Fracture evaluation of ultra-high-performance fiber reinforced concrete (UHPFRC)

Abstract The development of numerical simulation for Ultra-high-performance concrete (UHPC) and Ultra-high-performance fiber-reinforced concretes (UHPFRC) is fundamental for the design and construction of related structures. The simplified engineering stress-strain relationship and the input values are necessary in the finite element modeling. Four-linear curves and modified Kent–Park model were proposed to describe the engineering tensile and compressive stress-strain relationship, respectively. An attempt was made to simulate the fracture of UHPC and UHPFRC using concrete damaged plasticity model and element deletion strategies. The predicted tensile and compressive behaviors of UHPC and UHPFRC were successfully validated by the test results in the literature. For a better understanding of the mechanical behavior of UHPC and UHPFRC exposed to biaxial loadings, mixed-mode crack propagation simulation on the double-notched specimens exposed to combined shear-tensile and shear-compressive forces was discussed.

Recently, several studies have been reported related to the mechanical properties and numerical simulation of UHPC/UHPFRC. Shafieifar et al. [3] experimentally determined the compressive ultimate capacity of UHPC through the cylinder and cube compressive test, and tensile ultimate capacity using flexural, briquette, and splitting tension tests, respectively. The results showed that the compressive strength of commercial UHPC was three to four times greater than normal strength concrete, the tensile strength and ductility of UHPC was two to four times greater than normal strength concrete. Graybeal et al. [9] investigated the flexural behavior of a prestressed I-shaped UHPC girder. The cracking, flexural stiffness, and moment capacity are discussed and compared to predictions from the AASHTO standard. Soetens & Matthys [29], and López et al. [30] investigated the cracking strength and the post-cracking strength of UHPFRC using empirical formulations and cohesive models, respectively. Yoo et al. [30] investigated the effects of steel fibers types (short, medium-length, and long) on the flexural behaviors of UHPFRC. Rios et al. [31] experimentally investigated the tensile properties of UHPFRC manufactured with short and long fibers. The results showed that the type of fibers will affect the porosity distribution, and the tensile properties. Wang et al. [10] employed the modified Kent-Park model to describe the compressive stressstrain relationship of UHPFRC and evaluated the seismic performance of bridge pier made of UHPFRC. Mao et al. [31] simulated the performance of UHPFRC subjected to blast loading, the performance of the numerical models were verified by comparing modeling results to the data from corresponding full-scale blast tests. Li et al. [32] conducted a series of tests to investigate the performance of the UHPC slab exposed to explosive loading. Numerical simulation was conducted, and the feasibility and validity of the numerical predictions of UHPC slab responses were validated by test results.
The development of numerical simulation is fundamental for the design and construction of structures made of UHPC/UHPFRC. The simplified engineering stress-strain relationship and the input values in the finite element models are necessary for UHPC/ UHPFRC applications in the civil engineering structures. However, the literature related to simplified engineering stress-strain relationship and the simulation of UHPC/UHPFRC was not sufficient as far as the authors' knowledge. It is also noted that the UHPC fracture is modelled based on the average value of the test results, and the combination of UHPC fracture simulaton with probabilistic analysis [33][34][35] is an very interesting topic, which will be further investigated in the future.
In this paper, an attempt was made to simulate the fracture of UHPC/UHPFRC using concrete damaged plasticity model and element deletion strategies. The simplified model was proposed to describe the engineering stress-strain relationship. The input values to simulate the fracture of UHPC and UHPFRC based on the commercial finite element software [36] were discussed. The simulation results were validated by the test results in the literature. The crack propagation under combined axial (tensile and compressive)-shear forces was investigated based on the validated material model.

Simplified Engineering Stress-Strain Relationship
The simplified model of engineering stress-strain relationship for UHPC and UHPFRC is discussed in this section based on the test results reported in the literature [2]. The mixed constitution of UHPC is listed as below [2]: 657 kg/m 3 cement, 418 kg/m 3 Ground Granulated Blast Furnace Slag (GGBS), 119 kg/m 3 silica fume, 1051 kg/m 3 silica sand with a average size of 0.27 mm, 40 kg/m 3 superplasticizers, and 185 kg/m 3 water. Compared with UHPC, steel fibers with the volume ratio of 2% (157 kg/m 3 ) were added for the UHPFRC. The length and diameter of steel fibers were 13 mm and 0.2 mm, respectively. The tensile behavior of UHPC and UHPFRC was obtained by a direct tensile test using dog-bone specimens with a size of the cross-section of 26 mm by 50 mm, and the compressive behavior of UHPC and UHPFRC was obtained by uniaxial compression tests using a cylinder specimens with a diameter of 50 mm and a length of 100 mm [2].

Tensile behaviors
To describe the engineering tensile stress-strain relationship of UHPC/UHPFRC with both accuracy and simplicity, a model with four-linear curves was proposed in this paper, as shown in Fig. 1. The point (ε t1 , σ t1 ) is the corresponding strain and stress at end of the linear curve, (ε t2 , σ t2 ) is the corresponding strain and stress when the tensile stress reaches the peak; (ε t3 , σ t3 ) is the corresponding strain and stress at the stiffness turning point in the softening stage; and (ε t4 , σ t4 ) is the corresponding strain and stress at the rupture.
The parameters of simplified engineering tensile stress-strain of UHPC and UHPFRC at 7, 14, and 28 days after casting are calibrated in Table 1. The comparisons between the simplified model and experimental results in [2] are shown in Fig. 2. A good agreement is observed, indicating that the simple four-linear curves could successfully describe the engineering tensile stress-strain relationship of UHPC/UHPFRC.

Compressive behaviors
The modified Kent-Park model [37,38] was proposed to describe the engineering compressive stress-strain relationship of UHPC and UHPFRC as shown in Eq. (1).
Where: ε c and σ c are the compressive strain and stress of the concrete, respectively; σ pk and ε 0 are the peak compressive stress and the corresponding strain of the concrete, respectively; λ rs is the ratio of the residual strength to the peak stress; ε 20 is the beginning strain point of the residual strength λ rs σ pk . The ε 20 could be determined based on the following equation:  For UHPC/UHPFRC, the engineering compressive stress and strain relationship could be determined by Eq. (1) when four parameters σ pk , ε 0 , λ rs and ε 20 are known. Those parameters are calibrated as shown in Table 2. The comparisons between the modified Kent-Park model and experimental results in [2] are shown in Fig. 3. A good agreement was observed, indicating that the modified Kent-Park model could successfully describe the engineering compressive stress-strain relationship of UHPC and UHPFRC.

Concrete damaged plasticity (CDP) model
The concrete damage plasticity model (CDP) model was employed to simulate the mechanical behavior of UHPC/UHPFRC. The yield function of the CDP model [36] is expressed in Eq.(3). The evolution of the yield surface is controlled by the equivalent plastic strain. The yield surface in-plane stress of the CDP model is shown in Fig. 4. With: Where: q is the Mises equivalent effective stress; p is the hydrostatic pressure stress; σ max is the maximum principal effective stress; σ b0 / σ c0 is the ratio of initial equi-biaxial compressive yield stress to initial uniaxial compressive yield stress; K c is the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian at initial yield; σ t is the effective tensile cohesion stress; σ c is the effective compressive cohesion stress. No associated potential plastic flow is used in the concrete damaged plasticity model. The Drucker-Prager hyperbolic function is used as flow potential G: Where: ψ is the dilation angle measured in the p-q plane at high confining pressure; σ t0 is the uniaxial tensile stress at failure; ε is eccentricity parameter.
Due to lack of sufficient experimental data of UHPC/UHPFRC, the default values of dilation angle ψ (30 0 ), eccentricity parameter ε(0.1), the ratio of equi-biaxial compressive yield stress to initial uniaxial compressive yield stress σ b0 /σ c0 (1.16), the second stress invariant on the tensile meridian to that on the compressive meridian at initial yield K c (0.667) are used. The UHPC/UHPFRC fails and all stress components are set to zero when either of the following failure criterion is met: (1) the tensile cracking strain ε ck t or tensile cracking displacement u ck t reaches the critical value; . The element is deleted from the model when either of the above failure criterion is met.

Simulation of uniaxial tensile behaviors
The dog-bone tensile specimens were built in the software to simulate the behavior of UHPC and UHPFRC exposed to tensile load. The geometry, FE models and boundary conditions are presented in Fig. 5. Elements C3D8 are used for the tensile model. The bottom surfaces and top surface of tensile specimens are connected to reference point RB and RT using multi-point constraints "MPC", respectively. All degree freedoms of RB are fixed, and all rotation degree freedom and the horizontal displacement UY, UZ, of RT are fixed. The tensile load is applied through a displacement along the vertical direction. ABAQUS/EXPLICIT is used for the calculation with a total step time 1 s and time increment 1 × 10 -5 s.

Ux=Uy=Uz=0 Rotx=Roty=Rotz=0
Load: Uz Ux=Uy=0 Rotx=Roty=Rotz=0 (a) Geometry (b) FE model and Boundary  The true tensile stress-strain curve is obtained through Eqs. (8) and (9). To alleviate the mesh size effects, the stress-crack displacement curves were used to describe the uniaxial tensile behaviors. The crack displacement could be simply obtained by the product of plastic strain ε p t and the characteristic element length l eq . However, as shown in Fig. 6, the FE simulation results FEA ( η 1 l eq = 4 η 2 l eq = 4 ) are smaller than the experimental results. This is because microdamage happened after the peak point and Eqs. (8) and (9) could not effectively convert the engineering stress-strain curve to the true stress-strain curve.
In order to successfully predict the tensile behaviors after the peak stress, we proposed a simplified tensile yield stress-crack displacement relationship (see Fig. 7) based on the four-linear engineering stress-strain curves proposed in Fig. 1. The true stress, σ tr ti (i = 1…4), could be easily obtained through engineering stress using Eq. (9). The crack displacement could be obtained by equivalent plastic strain, the characteristic element length l eq , and the empirical revision parameter η i (i = 1, 2). The empirical revision parameters η i (i = 1, 2) could be obtained through the calibration. The calibration process of UHPFRC at 7 days is presented in Fig. 6. The FE prediction agreed best with the experimental results when the η 1 l eq = 40 and η 2 l eq = 48. The parameters of the simplified tensile yield stress-crack displacement relationship of UHPC and UHPFRC are calibrated as shown in Table 3. The tensile behavior comparisons between FE prediction and test results are shown in Fig. 8. The good agreement indicates that the proposed simplified tensile yield stress-crack displacement relationship is validated. The fracture displacement w 4 of UHPFRC at 7, 14, and 28 days is 159.4, 168.0, and 333.0 times larger than that of UHPC at 7, 14, and 28 days, respectively.

Simulation of uniaxial compressive behaviors
The FE analysis was conducted to predict the uniaxial compressive behavior of UHPC and UHPFRC. The FE model and boundary conditions used in the simulation are shown in Fig. 9. The dimension of the load and support plates is 100 mm × 100 mm × 40 mm. The diameter of the cylinder specimen is 26 mm, and the length of the cylinder specimen is 50 mm. All degree freedoms of the bottom surfaces of the support plate are fixed. The top surface of the load plate is connected to a reference point RP through multi-point constraints "MPC". The tensile load is applied through the reference point RP with a displacement along the vertical direction. ABAQUS/EXPLICIT is used for the calculation with a total step time 1 s and time increment 1 × 10 -5 s. Element type C3D8 was used for the compressive model. Surface to surface contact is built between cylinder specimen and support/load specimen, with a "hard" property for normal direction and 0.1 friction coefficient for tangential direction.  The uniaxial compressive true stress-strain relationship was directly used in the FE simulation based on the engineering compressive stress-strain relationship presented in Section 2.2. The comparisons between FE prediction and experimental results are shown in Fig. 10. A good agreement was observed, except that the fracture engineering strain of UHPC is relatively smaller than the experimental results.

Fracture parameters
The elements will be deleted from the model when u ck t ⩾   Figs. 11 and 13, the failure mode of UHPC exposed to uniaxial compressive from both tests and FE simulation presented an explosive pattern; in contrast, the failure mode of UHPFRC exposed to uniaxial compressive from both tests and FE simulation presented a major incline crack. As shown in Figs. 12 and 14, the tensile failure modes of UHPC and UHPFRC are quite similar, which present a horizontal crack pattern. Noted that the failure position in FE simulation is different from that in the experiment, which is suggested to be improved in the future.

Mixed Mode Crack Propagations
For a better understanding of the behavior of UHPC and UHPFRC exposed to biaxial loading, numerical simulation on the doublenotched specimens exposed to combined shear-tensile and shear-compressive forces was carried out using validated material model presented in Section 3. As shown in Fig. 15-a, the geometry of the double-notched specimen is 200 mm × 200 mm × 50 mm; the length and height of pre-notched crack are 20 mm and 10 mm, respectively. As shown in Fig. 15(b), two types of loading were applied, namely shear-tensile and shear-compressive loadings. All degree freedoms of the bottom surface of double-notched specimens were fixed, and the horizontal displacement UZ at the top surface was also fixed. The proportional displacement UX and UY were applied to the top surface of double-notched specimens to generate the combined shear-tensile and shear-compressive loading status. ABAQUS/ EXPLICIT was used for the calculation with a total step time 1 s and time increment 1 × 10 -5 s. Element type C3D8 was used for the mixed crack propagation model. The load-displacement curves of double notched specimens made of UHPC and UHPFRC exposed to the combined tensile-shear loading are shown in Fig. 16. The double notched specimens made of UHPC is relatively brittle, and the specimens fail when the displacement reaches approximately 0.5 mm. The force F x of UHPC specimen reaches the peak when the displacement is 0.19 mm while the force F y of UHPC specimen reaches to the peak when the displacement is 0.02 mm. Both the forces F x and F y of UHPC specimen drop quickly after reaching the peak point. The double notched specimens made of UHPFRC is more ductile, the failure displacement is 1.8 mm. Both the forces F x and F y of UHPFRC specimen reach the peak when the displacement is around 0.05 mm and gradually decreases after the peak point. Fig. 17 and Fig. 18 present the crack propagation process of UHPC and UHPFRC specimens exposed to combined tensile-shear loading. The crack presents a curved path for the UHPC double notched specimens, and an almost straight line pattern for the UHPFRC double notched specimens. The load-displacement curves of double notched specimens made of UHPC and UHPFRC exposed to the combined compressiveshear loading are shown in Fig. 19. Both the forces F x and F y of UHPC specimen reach the peak when the displacement increases to 0.09 mm and drop to almost zero when the displacement increases to around 0.16 mm. The force F x of UHPFRC specimen reaches the peak when the displacement is 0.25 mm while the force F y of UHPC specimen reaches the peak when the displacement is 0.22 mm. Both the forces F x and F y of UHPFRC specimen drop to almost zero when the displacement increases to around 0.42 mm.
The failure displacement of double notched specimens made of UHPC and UHPFRC exposed to the combined compressive-shear loading is smaller than that exposed to the combined tensile-shear loading. The maximum forces F x and F y of double notched specimens made of UHPC and UHPFRC exposed to the combined compressive-shear loading is larger than that exposed to the combined tensile-shear loading. Fig. 20 and Fig. 21 show the crack propagation process of UHPC and UHPFRC specimens exposed to combined compressive-shear loading. The crack of UHPC and UHPFRC double notched specimens both presented a curved path. The crack of UHPC and UHPFRC specimens initiated near the notch and final failure happened when the crack from two notches coalesced together by an incline major crack. Several crack branches were observed for the UHPC double notched specimens during crack propagation. The UHPFRC specimens only form one major crack during combined compressive-shear loading.

Conclusions
The development of numerical simulation is fundamental for the design and construction of structures made of ultra-highperformance concrete (UHPC) and ultra-high-performance fiber-reinforced concrete (UHPFRC). An attempt was made to simulate the fracture of UHPC and UHPFRC using the CDP model and element deletion strategies in this paper. For a better understanding of the behavior of UHPC and UHPFRC exposed to biaxial loading, numerical simulation on the double-notched specimens exposed to the combined shear-tensile and the shear-compressive forces was carried out using a validated material model. Following conclusions are  (1) Four-linear curves and the modified Kent-Park model were proposed to describe the engineering tensile and compressive stressstrain relationships, respectively. A good agreement was observed between the simplified model and test results regarding the engineering stress-strain relationship. (2) The tensile and compressive behaviors of UHPC and UHPFRC were predicted using the CDP model. The method to obtain the input values, including the tensile-crack displacement relationship, compressive stress-strain relationship, and fracture parameters was suggested. The simulation results were successfully validated by the test results in the literature. (3) The failure displacement of double notched specimens made of UHPC and UHPFRC exposed to the combined compressive-shear loading is smaller than that exposed to the combined tensile and shear loading. The maximum force of double notched specimens made of UHPC and UHPFRC exposed to the combined compressive-shear loading is larger than that exposed to the combined tensile-shear loading. (4) The crack presented a curved path for the UHPC double notched specimens but presented an almost straight line pattern for the UHPFRC double notched specimens when exposed to tensile-shear loading. The crack of UHPC and UHPFRC double notched specimens presented a curved path, and several crack branches were observed for the UHPC specimens but the UHPFRC specimens only form one major crack when exposed to combined compressive-shear loading.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.