Strain-Life (E-N) Approach

The Jiang-Sehitoglu model is one of the more successful incremental multiaxial cyclic plasticity models. ¹ In HyperLife, isotropic hardening part is removed from Jiang-Sehitoglu's original model, which is best described in deviatoric stress space.(1)

S_{i j} = σ_{i j} - α_{i j}

Yield Function

The Mises yield function,

F

, is expressed as:(2)

F = (\tilde{S} - \tilde{α}) : (\tilde{S} - \tilde{α}) - 2 k^{2} = 0

The notation $\tilde{S}$ is used for the deviatoric stress tensor, $\tilde{a}$ is the backstress tensor and $k$ is the yield stress is simple shear.

Flow Rule

The normality rule is given by:(3)

d {\tilde{ε}}^{p} = H 〈 d \tilde{S} : \tilde{n} 〉 \tilde{n}

Where,

{\tilde{ε}}^{p}

is the exterior unit normal to the yield surface at the loading point, as:(4)

\tilde{n} = \frac{\tilde{S} - \tilde{α}}{(\tilde{S} - \tilde{α}) : (\tilde{S} - \tilde{α})}

Hardening Rule

The total backstress

\tilde{α}

is divided into

M

parts.(5)

\tilde{α} = \sum_{i = 1}^{M} {\tilde{α}}^{i}

The evolution of the backstress (translation of the yield surface) for each of the parts is given by:(6)

d {\tilde{α}}^{i} = c^{i} r^{i} (\tilde{n} - \frac{{| {\tilde{α}}^{i} |}^{X^{i + 1}}}{r^{i}} \tilde{L}) d p

Where,

d p

is the equivalent plastic strain increment and

c^{i}

,

r^{i}

, and

X^{i}

are three sets of non-negative and single valued scalar functions:(7)

L^{i} = \frac{{\tilde{α}}^{i}}{| {\tilde{α}}^{i} |} (i = 1, 2, 3, ... M)

(8)

| {\tilde{α}}^{i} | = \sqrt{{\tilde{α}}^{i} : {\tilde{α}}^{i}} (i = 1, 2, 3, ... M)

(9)

d p = \sqrt{d ε^{p} : d ε^{p}}

Material memory is contained in the

a

terms. The plastic modulus function,

H

, is derived from the consistency condition which requires that the stress state be on the yield surface when plastic deformation is occurring.(10)

H = \sum_{i = 1}^{M} c^{i} r^{i} [1 - \frac{{| {\tilde{α}}^{i} |}^{X^{i + 1}}}{r^{i}} {\tilde{L}}^{i} : \tilde{n}] + \sqrt{2} \frac{d k}{d p}

There are four material parameters in this model, $c$ , $r$ , $X$ , and $k$ . All of these constants are computed from the cyclic stress strain curve of the material (Equation 13). A simple power function is fit to this curve to obtain three material properties; cyclic strength coefficient, $K'$ , cyclic strain hardening exponent, $n'$ , and elastic modulus, $E$ .

The shear yield strength, $k$ , is obtained by setting the plastic strain to 0.0002 (0.02%) and dividing by $\sqrt{3}$ . Both $c$ and $r$ are obtained by selecting a series of stress strain pairs along the material cyclic stress strain curve and describe the shape of the curve. Ratcheting rate is controlled by $X$ which is set at a fixed value of 5. The number of components ( $M$ ) is set to 10.

Non-Proportional Hardening

Non-proportional hardening is the term used to describe loading paths where the principal strain axes rotate during cyclic loading. The simplest example is a bar subjected to alternating cycles of tension and torsion loading. Between the tension and torsion cycles the principal axis rotates 45 degrees. Out-of-phase loading is a special case of non-proportional loading and is used to denote cyclic loading histories with sinusoidal or triangular waveforms and a phase difference between the loads. Materials show additional cyclic hardening during this type of loading that is not found in uniaxial or any proportional loading path.

The 90 degrees out-of-phase loading path has been found to produce the largest degree of non-proportional hardening. The magnitude of the additional hardening observed for this loading path as compared to that observed in uniaxial or proportional loading is highly dependent on the micro-structure and the ease with which slip systems develop in a material. A non-proportional hardening coefficient,

a

, can be introduced which is defined as the ratio of equivalent stress under 90 degrees out-of-phase loading/equivalent stress under proportional loading at high plastic strains in the flat portion of the stress-strain curve. This term reflects the maximum degree of additional hardening that might occur for a given material. HyperLife measures the non-proportional hardening coefficient,

a

, at 0.003 strain amplitude using:(11)

α = \frac{K^{'} 90 \times {0.003}^{n^{'} 90}}{K^{'} \times {0.003}^{n^{'}}}

Where,(12)

K^{'} 90 = K^{'} * c o e f k p 90

(13)

n^{'} 90 = n^{'} * c o e f n p 90

You will need to input the coefkp90 and coefnp90. The default value of coefkp90 is 1.2. Default value of coefnp90 is 1.0.

The degree of non-proportionality,

C

, is a fourth ranked tensor which is a function of the plastic strain and the direction of the plastic strain increment.(14)

d C_{i j k l} = C_{c} (n_{i j} n_{k l} - C_{i j k l}) d p

The amount of non-proportional hardening is computed from Tanaka's parameter. ²

(15)

A = \sqrt{1 - \frac{n_{α β} C_{ξ σ α β} C_{ξ σ γ η} n_{γ η}}{C_{i j k l} C_{i j k l}}}

For any proportional loading

A = 0

and for 90 degrees out-of-phase loading A = 1/sqrt(2). The material constants,

r^{i}

and the yield stress

k

are related to the non-proportional hardening parameter as:(16)

\begin{array}{l} d k = b (k_{c} - k) d p \\ d r^{i} = b (r_{c}^{i} - r^{i}) d p \\ k_{c} = k_{o} ((N_{p} - 1) A + 1) \\ r_{c}^{i} = r_{o}^{i} ((N_{p} - 1) A + 1) \\ N_{p} = \sqrt{2} (α - 1) + 1 \end{array}

The rate of hardening is controlled by the constant $b$ . Since you are interested in the stable solution for fatigue calculations, $b$ = 5 is selected for numerical stability. These equations are now sufficient to solve for the stresses and strains under any arbitrary loading path. The solution now proceeds by incrementally solving these equations, which can be solved in either stress, or strain control. The initial conditions are that the stress and backstress start at. All material constants start at their initial values as well and are updated after each loading increment.

Notch Correction

In uniaxial loading approximate solutions such as Neuber's rule are used to compute the stresses and strains during plastic deformation. Unfortunately, Neuber's rule cannot be directly extended to multiaxial loading because there are six unknowns and only five equations. To overcome this problem Koettgen ³ proposed a structural yield surface to obtain local elastic-plastic stresses and strains. Having defined a "yield surface," standard cyclic plasticity methods can be used to solve the unknown stresses and strains at the notch. The material memory effects are built directly into standard cyclic plasticity calculations. The method is based on the same concept at the analytical or experimental nominal stress - notch strain curves used in uniaxial fatigue analysis such as the one shown below.

Cyclically, nominal stress - notch root strain response has all the features associated with stress-strain response such as hysteresis, memory, cyclic hardening and softening. The concept of pseudo stress,

σ^{e}

, or strain,

ε^{e}

, is defined as theoretical elastic stress or strain computed using elastic assumptions. Neuber's rule may be written in terms of pseudo stress as:(17)

σ ɛ = σ^{e} ɛ^{e}

These concepts can be generalized to three-dimensional stress and strain states with a structural yield surface at a notch. Having defined a structural yield surface, standard cyclic plasticity methods can be used to solve the unknown stresses and strains at the notch. For multiaxial loading, the total stress or strain is obtained by superposition of the individual load cases. A relationship between pseudo stress and notch strain is described with a power function that has the same form as a stress-strain curve.(18)

ε = \frac{σ^{e}}{E} + {(\frac{σ^{e}}{K^{*}})}^{\frac{1}{n^{*}}}

Where, the constants $K *$ and $n *$ represent the behavior of the structure rather than the material. Uniaxial Neuber's rule is employed to establish the constants $E *$ , $K *$ and $n *$ .

For plane stress on the surface,

σ_{z}^{e}

,

τ_{y z}^{e}

and

τ_{x z}^{e}

are all zero and this yield function can be written as:(19)

\frac{1}{\sqrt{2}} \sqrt{{(σ_{x}^{e} - σ_{y}^{e})}^{2} + {(σ_{x}^{e})}^{2} + {(σ_{y}^{e})}^{2} + 6 {(τ_{x y}^{e})}^{2}} = σ_{y s}

This equation defines a structural yield function,

F_{o}

, which has the same form as the yield functions used in the plasticity models for stress-strain calculations, that is, a Mises yield function in terms of pseudo stress.(20)

F_{o} (σ_{i j}^{e}) = σ_{y s}

The process to calculate stress from pseudo stress is summarized as:

Apply uniaxial Neuber's rule to get $K *$ and $n *$ (Equation 19).
Run Jiang-Sehitoglu plasticity model in pseudo stress control with pseudo constants $K *$ and $n *$ to obtain the pseudo stress-local strain response. Now total strain is available.(21)
$△ ε = [f (K^{*}, n^{*}, E^{*})] △ σ^{e}$
Run Jiang-Sehitoglu plasticity model in strain control with material constants $K$ and $n$ to obtain the local strain - local stress response. Finally, stress is also available.(22)
$△ σ = [f (K, n, E)] △ ε$