Tension Bar Validation — Uniaxial Tension · Indux Mechanical

Last updated: 2026-09-10 5 min read

This case exercises the basic structural-analysis path in Indux Mechanical using one of the simplest three-dimensional linear-elastic problems. Its spatially uniform, unambiguous closed-form solution isolates the material law, surface loading, constraints, and stress/displacement post-processing from geometric complexity and stress concentrations.

What this case verifies

Hooke's law for isotropic linear elasticity
Transfer of a surface traction into a uniform axial stress
Fixed and free-surface boundary conditions
Axial elongation and lateral contraction caused by Poisson's effect
The complete Gmsh → SU2 → Indux Mechanical input, solve, and post-processing pipeline

Problem definition

Tension Bar boundary conditions

One end of a three-dimensional prismatic bar of length $L$ and cross-sectional area $A$ is fully fixed. A uniform tensile traction of $1$ MPa is applied to the opposite end. Taking the loading direction as $x$ , the region sufficiently far from the constrained end approaches a uniform uniaxial stress state by Saint-Venant's principle.

Reference solution

For a uniform bar subjected to axial force $F$ , the stress and free-end displacement are

\sigma_x = \frac{F}{A}, \qquad u_x(L) = \frac{FL}{AE} = \frac{\sigma_x L}{E}

where $E$ is Young's modulus. Poisson's ratio $\nu$ determines the lateral contraction:

\varepsilon_y = \varepsilon_z = -\nu\frac{\sigma_x}{E}

For a uniaxial stress state, the von Mises equivalent stress is also $|\sigma_x|$ . The same problem therefore checks the stress, axial-displacement, and lateral-displacement paths independently.

Results

Tension Bar axial displacement

Tension Bar lateral displacement

Quantity	Theory	FEM	Error
Mean von Mises stress $\sigma$	1.000e6 Pa	9.95e5 Pa	−0.50%
Axial displacement $\delta_x$	5.00e-7 m	4.98e-7 m	−0.40%
Lateral displacement $\delta_y, \delta_z$	±7.50e-9 m	±7.63e-9 m	+1.73%

The primary quantities—the mean stress and axial displacement—agree with theory to within 1%. The lateral displacement is only a few nanometers, so a small absolute difference produces a larger relative error; it is treated as a secondary check. An interior average is used instead of the peak stress at the constrained-end corners to exclude local boundary-condition singularities from the acceptance decision.

Engineering significance

The result establishes a baseline for tensile members, rods, and bolts under axial load. It is also a useful minimal case for diagnosing units, load direction, material properties, mesh input, or post-processing before investigating a more complex failed verification.

Reference

Beer, F. P. et al. Mechanics of Materials — axial loading, Hooke's law, and Poisson's effect.

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