Explicit double-phase-field formulation and implementation for bending behavior of UHPC-NC composite beams
Introduction
The past decade witnessed the emergence of ultra-high performance concrete (UHPC) as the innovative and competitive cementitious material [1,2], due to its excellent mechanical properties including greater tensile and compressive strengths, excellent impact and fatigue resistance [3], and enhanced fracture energy (toughness, [4]. Despite achievements, it is acknowledged that UHPC is still expensive hitherto and one compromised approach is the combination of UHPC and normal-strength concrete (NC). The so-called UHPC-NC composite members gained a lot of attraction in the fields of retrofitting existing structures [5,6], enhancing the durability [7], accelerating construction speed [8], etc.
Compared to the NC material, steel fibers can effectively inhibit the cracking propagation of UHPC [9]. This unique post-peak behavior renders the behavior of UHPC-NC composite beams different from the conventional NC beams. Using the UHPC layer to strengthen existing NC structures steamed from Switzerland and North America [10,11]. Experimental works regarding the flexural behavior of UHPC-NC composite beams were carried out by Refs. [[6], [10], [11], [12], [13], [14], [15], [16], [17]]. The potential influence of various parameters on the ultimate load-carrying capacity, stiffness and crack resistance of UHPC-NC beams was investigated, i.e., thickness of UHPC layer, reinforcement ratio, yield strength of rebars, tensile strength and restrained shrinkage of UHPC. In these tests, the interface between UHPC and NC layers performed well, without debonding before the failure of the specimens, provided with the interfacial rebars [18]. The UHPC layer was experimentally verified to improve the ultimate bearing capacity, ductility and rotation capacity of the UHPC-NC beams subject to combined flexure and shear forces [19]. Moreover, flexural fatigue tests on the UHPC-NC composite structures were performed by Refs. [20,21]. They pointed out that the UHPC layer was effective in strengthening the NC beam and improving the fatigue resistance of the NC beam, without interfacial debonding [17]. enhanced the pre-damaged NC beams with a reinforced UHPC layer. The test witnessed the effectiveness of the UHPC layer in suppressing the cracking evolution and crack width of the NC beam. However, the severer the pre-damage degree of the NC beam, the smaller improvement of the flexural performance of the UHPC layer. To sum up, these tests demonstrated the combination of UHPC and NC materials in improving the cracking and flexural performance. However, it is admitted that relevant experiment works are far from sufficient to understand the flexural behavior of UHPC-NC composite beams thoroughly.
The complex behavior of UHPC-NC composite beams calls for a reliable numerical model. As listed in Table 1, many literatures resort to commercial software with default constitutive models for the simulations. Among them, the concrete damage plasticity (CDP) model provided by Abaqus is widely adopted [22,20,23,24]. Although agreed numerical results are reported, inherent shortfalls are inevitable. First, model parameters in these constitutive models are purely empirical and an ad hoc calibration is required [25,26]. Taking the CDP model for illustration, the relationship between plastic strain and stress is determined artificially, upon fracture energy and mesh size [9]. Furthermore, these pointwise (local) material models are inappropriate and inaccurate which would lead to notorious mesh-dependent solutions [9].
Fortunately, the phase-field technique [30,31] for the progressive events of brittle and quasi-brittle fracture is a promising model to address the aforementioned shortfalls. It is able to represent all stages of fracture, such as crack initiation, propagation, coalescence, and branching [32]. Derived from the variational re-visitation of Griffith's brittle fracture model, the fracture problem solution is obtained by minimizing an energy functional where sharp cracks are captured and described by a smeared field, the phase-field. This function consists of two terms, one associated with the elastic strain energy of the damageable material and the other with the fracture dissipation. The original phase-field model was developed for general brittle fracture in isotropic homogeneous materials [33]. The pioneering work of [34] propelled the phase-field approach to model the onset and process of different types of fractures: brittle [35,36], quasi-brittle [37], ductile [38], thermo-elastic-plastic [39], etc. Above phase-field models do not distinguish tensile (mode I) and shear (mode II) fractures. However, the shear fracture energy is assumed to be at least three times greater than the tensile fracture energy, for concrete- or rock-like materials [40]. [41] split the strain energy into tensile, shear and compressive ones, by crack-based directional decomposition, and different tensile and shear fracture energies are assigned. Similar works can be found in Refs. [42,43]. Another, the phase-field model, hitherto, is rarely applied to study the behavior of specimens at the structural level (i.e., slab, beam, and column), which is more complex due to the interaction between cementitious materials and steel rebars.
The phase-field problems are, in general, solved by monolithic [35] or staggered (operator split) schemes [44]. The displacement field and phase-field are implicitly solved by the monolithic algorithm, or separately and interactively in the staggered scheme. However, an implicit solver requires very small load increments for convergence, and also the complex derivation of the tangent stiffness matrix (Ernesti et al., 2020). Even so, the convergence of implicit algorithms depends on many factors, and additional techniques are necessary, e.g., line search [45] and Anderson acceleration [46]. Compared to implicit algorithms, the explicit solver has the merits of massive parallel computing, stiffness matrix-free, and robust implementation [47]. Solving phase-field problems with an explicit solver is demonstrated by several scholars, but still limited [48]. developed a massively parallel algorithm for phase-field modeling of dynamic quasi-brittle fracture, on the graphical processing unit (GPU), to alleviate the expensive computational cost caused by the fine mesh size inherently required in implicit phase-field modeling [18]. applied the phase-field method to simulate the dynamic and quasi-static brittle fracture of thermo-elastic materials within general software Abaqus/Explicit and the source code was provided. Similar work can be found in Ref. [49] for the progressive tensile failure of the composite materials.
This study aims to investigate the bending behavior of UHPC-NC composite beams experimentally, with varying thick ratios. Furthermore, an explicit double-phase-field model is proposed and implemented for the numerical simulations. The study is organized as follows. In Sect. 2, the theoretical background of the double-phase-field is presented for cohesive tensile (mode I) and cohesive shear (mode II) fractures, together with the directional decomposition of strain energy with respect to existing cracks. A fully explicit finite element scheme is employed for the proposed model. Sect. 3 outlines the detailed implementation of the model within the general software Abaqus/Explicit. In Sect. 4, the sensitivity analysis is performed to probe the effects of critical parameters with a 1D numerical tensile bar. Experimental and numerical investigations regarding the flexural behavior of five UHPC-NC composite beams are conducted in Sect. 5. Finally, concluding remarks are given in Sect. 6.
Section snippets
Potential energy density
Considering an arbitrary domain in the general Euclidean space of dimension being 2 in this study, see Fig. 1, the boundary can be decomposed into the Dirichlet boundary and the Neumann boundary . The domain is subject to the prescribed displacements on , and prescribed tractions on . The domain has two sets of fractures, namely mode I and mode II .
The potential energy functional of a material point can be formulated to derive the evolutions of these
Finite element discretization
The explicit double-phase-field model is implemented within the conventional finite element framework. The displacement field and the corresponding strain vector of an arbitrary internal point are interpolated as:where is the shape function matrix for the displacement field, is its derivatives with respect to global spatial coordinate, and is the nodal displacement vector.
Similarly, the tensile and shear phase-field variables and and their spatial derivatives
Parametric study with 1D tensile bar
To illustrate the proposed explicit phase-field model in a straightforward way, a numerical 1D bar of 110 mm in length and 1.0 mm2 cross-sectional area is selected to present the effect of mesh size, length scale parameter, and viscosity. As shown in Fig. 4, The left end of the rebar is fixed, and a prescribed displacement is applied to the right end. An artificial ad hoc imperfection is assigned to the middle element of the rebar, with a 1% decrease of the tensile strength to initialize the
Bending test of UHPC-NC composite beams
The experimental investigation involves five UHPC-NC composite beams, whose geometries and reinforcing details are given in Table 3. All these beams were 1800 mm in length, 400 mm in height, and 200 mm in width. As illustrated in Fig. 8, specimens B–N and B–U were fabricated with merely NC and UHPC materials, respectively. The remaining three composite beams owned the UHPC substrate and NC overlay at the tensile and compressive sides, respectively. The 100 mm-height UHPC substrate was set for
Conclusion
The explicit double-phase-field model is proposed and implemented in Abaqus/Explicit to simulate the damage evolution of cementitious materials, considering both tensile and shear cracks. Sensitivity analysis on critical parameters is performed, including mesh size, length scale parameter and viscosity. The model is further applied to study the bending behaviors of UHPC-NC composite beams with varying UHPC thickness. The following conclusions can be tentatively drawn:
- (1)
The double-phase-field
CRediT author statement
Siqi Yuan: Conceptualization, Original draft preparation, Analysis. Teng Tong: Conceptualization, Methodology, Writing- Reviewing and Editing. Zhao Liu: Methodology, Writing- Reviewing and Editing. Peng Yang: Experiment.
Availability of data and materials
User subroutines are disclosed for potential users on GitHub repository: https://github.com/TengTongSEU/Esplicit-PHZ-for-UHPC-NC.
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.
Acknowledgment
This research is supported by the National Key Research and Development Program of China (2019YFE0119800), the National Natural Science Foundation of China (51778137), National Natural Science Foundation for Young Scientists of China (51808113), the National Natural Science Foundation for Young Scientists of Jiangsu Province (BK20180389), and Zhishan Youth Scholar Program of Southeast University (2242021R41172).
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