DIN 2093 Thermal-Mechanical Coupled FEA: Temperature-Stress Analysis of High-Temperature Disc Springs

1. Necessity of Thermal-Mechanical Coupled Analysis

When disc springs operate under high-temperature conditions (typically > 150 °C), not only do the material's elastic modulus and yield strength decrease, but significant thermal strain and thermal stress also occur. Due to the inherent geometric nonlinearity of disc springs and possible contact, non-uniform temperature fields or temperature changes can lead to:

• Preload changes caused by thermal expansion: Combined with the differential thermal expansion of bolts and connected parts, the actual preload may deviate from the design value.
• Superposition of thermal and mechanical stress: Affects the compressive stress level at the OM point, thereby altering fatigue life.
• Material property gradient: If a temperature gradient exists across the disc spring cross-section, the stiffness and strength vary locally, resulting in a complex stress state.

Therefore, isothermal mechanical analysis alone is insufficient to accurately evaluate the safety and functionality of high-temperature disc springs. Thermal-mechanical coupled finite element analysis is required, simultaneously solving the temperature field and displacement field while accounting for temperature-dependent material properties.

2. Governing Equations for Thermal-Mechanical Coupling

The thermal-mechanical coupled problem is described within the continuum mechanics framework by the simultaneous formulation of the heat conduction equation and the force equilibrium equation.

2.1 Heat Conduction Equation (Temperature Field)

For an isotropic solid, the transient heat conduction equation is:

• $T = T(\mathbf{x}, t)$ — Temperature field (K or °C)
• $\rho$ — Density (kg/m³)
• $c_p$ — Specific heat capacity (J/(kg·K))
• $k$ — Thermal conductivity (W/(m·K)), typically 40~50 W/(m·K) for spring steel
• $\dot{q}_{vol}$ — Volumetric heat source (W/m³), e.g., heat from plastic work, friction (often neglected in disc spring analysis)

In steady-state thermal analysis, the temperature does not change with time, and the equation simplifies to:

Boundary conditions can be specified as fixed temperature (Dirichlet) or convection/radiation (Neumann).

2.2 Force Equilibrium Equation (Displacement Field)

The quasi-static force equilibrium equation considering thermal expansion effects is:

• $\boldsymbol{\sigma}$ — Cauchy stress tensor (MPa)
• $\mathbf{b}$ — Body force (N/mm³), typically only gravity, often negligible.

2.3 Thermoelastic Constitutive Relation

In thermal-mechanical coupled analysis, the total strain $\boldsymbol{\varepsilon}$ is the linear superposition of elastic strain $\boldsymbol{\varepsilon}^e$ and thermal strain $\boldsymbol{\varepsilon}^{th}$:

where:

• $\alpha(T)$ — Instantaneous coefficient of thermal expansion (1/K), typically 11×10⁻⁶ ~ 13×10⁻⁶ /K for steel, varying slightly with temperature
• $T_{ref}$ — Reference temperature (usually room temperature at assembly, 20 °C)
• $\mathbf{I}$ — Second-order identity tensor, indicating isotropic volumetric expansion

The stress-elastic strain relationship still follows Hooke's law, but the elastic modulus $E$ and Poisson's ratio $\nu$ may both be functions of temperature:

where $\mathbf{D}(T)$ is the temperature-dependent elasticity matrix. Therefore, the temperature field not only directly induces stress through thermal strain but also indirectly affects the stress distribution by altering material stiffness.

3. Temperature-Dependent Material Properties

Before performing a thermal-mechanical coupled analysis, the variation of key material parameters with temperature must be defined as curves or tables:

• Elastic Modulus $E(T)$: Decreases with increasing temperature. A common relation is $E(T) = E_{20}[1 - \beta (T-20)]$, with $\beta \approx 2.0\times10^{-4} \text{ K}^{-1}$ for spring steel.
• Yield Strength $R_{p0.2}(T)$: Decreases more significantly at high temperatures, typically obtained from material data sheets. When stress exceeds yield, elastic-plastic analysis can be further incorporated.
• Coefficient of Thermal Expansion $\alpha(T)$: Often taken as constant for steel; fine adjustments with temperature can be considered for precise analysis.
• Thermal Conductivity $k$ and Specific Heat Capacity $c_p$: For steady-state thermal analysis only, $k$ is needed to solve the temperature field; transient heat conduction (e.g., rapid heating/cooling) requires $c_p$ and density.

4. Finite Element Implementation of Thermal-Mechanical Coupling

Within the finite element framework, thermal-mechanical coupling is implemented via sequential coupling or fully coupled methods.

4.1 Sequential Coupling (Temperature → Stress)

1. First, solve the pure heat conduction problem to obtain the nodal temperature field $\mathbf{T}$.
2. Import the temperature field as a predefined field into the mechanical analysis, calculating thermal strain and temperature-dependent material properties.
3. Solve the mechanical equilibrium equation to obtain displacements, strains, and stresses.

This method neglects the influence of mechanical deformation on the temperature field (i.e., the thermoelastic coupling effect is minimal and usually negligible), offering high computational efficiency. It is suitable for most steady-state and slow transient problems.

4.2 Fully Coupled (Simultaneous Solution)

When deformation generates significant heat (e.g., plastic work converted to heat, friction heat), or when the temperature field and displacement field are strongly interdependent, the temperature and displacement fields must be solved simultaneously. The discretized coupled system of equations is:

• $\mathbf{K}_{uu}$ — Mechanical displacement stiffness matrix (including thermal stress contributions)
• $\mathbf{K}_{TT}$ — Heat conduction matrix
• $\mathbf{K}_{uT}, \mathbf{K}_{Tu}$ — Coupling terms, representing the effect of temperature change on forces and the effect of deformation on heat conduction
• $\mathbf{R}_u, \mathbf{R}_T$ — Force and thermal residual vectors

For disc springs, the coupling terms are typically small, and the sequential coupling method can be used. Newton‑Raphson iteration is still required for geometric nonlinearity.

5. Special Considerations for High-Temperature Disc Spring Analysis

• Relaxation and Creep: Thermoelastic coupled analysis does not account for time-dependent relaxation. Under prolonged high temperatures, creep can cause a continuous decrease in preload, requiring additional coupling of a creep constitutive model.
• Contact Thermal Resistance: When disc springs are stacked, the contact between discs is not ideal for heat conduction; contact thermal resistance exists, affecting the temperature distribution, especially during rapid heating.
• Boundary Conditions: Accurate specification of heat exchange between the disc spring and the environment (convection coefficient, radiation) is necessary to obtain the correct temperature field.
• Safety Factor: At high temperatures, material strength decreases. Allowable stress must be derated according to temperature, and thermal stress should not be ignored.

6. Calculation Example (Conceptual)

A disc spring stack operates in a 200 °C environment, uniformly heated from an initial 20 °C. Using sequential coupling:

1. Thermal analysis: The inner and outer surfaces of the disc spring are in contact with a 200 °C fluid. A convective heat transfer coefficient is specified, and the steady-state temperature field is calculated (uniform 200 °C).
2. Mechanical analysis: The reference temperature is set to $T_{ref}=20$ °C. The entire disc spring temperature rises to 200 °C. The material elastic modulus decreases to approximately 198 GPa, and the yield strength decreases to approximately 1275 MPa (50CrV4). The disc spring compression is $s = 0.6 h_0$. Considering thermal expansion: if the disc spring is constrained in a fixed space, thermal compressive stress will develop; if free expansion is allowed, thermal strain does not generate stress, but it changes the disc spring cone height (free state $h_0$ changes slightly), thereby affecting the load.

Result: If the disc spring is preloaded during installation, the preload may increase or decrease after heating due to the differential thermal expansion between the disc spring and surrounding structures. The analysis must model the bolts, connected parts, etc., together to accurately predict the outcome.

7. Conclusion

Thermal-mechanical coupled FEA is an indispensable tool for the design of high-temperature disc springs. By combining the heat conduction and force equilibrium equations with temperature-dependent material data, it enables quantitative evaluation of the effects of thermal expansion and material softening on stress, load, and life. For operating conditions exceeding 150 °C, neglecting thermal effects leads to erroneous preload predictions and inaccurate strength assessments. Therefore, temperature-displacement coupled analysis must be employed, and final verification should be performed using the temperature derating factors required by DIN 2093.