3. Rice and Tracey’s model#

We are now interested in the case of ductile priming. Considering an element with an initially healthy volume, the ductile tear of this element results from the following elementary mechanisms:

  • nucleation of cavities caused by the decohesion of inclusions present in the material,

  • growth then coalescence of these cavities.

3.1. Isolated cavity in an infinite rigid plastic matrix#

In an analytical understanding approach, Rice and Tracey studied the behavior of a cavity, initially spherical (surface \({S}_{v}\)), isolated in an infinite isotropic medium (volume \(V\)), whose behavior is rigid plastic (elastic limit \({\sigma }_{0}\)), incompressible, incompressible, subjected to infinity at any rate of deformation \({\dot{\varepsilon }}^{\infty }\) (stress noted \({\sigma }^{\infty }\) at infinity). They show that the movement speed field, the solution of the mechanical problem posed, minimizes the functional:

(3.1)#\[ Q (\ dot {u}) =\ underset {V} {\ int} {\ int}\ left [{s}} _ {\ text {ij}} (\ dot {\ mathrm {\ epsilon}}}) - {s} _ {\ text {ij}}} {\ int}}\ left [{\ int}}}\ left [{s}} _ {\ epsilon}}} _ {text {ij}}\ text {dV} - {\ mathrm {\ sigma}}} _ {\ text {ij}} ^ {\ mathrm {\ infty}}\ underset {{S} _ {v}} {\ v}}} {\ int}}} {\ int} {v}}} {\ int} {v}}} {\ int} {v}}} {\ int} {n}} _ {j}\ text {v}}} {\ int} {v}}} {\ int} {v}}} {\ int} {v}}} {\ int} {n} {n} _ {d}}\]

3.2. Approximate law of cavity growth#

By managing to minimize this functional in various situations, Rice and Tracey then showed the predominant influence of the triaxiality rate \(\frac{{\mathrm{\sigma }}_{m}^{\mathrm{\infty }}}{{\mathrm{\sigma }}_{\text{eq}}^{\mathrm{\infty }}}\) (with \({\mathrm{\sigma }}_{m}^{\mathrm{\infty }}\) the trace and \({\mathrm{\sigma }}_{\text{eq}}^{\mathrm{\infty }}\) the VonMises equivalent of the stress imposed on the volume element in question) on the growth rate of the cavities.

They even show a law of cavity growth, which is certainly similar, but very close to the results of the previous model. Thus, in each of the main directions \((K)\) associated with the deformation rate \({\dot{\mathrm{\epsilon }}}^{\mathrm{\infty }}\), the rate of elongation of a cavity amounts to:

(3.2)#\[ {\ dot {R}} _ {K} =\ left\ {\ mathrm {\ mathrm {\ chi}} {\ dot {\ mathrm {\ epsilon}}}} _ {K} ^ {\ mathrm {\ infty}}} + {\ dot {\ mathrm {\ infty}}} + {\ dot {\ mathrm {\ infty}}} + {\ dot {\ mathrm {\ infty}}} + {\ dot {\ mathrm {\ infty}}} + {\ dot {\ mathrm {\ infty}}} + {\ dot {\ mathrm {\ infty}}} + {\ dot {\ mathrm {\ infty}}} + {\ dot {\ mathrm {\ infty}}\ right\} {R} _ {K}\]

For a main direction \(K\), the parameter \({R}_{K}\) is the radius of the cavity, \({\dot{\varepsilon }}_{K}^{\infty }\) the main value of the deformation rate imposed at infinity and \({\dot{\varepsilon }}_{\text{eq}}^{\infty }\) the equivalent von Mises value of the deformation rate imposed at infinity. This relationship in which the coefficients \(\khi\) and \(D\) depend on the situation under consideration:

  • \(\khi =\frac{5}{3}\) for a linear work-hardening matrix or a perfectly plastic matrix with a low triaxiality rate or \(\khi =2\) in the case of a perfectly plastic matrix with a high triaxiality rate,

  • \(D=\mathrm{\alpha }\text{exp}(\frac{3{\sigma }_{m}^{\infty }}{2{\sigma }_{0}})\) for a perfectly plastic die or \(D=\frac{{\sigma }_{m}^{\infty }}{4{\sigma }_{{}^{\text{eq}}}^{\infty }}\) for a linear work hardening die.

\(\alpha =\mathrm{0,}\text{283}\) is the value given by Rice and Tracey whereas more precise calculations (cf. [bib4]) have shown that this coefficient is higher (\(\mathrm{\alpha }=\mathrm{1,}\text{28}\)).

Mudry then proposed to apply these theoretical results to the case of tank steel, i.e.:

  • intermediate behavior between the extreme cases of behavior studied by Rice and Tracey with non-zero but reasonable work hardening,

  • cracked structures (high triaxiality rate).

From this, he deduced the following approximate law, valid for sufficiently high triaxiality rates (greater than 0.5):

(3.3)#\[ \ dot {R} =\ mathrm {\ alpha} {\ alpha} {\ dot {\ varepsilon}}} _ {\ text {eq}} ^ {\ infty}}\ text {exp} (\ frac {3 {\ sigma} (\ frac {3 {\ sigma}} {\ sigma}}}\ text {exp}} (\ frac {3 {\ sigma}}) (\ frac {3 {\ sigma}} _ {\ exp} (\ frac {3 {\ sigma}}) (\ frac {3 {\ sigma}} _ {\ text {eq}}} (\ frac {3 {\ sigma}}) ^ {\ infty}}) R\]

In which:

  • \({\dot{\varepsilon }}_{\text{eq}}^{\infty }\) has been replaced by \({\dot{\varepsilon }}_{\text{eq}}^{{p}^{\infty }}\) (equivalent (von Mises) of the plastic part of the deformation rate) in order to extend Rice and Tracey’s law to the elastoplastic case,

  • the elastic limit \({\mathrm{\sigma }}_{0}\) has been replaced by \({\sigma }_{\text{eq}}^{\infty }\) in order to take into account the hardening of the matrix around the cavity.

Experimental measurements of porosity growth for various levels of triaxiality made it possible to validate this expression. These results show that, when the initial porosity level remains low, the exponential nature of the relationship between the radius of the cavities and the triaxiality rate is well confirmed. On the other hand, the coefficient \(\alpha\) depends on the material in question as well as on the initial porosity fraction.

3.3. Ductile priming criterion#

\({R}_{0}\) and \(R(t)\) designating the radius of the initial cavities and at the instant \(t\) in question, the ductile priming criterion adopted here is:

(3.4)#\[ \ frac {R (t)} {{R} _ {0}}} = {(\ frac {R} {{R} _ {0}})}}} _ {c}\]

The first member of this expression results from the integration of the law of growth, in accordance with the indications in the previous paragraph.

Several arguments of principle can be objected to the direct use of this Rice and Tracey cavity growth law as a criterion for ductile priming. So:

  • the inclusions, and therefore the cavities, are not in reality isolated. Worse, they are often grouped together in clusters,

  • the coalescence of cavities is probably the result of interactions that, too, are not described in the established model,

  • in a cracked structure, the presence of gradients at the bottom of the crack makes the previous analysis, which concerns an infinite medium subject to homogeneous boundary conditions, less directly applicable.

However, using the previous criterion, it is hoped that this law remains realistic, on average, even in clusters or in areas with strong gradients (averaged over an element with dimensions comparable to that of the Beremin model). Moreover, it is hypothesized that the critical size selected, in general adjusted to given geometries (CT test piece, for example), reflects coalescence, which is equivalent to assuming that coalescence does not depend too much on the nature of the mechanical stresses imposed on the volume element (triaxiality, shear, etc.).

Finally, note that the Rice and Tracey model is only an approximate law, valid for significant triaxiality rates (i.e. greater than 0.5).

3.4. Implementation in code_aster#

Let us consider domain \({\Omega }_{c}\) of the studied structure, which can be the whole of the studied mesh, a group of elements or a mesh. Following an elastoplastic thermomechanical calculation, the evolution of stress, deformation and plastic deformation fields in this field is known and it is desired to determine the spatial and temporal variations in the growth rate of cavities in this domain.

To do this, we use the RICE_TRACEY keyword from the POST_ELEM command.

At each Gauss point in domain \({\Omega }_{c}\), the stresses and rates of deformation calculated at each moment are assimilated to the quantities applied to the infinite medium considered previously. Rice and Tracey’s growth law is thus integrated step by step using the following approximate formula:

(3.5)#\[ \ text {Log} (\ frac {R ({t} _ {n})} {{R} _ {0}}) =\ text {Log} (\ frac {R ({t} _ {n-1})} {{R} _ {n-1})}} {{R} _ {n})}} {{R} _ {n})}} {{R} _ {n})} +\ mathrm {0}}}) +\ mathrm {0}}}) +\ mathrm {0}}}) +\ mathrm {0}}}) +\ mathrm {0}}}) +\ mathrm {0}}}) +\ mathrm {0}}}) +\ mathrm {0}}) +\ mathrm {} ({t} _ {n})} {{\ sigma} _ {\ text {eq}}} ({t} _ {n})})\ text {Exp} (\ mathrm {1.5}\ text {.} \ mid\ frac {{\ sigma} _ {m} ({t} _ {n})} {{\ sigma} _ {\ sigma}} ({t} _ {n})}\ mid) ({\ varepsilon} ({\ varepsilon}} _ {\ silon}} _ {n} _ {n}) - {\ varepsilon} ({\ varepsilon}} _ {\ text}} _ {\ text}} {eq}} ^ {p} ({t} _ {n-1}))\]

The values of the \(\frac{R}{{R}_{0}}\) ratio are thus obtained at each moment at each Gauss point in the \({\Omega }_{c}\) domain, the sign of the triaxiality ratio allowing changes in both tension and compression to be taken into account. Two functionalities are then offered in code_aster: the maximum value and the average value of the growth rate.

3.4.1. Finding the maximum value of the growth rate#

At each moment, we search for the entire domain \({\Omega }_{c}\) for the Gauss point (and the volume of the associated subgrid) maximizing \(\frac{R}{{R}_{0}}\).

3.4.2. Calculating the average value of the growth rate#

By quadrature on each cell and then averaging over the target domain \({\Omega }_{c}\), the average value of \(\frac{R}{{R}_{0}}\) over \({\Omega }_{c}\) is deduced at each moment.

As in the case of the Weibull model, a variant is introduced: the previous temporal integration is then carried out on the basis of the stress and the mean plastic deformation per mesh.