6. Variable amplitude criteria

Variable amplitude criteria are implemented when loading is not periodic. When the loading is not periodic, it is necessary to decompose the loading path undergone by the structure into elementary sub-cycles using a cycle counting method. In the case where the load is non-radial, there is no proven multiaxial counting method. Consequently we choose, as in the literature, to use the counting method RAINFLOW [bib7] which requires a scalar as input. This is why we reduce the split, which is the orthogonal projection of the stress vector onto a plane, to one dimension, by projecting the tip of the split vector onto one or two axes. Another important difference with the critical plane criteria is that it is not the shear amplitude that makes it possible to select the critical plane but the accumulation of damage that results from the elementary sub-cycles.

The projection method we use is explained in chapters 7 and 8. In the following we describe how we have evolved the criteria of MATAKE and DANG VAN to adapt them to cases where the load is non-periodic.

6.1. MATAKE criterion changed

In the context of cumulative damage and periodic loading, the criterion of MATAKE [bib6], is written as follows:

\({\sigma }_{\text{eq}}\mathrm{=}{c}_{p}\frac{\Delta \tau ({\overrightarrow{n}}^{\text{*}})}{2}+a{N}_{\text{max}}({\overrightarrow{n}}^{\text{*}})\) eq 6.1-1

where \({\sigma }_{\text{eq}}\) represents the equivalent stress in the sense of the MATAKE criterion and with:

\({\overrightarrow{n}}^{\text{*}}\)	normal to the plane for which the shear amplitude is maximum;
\(\Delta \tau ({\overrightarrow{n}}^{\text{*}})\mathrm{/}2\)	maximum shear half-amplitude;
\(a\)	constant which can be defined by an alternating pure shear and alternating traction-compression test or by an alternating traction-compression and non-alternating traction-compression test;
\({N}_{\text{max}}({\overrightarrow{n}}^{\text{*}})\)	maximum normal stress on the plane of normal \({\overrightarrow{n}}^{\text{*}}\) during the cycle;
\({c}_{p}\)	harmful effect of pre-work hardening in controlled deformation \({c}_{p}\mathrm{\ge }1\).

To calculate the cumulative damage in the case where the load is non-periodic, the first step is to determine the split (shear vector) in a plane of normal \(\overrightarrow{n}\) at all times of the load. The technique that is used to do this is described in reference [bib6]. In the second step we start by reducing the history of the split to a one-dimensional function of time by projecting the tip of the split vector onto one or two axes defined in the normal plane \(\overrightarrow{n}\) under consideration, cf. chapters 7 and 8. Thus, the evolution of the projected split comes down to the relationship: \({\tau }_{p}\mathrm{=}f(t)\), which makes it possible to use the RAINFLOW counting method. In figure [Figure 6.1-a] we show the values reached by the end of the shear vector in a plane of normal \(\overrightarrow{n}\) before projection on one axis or two axes and in figure [Figure 6.1-b] these same values after projection on one axis. At this point we need to introduce the concept of elementary equivalent constraint \({\sigma }_{\text{eq}}^{i}\). Practically this concept has the same meaning as the concept of equivalent stress defined by the relationship [éq 6.1-1], but it applies to the elementary subcycles resulting from the counting method RAINFLOW. So from the projected split \({\tau }_{p}\) we calculate elementary equivalent constraints \({\sigma }_{\text{eq}}^{i}\).

Figure 6.1-a: Points of the cission vector before projection

Figure 6.1-b: Spikes of the cission vector after projection onto an axis

Method RAINFLOW breaks \({\tau }_{p}\mathrm{=}f(t)\) down into periodic elementary subcycles and breaks up the loading history, as shown in figure [Figure 6.1-c]. Thus, for a given normal \(\overrightarrow{n}\), the RAINFLOW method provides for each elementary sub-cycle two values, high and low points, of the tip of the cission vector \({\tau }_{{p}_{1}}^{i}(\overrightarrow{n})\) and \({\tau }_{{p}_{2}}^{i}(\overrightarrow{n})\) associated with two maximum normal stress values \({N}_{1}^{i}(\overrightarrow{n})\) and \({N}_{2}^{i}(\overrightarrow{n})\).

Figure 6.1-c: The fifteen elementary subcycles after treatment by method RAINFLOW

For the criterion of MATAKE we define the elementary equivalent stress in the following way:

\({\sigma }_{\text{eq}}^{i}(\overrightarrow{n})\mathrm{=}{c}_{p}\frac{\text{Max}({\tau }_{{p}_{1}}^{i}(\overrightarrow{n}),{\tau }_{{p}_{2}}^{i}(\overrightarrow{n}))\mathrm{-}\text{Min}({\tau }_{{p}_{1}}^{i}(\overrightarrow{n}),{\tau }_{{p}_{2}}^{i}(\overrightarrow{n}))}{2}+a\text{Max}({N}_{1}^{i}(\overrightarrow{n}),{N}_{2}^{i}(\overrightarrow{n})\mathrm{,0})\) eq 6.1-2

For the accumulation of damage, this elementary equivalent stress is to be used with a shear fatigue curve. If we use a fatigue curve in tensile compression we must multiply [éq6.1-2] by a corrective coefficient which corresponds to the ratio of the limits of endurance in alternating flexure and twisting and which we note \(\alpha\):

\({\sigma }_{\text{eq}}^{i}(\overrightarrow{n})\mathrm{=}\alpha ({c}_{p}\frac{\text{Max}({\tau }_{{p}_{1}}^{i}(\overrightarrow{n}),{\tau }_{{p}_{2}}^{i}(\overrightarrow{n}))\mathrm{-}\text{Min}({\tau }_{{p}_{1}}^{i}(\overrightarrow{n}),{\tau }_{{p}_{2}}^{i}(\overrightarrow{n}))}{2}+a\text{Max}({N}_{1}^{i}(\overrightarrow{n}),{N}_{2}^{i}(\overrightarrow{n})\mathrm{,0}))\) eq 6.1-3

From \({\sigma }_{\text{eq}}^{i}(\overrightarrow{n})\) and a Wöhler curve we deduce the number of cycles at break \({N}^{i}(\overrightarrow{n})\) and the elementary damage \({D}^{i}(\overrightarrow{n})\mathrm{=}1\mathrm{/}{N}^{i}(\overrightarrow{n})\) corresponding to an elementary subcycle. We use a linear accumulation of damage. Let \(k\) be the number of elementary subcycles, for a fixed normal \(\overrightarrow{n}\), the cumulative damage is equal to:

\(D(\overrightarrow{n})\mathrm{=}\mathrm{\sum }_{i\mathrm{=}1}^{k}{D}^{i}(\overrightarrow{n})\) eq 6.1-4

To determine the normal vector \({\overrightarrow{n}}^{\mathrm{\ast }}\) corresponding to the maximum cumulative damage simply vary \(\overrightarrow{n}\) and calculate [éq 6.1-4]. The normal vector \({\overrightarrow{n}}^{\mathrm{\ast }}\) corresponding to the maximum cumulative damage is then given by:

\(D({\overrightarrow{n}}^{\text{*}})\mathrm{=}\underset{\overrightarrow{n}}{\text{Max}}(D(\overrightarrow{n}))\)

6.2. Criteria of DANG VAN modified

In the context of damage and periodic loading, the criterion of DANG VAN is written as:

\({\sigma }_{\text{eq}}({\overrightarrow{n}}^{\text{*}})\mathrm{=}{c}_{p}\frac{\Delta \tau ({\overrightarrow{n}}^{\text{*}})}{2}+aP\)

where \({\sigma }_{\text{eq}}\) represents the equivalent stress in the sense of the DANG criterion VAN and with:

\({\overrightarrow{n}}^{\text{*}}\)	normal to the plane for which the shear amplitude is maximum;
\(\Delta \tau ({\overrightarrow{n}}^{\text{*}})\mathrm{/}2\)	maximum shear half-amplitude;
\(a\)	constant which can be defined by an alternating pure shear and alternating traction-compression test or by an alternating traction-compression and non-alternating traction-compression test;
\(P\)	maximum hydrostatic pressure during the cycle;
\({c}_{p}\)	harmful effect of pre-work hardening in controlled deformation \({c}_{p}\ge 1\).

When the load is non-periodic, we calculate the damage using the same process as that used for the MATAKE criterion. The only difference is in the definition of the elementary equivalent stress:

\({\sigma }_{\text{eq}}^{i}(\overrightarrow{n})\mathrm{=}\frac{\text{Max}({\tau }_{{p}_{1}}^{i}(\overrightarrow{n}),{\tau }_{{p}_{2}}^{i}(\overrightarrow{n}))\mathrm{-}\text{Min}({\tau }_{{p}_{1}}^{i}(\overrightarrow{n}),{\tau }_{{p}_{2}}^{i}(\overrightarrow{n}))}{2}+a\text{Max}({P}_{1}^{i}(\overrightarrow{n}),{P}_{2}^{i}(\overrightarrow{n})\mathrm{,0})\) eq 6.2-1

where \({P}_{1}^{i}\) and \({P}_{2}^{i}\) represent the two hydrostatic pressure values associated with each elementary subcycle. This elementary equivalent stress is to be used with a shear fatigue curve. If one must use a fatigue curve in tensile compression it is necessary to multiply [éq 6.2-1] by the corrective coefficient \(\alpha\):

\({\sigma }_{\text{eq}}^{i}(\overrightarrow{n})\mathrm{=}\alpha ({c}_{p}\frac{\text{Max}({\tau }_{{p}_{1}}^{i}(\overrightarrow{n}),{\tau }_{{p}_{2}}^{i}(\overrightarrow{n}))\mathrm{-}\text{Min}({\tau }_{{p}_{1}}^{i}(\overrightarrow{n}),{\tau }_{{p}_{2}}^{i}(\overrightarrow{n}))}{2}+a\text{Max}({P}_{1}^{i}(\overrightarrow{n}),{P}_{2}^{i}(\overrightarrow{n})\mathrm{,0}))\)

After having defined the criteria of MATAKE and DANG - VAN in the context of the accumulation of damage and non-periodic loading, we still have to specify the projection technique that we propose.

6.3. Criteria of FATEMI - SOCIE modified

6.3.1. Description

The criterion of FATEMI and SOCIE is a critical plan criterion [9], [10]. Initially formulated for periodic loads, we offer a version adapted to non-periodic loads.

In this criterion the parameter \(a\) is defined as follows: \(a\mathrm{=}k\mathrm{/}{\sigma }_{y}\) where where \({\sigma }_{y}\) is the elastic limit and \(k\) is a coefficient that depends on the material. We’ll come back to how to calculate \(k\). This criterion combines shear deformation and maximum normal stress. We propose to define an « elementary » equivalent deformation in the following way:

\({\varepsilon }_{\text{eq}}^{i}(\overrightarrow{n})\mathrm{=}\alpha ({c}_{p}\frac{\text{Max}({\gamma }_{{p}_{1}}^{i}(\overrightarrow{n}),{\gamma }_{{p}_{2}}^{i}(\overrightarrow{n}))\mathrm{-}\text{Min}({\gamma }_{{p}_{1}}^{i}(\overrightarrow{n}),{\gamma }_{{p}_{2}}^{i}(\overrightarrow{n}))}{2}\left[1+a\text{Max}({N}_{{1}_{}}^{i}(\overrightarrow{n}),{N}_{{2}_{}}^{i}(\overrightarrow{n}),0)\right])\)

where \({\gamma }_{{p}_{1}}^{i}\) and \({\gamma }_{{p}_{2}}^{i}\) represent the extreme shear deformations of subcycle number \(i\).

Apart from a different definition of the criterion, the approach used to calculate the damage is identical to the two previous criteria. Finally, it is also the maximum damage that makes it possible to select the critical plane.

Note that the shear deformations used in the criteria of FATEMI and SOCIE are distortions \({\gamma }_{\mathit{ij}}\) (\(i\ne j\)) *. If you use tensor-type shear deformations*\({ϵ}_{\mathit{ij}}\) (\(i\ne j\))**, you have to multiply them by a factor of 2 because**:math:`{gamma }_{mathit{ij}}=2{ϵ}_{mathit{ij}}`** . **

6.3.2. Identifying the coefficient k

The author proposes to identify the coefficient \(k\) from pure traction-compression and pure alternating torsional tests on a thin tube [9], [10]. In order not to introduce bias, both types of tests must be carried out on the same type of test piece. Before presenting the formula that defines \(k\) we introduce the following notations:

\(\nu\)	: Poisson’s ratio, (generally \(\mathrm{0,3}\) for our materials);
\({\nu }_{p}\)	: coefficient of incompressibility of plastic deformations (value \(\mathrm{0,5}\) in [9] and [10]);
\(E\)	: Young’s modulus of elasticity;
\(G\)	: Elastic shear modulus;
\({N}_{f}\)	: Number of cycles at break.

Contrary to the two previous criteria, this one can treat cases where elastoplastic areas remain in the structure. The coefficient \(k\) is defined by the following relationship:

\(k\mathrm{=}\left[\frac{\frac{{\tau }_{f}^{\text{'}}}{G}{({\mathrm{2N}}_{f})}^{{b}_{o}}+{\gamma }_{f}^{\text{'}}{({\mathrm{2N}}_{f})}^{{c}_{o}}}{(1+\nu )\frac{{\sigma }_{f}^{\text{'}}}{E}{({\mathrm{2N}}_{f})}^{b}+(1+{\nu }_{p}){\varepsilon }_{f}^{\text{'}}{({\mathrm{2N}}_{f})}^{c}}\mathrm{-}1\right]\frac{{k}^{\text{'}}{(\mathrm{0,}\text{002})}^{n\text{'}}}{{\sigma }_{f}^{\text{'}}{({\mathrm{2N}}_{f})}^{b}}\),

where the terms: \({\tau }_{f}^{\text{'}}\), \({b}_{o}\), \({\gamma }_{f}^{\text{'}}\), \({c}_{o}\), \({\sigma }_{f}^{\text{'}}\),,,,, \(b\),, \({\varepsilon }_{f}^{\text{'}}\), \(c\), \(k\text{'}\), and \(n\text{'}\) are defined by means of tests.

Pure traction-compression tests

Pure traction-compression tests make it possible to identify the coefficients: \({\sigma }_{f}^{\text{'}}\), \(b\),, \({\varepsilon }_{f}^{\text{'}}\), \(c\), \(k\text{'}\) and \(n\text{'}\).
\({\varepsilon }_{a}\mathrm{=}\frac{\Delta \varepsilon }{2}\mathrm{=}\frac{{\sigma }_{f}^{\text{'}}}{E}{({\mathrm{2N}}_{f})}^{b}+{\varepsilon }_{f}^{\text{'}}{({\mathrm{2N}}_{f})}^{c}\)
			The curves opposite use Log-Log scales.

Pure alternating torsional tests

Pure alternating torsional tests make it possible to identify the coefficients: \({\tau }_{f}^{\text{'}}\), \({b}_{o}\), \({\gamma }_{f}^{\text{'}}\), \({c}_{o}\).
\({\gamma }_{a}\mathrm{=}\frac{\Delta \gamma }{2}\mathrm{=}\frac{{\tau }_{f}^{\text{'}}}{G}{({\mathrm{2n}}_{f})}^{{b}_{0}}+{\gamma }_{f}^{\text{'}}{({\mathrm{2N}}_{f})}^{{c}_{0}}\)