3. Behavioral relationship of mechanisms#
The behavior of mechanisms 1 and 2 is similar. It is non-linear between a rigid initial tangent behavior and an asymptotic limiting behavior.
It is described by two essential parameters: the nonlinearity parameter and the boundary surface parameter.
The stop (mechanism 1) or the ruin (mechanism 2) are described by an associated kinematic criterion.
3.1. Unidirectional behavior#
We told [§2] that unidirectional behaviors during normal effort and around time are similar [Figure 2-b].
They can be described by the same relationship if the dimensionless quantities are used:
reduced forces: \(n=\frac{{N}_{x}}{\overline{N}}\text{et}m=\frac{{M}_{y}}{\overline{M}}\)
reduced trips: \({U}_{r}=\frac{U}{\overline{U}}\text{et}{\theta }_{r}=\frac{\theta }{\overline{\theta }}\)
The [Figure 3.1-a] represents unidirectional behavior in dimensionless form. Analytically, it can be written (it is a choice):
\(\begin{array}{c}\begin{array}{}{U}_{r}=h(n)\text{ou}{\mathrm{\theta }}_{r}=h(m)\\ \text{avec}h(x)=\frac{1}{\overline{d}}\frac{{x}^{a+1}}{1-{x}^{a}}\\ \overline{d}=\frac{{\overline{n}}^{a+1}}{1-{\overline{n}}^{a}}\end{array}\end{array}\)
\(a\) is the scalar nonlinearity parameter. \(\overline{n}\) and \(a\) are identified on unidirectional tests. \(\overline{n}\) that takes into account the variability of the trials generally takes the value \(0.95\).
Figure 3.1-a: assembly behavior relationship
Note that \(h(\overline{n})=1\) or \(h(\overline{m})=1\), that is to say: \({U}_{r}=1\) or \({\mathrm{\theta }}_{r}=1\), or again: \(U=\overline{U}\) or \(\mathrm{\theta }=\overline{\mathrm{\theta }}\).
The unidirectional kinematic criterion is therefore verified for \(n=\overline{n}\) or \(m=\overline{m}\).
3.2. Incremental two-dimensional behavior#
The boundary coupling is defined by the boundary surface:
\({(\frac{{N}_{x}}{\overline{N}})}^{2}+{(\frac{{M}_{y}}{\overline{M}})}^{2}=1\)
Unidirectional behavior in reduced variables is described by the [§3.1] relationship:
\(\begin{array}{c}\underline{d}\mathrm{=}h(\underline{f})\end{array}\)
where |
\(\underline{d}\) is the reduced displacement vector \((\begin{array}{}{U}_{r}\\ {\mathrm{\theta }}_{r}\end{array})\) |
\(\underline{f}\) is the reduced forces vector \((\begin{array}{}n\\ m\end{array})\) |
In two-dimensional behavior, isotropy is translated by a model with an internal variable scalar \(p\) such as:
\(\begin{array}{c}p=h(\text{feq})\text{en chargement}\end{array}\)
where \(\text{feq}\) is the equivalent reduced force (scalar).
\(\text{feq}\) is defined as:
\(\underline{F}=\text{feq}\ast \underline{\overline{F}}\ast\)
The expression for \(\text{feq}\) is derived from the expression for the boundary surface. \(\begin{array}{}\underline{\overline{F}}\ast \\ \end{array}\)”s membership in the boundary surface is written as:
\({(\frac{{\overline{N}}_{x}^{\text{*}}}{\overline{N}})}^{2}+{(\frac{{\overline{M}}_{y}^{\text{*}}}{\overline{M}})}^{2}=1\)
By the definition of \(\text{feq}\), we can write:
\({(\frac{{N}_{x}}{\text{feq}\overline{N}})}^{2}+{(\frac{{M}_{y}}{\text{feq}\overline{M}})}^{2}=1\)
that is to say according to the reduced forces \(n\) and \(m\):
\({(\frac{n}{\text{feq}})}^{2}+{(\frac{m}{\text{feq}})}^{2}=1\)
Hence \(\begin{array}{c}\text{feq}=\sqrt{{n}^{2}+{m}^{2}}\end{array}\)
The loading surface \(F\), which is homothetic to the boundary surface, is then defined by:
\(\begin{array}{c}\begin{array}{cc}F:& \text{feq}-R(p)=0\\ & \text{où}R(p)={h}^{-1}(p)\end{array}\end{array}\)
For a formalism similar to that of isotropic work hardening plasticity [bib2], we obtain the continuous behavior relationship expressed in reduced quantities:
\(\begin{array}{c}\begin{array}{}\underline{\stackrel{\text{.}}{d}}=\stackrel{\text{.}}{p}\frac{\partial F}{\partial \underline{f}}=\stackrel{\text{.}}{p}\frac{\underline{f}}{\text{feq}}\\ \stackrel{\text{.}}{p}=0\text{si}\text{feq}-R(p)<0\\ \stackrel{\text{.}}{p}=h\text{'}(\text{feq})\stackrel{\text{.}}{f}\text{eq}\text{si}\text{feq}-R(p)=0\end{array}\end{array}\)
The rigid - plastic behavior relationship without elasticity is finally written:
\(\begin{array}{}\underline{\stackrel{\text{.}}{D}}=\frac{\stackrel{\text{.}}{p}}{\text{feq}}\left[\overline{D}\right]{\left[\overline{F}\right]}^{\text{-}1}\underline{F}\\ \text{où}\underline{D}=(\begin{array}{}U\\ \theta \end{array})\mathrm{et}\underline{F}=(\begin{array}{}{N}_{x}\\ {M}_{y}\end{array})\\ \left[\overline{D}\right]=\left[\begin{array}{cc}\overline{U}& 0\\ 0& \overline{\theta }\end{array}\right]\mathrm{et}\left[\overline{F}\right]=\left[\begin{array}{cc}\overline{N}& 0\\ 0& \overline{M}\end{array}\right]\end{array}\)
The incremental behavior relationship in reduced quantities is obtained by integrating the continuous relationship between \(t\) (variables -) and \(\text{t+ dt}\) (variables +).
While loading, \(\mathrm{\Delta \; p}\) checks \(F\) = 0 to \(t\) + \(\text{dt}\):
\(\begin{array}{c}{\text{feq}}^{\text{+}}=R({p}^{\text{-}}+\mathrm{\Delta \; p})\end{array}\) eq 2.2-1
By introducing the behavioral relationship,
\(\begin{array}{c}\mathrm{\Delta }\underline{d}=\mathrm{\Delta \; p}\frac{{\underline{f}}^{\text{+}}}{{\text{feq}}^{\text{+}}}\end{array}\) eq 2.2-2
we deduct the value of \(\mathrm{\Delta \; p}\),
\(\begin{array}{c}\mathrm{\Delta \; p}=\parallel \mathrm{\Delta }\underline{d}\text{.}\mathrm{\Delta }\underline{d}\parallel =\sqrt{{\mathrm{\Delta \; U}}_{r}^{2}+{\mathrm{\Delta \; \theta }}_{r}^{2}}\end{array}\)
and we calculate the value of \({\text{feq}}^{\text{+}}\) by [éq 2.2-1]. The behavioral relationship [éq 2.2-2] results in reduced efforts:
\(\begin{array}{c}\begin{array}{}{n}^{\text{+}}=\frac{{\mathrm{\Delta \; U}}_{r}}{\mathrm{\Delta \; p}}R({p}^{\text{-}}+\mathrm{\Delta \; p})\\ {m}^{\text{+}}=\frac{{\mathrm{\Delta \; \theta }}_{r}}{\mathrm{\Delta \; p}}R({p}^{\text{-}}+\mathrm{\Delta \; p})\end{array}\end{array}\)
When unloading, \(\mathrm{\Delta \; p}\) = 0 and we have by [éq 2.2-2]:
\(\mathrm{\Delta }\underline{d}=0\)