Overview

Test Case

A cylindrical Uranium bar with length L₀ = 0.110625 m and radius r₀ = 0.0125 m has initial temperature T₀ = 0° C. The bar has constant thermal generation rate of 5000°C/s. The linear thermal expansion coefficient as a function of temperature is defined below:

Find the ratio of bar volume at t = 0.1 s to initial volume (V_f/V₀).

Figure 55: Problem Sketch

Analysis Assumptions and Modeling Notes

LS-DYNA Thermal Solver 11 is used. FWORK in *CONTOL_THERMAL_SOLVER is set to 1E–20 to turn off plastic deformation heating. The bar is made of solid elements with ELFORM 1. Constant thermal generation in the bar is defined in *MAT_THERMAL_ISOTROPIC. The coefficient of thermal expansion and other temperature dependent material properties are defined in *MAT_ELASTIC_PLASTIC_THERMAL.

An alternate method to *MAT_ELASTIC_PLASTIC_THERMAL is to reference a curve defining the thermal expansion coefficient as a function of temperature in *MAT_ADD_THERMAL_EXPANSION. Other material properties would be defined in *MAT_ISOTROPIC_ELASTIC_PLASTIC.

Symmetry is exploited so only a quarter of the bar is present.

The volumetric thermal expansion coefficient, β, as a function of temperature can be approximated by the equation:

The volumetric thermal expansion coefficient can be written as:

Combining equations gives the following:

Integrating LHS from V₀ to V_f and RHS from T₀ = 0°C to T_f = 500°C gives us the analytical solution of (V_f/V₀) = 2.53.

Figure 56: Fringe plot of xy-displacement at t = 0.1 s

Results Comparison

The error shown in the Results Table may be due to the fitting method of the thermal expansion coefficient as a function of temperature. LS-DYNA uses the temperature and thermal expansion coefficient points mentioned in the Test Case Description, while the analytical solution uses a best-fit line of those points.

Figure 57: Thermal Expansion Coef. versus Temperature