Explanation of the Coefficient of Thermal Expansion (CTE) For Use in NASTRAN

Dr. William CaseApril 14, 1994

By definition, thermal strain at temperature T is the change in length of a member, due to a change in temperature, ( T - Tref ), divided by the original length l of that member. Denoting thermal strain at temperature T as eT

(1).

The thermal strain is zero at the reference temperature, Tref .

Also, by definition, the coefficient of thermal expansion (CTE or ) at temperature T is is defined as

(2).

Therefore,

(3),

where t is a dummy variable of integration. Define to be the average value of alpha over the temperature range from Tref to T.

Then,

(4).

Then, the thermal strain can be written as

(5).

Generally, thermal expansion data is given as a curve of vs. T. Based on equations (2) and (5), it is seen that

(6),

whereas

(7).

NASTRAN requires that be input, not . However, the two are equal when is constant. An example of the calculation of , along with the NASTRAN material data input, is given below.

 Temp, deg K 103 Delta L/L 106 Alpha 106 Alpha-bar 0 -4.90 0.0 16.7 10 -4.90 0.1 17.3 20 -4.90 0.4 17.9 30 -4.89 1.4 18.6 40 -4.86 3.3 19.2 50 -4.83 5.7 19.9 60 -4.75 8.1 20.4 70 -4.66 10.3 20.9 80 -4.54 12.2 21.3 90 -4.41 13.9 21.7 100 -4.27 15.4 22.1 120 -3.93 17.6 22.7 140 -3.56 19.4 23.3 160 -3.16 21.0 23.8 180 -2.73 22.2 24.2 200 -2.27 23.2 24.4 220 -1.80 23.9 24.7 240 -1.32 24.4 24.9 260 -0.83 24.8 25.1 280 -0.33 25.2 25.3 300 0.18 25.5 25.4

NASTRAN Temperature Dependent Material Data Input for Magnesium
(using degrees Kelvin for temperature)

```MAT1    11      6.5E6   2.4E6           .066    25.35E-6 293.
MATT1                                           115
TABLEM1 115                                                             +T1151
+T1151  0.      16.7E-6 10.     17.3E-6 20      17.9E-6 30.     18.6E-6 +T1152
+T1152  40.     19.2E-6 50.     19.9E-6 60.     20.4E-6 70.     20.9E-6 +T1153
+T1153  80.     21.3E-6 90.     21.7E-6 100.    22.1E-6 120.    22.7E-6 +T1154
+T1154  140.    23.3E-6 160.    23.8E-6 180.    24.2E-6 200.    24.4E-6 +T1155
+T1155  220.    24.7E-6 240.    24.9E-6 260.    25.1E-6 280.    25.3E-6 +T1156
+T1156  300.    25.4E-6 ENDT
```

Dr. William Case
April 14, 1994

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