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

Dr. William Case
April 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

Equation 1 (1).

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

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

Equation 2 (2).

Therefore,

Equation 3 (3),

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

Then,

Equation 4 (4).

Then, the thermal strain can be written as

Equation 5 (5).

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

Equation 6 (6),

whereas

Equation 7 (7).

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

Thermal Expansion of Magnesium
(Delta L/L and Alpha data from NBS Monogram 29: Thermal Expansion of Technical Solids at Low Temperatures)
Temp, deg K103 Delta L/L106 Alpha106 Alpha-bar
0-4.900.016.7
10-4.900.117.3
20-4.900.417.9
30-4.891.418.6
40-4.863.319.2
50-4.835.719.9
60-4.758.120.4
70-4.6610.320.9
80-4.5412.221.3
90-4.4113.921.7
100-4.2715.422.1
120-3.9317.622.7
140-3.5619.423.3
160-3.1621.023.8
180-2.7322.224.2
200-2.2723.224.4
220-1.8023.924.7
240-1.3224.424.9
260-0.8324.825.1
280-0.3325.225.3
3000.1825.525.4

Graph of Thermal Expansion of Mg

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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