Simcenter Testing Solutions Neuber's Rule

Simcenter Testlab



Neuber’s Rule

Neuber’s Rule is used to calculate the corresponding elastic-plastic strain time history from an elastic stress time history (Figure 1).
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Figure 1: Neuber’s Rule can be used to derive the corresponding elastic-plastic strain from elastic stress.

Materials often exhibit both elastic and plastic behavior:
  • Elastic: At low load levels, a material acts like an elastic spring.  There is a linear relationship (expressed as Young’s Modulus) between stress and strain
  • Plastic: At higher load levels, the material can permanently deform under load, exhibiting plastic behavior.
An object exhibiting elastic-plastic behavior has a combination of both material areas.

This article has the following sections:
1. Why use Neuber’s Rule?
2. Simcenter Testlab Neo and Strain Life
   2.1 Load Time History
   2.2 Stress Time History
   2.3 Strain Time History

1. Why use Neuber’s Rule?

An elastic-plastic strain time history is needed to calculate fatigue life, but sometimes only an elastic stress time history is available.  For example, when using a linear finite element model and applying forces, the predicted stresses are elastic (Figure 2).

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Figure 2: Left – Because the stress time history from a linear finite element model is solely elastic, it is necessary to use Neuber’s Rule to derive the corresponding elastic-plastic strain time. Right – Strain gauges measure elastic-plastic strain directly.

For each time point in an elastic stress time history, Neuber’s Rule calculates the corresponding elastic-plastic strain by keeping the red areas (plastic) and blue areas (elastic only) equivalent as shown in Figure 3.

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Figure 3: Neuber’s Rule converts an elastic stress/strain pair (blue dot) using the Young’s Modulus (slope of dashed green line) and converts it to equivalent plastic strain (red dot) using the elastic-plastic Ramberg Osgood relationship (solid green curved line).

Neuber’s Rule works as follows:

  • For a given elastic stress value taken from a time history, the corresponding point (blue) can be plotted on the Young’s Modulus line (which is used to describe elastic behavior). 
  • A corresponding point (red) that has equivalent area can then be found on the Ramberg-Osgood line (which describes non-linear plastic behavior).  This point equates the equivalent plastic stress and strain based on the original elastic stress and strain.

Neuber’s Rule allows a linear model to predict non-linear behavior!

See the “Strain Life Approach” knowledge article for more information on the Ramberg-Osgood relationship, which is used in the mapping of elastic to plastic behavior.

2. Simcenter Testlab Neo Strain Life

When using the Strain-Life method of Simcenter Testlab Neo, several different starting inputs are possible. These include load, stress, and strain as shown in Figure 4.
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Figure 4: Possible input time history blocks for the strain life method include load (top – red), stress (middle – green), and strain (bottom – blue).

Any input must be converted to elastic-plastic strain in order to perform fatigue life calculations.  The “Input Type” field of the Simcenter Testlab Neo Strain Life method (Figure 5) needs to be set appropriately.
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Figure 5: In the Strain Life method properties, the “Input Type” parameter (lower right) needs to match the time histories being processed. “Elastic-plastic strain” can be used with strain gauge measurements, while “Load” is used with stresses or forces.

Either “Load” or “Elastic-plastic strain” can be selected as input type:
  • Load: Selected if force or stress is the input time history.  These inputs will be converted to elastic-plastic stress using Neuber’s Rule and other settings.
  • Elastic-plastic strain: Selected if input time history is measured strain.  No conversion is needed.

2.1 Load Time History

If starting with an input type of load (force), conversion to stress (force over area) can be done two different ways (Figure 6):
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Figure 6: In the Strain Life Method of Simcenter Testlab Neo, if “Input Type” is “Load”, then “Load Influence” can be set to “Manual” or “Input base unit -> Material data base unit”.

The software has two settings: “Manual” and “Input Base Unit -> Material Base Unit”:

Manual (Figure 6)

If the load influence is set to Manual, “Influence factors” can be entered to scale a force time history.  There is both a stress influence factor and a load influence factor.

For example, consider a Finite Element Model. When a unit load is applied, a critical element has a stress to force sensitivity 5.03983 MPa/N as shown in Figure 7.
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Figure 7: Setting the Stress influence and Load influence based on Finite Element model properties.

If a force time history to be applied at the node is available for analysis in Simcenter Testlab Neo, then the Stress Influence factor should be set to 5.03983 MPa and the Load Influence factor should be set one Newton.  This force time history is converted to elastic stress for use with Neuber’s Rule.

Input Base Unit -> Material Data Base Unit (Figure 6)

When the influence coefficients are not known, the “Input Base Unit -> Material Base Unit” (Figure 6) can be used as a fallback.  The units of the input time history are reassigned the corresponding material database unit.  For example, 2,000,000 N load cycle will be converted to a 2,000,000 Pa (= 2 MPa) load cycle. This will not yield correct results in absolute terms but might be used for a relative comparison.

2.2 Stress Time History

If the input time block history is elastic stress (units of MPa) then the input type can be set to Load and Influence to Manual (Figure 8).
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Figure 8: Left – Simcenter Testlab Neo Material Data in MPa, Right - Settings for using MPa stress time history

The stress used in Simcenter Testlab Neo needs to be in units of MPa to match the material data information used in the Strain-Life method. If the stress time history is in MPa, the two influence factors are set to 1 (with units set to [MPa]).

Neuber’s Rule can then be used to calculate the corresponding elastic-plastic strain from this stress.

2.3 Strain Time History

If using a directly measured strain time history, the “Input Type” should be set to “Elastic-Plastic Strain”.  This can be used directly in the calculation of fatigue life using the Strain Life approach.

Hope this is helpful! Questions?  Email or contact Siemens Support Center.

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KB Article ID# KB000044429_EN_US



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