Local stress-strain behavior

Modeling inelastic stress-strain behavior is a key step in determining fatigue damage in the strain-life approach.

For uniaxial loading of metals basic observations include yield point, unloading and reloading, and Masing behaviors. Multiaxial loading and material behavior will be discussed in Advanced topics.

For monotonic loading of the material, the yield point is the stress level beyond which there is permanent, plastic deformation. The post-yield behavior can often be described by two parameters, K and n, which are the hardening coefficient and hardening exponent of the material. The strain can be determined by the Ramberg-Osgood relation for monotonic loading:

The first term on the right hand side of the equation is the elastic or recoverable strain, and the second term represents the plastic or permanent strain.

ε = εe + εp

Upon unloading from the stress-strain curve, followed by subsequent re-loading, the material re-joins the initial stress strain curve as the stress level is further increased. This process is described as material memory or hysteresis, because the material has "remembered" where it left the stress-strain curve. Monotonic loading interrupted by two changes of the loading direction forming a hysteresis loop are illustrated in the following figure.

Hysteresis Loop

Most metals approach a cyclically stable state after a certain amount of cycles, and usually cyclically stable (or half-life) material properties are used in a fatigue analysis.

The cyclic stress-strain curve is constructed by connecting the tips of stable hysteresis loops and using the Ramberg-Osgood relation to determine the cyclic constants, K' and n', the cyclic hardening coefficient and cyclic hardening exponent:

Cyclic Stress-Strain Curve

These cyclic material properties and the cyclic yield strength are often different from the monotonic values. Together with the elastic modulus of the material and the Masing and memory model, they are used to define the cyclic uniaxial response of the material, which determine the stress and strain range of the closed hysteresis loops and the mean stress associated with each loop.

The Masing model

Masing behavior is exhibited by materials that behave in a similar fashion in tension and compression. Masing’s hypothesis is that the branches of a hysteresis loop can be described by doubling the cyclic stress-strain curve. That is, if the cyclic stress-strain curve can be expressed by

ε = g(σ)

then the hysteresis branch can be described by

Δε = 2g(Δσ / 2)

with the origin of the hysteresis branch shifted to the tips of the loops. For the specific Ramberg-Osgood form of the cyclic stress-strain curve,

The hysteresis branches described by the Masing hypothesis are illustrated in the following figure.

Masing Hypothesis

The memory model

Another hypothesis that we assume for the identification of cycles are the memory rules. These rules are the same that are assumed for the rainflow counting algorithm:

Memory Rules

• Memory1 (M1)Hysteresis at the basic curve, not nested inside another hysteresis loop: After reaching the basic curve again (closing the loop) the path then follows the basic curve again.

• Memory2 (M2)Hysteresis nested inside another hysteresis loop: After reaching the outside hysteresis loop again (closing the loop) the path follows the outside hysteresis loop again.

• Memory3 (M3)Hysteresis branch that started from the basic curve touches the basic curve in the opposite quadrant. In this case no loop is built up but the path follows the basic curve.

Parameter	Meaning	Unit
E	Young's Modulus	MPa
K'	Cyclic Hardening Coefficient	MPa
n'	Cyclic Hardening Exponent	1

Learn more

Constant amplitude life curves

Endurance limit and static failure

Determining material properties

Mean stress and damage parameters

Axial versus torsion tests

Notch analysis

The strain-life analysis in Specialist Durability

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