The mechanical behaviour of many non-metallic materials such as soils, rocks, but also concrete and masonry, is linked to resistant frictional phenomena dependent on the hydrostatic component of the stress tensor. Consequently, the breaking surface, which can be expressed in forms
or
, will be dependent on the first invariant
or on the coordinate
. In addition, to describe the surface in the three-dimensional space of tensions, a single deviatoric cross-section will not be sufficient, given the variability of this along the hydrostatic axis. From this perspective, the study of surface meridians in the meridian plane is fundamental
varying the angle
. For isotropic materials, the conformation of the deviatoric section is characterized by triple symmetry, as shown in
Figure 3. Consequently, both experimental and analytical characterization can be carried out on the single sector with an angle θ between 0° and 60°.
By sorting the principal tension by
, it is possible to identify the following limit cases:
corresponding to the sectors
and
, respectively. The meridian corresponding to the angle
is called “meridian of compression”, being characterized by compressive stress superimposed on the spherical stress state, in one direction, while the meridian corresponding to
is called “meridian of traction” being characterized by tensile stress superimposed on the deviatoric stress state, in one direction.
3.2.1. Mohr-Coulomb Failure Criterion
The Mohr-Coulomb failure criterion, also called the internal friction criterion, can be seen as a generalization of the Tresca criterion [
13]. Both identify the cause of the failure in reaching the maximum shear stress, but in the first case, the limit value is constant, while in the second the limit shear stress is a function of the normal stress
. Consequently, it can be assumed that the crisis occurs when, fixed a value of the normal stress
on a plane, the tangential stress reaches the value:
where
c is the cohesion and
is the internal friction angle (
Figure 4); both are constants of the material to be determined experimentally.
Referring to a pluriaxial stress state, it is evident that the maximum value of the tangential stress τ is drawn from the plane representative of the greater of the three Mohr circles. Sorting the main stresses so that it turns out , and are respectively maximum and minimum main tension and the representative circle is the one characterized by a radius equal to . For the considerations and assumptions made, the influence of the intermediate main stress is excluded .
From
Figure 4 the radius of the circle can also be expressed as:
where
. The failure criterion can therefore be rewritten in the following form:
The equations of six planes can be defined in the space of principal stresses
, whose envelope is a pyramid with a vertex in the point
, representative of the triaxial tensile strength and axis coinciding with the hydrostatic axis. Considering all the possible differences we obtain:
From Equation (24) further information can be obtained, for example by referring to the limit circles corresponding to the uniaxial tensional states of traction and compression (
. Highlighting the main tensions in the (24) it consequences that:
and, therefore:
from which results:
Consequently, Equation (29) can be rearranged in the form:
from which it is deduced the dependence of the conformation of the domain, in the case of a plane state of tension (in the specific case
from the relationship between the monoaxial tensions of traction and compression
, as reported in
Figure 5. From
Figure 5, it is observed that for values of
equal to 1, the Tresca criterion and the Mohr-Coulomb criterion are coincident and defined by the same symmetrical hexagon with respect to the bisectors of the orthogonal reference system considered.
The equation of the failure surface in the stress space can also be rewritten as a function of the invariants
and of the angle θ or also in the coordinates
remembering that
represent the axes of the plane of meridians. To do this, Equations (13) and (14) can be substituted in Equation (28), obtaining:
where
.
These relations are able to represent the irregular pyramid with a hexagonal base, exploiting the symmetry of the domain, making it sufficient to study a single sector of . Once the apex of the pyramid is known, with coordinates , it is sufficient to define the deviatoric section for ξ, defined by the radii and obtained through the intersection of the meridians in traction and compression with the axis of equation .
By considering that
and
in the Equation (34):
and the ratio of these lengths (also called flow stress ratio) is given by:
The ratio
is fundamental for the Drucker-Prager criterion to calibrate the parameters on the base of those characterising the Mohr-Coulomb criterion (
Figure 6).
3.2.2. Drucker-Prager Failure Criterion
To exploit the analytical and computational advantages of a representation of the breaking domain via a regular surface, the Drucker-Prager (DP) criterion was formulated in 1952, which can be seen as a generalized version of the Von Mises criterion, in which the dependence on the hydrostatic component of the stress tensor is manifested by adding a term to the relation proposed by Von Mises [
11,
13]. Consequently, the criterion can be written in the following form:
or by using the variables
as:
where
and
k depend on the material and must be determined experimentally. If the parameter
is equal to 0 the criterion coincides with the Von Mises one. In terms of main stresses, the criterion can be written in the following form:
The relationship defines a cone with a vertex at the point in the principal coordinate space
and characterized by circular sections on the deviatoric planes with radii linearly dependent on the hydrostatic tension (
Figure 7). The parameters
e
k can be related to the monoaxial tension state in traction and compression according to the following relations:
The second relationship, in particular, places limitations on the value assumed by the parameter α, which must be less than √3.
A fundamental parameter indicative of the asymmetry of the domain, for flat stress states, is the ratio
between compression and uniaxial traction:
From the representation of a plane (biaxial) stress state, it is immediately possible to notice how for and therefore for , the criterion coincides with that of Von Mises.
3.2.3. Calibration of Drucker-Prager Parameters according to Mohr-Coulomb Criterion
Since the Drucker-Prager model turns out to be a smooth version of the Mohr-Coulomb criterion to avoid any numerical problems deriving from stress states located along the intersections of the faces constituting this domain, it may be useful to define the parameters of the Drucker model-Prager
as a function of cohesion
and the internal friction angle
characterizing the Mohr-Coulomb criterion. Considering Equations (38) and (39), it is possible to obtain a Drucker-Prager failure surface that circumscribes or inscribes that of Mohr-Coulomb, as shown in the deviatoric section in
Figure 8. In the first case, that is when the of Drucker-Prager is tangent to the outermost apexes of the Mohr-Coulomb hexagon, the two surfaces coincide along the compression meridian
, with
. For ease of calculation, we consider the meridian radius
, characterized by
. Obtaining
for
θ = 60° and
from Equation (39) and from equality with Equation (36) we obtain:
If, on the other hand, the Drucker-Prager circle coincides with the innermost vertices of the Mohr-Coulomb hexagon, the two surfaces coincide along the meridian of traction
, with
. Obtaining
for
and
from the Equation (39) and from equality with the Equation (35), it is possible to get:
A further modification to the model can be made to make the adherence between the two breaking models even more effective and therefore the yield for more complex load paths. In this case, the Drucker-Prager criterion can also depend on the third invariant of the deviatoric stress tensor (Extended Drucker-Prager), obtaining a function of the type
, which allows to obtain a deviatoric section passing through all the corners of the Mohr-Coulomb hexagon, as shown in
Figure 9. The fracture surface and consequently its deviatoric section are dependent on the parameter
called flow stress ratio, already explained above through the Equation (37) as the ratio between the meridians
,
. For
(upper limit), the deviatoric section will coincide with the circle tangent to the external vertices of the Mohr-Coulomb hexagon, while for decreasing values of
, the section will have the shape shown in
Figure 9 with increasingly sharp vertices. The value of
will be limited below the value 0.778 (lower limit) to guarantee a breaking surface that is always convex.
In this case, the values of the parameters
as a function of cohesion
and the internal friction angle
characterizing the Mohr-Coulomb criterion can be obtained under the following relations:
3.2.4. Concrete Damaged Plasticity Failure Criterion (CDP)
The Concrete Damaged Plasticity Model (CDP) is a criterion that was created for frictional materials, rocks and therefore for almost fragile materials such as concrete as an evolution of the Mohr-Coulomb and Drucker-Prager criteria, from which it takes the basis for the definition of the yield function [
1]. The latter together with the flow rule and the hardening rule, represent the fundamental elements for the definition of a plasticity model. In the specific case, the innovative aspect lies in replacing the hardening variable with a plastic-damage variable (
k), always increasing like the previous one (stability) and directly proportional to the plastic deformation
. The plastic-damage variable, dimensionless, can assume values between 0 and 1 which correspond to the limit cases of zero-damage and total damage, associated with the formation of macroscopic fractures. The damage is applied to the cohesion c which will no longer be a constant parameter as in the case of the Mohr-Coulomb and Drucker-Prager criteria but will depend on the value assumed by the plastic-damage variable. The value of the cohesion c will initially be equal to
, that is the uniaxial yield stress (compression-traction), which corresponds to a damage function
while it will be equal to 0 if
(total damage). In this way, it will be possible to follow the evolution of the yield surface as the plastic deformation varies [
10,
11,
12].
The damage parameter can be defined starting from the uniaxial stress states in traction and compression. Assuming to have the experimental stress-strain diagrams and converting them into plastic stress-strain diagrams
(
Figure 10).
For the traction it is defined:
with
equal to the area under the curve in traction. By setting
as an independent variable, it is possible to express the tension according to a function of the type
, respecting the conditions
and
, with
yield strength in traction. Similarly, for compression we can write:
with
equal to the area under the curve in traction. By setting
as an independent variable, it is possible to express the tension according to a function of the type
, fulfilling the conditions
and
, with
yield strength in compression.
A possible function
compatible with the experimentally observed behaviour with asymptotic tendency to zero in terms of tension is the following (valid for both the branches of
Figure 10):
where
a e
b are constant if the function
, that is, the area underlying the function that defines the
link and its derivative
for
, are provided according to the relationships:
A value of implies an initial hardening phase, whereas a value of implies a softening branch after yielding. From the integration of Equations (50) and (51), considering (52), we obtain a final expression of the damage parameter as a function of the plastic deformation.
The extension of the previous definitions to the case of pluriaxial tension is provided in differential terms starting from the cases of biaxial compression and traction, according to the following relationships:
A more general formulation, valid for any state of tension that is neither pure tension nor pure compression (for instance
) can be expressed according to the following relationship:
where
is a factor included between 0 e 1. For
, the tensions are
for each
and Equation (57) is coincident with Equation (55), for biaxial traction. For
, the tensions
for each
and Equation (57) is coincident with Equation (56), which is the case of biaxial compression. The parameter
can be expressed in the following form:
where (the Macauley bracket)
.
The yield function starts from the extended Drucker-Prager formulation, replacing the dependence on the third invariant of the
deviatoric stress tensor with the algebraically largest principal stress. This term provides greater adherence in the representation of differences in behaviour in the tension and compression regions. The function can be expressed as follows:
where
must be provided experimentally. For
, namely in the case of biaxial compression, the criterion essentially takes the form of Drucker-Prager, and the biaxial tension can be expressed in the following way (the tensions are reported with their own sign):
By considering the initial cohesion equal to
and the initial equibiaxial compressive stress
, it is possible to write [
14,
15]:
and as a consequence:
Experimentally, the ratio reported in Equation (61) evidenced a value included between 1.10 and 1.16, and therefore the value of
is included between 0.08 and 0.12. In case of uniaxial tension:
The parameter
is present only in case of triaxial compression. In order to evaluate it, we can exploit the TM (tensile meridian) for which it results that
and CM (compressive meridian) for which it results that
. By Equation (14) which allows to express the principal stresses as a function of the invariants
and of the angle in the deviatoric plane
, it can be written for the first case:
and for the second case:
The expressions of the respective meridians can be written as follows:
where
is the critical stress in uniaxial compression, defined as yield stress or ultimate stress depending on the location in the domain (yield surface or Failure surface). By defining
the ratio between the meridians TM and CM (
Figure 11), it is possible to write:
and
The constants as the coesion , through the function and change their value and determine the evolution of the failure domain as a function of the level of damage reached.
As for the flow rule, it is non-associative to control the high plastic expansion that characterizes the frictional materials. Lubliner [
15] proposes to use the Mohr-Coulomb yield function as a potential function
by replacing the internal friction angle
with the dilation angle
:
Alternatively, Lee in 1998 proposes the Drucker-Prager yield function as Flow potential function G (Equation (38)) [
16,
17,
18,
19].
The damage function can also be applied to consider the variation of the elastic modulus
as the plastic deformation
increases. In terms of tensions, it can be written:
which became for the elastic modulus
E in the following function: