Stress-Strain Curve & Material Nonlinear Models

The Ramberg-Osgood model is a widely used empirical equation that neatly combines elastic and plastic strain into one expression. It's excellent for metals that show a smooth, rounded transition at yielding.

Here, $\varepsilon$ is total strain, $\sigma$ is stress, $E$ is Young's Modulus, $\sigma_y$ is the yield stress (where plastic strain of 0.002 or 0.2% is defined), and $n$ is the hardening exponent. The first term is elastic strain, the second is plastic strain.

The Power-Law (or Hollomon) model is simpler and is often used for the fully plastic region, assuming a power-law relationship between stress and plastic strain.

Here, $\sigma$ is true stress, $\varepsilon_p$ is true plastic strain, $n$ is the strain hardening exponent (same concept as before), and $K$ is the strength coefficient. In the simulator, this model defines the curve after the yield point.

The tensile $\sigma$–$\varepsilon$ curve is a material's "fingerprint." For a ductile metal these regions appear:

• Elastic region: Hooke's law $\sigma = E\varepsilon$ holds and the material returns on unloading. The slope is Young's modulus $E$.
• Yield point: the stress at which plastic deformation begins, yield strength $\sigma_y$ (mild steel shows upper and lower yield points).
• Plastic region (strain hardening): stress rises with deformation; the peak is the ultimate tensile strength $\sigma_u$ (UTS).
• Necking and fracture: beyond UTS the section necks locally and fails at the fracture strain $\varepsilon_f$ (elongation).

The area under the curve is the toughness (energy absorbed to fracture); the elastic area is the resilience. Brittle materials (cast iron, concrete) have almost no yielding and fracture suddenly while elastic.

0.2% offset yield: for materials without a sharp yield point (e.g. aluminum alloys), the stress at 0.2% permanent strain ($\varepsilon=0.002$) is taken as the yield strength — shift the elastic line by $0.002$ and read its intersection with the curve.

Engineering vs. true stress/strain: the usual curve uses the original area and gauge length (engineering values). True stress/strain, based on the instantaneous section, convert as (valid up to necking):

$\sigma_{true} = \sigma_{nom}(1+\varepsilon_{nom}), \qquad \varepsilon_{true} = \ln(1+\varepsilon_{nom})$

True stress exceeds engineering stress and keeps rising past UTS. Plasticity analysis and FEM use true stress/strain.

Analysis approximates the real curve with mathematical models. The models supported here:

A larger hardening exponent $n$ gives more uniform elongation (better formability), and a larger $\sigma_u/\sigma_y$ ratio indicates greater ductility. Switch presets to compare steel, aluminum, titanium, and rubber-like polymers.

Automotive Crash Simulation: When simulating a car crash in LS-DYNA (using MAT_024), engineers input a stress-strain curve from a tensile test. The shape defined by $\sigma_y$, $\sigma_u$, and $n$ determines how the car's frame crumples and absorbs energy, which is vital for passenger safety.

Pressure Vessel Design (ASME Code): For fatigue assessment of nuclear or chemical plant components, the Ramberg-Osgood parameters ($\sigma_y$, $n$) are direct inputs. They help predict how many pressure cycles the vessel can withstand before cracking, ensuring long-term operational safety.

Consumer Product Design: The hyperelastic curve for rubber is used in FEM software like ABAQUS to design products like shoe soles, seals, or phone cases. Simulating the large, recoverable stretch ensures the product is durable and comfortable.

Civil Engineering & Seismic Analysis: The nonlinear stress-strain model for concrete, with its low fracture strain, is crucial for designing earthquake-resistant buildings. Engineers use it in software to see how concrete columns will crack and yield during an earthquake, informing reinforcement placement.

First, mistaking yield stress for the material's maximum strength. In reality, yield stress is the indicator for when a material begins to deform permanently, while the maximum strength is the "Tensile Strength (UTS)" which increases afterwards due to work hardening. For example, structural steel may have a yield stress of 350MPa but a UTS exceeding 500MPa. If you input only the yield stress in CAE and judge that "the part will fail beyond this point," you risk underestimating the component's reserve capacity, leading to an unnecessarily heavy design.

Next, overgeneralizing model selection. It is dangerous to use the bilinear model for all nonlinear analyses simply because it's computationally fast. For instance, materials like rubber and some polymers exhibit "hyperelastic" behavior without a clear yield point. If you select "Rubber" in this tool, you'll notice the curve is curved from the start. Applying a bilinear model to such materials leads to significant misinterpretation of their deformation behavior. The first step is always to observe the material's intrinsic behavior.

Finally, misinterpreting the parameter 'n'. The 'n' in the Power Law hardening rule and the 'n' in Ramberg-Osgood have the same name but different meanings. The former primarily represents the "ease" of work hardening, while the latter mainly indicates the "gradualness" of yielding. If you set both models to 'n=0.2' and compare them, you'll confirm the curve shapes are completely different. Always check the physical definition of each parameter within the material model in your CAE software manual.