Nonlinear material properties

When adding nonlinear materials to a personal library, you can characterise them as Elastic, Plastic, or Elasto-plastic.

• Elastic: This option allows you to define points on a stress-strain plot for both compressive (negative) and tensile (positive) response. Do not use this option when plastic deformation is expected (stresses cannot exceed the yield strength of the material). However, unlike the Young's modulus, the points do not have to lie along a straight line, nor do the compressive and tensile stiffness have to be identical. The slope of the curve at all points must remain positive.
• Plastic: Enter points in a stress-strain table to define the material behavior at stresses above the initial yield strength. The first two data points are automatically generated based on the no-stress and initial yield-stress points. The additional data points define how the material work hardens as it is strained in tension beyond the initial yield strength. While the points can define a curve of various shapes beyond yield (that is, nonlinearly varying stiffness), values defining a negative slope should be avoided. Such materials can cause instability and prevent the solution from converging.
• Elasto-plastic (bi-linear): This option approximates material behavior using a simple bilinear stress-strain assumption. The initial slope is the Young's Modulus (elastic range). A second slope, the Tangent Modulus, defines the stiffness and work-hardening effect beyond yield.

True Stress-Strain versus Engineering Stress-Strain Data

When specifying stress-strain data for either Elastic or Plastic nonlinear materials, True stress-strain data is preferred. Engineering stress-strain data is produced from tensile tests of materials, in which case stresses are based on the original cross sectional area of the specimen. That is, the data is not corrected to account for the change in the cross-sectional area of the test specimen due to lateral or radial deformation. It is difficult to measure the change is cross sectional area. Therefore, true stress-strain data is typically derived from engineering stress-strain data by making appropriate calculations.

When specifying ductile materials using engineering stress-strain data, you should understand the following limitations:

• Engineering stress-strain data is acceptable for elastic, small strain situations. Differences between true and engineering stress-strain data are negligible at or close to yield.
• Engineering stress-strain data introduces progressively more error for large displacement, plastic strain scenarios.
• Engineering stress-strain curves are more likely to contain a negative-slope region, which is problematic for solving. When the reduced cross section is taken into account, the resultant true stress values are greater. Therefore, true stress-strain curves are less likely to include a negative-slope region.

Tip:

Post-yield behavior of ductile materials can be represented with a straight-line segment based on two data points. The first data point of the plastic (post-yield) region is the yield point, which is also the end point of the elastic range. The second point is the ultimate tensile strength (UTS), which is the maximum stress achieved before necking and failure begin. This method has a clear basis since it involves two well defined and measurable data points. Of course, when you define the material curve this way, stress increases linearly with strain between the yield and UTS points.

Ideally, adjust the UTS and corresponding strain value to account for the reduction of the cross section. In the plastic region, the volume change of the material is negligible (Poisson's ratio is approximately 0.5). You can use the following equations to convert the engineering strain and stress to true strain and stress at the UTS:

εt = ln(1+εe) σt = σe (1+εe)

where:

• εt = True Strain
• εe = Engineering Strain
• σt = True Stress
• σe = Engineering Stress

Beyond the UTS the test specimen begins rapid necking, the tensile force drops off, and failure is imminent. Therefore, test data beyond UTS is not very meaningful. However, you may have to extend the stress-strain curve beyond the UTS to cover the range of strain encountered in a nonlinear simulation. If so, use a flat curve (zero slope) beyond UTS, to minimize solution difficulties.

Hardening

The Hardening options affect the way the material behaves when the direction of strain changes after yielding has occurred. To help you understand the hardening options, visualize a 3D strain plot. The plot origin is the zero-strain state. Any other point in 3D space represents a strain vector acting on the subject material, indicating both the magnitude and the direction of strain. Isotropic materials have properties that are the same regardless of the direction of strain. Now imagine drawing thousands of vectors in various directions from the plot origin, each of them just great enough to reach the initial yield strength of the material. Each vector would have the same magnitude (that is length), and the tips of each vector would be located at the same radial distance from the origin. Therefore, all of these points of yield strain lie on a sphere centered over the plot origin. This sphere is called the yield surface, since it represents the strain threshold in any direction at which initial yielding occurs. This illustration applies to all three hardening options. The difference between them is what occurs as the strain is increased beyond yield, and the direction of strain is subsequently changed.

Exceeding the yield strength typically work-hardens a material, increasing the yield strength. The new, work-hardened yield strength is what we will call the maximum stress. How the work-hardening process affects the spherical yield surface is what differentiates the three hardening models:

• Isotropic: The radius of the spherical yield surface increases, and the sphere remains centered at the original location. In other words, the strain vector magnitude is no longer defined by the initial yield stress of the material. Instead, the strain vectors are now based on the maximum stress magnitude, regardless of the strain direction. Therefore, the sphere has expanded in size.

  Example: Assume the material is strained in the +X direction (tensile strain) until it has yielded and work-hardened to a maximum stress (SM) equal to 1.05 times the initial yield strength. If the strain is then reversed, the strain magnitude in the -X direction (compressive strain) would have to produce a stress of -SM for additional yielding to commence. In other words, the yield strengths in tension and compression are initially equal, and they will remain equal as plastic strain and work-hardening evolves, regardless of the strain direction.

  The Isotropic Hardening option is generally recommended for situations where only one-way bending occurs.

• Kinematic: The radius of the spherical yield surface remains unchanged, but the sphere location is offset in the direction of strain. The point on the yield surface (the tip of the strain vector) coincides with the new work-hardened material strength (the maximum stress), but only in the direction of the current strain. The distance from the original centroid (the zero-strain condition) to any point on the translated sphere is no longer a constant.

  Example: Assume again that the material is strained in the +X direction (tensile strain) until it has yielded and work-hardened to a maximum stress (SM) equal to 1.05 times the initial yield strength (Sy). Now, reverse the direction of strain. Yielding will not recommence at a stress of -SM, as would occur for the Isotropic hardening option. Instead, the stress would be a lesser value equal to SM - (2 * Sy). The radius of the sphere is unchanged, and it remains related to the initial yield stress (Sy). This phenomenon reduces the stress required to resume yielding when the direction of strain changes after work-hardening. Put simply, this option is designed to capture the case where hardening in tension can lead to later softening in compression.

  The Kinematic Hardening option is generally recommended for situations where reverse bending cycles occur.

• Isotropic + Kinematic - This method combines the effects of the other two hardening options previously described. The spherical yield surface expands somewhat (but less than for the Isotropic option). Similarly, the sphere also translates somewhat (but less than for the Kinematic option). Therefore, the calculated results are between the results predicted by the other two methods.

Yield Criterion

• von Mises: The von Mises stress is an equivalent stress that combines all stress tensors or principal stresses to a positive equivalent stress magnitude. This magnitude represents the distortion energy in the material. Yielding begins when the maximum distortion energy in the material is reached. This criterion is typically used for ductile materials like metals.
• Tresca: Yielding begins when the maximum shear stress in the material is reached. This criterion is typically used for ductile materials like metals.
• Mohr-Coulomb: Similar to the Tresca criterion, but with additional parameters to account for materials that have different tensile and compressive yield strengths. This criterion is typically used for granular materials like soils.
• Drucker-Prager: Similar to the von Mises criterion, but with additional parameters to account for materials that have different tensile and compressive yield strengths. This criterion is typically used for concrete materials.