3. Structural mechanics essentials for thermo-mechanical problems

Thermo-mechanical FEA rests on a small set of continuum-mechanics quantities and relations. If these are not set up consistently, the thermal part of the model can be perfectly converged and the structural results will still be wrong. This chapter establishes the definitions of stress and strain, the role of thermal expansion, and the constitutive laws that connect them, with the sign and tensor conventions used throughout the rest of the course.

3.1 Stress and Strain: Definitions and Conventions

Stress is the internal force intensity transmitted across a material surface. At a point, it is a second-order tensor with nine components in a Cartesian frame, written as σ_ij. The first index denotes the normal direction of the face on which the traction acts, and the second denotes the direction of the traction component. For a body in equilibrium with no distributed body couple, angular momentum balance gives σ_ij = σ_ji, so only six independent components remain. In Voigt notation these are commonly ordered as

σ = [σ_xx, σ_yy, σ_zz, σ_yz, σ_xz, σ_xy]^T

Strain is the symmetric part of the displacement gradient:

ε_ij = 1/2 (∂u_i/∂x_j + ∂u_j/∂x_i)

The antisymmetric part is rigid-body rotation and produces no stress in a classical continuum. The same Voigt ordering is used for strain, with the engineering shear strains γ_yz = 2ε_yz, γ_xz = 2ε_xz, γ_xy = 2ε_xy. Mixing tensor shear strain and engineering shear strain is one of the most common sources of factor-of-two errors in hand checks and in user-defined material routines.

3.2 Thermal Expansion and the Reference State

A free body subjected to a uniform temperature change ΔT = T − T_ref expands without generating stress. The thermal strain is

ε_th = α (T - T_ref)

where α is the coefficient of thermal expansion (CTE), generally temperature dependent. For an isotropic material, ε_th is the same in all three normal directions and the shear thermal strains are zero. For orthotropic or anisotropic materials, α becomes a tensor and the thermal strain vector must be transformed with the material orientation.

The reference temperature T_ref is not a numerical detail. It defines the stress-free configuration. If a part is stress-free at 20 °C and the analysis sets T_ref = 0 °C, the model will predict a spurious compressive pre-stress at 20 °C. In practice, T_ref should be the temperature at which the as-built or as-assembled geometry is stress-free, and this value must be documented in the model setup.

Total strain decomposes additively into mechanical and thermal parts for small-strain thermo-elasticity:

ε_total = ε_mech + ε_th

Only the mechanical part generates stress. This decomposition is the conceptual core of every thermo-mechanical analysis: the solver computes total strain from displacements, subtracts thermal strain, and applies the constitutive law to the remainder.

3.3 Constitutive Laws

For linear isotropic thermo-elasticity, the stress-strain relation is

σ_ij = λ ε_kk δ_ij + 2 μ ε_ij - (3λ + 2μ) α ΔT δ_ij

with Lamé parameters λ and μ, Kronecker delta δ_ij, and ε_kk the volumetric strain. The equivalent engineering form is

σ = D (ε - ε_th)

where D is the 6×6 isotropic elasticity matrix built from Young's modulus E and Poisson's ratio ν. This is the form used by most commercial FEA codes internally.

When temperature varies significantly, E, ν, and α should be entered as functions of temperature. The solver then evaluates D and ε_th at each integration point. Two practical consequences follow. First, a single room-temperature elastic modulus is rarely adequate above roughly 0.3 T_melt. Second, if α is temperature dependent, the thermal strain must be integrated rather than multiplied:

ε_th(T) = ∫_{T_ref}^{T} α(τ) dτ

Most solvers handle this internally, but user subroutines and hand calculations must respect it.

3.4 Yield, Plasticity, and the Thermo-Mechanical Coupling

When the equivalent von Mises stress exceeds the temperature-dependent yield stress σ_y(T), the response becomes plastic. The total strain then decomposes further:

ε_total = ε_elastic + ε_plastic + ε_th

Plastic strain is incompressible and does not contribute to volumetric change, so thermal expansion remains the dominant source of volume change in most heated metallic components. The yield surface, hardening rule, and flow rule must all be defined as functions of temperature for a credible thermo-mechanical simulation. Isotropic hardening with a bilinear or multilinear stress-strain curve is the usual starting point; kinematic hardening is required when cyclic loading and the Bauschinger effect matter.

Two failure modes deserve early attention. Constrained thermal expansion against stiff surroundings produces compressive stresses that can drive buckling or ratcheting. Repeated thermal cycles with local plasticity can produce low-cycle fatigue; the strain range, not the stress range, is then the governing quantity.

3.5 A Worked Hand Check: Constrained Bar

Consider a steel bar of length L = 200 mm, cross-section 20 mm × 20 mm, fully constrained at both ends, heated uniformly from 20 °C to 120 °C. Take E = 210 GPa, ν = 0.3, α = 12 × 10⁻⁶ /°C, and assume the response stays elastic.

1. Free thermal strain: ε_th = α ΔT = 12 × 10⁻⁶ × 100 = 1.2 × 10⁻³.
2. Because the ends cannot move, total axial strain ε_xx = 0.
3. Mechanical strain: ε_mech = ε_total − ε_th = −1.2 × 10⁻³.
4. Axial stress: σ_xx = E ε_mech = 210 × 10⁹ × (−1.2 × 10⁻³) = −252 MPa (compression).
5. Reaction force: F = σ_xx A = −252 × 10⁶ × 400 × 10⁻⁶ = −100.8 kN.

If the material yield stress at 120 °C is 220 MPa, the elastic prediction of 252 MPa is not admissible and the bar yields. A nonlinear analysis with temperature-dependent plasticity is then required, and the residual stress after cool-down will not return to zero. This simple check is a useful sanity test for any thermo-mechanical model: run the same case in the solver and compare the reaction force at the supports.

3.6 How the Pieces Fit Together

The diagram below summarizes the flow from thermal input to structural response, showing where each constitutive ingredient enters.

3.7 Practical Checklist Before Running a Thermo-Mechanical Model

With these definitions and checks in place, the thermal and structural solvers can be coupled with confidence. The next chapters build on this foundation to address one-way and two-way coupling strategies, element selection, and convergence behavior in nonlinear thermo-mechanical analyses.