The Piecewise function named alpha_solid_1 is the measured thermal expansion coefficient αm(T).
Using Functions in Materials in the COMSOL Multiphysics Reference Manual
The Material Contents section in Figure 2-19 shows the material property alpha, which is the redefined thermal expansion coefficient αr(T). The complete expression for alpha is as follows:
(alpha_solid_1(T[1/K])[1/K]+(Tempref-293[K])*
if(abs(T-Tempref)>1e-3,(alpha_solid_1(T[1/K])[1/K]
-alpha_solid_1(Tempref[1/K])[1/K])/(T-Tempref),
d(alpha_solid_1(T[1/K]),T)[1/K]))/
(1+alpha_solid_1(Tempref[1/K])[1/K]*(Tempref-293[K]))
This is essentially Equation 2-37, but with a small modification to avoid problems if T=Tref.
Figure 2-19: An example in COMSOL Multiphysics showing the Materials branch and where to find the temperature-dependent thermal expansion coefficient.
In the definition of alpha (to be more specific: <material_tag>.def.alpha) in COMSOL Multiphysics, Tm is set as 293 K and Tref is obtained from the variable Tempref, which typically fetches its value from the physics interface.
Using Your Own Material Data
If you use our own material data in COMSOL Multiphysics (via an interpolation or any other function), you can still copy-paste the built-in expression for alpha into your New Material. You just need to:
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Replace the function name alpha_solid_1(T[1/K])[1/K], with the function name that you have assigned to the temperature-dependent measured thermal expansion coefficient and use the correct temperature units.
Take care when describing the units. Temperature unit conversions can be the cause of subtle errors because of the shift in zero-point value. Use kelvin (K) as the temperature unit to the largest possible extent. As an alternative, you can use the other absolute temperature scale, Rankine (R). Avoid using Fahrenheit and Celsius unless you are completely familiar with how the temperature unit conversion works.
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Ensure that the value of Tm is changed from 293 K to the actual value of temperature that was used as the strain-free temperature to compute αm(T) in the function that you created.
User-Defined Materials and Libraries in the COMSOL Multiphysics Reference Manual