Stress Relaxation and Creep Relaxation
Theory and Physics
What is Stress Relaxation?
Professor, how is stress relaxation different from creep?
Creep is an increase in strain over time under constant stress. Stress relaxation is a decrease in stress over time under constant strain. They are different aspects of the same viscoelastic phenomenon. Typical examples include the decrease in bolt preload and the loss of sealing force in rubber.
Summary
The Deep Relationship Between Relaxation and Displacement Control
Creep and relaxation are merely the "stress-controlled version" and "displacement-controlled version" of the same material phenomenon. "Stress relaxation" under constant displacement, where stress decreases, becomes a problem in applications like bolted joints. For example, the JIS B 1083 fastening design guidelines recommend evaluating the residual axial force ratio after 10 years using relaxation calculations based on Norton's law when using steel bolts at 200°C or higher. There are cases where retightening is mandated when the force falls below 80% of the initial axial force.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but ending up with "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because the vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum Assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small Deformation Assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic Material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static Assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable Cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
FEM Settings
Abaqus: VISCOELASTIC, TIME=PRONY + VISCO step. Define relaxation using Prony series.
Summary
Collaboration with Prony Series
In creep relaxation analysis, long-term behavior is often represented by Prony series (parallel combination of Maxwell elements). Time step selection greatly influences accuracy. For example, the concrete creep coefficient φ in DIN EN 1992-1-1 (Eurocode 2) covers 1 to 50 years on a logarithmic time axis with about 5 to 7 steps. In Abaqus, data can be input directly into "*VISCOELASTIC, TIME=PRONY" or automatic fitting is possible using the Time-Temperature Superposition tool.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Midside Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (increasing order).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix every iteration. Provides quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying the full load at once, it is applied in small increments. The arc-length method (Riks method) can track beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
Practical Checklist
Relaxation in Bolted Joints: Practice in Nuclear Power Piping
Creep relaxation is a serious issue in bolt fastening design for nuclear power plants. For example, in GE BWR piping flange sections, it was confirmed that 316L stainless steel bolts lost 35% of their initial tightening force after 10 years of operation at 300°C. The ASME Section III standard specifies a fatigue evaluation procedure that explicitly considers creep relaxation. This mechanism can be reproduced with ANSYS's CREEP module.
Analogy for the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If the mesh quality is poor, the results will be a mess no matter how good the solver is.
Pitfalls Beginners Often Fall Into
Are you checking for mesh convergence? Do you think "the calculation ran = the results are correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer can be far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
How to Think About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling the real-world constraint conditions is often the most important step in the entire analysis.
Software Comparison
Tools
All solvers support Prony series. Abaqus's *VISCOELASTIC is the most flexible.
Implementation of Creep Laws by Vendors: From Norton's Law to Dorn's Law
Implementation of creep laws varies significantly between solvers. MSC Nastran standardly implements the Norton-Bailey law in time-hardening form, while ABAQUS also allows strain-hardening selection. ANSYS supports six types of creep laws via Implicit Creep Equations. In a comparative case for high-pressure steam turbine blade design, the creep strain after 1,000 hours differed by 20% between solvers even with the same material constants.
The Three Most Important Questions for Selection
- "What are you solving?": Does the required physical model and element type for stress relaxation and creep relaxation correspond? For example, in fluids, the presence of LES support; in structures, the capability for contact and large deformation makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between an automatic transmission car (GUI) and a manual transmission car (script).
- "How far will you expand?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technologies
Advanced
Pioneers in Creep Research: High-Temperature Deformation of Gas Turbine Blades
Engineering research on metal creep became full-scale in the 1950s during the development of the Rolls-Royce Avon turbojet engine. To explain the phenomenon where IN-100 nickel alloy blades elongated by 0.3mm after 1,000 hours of operation at a turbine inlet temperature of 1,050°C, the Norton-Bailey creep law was optimized against measured data. This is now inherited in ABAQUS as the CREEP material option.
Troubleshooting
Troubles
Convergence Difficulties in Creep Analysis: Practical Time Increment Control
The main cause of convergence failure in creep/stress relaxation analysis is improper time increment settings. In ABAQUS/Standard, the default `CETOL` (creep strain error tolerance) of 0.005 is coarse, causing frequent Increment Cutback in rapid creep regions at high temperatures. In practice, a two-stage setting is recommended: using a fine `CETOL=0.0001` only in the primary creep region and larger increments in the secondary creep region.
When You Think "The Analysis Doesn't Match"
- First, take a deep breath—Panicking and randomly changing settings will only complicate the problem further.
- Create a minimal reproducible case—Reproduce the stress relaxation and creep relaxation problem in its simplest form. "Subtractive debugging" is the most efficient.
- Change one thing and re-run—Making multiple changes simultaneously makes it impossible to know what worked. It's the principle of a "controlled experiment", just like in science.
- Return to the physics—If the calculation result is non-physical, like "an object floating against gravity", suspect a fundamental error in the input data.
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