6.2 Two dimensional mixed-mode fracture smeared crack model
6.2.3 Model implementation in FEMIX V4.0
The MMFSCM approach relies on the strain decomposition of the incremental strain vector by splitting the strains of the cracked and uncracked concrete Sena-Cruz [220]. Hence, the incremental strain vector according to this formulation is given by Equation (6.1),
Δ𝜀=Δ𝜀𝑐𝑟 +Δ𝜀𝑐𝑜 (6.1)
where𝜀𝑐𝑟 and 𝜀𝑐𝑜 are the crack incremental strain vector and uncracked incremental strain vector, re-spectively.
The crack normal and tangential strains are defined by𝜀𝑐𝑟𝑛 and𝛾𝑡𝑐𝑟, respectively, and the increment of strains at the crack local coordinate system (LCS) is given by Equation (6.2).
Δ𝜀𝑐𝑟ℓ =
h Δ𝜀𝑐𝑟𝑛 Δ𝛾𝑐𝑟𝑡 i𝑇
(6.2) On the other hand, the strains in the global coordinate system (GCS) are comprised of three compo-nents as defined by Equation (6.3).
Δ𝜀𝑐𝑟 =
h Δ𝜀𝑐𝑟1 Δ𝜀𝑐𝑟2 Δ𝛾𝑐𝑟12 i𝑇
(6.3) The increment of crack strain components in the LCS and GCS are related from the transformation matrix according to Equation (6.4),
Δ𝜀𝑐𝑟 = 𝑇𝑐𝑟𝑇
Δ𝜀𝑐𝑟ℓ (6.4)
that depends on the angle𝜃𝑐𝑟 between the x axis and the orthogonal to the plane of the crack.
The increment of stress components at a given crack can be defined as given by Equation (6.5), Δ𝜎𝑐𝑟ℓ =
h Δ𝜎𝑛𝑐𝑟 Δ𝜏𝑡𝑐𝑟 i𝑇
(6.5) that are in equilibrium with the increment of the stress components in the GCS,Δ𝜎𝑐𝑟ℓ , through Equa-tion (6.6),
Δ𝜎𝑐𝑟ℓ =𝑇𝑐𝑟Δ𝜎 (6.6)
6.2. TWO DIMENSIONAL MIXED-MODE FRACTURE SMEARED CRACK MODEL
The concrete between cracks is assumed to be undamaged and therefore to behave elastically in the present phase of the model development. Thus, the constitutive relationship between the increment of strains and stresses in the undamaged concrete is given by Equation (6.7),
Δ𝜎 =𝐷𝑐𝑜Δ𝜀𝑐𝑜 (6.7)
where𝐷𝑐𝑜 is the elasticity matrix.
The present model is formulated in terms of normal (w𝑐𝑟𝑛) and tangential (w𝑐𝑟𝑡 ) displacements at the crack level and, consequently, it is necessary to convert the strains into displacements. This is achieved by means of the crack band width (CBW) parameter, which represents the length of the material band in which the crack displacements of the smeared cracks are related to the crack strains (Equation (6.8)),
Δw𝑐𝑟ℓ = ℓ𝑐𝑟𝑏 −1
Δ𝜀𝑐𝑟ℓ (6.8)
whereℓ𝑐𝑟𝑏 is a matrix containing the inverse of the CBW parameter:
ℓ𝑐𝑟𝑏 =
1 ℓ𝑏 0
0 1 ℓ𝑏
(6.9) therefore it is assumed the same CBW for crack opening and crack sliding, which is arguable, but with-out reliable data from comprehensive research in this topic, it was decided to adopt this assumption in the present phase of the model development. The crack constitutive law at the incremental level of displacements and stresses is:
Δ𝜎𝑐𝑟ℓ =𝐷𝑐𝑟Δw𝑐𝑟ℓ (6.10)
with,
𝐷𝑐𝑟 =
𝐷𝑐𝑟11 𝐷𝑐𝑟12 𝐷𝑐𝑟21 𝐷𝑐𝑟22
(6.11) where𝐷𝑐𝑟11,𝐷𝑐𝑟12,𝐷𝑐𝑟21and𝐷𝑐𝑟22are the terms of the crack constitutive matrix, whose values can be positive or negative, depending on the hardening or softening nature of the FRC.
These coefficients are obtained by computing the Jacobian matrix of theDmaxmatrix:
𝐷𝑐𝑟 =
𝜕Δ𝜎𝑛𝑐𝑟
𝜕Δw𝑐𝑟𝑛
𝜕Δ𝜎𝑛𝑐𝑟
𝜕Δw𝑐𝑟𝑡
𝜕Δ𝜏𝑡𝑐𝑟
𝜕Δw𝑐𝑟𝑛
𝜕Δ𝜏𝑡𝑐𝑟
𝜕Δw𝑐𝑟𝑡
(6.12)
The elements that compose theDmaxare obtained according to the procedure described in Annex II.
When a strain increment (Δ𝜀𝑚) occurs at a given cracked integration point, the stress state has to be updated accordingly (𝜎𝑚). Thus, the total stresses can be obtained from:
𝜎𝑐𝑟ℓ,𝑚 =𝑇𝑐𝑟𝑚𝜎𝑚 (6.13)
𝜎𝑐𝑟ℓ,𝑚−1+Δ𝜎𝑐𝑟ℓ,𝑚=𝑇𝑐𝑟𝑚 𝜎𝑚−1+Δ𝜎𝑚
(6.14) Substituting Equations (6.4), (6.7) and (6.8) in Equation (6.14) it is obtained:
𝜎𝑐𝑟ℓ,𝑚−1+Δ𝜎𝑐𝑟ℓ,𝑚
Δw𝑐𝑟ℓ,𝑚
+𝑇𝑐𝑟𝑚𝐷𝑐𝑜 𝑇𝑚𝑐𝑟𝑇
ℓ𝑐𝑟𝑏 Δw𝑐𝑟ℓ,𝑚−𝑇𝑐𝑟𝑚𝜎𝑚−1−𝑇𝑐𝑟𝑚𝐷𝑐𝑜Δ𝜀𝑚 =0 (6.15) where the vector of incremental crack stress (Δ𝜎𝑐𝑟ℓ,𝑚) depends on the vector of the incremental crack displacements (Δw𝑐𝑟ℓ,𝑚), which are the unknown variables to be determined. After obtaining Δw𝑐𝑟ℓ,𝑚, the vector of incremental crack strains (Δ𝜀𝑐𝑟ℓ,𝑚) are obtained by means of Equation (6.8).
The Newton-Raphson method is employed to solve the system of nonlinear equations given by Equa-tion (6.15):
𝑓
Δw𝑐𝑟ℓ,𝑚
=𝜎𝑐𝑟ℓ,𝑚−1+𝐷𝑐𝑟Δw𝑐𝑟ℓ,𝑚+𝑇𝑐𝑟𝑚𝐷𝑐𝑜 𝑇𝑚𝑐𝑟𝑇
ℓ𝑐𝑟𝑏 Δw𝑐𝑟ℓ,𝑚−𝑇𝑐𝑟𝑚𝜎𝑚−1−𝑇𝑐𝑟𝑚𝐷𝑐𝑜Δ𝜀𝑚 (6.16) Considering that:
𝜎𝑐𝑟ℓ,𝑚−1 =
"
𝜎𝑛,𝑚−𝑐𝑟 1 𝜏𝑡,𝑚−𝑐𝑟 1
#
(6.17)
Δ𝜎𝑐𝑟ℓ,𝑚 =
𝐷𝑐𝑟11Δw𝑐𝑟𝑛,𝑚+𝐷𝑐𝑟12Δw𝑐𝑟𝑡,𝑚
𝐷𝑐𝑟21Δw𝑐𝑟𝑛,𝑚+𝐷𝑐𝑟22Δw𝑐𝑟𝑡,𝑚
(6.18)
𝑇𝑐𝑟𝑚𝜎𝑚−1 =
"
𝐴1
𝐴2
#
(6.19)
𝑇𝑐𝑟𝑚𝐷𝑐𝑜Δ𝜀𝑚 =
"
𝐵1
𝐵2
#
(6.20)
𝑇𝑐𝑟𝑚𝐷𝑐𝑜 𝑇𝑐𝑟𝑚𝑇
ℓ𝑐𝑟𝑏 =
"
𝐶11𝐶12
𝐶21𝐶22
#
(6.21) where A, B, C are constants, Equation (6.16) can get the following configuration:
6.2. TWO DIMENSIONAL MIXED-MODE FRACTURE SMEARED CRACK MODEL
𝑓ˆ1
𝑓ˆ2
=
𝜎𝑛,𝑚−1𝑐𝑟 𝜏𝑡,𝑚−𝑐𝑟 1
+
𝐷𝑐𝑟11Δw𝑐𝑟𝑛,𝑚 +𝐷𝑐𝑟12Δw𝑐𝑟𝑡,𝑚 𝐷𝑐𝑟21Δw𝑐𝑟𝑛,𝑚 +𝐷𝑐𝑟22Δw𝑐𝑟𝑡,𝑚
(6.22)
+
𝐶11𝐶12
𝐶21𝐶22
Δw𝑐𝑟𝑛,𝑚
Δw𝑐𝑟𝑡,𝑚
−
𝐴1
𝐴2
−
𝐵1
𝐵2
whose unknowns can be obtained from:
𝜕𝑓
Δw𝑐𝑟ℓ,𝑚
𝜕Δw𝑐𝑟ℓ,𝑚 =
𝜕𝑓ˆ1
𝜕Δw𝑐𝑟𝑛,𝑚
𝜕𝑓ˆ1
𝜕Δw𝑐𝑟𝑡,𝑚
𝜕𝑓ˆ2
𝜕Δw𝑐𝑟𝑛,𝑚
𝜕𝑓ˆ2
𝜕Δw𝑐𝑟𝑡,𝑚
(6.23)
where,
𝜕𝑓ˆ1
𝜕Δw𝑐𝑟𝑛,𝑚 =𝐷11+ 𝜕𝐷11
𝜕Δw𝑐𝑟𝑛,𝑚 ·Δw𝑐𝑟𝑛,𝑚+ 𝜕𝐷12
𝜕Δw𝑐𝑟𝑛,𝑚 ·Δw𝑐𝑟𝑡,𝑚 +𝐶11 (6.24)
𝜕𝑓ˆ1
𝜕Δw𝑐𝑟𝑡,𝑚 = 𝜕𝐷11
𝜕Δw𝑐𝑟𝑡,𝑚 ·Δw𝑐𝑟𝑛,𝑚 +𝐷12+ 𝜕𝐷12
𝜕Δw𝑐𝑟𝑡,𝑚 ·Δw𝑐𝑟𝑡,𝑚 +𝐶12 (6.25)
𝜕𝑓ˆ2
𝜕Δw𝑐𝑟𝑛,𝑚 =𝐷21+ 𝜕𝐷21
𝜕Δw𝑐𝑟𝑛,𝑚 ·Δw𝑐𝑟𝑛,𝑚+ 𝜕𝐷22
𝜕Δw𝑐𝑟𝑛,𝑚 ·Δw𝑐𝑟𝑡,𝑚 +𝐶21 (6.26)
𝜕𝑓ˆ2
𝜕Δw𝑐𝑟𝑡,𝑚 = 𝜕𝐷21
𝜕Δw𝑐𝑟𝑡,𝑚 ·Δw𝑐𝑟𝑛,𝑚 +𝐷22+ 𝜕𝐷22
𝜕Δw𝑐𝑟𝑡,𝑚 ·Δw𝑐𝑟𝑡,𝑚 +𝐶22 (6.27)
which can be rearranged into Equation (6.28),
𝜕𝑓
Δw𝑐𝑟ℓ,𝑚
𝜕Δw𝑐𝑟ℓ,𝑚 =𝐷𝑐𝑟 +𝐷ˆ𝑐𝑟Δwˆ𝑐𝑟𝑙,𝑚+𝑇𝑐𝑟𝑚𝐷𝑐𝑜 𝑇𝑐𝑟𝑚𝑇
ℓ𝑐𝑟𝑏 (6.28)
where𝐷ˆ𝑐𝑟 andΔwˆ𝑐𝑟𝑙,𝑚 are given by Equations (6.29) and (6.30),
𝐷ˆ𝑐𝑟 =
𝜕𝐷11
𝜕Δw𝑐𝑟𝑛,𝑚
𝜕𝐷12
𝜕Δw𝑐𝑟𝑛,𝑚
𝜕𝐷11
𝜕Δw𝑐𝑟𝑡,𝑚
𝜕𝐷12
𝜕Δw𝑐𝑟𝑡,𝑚
𝜕𝐷21
𝜕Δw𝑐𝑟𝑛,𝑚
𝜕𝐷22
𝜕Δw𝑐𝑟𝑛,𝑚
𝜕𝐷21
𝜕Δw𝑐𝑟𝑡,𝑚
𝜕𝐷22
𝜕Δw𝑐𝑟𝑡,𝑚
(6.29)
Δwˆ𝑐𝑟𝑙,𝑚 =
Δw𝑐𝑟𝑛,𝑚 0 Δw𝑐𝑟𝑡,𝑚 0
0 Δw𝑐𝑟𝑛,𝑚 0 Δw𝑐𝑟𝑡,𝑚
(6.30)
The 𝐷ˆ𝑐𝑟 matrix is obtained by computing the derivatives of each𝐷𝑐𝑟 element with respect to each component of the incremental crack displacement vector (Δw𝑐𝑟ℓ ). However, due to the complexity of the expressions that constitute theDmax, computing the second derivative of such expressions analytically would be impractical and error prone. Therefore, the elements of𝐷ˆ𝑐𝑟 are obtained in a numeric fashion by means of a finite differences procedure included in the GNU scientific library (GSL).
Crack status management
During the loading process a crack can undergo different statuses either due to the formation of new cracks in the same integration point (IP) of the existing crack, or in the neighbourhood IPs, or due to stress redistribution in consequence of damage evolution or alteration of the loading conditions.
The crack status of each crack is stored as an historical variable so that it can be used throughout the nonlinear analysis to correctly accomplish the assumptions adopted for the crack constitutive law for representing the cracking process.
In the MMFSCM, a crack can be assigned 6 distinct crack statuses in mode I following the approach proposed by Sena-Cruz [220], namely,1. initiation,2. opening, 3. closing,4. reopening,5. closedand 6. fully open. On the other hand, 5 crack status are adopted for mode II, 1. stiffening,2. Softening,3.
unloading,4. reloading,5. free-sliding. In addition, it is considered that the fracture mode I governs the crack state, which means that a crack only exists until aclosedstatus is assigned regardless of the status in fracture mode II.
In Figure 122 a schematic representation of the crack normal and shear stress surfaces and the correspondent possible crack status for each fracture mode is presented.
Regarding fracture mode I, a new crack is assigned theinitiation status as the principal stress (𝜎𝐼) attains 𝑓𝑐𝑡 at less than an adopted tolerance, and the angle formed by this new crack and the already existing cracks in the IP exceeds the adopted threshold angle.
Asw𝑐𝑟𝑛 increases the crack is assigned anopeningstatus, provided thatw𝑐𝑟𝑛 is lower than the ultimate crack opening displacement (w𝑐𝑟𝑢𝑙𝑡,𝐼).
Whenw𝑐𝑟𝑛 exceedsw𝑢𝑙𝑡,𝐼𝑐𝑟 the crack is attributed afully openstatus. If however w𝑐𝑟𝑛 decreases, but is still positive, aclosingstatus is considered and the crack stress state is updated according to a secant
6.2. TWO DIMENSIONAL MIXED-MODE FRACTURE SMEARED CRACK MODEL
(a) (b)
Figure 122: Schematic representation of possible crack status a crack can assume in (a) mode I and (b) mode II.
approach based on the maximum crack opening displacement (w𝑐𝑟𝑛,𝑚𝑎𝑥) as shown in Figure 122a.
On the other hand, if aclosing crack undergoes a positivew𝑐𝑟𝑛 increment, the status is changed to reopeningprovided that thew𝑐𝑟𝑛 is lower thanw𝑐𝑟𝑛,𝑚𝑎𝑥, otherwise anopeningstatus is assigned again. If w𝑐𝑟𝑛 exceedsw𝑢𝑙𝑡,𝐼𝑐𝑟 or decreases to 0, afully-openorclosedstatus is considered, respectively.
Regarding mode II, astiffeningstatus is assigned to a new crack until the sliding peak displacement (w𝑐𝑟𝑡,𝑝 =𝐷𝑚𝑎𝑥/100) is attained. During this stage, the crack status will remain instiffeningstatus regard-less of the type of increment ofw𝑐𝑟𝑡 .
The mode II crack status is changed to softeningwhen w𝑐𝑟𝑡 is larger than w𝑐𝑟𝑡,𝑝, but lower than the ultimate crack sliding displacement (w𝑐𝑟𝑢𝑙𝑡,𝐼𝐼).
Theunloadingstatus is assigned to a crack when occurring a negative increment ofw𝑐𝑟𝑡 , and similarly to mode I, the stress state is updated based on a secant trajectory as shown in Figure 122b. Anunloading crack will be set to areloadingstate if a positivew𝑐𝑟𝑡 increment is observed, provided thatw𝑐𝑟𝑡 does not exceedsw𝑐𝑟𝑡,𝑚𝑎𝑥. Ifw𝑐𝑟𝑡 orw𝑐𝑟𝑡 exceedsw𝑐𝑟𝑢𝑙𝑡,𝐼𝐼, the crack is considered to be in afree-slidingstate.
Since the MMFSCM is formulated based on the fibre pullout and aggregate interlock resisting mech-anisms, the concept of fracture energy of the FRC is not explicitly used and, consequently, it is not used to determinew𝑐𝑟𝑢𝑙𝑡,𝐼 andw𝑐𝑟𝑢𝑙𝑡,𝐼𝐼.
Hence, for fracture mode I the value of w𝑐𝑟𝑢𝑙𝑡,𝐼 is assumed to be equal to the ultimate fibre pullout value, which is typically considered equal to𝑙𝑓/4[265, 266], whilst for mode IIw𝑐𝑟𝑢𝑙𝑡,𝐼𝐼 is assumed equal to𝐷𝑚𝑎𝑥/2, which corresponds to thew𝑐𝑟𝑛 value to which the aggregate interlock component is null.