Design principles

Since the whole idea of the model is to tailor a function to experimental data, the authors have defined a certain group of features as desirable, even if not essential, among those:

• smoothness of the surface;
• possibility of changing the shape and thus the interpolation on a broad class of experimental data for different materials;
• possibility to represent known criteria with limit set of parameters;
• convexity of the surface.

Parametric function

Representation of coordinates ${\displaystyle p,q}$ and ${\displaystyle \theta }$ of the BP yield surface in Haigh Westergaard stress space.

Deviatoric sections of a BP yield surface.

Meridian sections of a BP yield surface.

The Bigoni–Piccolroaz yield criterion is a seven-parameter surface defined as:

${\displaystyle f(p,q,\theta )=F(p)+{\frac {q}{g(\theta )}}=0,}$

where p, q and ${\displaystyle \theta }$ are invariants dependent on the stress tensor, while ${\displaystyle F(p)}$ is the "meridian" function:

${\displaystyle F(p)=\left\{{\begin{array}{ll}-Mp_{c}{\sqrt {(\phi -\phi ^{m})[2(1-\alpha )\phi +\alpha ]}},&\phi \in [0,1],\\+\infty ,&\phi \notin [0,1],\end{array}}\right.}$

${\displaystyle \phi ={\frac {p+c}{p_{c}+c}},}$

describing the pressure-sensitivity and ${\displaystyle g(\theta )}$ is the "deviatoric" function:^[3]

${\displaystyle g(\theta )={\frac {1}{\cos[\beta {\frac {\pi }{6}}-{\frac {1}{3}}\cos ^{-1}(\gamma \cos 3\theta )]}},}$

describing the Lode-dependence of yielding.

The mathematical definitions of the parameters ${\displaystyle p,q}$ and ${\displaystyle \theta }$ are:

${\displaystyle p=-{\frac {tr~\sigma }{3}},\quad q={\sqrt {3{\textit {J}}_{2}}},\quad \theta ={\frac {1}{3}}cos^{-1}\left({\frac {3{\sqrt {3}}}{2}}{\frac {J_{3}}{J_{2}^{3/2}}}\right)}$

With:

${\displaystyle J_{2}={\frac {1}{2}}{\textbf {S}}\cdot {\textbf {S}},\quad {\textbf {S}}=\sigma -{\frac {tr\ \sigma }{3}}{\textbf {I}},\quad J_{3}={\frac {1}{3}}tr\ {\textbf {S}}^{3}}$

Where ${\displaystyle {\textbf {S}}}$ is the deviatoric stress, ${\displaystyle {\textbf {I}}}$ is the identity tensor, ${\displaystyle \sigma }$ is the stress tensor and the dot indicates the scalar product.

A better understanding of those important parameters can be grasped by using their geometrical representation in the Haigh–Westergaard stress space.^[4] Considering the tern of principal stresses and the deviatoric plane ${\displaystyle \pi }$, orthogonal to the trisector of the first quadrant and passing through the origin of the coordinate system, the tern ${\displaystyle p,q}$ and ${\displaystyle \theta }$ unequivocally represents a point in the space acting as a cylindrical coordinate system with the trisector as an axis:

• ${\displaystyle -{\sqrt {3}}~p}$ is the distance of the point from the deviatoric plane ${\displaystyle \pi }$;
• ${\displaystyle {\sqrt {2/3}}~q}$ is the distance from the trisector;
• ${\displaystyle \theta }$ represents the angle between the projections of ${\displaystyle q}$ and the axis ${\displaystyle \sigma _{1}}$ on the deviatoric plane ${\displaystyle \pi }$.

The usage of p and q instead of the correct cylindrical coordinates ${\displaystyle \xi }$ and ${\displaystyle \rho }$:

${\displaystyle \xi ={\frac {1}{\sqrt {3}}}tr(\sigma )=-{\sqrt {3}}~p,\quad \rho ={\sqrt {2~J_{2}}}={\sqrt {\frac {2}{3}}}q}$

is justified by the easier physical interpretation:^[5] p is the hydro-static pressure on the material point, q is the Von Mises equivalent stress.

The described yield function corresponds to the yield surface:

${\displaystyle q=-f(p)g(\theta ),\quad p\in [-c,p_{c}],\quad \theta \in [0,\pi /3]}$

which makes explicit the relation between the two functions ${\displaystyle f(p)}$ and ${\displaystyle g(\theta )}$ and the shape of the meridian and deviatoric sections, respectively.

The seven, non-negative material parameters:

${\displaystyle \underbrace {M>0,~p_{c}>0,~c\geq 0,~0<\alpha <2,~m>1} _{{\mbox{defining}}~\displaystyle {F(p)}},~~~\underbrace {0\leq \beta \leq 2,~0\leq \gamma <1} _{{\mbox{defining}}~\displaystyle {g(\theta )}},}$

define the shape of the meridian and deviatoric sections. In particular, some of the parameters are easily relatable to mechanical properties: ${\displaystyle M}$ controls the pressure sensitivity, ${\displaystyle c}$ and ${\displaystyle p_{c}}$ are the yield strength under isotropic conditions of tension and compression. The other parameters define the shape of the surface when intersected by the meridian and deviatoric planes: ${\displaystyle \alpha }$ and ${\displaystyle m}$ define the meridian section, ${\displaystyle \beta }$ and ${\displaystyle \gamma }$ define the deviatoric section.