Thermal stress 03: temperature-dependent material validation

Verify mechanical and thermal deformation at various temperatures for a temperature-dependent material.

Case Description

Three identical round rods are subjected to a common tensile force of 6,000 pounds with each rod at a different temperature—100, 300, and 500° F. The rods are constrained in a manner that keeps them statically stable but does not impede the natural thermal expansion or tensile strain that is expected. The initial cross sectional area of each rod is 1 square inch, and each rod is initially 10 inches in length. Temperature-dependent material properties are specified. The thermal expansion coefficient, Young's modulus, and strength (yield and ultimate) varying linearly as a function of temperature. Each parameter is defined for a 0 to 600° F temperature range. For simplicity, all properties are assumed to vary linearly between 0 and 600° F. Therefore, you can determine the properties for any temperature within this range by simple linear interpolation.

Both the axial (Z) and diametral displacement include a combination of thermal and mechanical load effects. The theoretical change in length and diameter of each rod is calculated and compared to the Fusion simulation results. The model faces have been split to provide vertices at each point of interest. Point probes are defined at each result comparison location. Specifically, the axial elongation is measured at the centroid of the end faces where the 6,000 pound loads are applied. Also, change in diameter is measured at the top dead center and bottom dead center points at the mid-length location along each rod. In this case, the Y-displacements are used to determine the change in diameter.

Figure 1: Model Diagram

Dimensions (inch)

• Diameter of each rod: 1.128378 (produces a cross sectional area of 1.000 inch2)
• Length of each rod: 10.0

Study Type and Parameters

• Study Type: Thermal Stress
• Stress-Free Reference Temperature: 0° F

Mesh Parameters

• Mesh Type: Solid, Tetrahedral
• Mesh Size: 0.19 inch, absolute
• Element Order: Parabolic
• Create Curved Mesh Elements: Enabled
• Adaptive Mesh Refinement: None

Material Properties

The following image is a screen capture of the Advanced > Temperature-Dependent properties as defined in Fusion:

Figure 2: Temperature-Dependent Material Properties

Constraints

• Fixed Constraint, Z direction only (UZ): -Z circular end face of each rod
• Pin Constraint, Tangential direction only: All cylindrical faces of each rod

Loads

Structural

• Force, 6,000 pounds in Z direction: At each +Z end face of each rod.

(Each of these faces is divided into four quarters. Therefore, the program applies 1,500 lbs/quarter for a total load of 6,000 lbs/rod.)

Thermal

• Applied Temperature, 100° F: First body (rod in the -X position)
• Applied Temperature, 300° F: Second body (rod in the middle position)
• Applied Temperature, 500° F: Third body (rod in the +X position)

Theoretical Solution

The variables used in the equations and table on this page are defined as follows:

• δz: Z (axial) displacement. Additional subscripts indicates whether the effect is thermal (δz_t) or due to structural force (δz_f).
• α: Coefficient of thermal expansion (varies with temperature)
• T: Temperature
• E: Young's modulus (varies with temperature)
• ν: Poisson's ratio
• L: Initial length of rod
• D: Initial diameter of rod
• ΔD: Change in rod diameter. Subscripts indicate whether the effect is thermal (ΔDt) or due to structural force (ΔDf).
• F: Applied axial tensile force (6,000 lbs.)
• A: Initial cross sectional area of each rod (1 in2).

For all equations, the material properties are linearly interpolated between the two specified data points (0° F and 600° F). For example, Young's modulus (E) at 500° F is determined as follows:

E0 = 6 x 106 psi E600 = 4.5 x 106 psi E500 = E0 + (500° F / 600° F)(E600 - E0) = 6 x 106 psi + (5/6)(4.5 x 106 psi - 6 x 106 psi) = 4.75 x 106 psi

Use the same technique for all material properties, since all are defined with a straight line (constant slope) between 0 and 600°F.

Axial Displacement

The axial displacement is the combination of the thermal expansion due to the applied temperature and the structural elongation due to the applied force. In all cases, these two effects act in the same direction and are directly additive.

Thermal Component

δz_t = T * α * L

Structural Component

δz_f = (F/A) / E * L

Change in Diameter

Thermal Component

ΔDt = T * α * D

Structural Component

ΔDf = (F/A) / E * ν * D

Comparison of Results

Axial Displacement

Three point probes were defined to determine the Z displacement at the centroid of the end face of each rod (+Z, free end):

Figure 3: Z Displacement Results

Change in Diameter

Six point probes were defined to determine the Y displacement at the top dead center (TDC) and bottom dead center (BDC) points at the mid-length location along each rod. The combined Y displacement values equal the change in diameter of the rods according to the following equation:

ΔD = δy(TDC) - δy(BDC)

Figure 4: Y Displacement Results

Results Comparison Tables

The following tables compare theoretical and Fusion simulation results for the Z displacement (elongation) and change in diameter of the rods at the three test temperatures.