A Numerical Simulation Investigation on the Mechanical Constitutive Model of Lithium Slag UHPC and the Bending Behavior of Its Prefabricated Connection Components

Abstract

Using industrial by-product lithium slag (LS) as a raw material for ultra-high performance concrete (UHPC) is an important way to achieve low-carbon prefabricated structures. However, existing studies lack a constitutive model for LS-UHPC and its application in prefabricated beam connection nodes. To fill this gap, this paper first established a tensile-compressive constitutive model for LS-UHPC through mechanical tests; then it was embedded into the finite element model to simulate the bending performance of the connection nodes of the post-cast LS-UHPC prefabricated beams and verified by the test results. Finally, parameter analysis is carried out. The results show that moderately increasing the diameter of longitudinal reinforcement can significantly improve the flexural bearing capacity of the connection node, but when the diameter exceeds 18 mm and HRB500 high-strength steel bars are used, the node exhibits over-reinforced failure characteristics; increasing the strength grade of ordinary concrete has a limited effect on the improvement of flexural bearing capacity (<5%). This study clarified the mechanical constitutive relationship of LS-UHPC, revealed the failure mechanism and bearing capacity evolution law of its prefabricated connection nodes under parameter changes, and provided a theoretical basis and design suggestions for the application of low-carbon lithium slag UHPC in prefabricated assembly structures.

1. Introduction

Prefabricated concrete structures have attracted increasing attention because of their high construction efficiency, improved quality control, reduced on-site wet work, and potential environmental benefits. However, the structural integrity of prefabricated systems strongly depends on the reliability of their connections. Existing connection methods mainly include grouted sleeve connections [1,2], post-cast monolithic connections [3,4], bolted connections [5,6], welded connections [7], and prestressed connections [8,9]. Although these methods can provide force transfer between precast members, they are often associated with complicated detailing, strict construction tolerance requirements, difficulty in inspecting reinforcement connections, cracking in the joint region, and insufficient interfacial durability [1,2,10,11]. Therefore, developing a simplified, reliable, and crack-resistant connection system remains an important issue for improving the safety and applicability of prefabricated concrete structures.

Ultra-high performance concrete (UHPC) exhibits high compressive strength, high tensile toughness, excellent durability, and superior bond performance with reinforcing bars, making it effective in reducing reinforcement anchorage or lap-splice length and improving crack control in post-cast connection regions [12,13]. Consequently, the use of UHPC in prefabricated member connections has become an important research topic. Ma et al. [3] investigated interior precast beam–column connections with lap-spliced steel bars embedded in field-cast reactive powder concrete, demonstrating the potential of high-performance cementitious materials in improving the seismic behavior of lap-spliced connections. Lin et al. [14] proposed precast beam–column connections with UHPC core shells and reported that UHPC could delay joint cracking and improve hysteretic behavior. Xu et al. [4] further investigated the flexural behavior of precast concrete beams in-span joined with UHPC, confirming the feasibility of post-cast UHPC for connecting standardized short precast beams. These studies demonstrate the advantages of UHPC in prefabricated connections. However, most existing studies have focused on conventional UHPC or reactive powder concrete, while the structural application and numerical modeling of industrial-waste-modified UHPC, especially lithium slag UHPC (LS-UHPC), in prefabricated connections remain insufficiently explored.

In recent years, the combined use of fiber-reinforced polymer (FRP) and UHPC has also received considerable attention. Zeng et al. [15] investigated the flexural behavior of FRP grid-reinforced UHPC composite plates with different types of fibers, showing that the synergy between FRP grids and UHPC can improve the load-bearing and deformation capacities of thin plate members. Liao et al. [16] studied the axial compressive behavior of FRP-confined UHPC columns and reported that FRP confinement can enhance the post-peak ductility and compressive deformation capacity of UHPC columns. For prefabricated connection systems, Zeng et al. [17] developed FRP bar-reinforced UHPC plates with grouting sleeve connections and evaluated their flexural behavior. Subsequently, Zeng et al. [18] proposed prefabricated FRP bar-reinforced UHPC shells for column strengthening and verified their effectiveness through axial compression tests. These studies have advanced the development of FRP-UHPC composite members and connection systems. Nevertheless, such methods generally rely on FRP bars, FRP grids, grouting sleeves, or prefabricated shells, which may increase material cost and construction complexity. In contrast, conventional steel bar lap splicing combined with post-cast UHPC still offers advantages in construction adaptability, detailing simplicity, and engineering applicability. Therefore, further investigation is needed into steel-reinforced post-cast UHPC connections, particularly when low-carbon UHPC materials are used.

Despite its excellent mechanical properties, conventional UHPC usually requires a high cement content, a very low water-to-binder ratio, and a large amount of silica fume, leading to high material cost, considerable autogenous shrinkage, and high carbon emissions. To reduce the environmental impact of UHPC, lower-carbon UHPC has become a major research direction. Recent reviews have indicated that the main strategies for reducing the carbon footprint of UHPC include the use of supplementary cementitious materials [19], industrial solid wastes [20,21,22], alkali-activated binders [23,24,25], particle packing optimization [26,27], and alternative fibers [28]. Lithium slag, a by-product of the lithium salt industry, contains reactive SiO₂, Al₂O₃, and a certain amount of CaO, and therefore has potential pozzolanic and filler effects. He et al. [29] reported that the incorporation of lithium slag can improve the hydration degree of UHPC and enhance the properties of the interfacial transition zone. Yang et al. [30] further showed that incorporating 10–40% lithium slag can improve the 28-day compressive strength of UHPC, and that 20% lithium slag is beneficial for pore refinement while significantly reducing the carbon emissions of UHPC. These studies provide a material basis for the green production of LS-UHPC. However, existing investigations have mainly focused on hydration, microstructure, compressive strength, shrinkage, and environmental benefits. Limited attention has been paid to the tensile and compressive constitutive behavior of LS-UHPC, its finite element implementation, and its structural performance in prefabricated connection members.

For the numerical simulation of UHPC structures, appropriate constitutive models, interface models, and bond–slip models are essential for ensuring reliable finite element predictions. Previous studies have systematically investigated the tensile constitutive behavior, compressive constitutive behavior, and concrete damage plasticity parameters of UHPC. Hiew et al. [31] proposed a unified tensile constitutive model for mono- and hybrid-fiber-reinforced UHPC, emphasizing the significant influence of fiber type and fiber volume fraction on the tensile softening branch. Fakeh et al. [32] calibrated the concrete damage plasticity parameters and constitutive inputs for UHPC in ABAQUS (2023 Version), indicating that UHPC should not be modeled directly using parameters developed for normal-strength concrete. In addition, the UHPC–normal concrete interface and the rebar–UHPC bond–slip behavior can significantly influence the stiffness, cracking process, and bearing capacity of connection members. Shao and Ostertag [33] investigated the bond–slip behavior of steel-reinforced UHPC under flexure and pointed out that parameters obtained from pull-out tests may not fully represent the bond mechanism in flexural members. He et al. [34] studied the bond performance, constitutive model, and size effect of UHPC-to-normal concrete interfaces, showing that interface roughness, interface size, and stress state affect interfacial mechanical response. Therefore, using conventional UHPC or normal concrete models alone may not accurately capture the coupled effects of material constitutive behavior, interface force transfer, and reinforcement bond in LS-UHPC precast connections.

Based on the above analysis, the current research still has the following shortcomings: Firstly, the research on low-carbon LS-UHPC mainly focuses on material preparation and microscopic mechanisms, lacking a tensile-compressive constitutive model that can be used for structural finite element analysis. Secondly, the existing research on prefabricated connection of UHPC or FRP-UHPC mostly relies on ordinary UHPC, RPC, FRP bar, grouting sleeve or prefabricated shell, and does not pay enough attention to the ordinary rebar lapp-post-cast LS-UHPC connection system. Thirdly, the influence of the UHPC-ordinary concrete interface force transmission in LS-UHPC connection components on the finite element results is still lacking in systematic discussions. In order to make up for the above shortcomings, LS-UHPC is prepared by using lithium slag, an industrial by-product, and it is applied to the post-casting connection area of a precast beam with a common steel bar lap. Firstly, the tensile and compressive constitutive model of LS-UHPC is established through uniaxial compression and uniaxial tensile tests. Secondly, the constitutive model is embedded into the finite element model to simulate the flexural performance of the post-cast LS-UHPC prefabricated beam connection components, and the reliability of the model is verified in combination with the test results. The analytical framework of “material constitutive, interfacial force transfer and component response” connected by LS-UHPC precast beams is established in this paper, which provides a theoretical basis and design reference for the application of low-carbon LS-UHPC in prefabricated concrete structures.

2. Overview of the Experiment

2.1. Raw Materials for the Experiment

Cement (C): Anhui Conch brand P·II 52.5 cement (Anhui Huaguo Cement Co., Ltd., Wuhu, China); Silica fume (SF): Grey powder produced by Elkem (Eiken International Trade Co., Ltd., Shanghai, China); Fly ash (FA): Grade II fly ash from Platinum Refractory Materials Co., Ltd. (Zhengzhou, China); Lithium slag (LS): Light-yellow powder from Tangshan Xinfeng Lithium Industry Co., Ltd. (Tangshan, China), with a specific surface area of 600 m²/kg. The chemical compositions of the above four materials are presented in

Table 1. Chemical composition of raw materials (%).

	SiO2	Al2O3	CaO	Fe2O3	MgO	Na2O	K2O	SO3	TiO2	P2O5	Others
C	20.02	4.64	63.13	5.71	1.71	0.20	1.27	1.86	0.58	0.07	0.81
SF	96.24	0.49	0.85	0.12	0.48	0.23	0.89	0.28	0.05	0.18	0.19
FA	49.34	25.91	10.63	6.88	1.00	0.21	1.41	1.28	1.82	0.10	1.42
LS	38.72	13.14	27.17	0.71	0.89	3.02	1.49	13.21	0.11	1.22	0.32

. Three types of quartz sand are selected (Gongyi Xinde Water Purification Materials Sales Co., Ltd., Zhengzhou, China), with particle size ranges of 0.079~0.56 mm (fine sand), 0.11~0.89 mm (medium sand), and 0.22~1.42 mm (coarse sand), the cumulative particle size distribution curve is shown in Figure 1; Water-reducing admixture (Xiamen Jibang New Materials Co., Ltd., Xiamen, China): UHPC60-type modified polycarboxylate-based superplasticizer, with a water reduction rate of 40%; Steel fiber: Copper-coated straight steel fiber (Henan Zange Industrial Co., Ltd., Zhengzhou, China), 14 mm in length, 0.2 mm in diameter, its performance indicators are shown in

Table 2. Main technical indicators of steel fibers.

Density/kg/m3	Length/mm	Diameter/mm	Elastic Modulus/Gpa	Tensile Strength/MPa	Fracture Elongation/%
7800	14	0.25	200~220	≥2000	≥10

.

2.2. Mix Proportions

LS with a specific surface area of 1100 m²/kg exhibits the highest pozzolanic activity and delivers the most significant improvement in UHPC performance [35]. The influence of LS replacement rate (0~50%) on UHPC performance is comprehensively reported in reference [36]. To avoid redundant experiments and concentrate resources on the systematic characterization of the influencing patterns, this study focuses on an alternative range of 10% to 30%, which is the optimal and most significant range observed in previous studies.

Table 3. Mix proportions of LS-UHPC for uniaxial compressive strength testing (kg/m³).

Mixture	C	LS	FA	SF	FS	MS	CS	W	Steel Fiber	Water Reducer
LS10	688	76	113	207	317	286	493	195	156	28
LS20	622	142	113	207	317	286	493	195	156	28
LS30	535	229	113	207	317	286	493	195	156	28

lists the detailed mixture design parameters, and the specimen naming convention follows the format “LS-X”, where X denotes the replacement rate (in %).

2.3. Test Method

(1)

Specimen preparation

The cementite material (including cement, silica floss and lithium slag) and quartz sand are poured into the mixing pot and stirred for 120 s. The water-reducing agent and water are poured into the stirring pot, wet mix for 300 s, steel fiber is evenly added and stirring is continued for 180 s. They are respectively poured into a 100 mm × 100 mm × 100 mm test mold for the cube compression test, a 100 mm × 100 mm × 300 mm test mold for the axial compression test and a 60 mm × 30 mm × 330 mm test mold for the axial tensile test. Each group contains three mix ratios (LS10/LS20/LS30). A total of 27 pieces are placed on the shaking table for 60 s and then coated immediately. After standing at room temperature for 24 h, the specimens are demolded and subsequently cured under standard conditions for 28 days before performance testing.

(2)

Mechanical property testing

a.

Axial compression test

A UHPC prism specimen of 100 mm × 100 mm × 300 mm is used. After reaching the curing age, a compressive test is conducted. Strain gauges are mounted on the lateral surface of the UHPC specimen, as illustrated in Figure 2, to facilitate the recording of the lateral and axial deformations of the specimen. To obtain a complete stress–strain curve, force control is used initially during the test and then switched to displacement control. That is, the specimen is loaded at a rate of 5 kN/s until 80% of the peak stress is reached and then loaded at 0.2 mm/min until failure.

• b.

  Uniaxial tensile test

Dog-bone-shaped specimens with dimensions as shown in Figure 3 are used. After curing to the specified age, uniaxial tensile tests are performed. Two displacement transducers are mounted symmetrically on the gauge section of each specimen using fixtures to record axial displacement and applied load simultaneously. The test is conducted under displacement control at a steady crosshead speed of 0.2 mm/min, terminating when either the tensile force declines to 20% of its maximum value or catastrophic failure of the specimen is observed.

2.4. Establishment of the Finite Element Simulation

(1)

Geometric dimensions

Component designs are based on the existing literature [37]. The dimensions of the beams are all 150 mm × 250 mm × 2000 mm (b × h × l). The connecting beams use C40 concrete + UC150 LS-UHPC. The UHPC bond lengths are taken as 8d, 10d, and 12d. The strength grades of the longitudinal bars are HRB400 and HRB500, with a diameter of 16 mm. Longitudinal bars consist of both HRB400 and HRB500 steel, each having a nominal diameter of 16 mm, while the stirrups are made of HRB400 steel with a diameter of 8 mm. To enhance interfacial bond performance between the UHPC and conventional concrete, the contact surface is intentionally roughened and equipped with keyways—illustrated in Figure 4. The parameters of the components are shown in

Table 4. Component parameter settings.

	Longitudinal Reinforcement Strength Grade	Longitudinal Reinforcement Diameter/mm	Reinforcement Strength Grade	Rebar Diameter/mm	Concrete Strength Grade	UHPC Strength Grade	UHPC Bond Length
L1	HRB400	16	HRB400	8	C40	150	8d
L2	HRB400	16	HRB400	8	C40	150	10d
L3	HRB500	16	HRB400	8	C40	150	10d
L4	HRB500	16	HRB400	8	C40	150	12d

.

(2)

Finite element model

Both ordinary concrete and UHPC are discretized using the C3D8R element—a three-dimensional, eight-node hexahedral solid element with linear shape functions and reduced integration. Reinforcing bars are represented by the T2D3 element, a two-node linear truss formulation. Following the refinement process, the resulting grid cells are presented below: a uniform mesh size of 20 mm is employed for both conventional concrete and UHPC, whereas the reinforcement elements are discretized with a coarser mesh of 25 mm; the analysis of grid convergence can be found in Section 2.4 (5). Simply supported boundary conditions are applied, and displacement-controlled loading is used. Figure 5 presents the geometric configuration of the beam. To study the influence of parameters such as the diameter of longitudinal reinforcement (14/16/18/20 mm), strength grade HRB400/HRB500, and concrete strength grade (C30/C35/C40/C45) on the bending performance of the post-poured beam, numerical simulations are conducted using finite element software (2023 Version).

(3)

Parameter settings

a.

Constitutive model

Ordinary concrete adopts the constitutive model suggested by the code [38], and its plastic damage model parameters are taken from the literature [39]. The compressive constitutive relationship of UHPC is shown in Section 3.2, and its plastic damage model parameters are taken from the literature [32] and calibrated. The specific parameters can be found in

Table 5. Material parameters for the concrete damaged plasticity (CDP) model.

Concrete	Expansion Angle	Eccentricity	${\mathit{\delta }}_{\mathit{b}0}/{\mathit{\delta }}_{\mathit{c}0}$	${\mathit{K}}_{\mathit{c}}$	Viscosity Coefficient
C40	30	0.1	1.16	0.667	0.005
LS-UHPC	55	0.1	3.0	0.667	0.005

.

Constitutive model for reinforcing steel: the double-line synchronous reinforcement model [40] is adopted, and its expression is:

$$\sigma =\left\{\begin{array}{c}{E}_s\epsilon \\ f_y+0.01E_s(\epsilon -{\epsilon }_y)\end{array}\right. \begin{array}{c}\begin{array}{c}0\prec \epsilon \prec {\epsilon }_y\\ {\epsilon }_y\prec \epsilon \prec {\epsilon }_u\end{array}\end{array}$$

In the formula: $E_s$ denotes the elastic modulus of the reinforcement steel; $f_y$ represents its yield strength; ${\epsilon }_y$ corresponds to the yield strain; and ${\epsilon }_u$ indicates the ultimate tensile strain.

• b.

  Interface settings

The interface parameters of ordinary concrete and UHPC are obtained from reference [41], and are verified through experiments as shown in

Table 6. Parameters of the cohesive-coulomb friction model.

Knn	Kss	Ktt	tn0	ts0	tt0	σp	μ
2884	73	73	12.32	21.33	21.33	0.256	1.49

.

The parameters of the bond–slip element at the interface between reinforcement and concrete are taken from the literature [42]. The parameters of the bonding slip unit at the interface between reinforcing bars and UHPC are taken from reference [43] and verified. The damage parameter values are shown in

Table 7. The interface bond–slip parameters.

	gK1	gK2	gK3	gK4	gKL	gD1	gD2	gD3	gD4	gDL	gF1	gF2	gF3	gF4	gFL
Rebar-Concrete	0	0	0.3	0.2	0.1	0.5	0.5	2.0	2.0	0.5	1.5	1.0	1.0	2.2	0.8
Rebar-UHPC	0	0	0.3	0.2	0.1	0.5	0.5	2.0	2.0	0.5	2.2	2.0	0.8	2.2	0.7

.

(4)

Boundary conditions

The finite element model adopts boundary conditions and loading protocols identical to those used in the experimental study [37]. Specifically, the left support fully constrains translational degrees of freedom in all three directions (X, Y, Z) and rotational degrees of freedom about the Y- and Z-axes. The right support restricts translations along the X- and Y-axes and rotations about the X- and Y-axes.

(5)

Mesh Convergence Analysis

To ensure that the calculation results do not depend on the mesh size, the specimen L2 (HRB400 reinforced bar, lap length 10d) is selected for mesh convergence analysis. Three groups of concrete mesh schemes are examined: coarse mesh (50 mm), medium mesh (30 mm), and fine mesh (20 mm). The evaluation indicators included ultimate load, initial stiffness, and failure mode. The results show that the ultimate load difference is 0.08%, the initial stiffness difference is less than 2%, and the failure mode is almost the same between the medium mesh and the fine mesh, while the calculation time of the fine grid is 1.46 times that of the medium mesh. Therefore, the mesh size of 20 mm concrete and 25 mm steel bar is confirmed as the optimal solution on the premise of ensuring the calculation accuracy. The detailed convergence analysis results are presented in

Table 8. Evaluation indexes for grid convergence analysis.

Concrete Grid	Ultimate Load/KN	Initial Stiffness/N/mm	Destructive Mode	Calculate the Time	Convergence Analysis
50 mm	114.97	24,750	Consistent	20 min	Convergence
30 mm	114.76	24,772	Consistent	48 min	Convergence
20 mm	114.85	24,280	Consistent	70 min	Convergence

and Figure 6.

3. Results Analysis

3.1. Test Results

(1)

Axial compressive and tensile strength

Figure 7 and Figure 8 illustrate the axial compressive and tensile strengths of LS-UHPC, respectively. As indicated, both the compressive and tensile strengths exhibit a non-monotonic trend—initially rising and subsequently declining—as the LS replacement ratio increases.

The optimal comprehensive performance is achieved when LS replaces 20% of the cement by mass. Studies have demonstrated that lithium slag (LS) exhibits pozzolanic activity and reacts with Ca(OH)₂ to form C-S-H gel, which fills capillary pores and refines the pore structure of UHPC. With increasing LS replacement ratio, the early-age heat evolution and Ca(OH)₂ content of UHPC gradually decrease. The amount of cement hydration and the bound water content per unit volume first increase and then decrease, whereas the total porosity and the number of harmful pores first decrease and then increase. Similar trends are reported in reference [35].

(2)

Elastic modulus

The elastic modulus is determined in this study by the slope of the ascending segment of the stress–strain curve, from the origin to 0.4 times the peak stress [44]. Figure 9a displays the elastic modulus values for all UHPC specimen groups, ranging from 4.88 × 10⁴ MPa to 4.93 × 10⁴ MPa, and demonstrating a consistent increasing trend with rising axial compressive strength. Using the formula proposed in reference [45], the cube strength of UHPC is calculated as shown in Formula (1). Referring to the concrete elastic modulus formula proposed by the China Academy of Building Research, the elastic modulus of UHPC is fitted as shown in Figure 7b, as shown in Formula (2).

$$f_{cu}=0.94f_c+27.52$$

(1)

$$E_c={\displaystyle \frac{{10}^5}{1.19+\frac{143.24}{f_{cu}}}}$$

(2)

(3)

Peak strain

The compressive strain at peak load for LS-UHPC, as shown in Figure 10a, ranges from 3200 to 3500 µε and exhibits a positive correlation with the uniaxial compressive strength. The existing literature suggests that the tensile peak strain of UHPC is strongly influenced by steel fiber content and increases with higher fiber dosage [46]. All UHPC employed in this investigation incorporated steel fibers at a volumetric dosage of 2%; consequently, the peak tensile strain across all groups is nearly identical, with the maximum and minimum values differing by only 5.63%.

(4)

Poisson’s ratio

As shown in Figure 11, the Poisson’s ratio of LS-UHPC remains stable between 0.17 and 0.19 when the axial strain ranges from 1500 to 3000 µε. Accordingly, the mean Poisson’s ratio corresponding to an axial strain of 2000 µε is adopted as the representative value, corresponding to 0.18.

Compared with the existing UHPC studies, Graybeal et al. [47] reported that the elastic modulus of the traditional UHPFRC is approximately 5.0–5.3 × 10⁴ MPa, and Wille et al. [48] reported that the elastic modulus of the heat-treatment-free high-strength UHPC could reach 5.2–5.5 × 10⁴ MPa. The elastic modulus of the LS-UHPC in this study (4.45–4.97 × 10⁴ MPa) is slightly lower by about 5–8%, but the peak strain (3150–3500 με) is comparable to the reported value of Graybeal et al. [47] (3200–3400 με). This indicates that lithium slag replacing cement has a certain reducing effect on the stiffness of UHPC but has little influence on the deformation capacity.

3.2. Constitutive Relationship

(1)

Compressive constitutive behavior

A uniaxial compressive constitutive relationship for LS-UHPC is formulated using the above test results. Figure 12 displays the dimensionless stress–strain response of LS-UHPC in compression. The proposed constitutive model should meet the following geometric requirements: ① x = 0, y = 0; ② x = 0, ${\displaystyle \frac{dy}{dx}}\ge 1$; ③ 0 ≤ x < 1, ${\displaystyle \frac{d^{2}y}{d{x}^2}}<0$, the slope of the ascending section curve monotonically decreases; ④ x = 1, y = 1 and ${\displaystyle \frac{dy}{dx}}=0$; ⑤ ${\displaystyle \frac{d^{2}y}{d{x}^2}}=0$, $X_D>1.0$; ⑥ ${\displaystyle \frac{d^{3}y}{d{x}^3}}=0$, $X_E>X_D$; ⑦ $x\to \infty$, $y\to 0$ and ${\displaystyle \frac{dy}{dx}}\to 0$; ⑧ All the coordinates of the points on the curve should satisfy both $x\ge 0$ and $0\le y\le 1$.

As reported by Ma et al. [49], polynomial fitting yields superior performance for the ascending branch, that is, $y={\mathrm{a}}_0+{\mathrm{a}}_{1}x+{\mathrm{a}}_3{x}^3+{\mathrm{a}}_4{x}^4$. For the descending branch, a rational function is adopted for curve fitting, that is, $y={\displaystyle \frac{x}{\mathrm{\alpha }{x}^2+\mathrm{\beta }x+\mathrm{\gamma }}}$. Combining formulas ① to ⑥, the compressive constitutive model is as follows:

$$y=\left\{\begin{array}{c}\mathrm{a}x+(3-2\mathrm{a})x^2+(\mathrm{a}-2)x^3 0\le x\le 1\\ {\displaystyle \frac{x}{\mathrm{\alpha }{(x-1)}^2+x}} x\succ 1\end{array}\right.$$

In the formula, a controls the initial tangent stiffness and the stiffness degradation rate of the compression ascending segment. Its value is related to the dense matrix of LS-UHPC, the inhibition of micro-crack propagation by steel fibers, and the filling/nucleation effect of lithium slag; when a > 1, the curve shows the characteristic of being convex in the early stage and concave in the later stage, that is, the initial stiffness is higher, and then the stiffness gradually degrades. α controls the softening rate in the compression descent section and the post-peak energy dissipation capacity. Its value reflects the balance between the bridging effect of steel fibers and the brittleness of the matrix. The larger the α value, the milder the descent section, indicating that the material has better post-peak ductility, as shown in Figure 13.

It can be seen that when a = 1.2, the ascending section closely matches the experimental data, indicating that LS-UHPC has a relatively high tangent modulus in the initial stage of compression. This is related to the bridging effect of steel fibers—the fibers can effectively inhibit the expansion of cracks during the stage of micro-crack initiation, maintaining a high stiffness. Ma et al. [49] in their constitutive study of RPC200 obtained a value of approximately 1.1 to 1.3. This model’s a = 1.2 is completely consistent with this range. The descending section α = 4 is within a reasonable range compared to Ma Yafeng’s study (α = 3.5 to 5.0), indicating that although the strength of LS-UHPC slightly decreases due to lithium slag substitution, the ductility after the peak does not significantly deteriorate. The constitutive relationship of LS-UHPC under compression at this time is shown in Figure 14, and the formula is as follows:

$$y=\left\{\begin{array}{l}1.2x+0.4x^3-0.6x^4 0\le x\le 1\\ {\displaystyle \frac{x}{4{(x-1)}^2+x}} x\ge 1\end{array}\right.$$

(3)

To verify the rationality of the compression constitutive parameters of LS-UHPC proposed in this paper, they are compared with the existing uniaxial compression constitutive models of RPC/UHPC, as shown in Figure 15 and

Table 9. Comparison of uniaxial compression constitutive models of UHPC.

Literature	Material Object	Ascending Phase Form	Descending Phase Form	Ascending Phase Parameter	Descending Phase Parameter	Main Features	The Differences from the LS-UHPC Model in This Article
Ma et al. [49], 2006	RPC200	$\begin{array}{l}ax+(5-4a)x^4\\ +(3a-4)x^5\end{array}$	${\displaystyle \frac{x}{\alpha {(x-1)}^2+x}}$	a = 1.1~1.4	α = 6.0~10.0	For a 200 MPa-level RPC, provide a complete description of the compression ascending and descending segments	High-strength grade material is RPC20
Guo et al. [50], 2017	UHPC /RPC	${\displaystyle \frac{Ax}{1+(A-1)x^{\frac{A}{A-1}}}}$	${\displaystyle \frac{x}{B(x-1)x^2+x}}$	$A={\displaystyle \raisebox{1ex}{$E_c{\epsilon }_{cu}$}\!\left/ \!\raisebox{-1ex}{$f_c$}\right.}$, And provide the experience formula	B Control the softening of the descent section	Unified processing of multiple UHPC/RPC compression models	It is more suitable to serve as the direct comparison benchmark for the compression model in this article
Zhao et al. [51], 2024	UHPC	Compressive skeleton curve	Compression hysteresis/regression curve	-	-	Applicable to uniaxial compression cyclic loading	It can be used to illustrate the applicable boundaries of the model in this paper
This article	LS-UHPC	$\begin{array}{l}ax+(3-2a)x^2\\ +(a-2)x^3\end{array}$	${\displaystyle \frac{x}{\alpha {(x-1)}^2+x}}$	a = 1.2	α = 4	The test curves for LS10, LS20 and LS30 are established	It focuses on the characteristics of stiffness, peak strain and post-peak softening after lithium slag substitution

. Compared with the above model, the parameter a of the compression rising section of LS-UHPC in this paper is 1.2, which is within the common parameter range of RPC/UHPC, indicating that lithium slag substitution does not significantly change the initial stiffness evolution characteristics of LS-UHPC. However, the descending segment parameter α = 4 in this paper is not directly equivalent to the B parameter in the traditional RPC200 model, which mainly reflects the post-peak softening rate of LS-UHPC. Due to the influence of lithium slag substitution on the hydration of the matrix, the pore structure, and the crack expansion process after the steel fiber bridging, the descending segment parameters of LS-UHPC need to be recalibrated based on the experimental data of this paper. Guo et al. [50] also put forward the same suggestion. The differences in mix ratios and test methods lead to significant variations in the descending sections of the uniaxial compression stress–strain curve of ultra-high performance concrete, and the corresponding parameter values of the descending sections also have a large degree of dispersion. Therefore, a large number of experimental studies need to be conducted on the basis of determining the standard test methods.

(2)

Tensile constitutive behavior

Figure 16 shows the normalized tensile stress–strain curve of LS-UHPC. As can be seen from the figure, the curve comprises an elastic ascending branch, a plastic ascending branch, and a softening descending branch.

The elastic ascending branch: Experimental studies have shown that, prior to crack initiation in UHPC specimens, the stress–strain curve is approximately linear. Similar observations are reported in references [51,52]. Thus $0\le x\le {\displaystyle \frac{{\epsilon }_{te}}{{\epsilon }_{tu}}}$, $y=E_{t0}{\displaystyle \frac{{\epsilon }_{tu}}{{\sigma }_{tu}}}x$.

In the formula: $E_{t0}$ represents the initial tensile elastic modulus, calculated using Equation (2); ${\sigma }_{tu}$ and ${\epsilon }_{tu}$ denote the peak tensile stress and peak tensile strain of UHPC, respectively; ${\epsilon }_{te}$ is the strain at cracking of UHPC, taken as $200 \mathrm{\mu }\mathrm{\epsilon }$ based on experimental results.

The plastic ascending branch: After cracking of LS-UHPC, a turning point appears on the stress–strain curve. The UHPC curve exhibits an ascending branch characterized by a reduction in tensile stiffness. The fitted expression is $y=0.64+0.36x^{0.25}$, as shown in Figure 17a.

Softening descending branch: According to GB 50010-2010 [38], the descending branch of the UHPC’s tensile stress–strain behavior is modeled using a rational function, which is: $y={\displaystyle \frac{x}{{\alpha }_t{(x-1)}^{\beta }+x}}$. In the equation, ${\alpha }_t$ represents the softening section after the tensile peak, which is related to the volume fraction of steel fibers, fiber pull-out, localization of cracks, and the fracture energy of the matrix. The smaller the value, the steeper the softening section, indicating that the bearing capacity drops sharply after the peak stress. The fitted value of β is 3, as depicted in Figure 17b.

Figure 17b shows the comparison between the test curve and the theoretical curve of the tensile constitutive descent section of UHPC. It can be seen that the shapes of the three groups of test curves are consistent with the shape of the theoretical model. All the curves fall within the range of $\alpha =0.1$–$0.15$, and $\alpha =0.13$, which is consistent with the conclusion of Zhang et al. [52]: when the volume content of steel fiber is 2%, the softening section of UHPC is steeper, because the fiber pulling process is more concentrated and the bearing capacity decreases rapidly. And in the unified tensile constitutive model of single/mixed fiber UHPC proposed by Hiew et al. [31], it is clearly pointed out that the fiber content directly affects the slope of the tensile softening section, which is consistent with the observed results of this study that β is smaller and the softening is steeper. The tensile constitutive behavior of UHPC under tension is shown in Figure 18, and the formula is as follows:

$$y=\left\{\begin{array}{l}{E}_{t0}{\displaystyle \frac{{\epsilon }_{tu}}{{\sigma }_{tu}}}x0\le x\le {\displaystyle \frac{{\epsilon }_{te}}{{\epsilon }_{tu}}}\\ 0.64+0.36x^{0.25}{\displaystyle \frac{{\epsilon }_{te}}{{\epsilon }_{tu}}}<x<1\\ {\displaystyle \frac{x}{0.13{(x-1)}^3+x}}x\ge 1\end{array}\right.$$

(4)

To further verify the rationality of the LS-UHPC tensile constitutive model proposed in this paper, it is compared with the existing research on the tensile performance of UHPC and constitutive models, as shown in Figure 19 and

Table 10. Comparison of uniaxial tensile constitutive models of UHPC.

Literature	Material System	Curved Form	Main Parameters	The Physical Meaning of Parameters	The Differences from the LS-UHPC in This Article
Su et al. [54]	Mixed steel fiber UHPC	Mainly focuses on uniaxial tensile properties and the hybrid effect	Tensile strength, peak strain, tensile toughness, and mixed effect indicators	It reflects the effect of different steel fiber combinations on crack control and fiber bridging ability	The key point is the effect of fiber heterogeneity
Gao et al. [55]	Strain-hardening/strain-softening type UHPC	Monotonic envelope line + cyclic unloading/reloading path	Cracking strength, ultimate tensile strength, residual strain, unloading stiffness, cyclic degradation parameters	Reflecting the closure of cracks, the accumulation of damage and the degradation of stiffness under cyclic tensile loading	Applicable to cyclic loading
Wang et al. [53]	Mixed steel fiber UHPC	$\left\{\begin{array}{cc}{E}_{t0}{\displaystyle \frac{{\epsilon }_{tu}}{{\sigma }_{tu}}}x\hfill & \hfill 0\le x\le {\displaystyle \frac{{\epsilon }_{te}}{{\epsilon }_{tu}}}\\ A+a{e}^{bx}\hfill & \hfill {\displaystyle \frac{{\epsilon }_{te}}{{\epsilon }_{tu}}}\prec x\le 1\\ {\displaystyle \frac{x}{{\alpha }_t{(x-1)}^{\beta }+x}}\hfill & \hfill 1\le x\end{array}\right.$	A, a, b, ${\alpha }_t$, β	A, a and b control the ascending section after cracking: ${\alpha }_t$ and β control the softening section after the peak	It is closest to the form of this article, but the parameters are calibrated by ordinary/hybrid steel fiber UHPC
This article	LS-UHPC				Finite element analysis for LS-UHPC and post-poured connections; The scope of application is limited to monotonous tension at normal temperature

. In this paper, the LS-UHPC tensile constitutive model borrows the piecewise modeling idea of Wang Qiwei et al. [53], but the model parameters are recalibrated according to the experimental curves of LS10, LS20 and LS30. The reason is that the substitution of lithium slag will change the hydration products, pore structure, and fiber-bond interface properties of UHPC, thereby affecting the cracking strain, peak tensile strain, and post-peak softening rate. Therefore, the model in this paper is based on the same three-stage mechanical mechanism to establish a tensile constitutive expression applicable to LS-UHPC.

From the perspective of physical mechanism, in the elastic rising stage, the LS-UHPC matrix has not yet appeared obvious cracking stage. The plastic ascending section is a multi-crack development stage controlled by the bridging effect of steel fibers after the matrix cracks. At this time, the tensile stiffness decreases but the bearing capacity can still continue to rise. In the softening descent section, after the macroscopic main crack forms, it is the stage of fiber pull-out, local debonding and crack localization development. The softening parameter ${\alpha }_t$ reflects the rate of peak load decay after the peak, and its value is related to the volume fraction of steel fibers, the fiber pull-out performance, the degree of crack localization, and the fracture energy of the lithium slag modified matrix. Compared with the existing UHPC constitutive models, the main difference in this model lies in extending the material system from ordinary or mixed steel fiber UHPC to LS-UHPC and further applying this constitutive model to the finite element analysis of post-poured LS-UHPC prefabricated connection components.

3.3. Finite Element Simulation of the In-Span Connection in LS-UHPC Beams

This chapter presents numerical simulations and verification for four post-poured beams; the results are shown in Figure 20. From top to bottom: the experimental failure pattern of the beam, the tensile damage contour, the compressive damage contour, and the mises contour. The failure mode and cracking characteristics of the beam agree well with the experimental results.

For precast beams with post-poured UHPC connections, under bending loading, cracks initiate first at the interface between ordinary concrete and UHPC. Subsequently, multiple symmetrical short cracks develop in the ordinary concrete. As diagonal cracks form in the flexural-shear zone, they propagate progressively into the compression zone, ultimately leading to crushing failure of the ordinary concrete. As shown in the figure, the mid-span tensile region of the post-poured UHPC area exhibits slight damage, whereas the compression region remains essentially undamaged. Longitudinal reinforcement yields in the ordinary concrete area, while its stress in the UHPC area remains relatively low. This behavior arises because UHPC possesses higher compressive and tensile strengths than C40 concrete, enabling it to resist both tensile and compressive loads more effectively. Relevant discussions can be found in reference [56]. Increasing the strength grade of the longitudinal reinforcement significantly enhances the flexural capacity of the specimens. Figure 20e,f present the experimental and simulated load–displacement curves for precast beams with post-poured UHPC connections under different reinforcement strength grades. The linear ascending branch in the simulation is steeper than that in the test, indicating a higher initial stiffness in the numerical model, approximately 7% to 13%. This deviation mainly resulted from three factors: ① the bond–slip parameters are derived from standard pull-out tests, which differed from the bending stress state in the beam; ② in the simulation, it is assumed that the interface is perfectly bonded, while actual construction had microscopic defects; ③ the shrinkage difference between UHPC and ordinary concrete might cause micro-cracks at the interface. Despite these deviations, the predictions of the ultimate load and failure mode are in good agreement with the experiments (error < 5%), indicating that this model is suitable for ultimate bearing capacity design. However, in the serviceability limit state (such as deflection control), a reduction factor of 0.85 is recommended. Nevertheless, the finite element simulation reasonably captures the overall experimental response within acceptable limits; the discrepancies in crack evolution and load–deformation behavior remain within an acceptable range. These results demonstrate that the proposed model reliably predicts the flexural performance of UHPC connections and provides a sound foundation for future parametric analyses or optimization studies under diverse design conditions.

3.4. Parametric Analysis of Finite Element Simulation

(a) Influence of longitudinal reinforcement diameter (lap length = 10d; stirrup diameter = 8 mm; stirrup spacing = 100 mm)

Figure 21 and Figure 22 present the simulation results for post-poured UHPC-connected prefabricated beams with varying longitudinal reinforcement diameters. As the longitudinal reinforcement diameter increases, both compressive damage and tensile damage in the beam progressively intensify. For diameters of 14~18 mm, the failure mode remains dominated by diagonal crack propagation into the compression zone, culminating in concrete crushing within that zone. At 20 mm, pronounced diagonal cracking occurs, accompanied by severe compression-zone damage; meanwhile, the reinforcement stress on both sides of the interface reaches only the yield level—indicating an over-reinforced condition: the longitudinal bars remain unyielded while the compression-zone concrete crushes prematurely, particularly in the HRB500 case. Hence, a longitudinal reinforcement diameter below 18 mm is recommended for such connections. Figure 23 presents the experimentally measured and numerically predicted load–displacement responses of the tested beams. With increasing bar diameter, the flexural capacity rises markedly—a trend corroborated by reference [4]. Notably, for the HRB500/20 mm case, the load–displacement curve exhibits a distinct softening branch in the post-peak regime, reflecting premature concrete damage before full utilization of the reinforcement’s strength.

Table 11. Comparison of initial cracking and ultimate loads of post-poured UHPC-connected precast beams (HRB400) with different longitudinal reinforcement diameters.

Diameter of Longitudinal Reinforcement Bars/mm	Initial Cracking Load/kN		Relative Error/%	Ultimate Load/kN		Relative Error/%
	Test Value	Simulated Value		Test Value	Simulated Value
14	-	18.6	-	-	90.0	-
16	24	19.9	17.1	109.9	114.1	3.8
18	-	21.6	-	-	142.0	-
20	-	22.9	-	-	172.2	-

Note:”-” indicates that this condition is not tested in reference [37], and it is an extended condition for the parameter analysis of this study.

and

Table 12. Comparison of initial cracking and ultimate loads of precast beams (HRB500) with post-poured UHPC connections having different longitudinal reinforcement diameters.

Diameter of Longitudinal Reinforcement Bars/mm	Initial Cracking Load/kN		Relative Error/%	Ultimate Load/kN		Relative Error/%
	Test Value	Simulated Value		Test Value	Simulated Value
14	-	17.9	-	-	108.0	-
16	25	20.3	18.8	135.5	136.6	0.8
18	-	20.1	-	-	169.5	-
20	-	12.4	-	-	203.2	-

Note:”-” indicates that this condition is not tested in reference [37], and it is an extended condition for the parameter analysis of this study.

summarize the initial cracking and ultimate loads. For HRB400, both cracking and ultimate loads increase monotonically with bar diameter—consistent with reference. For HRB500, the cracking load first increases then decreases with diameter, whereas the ultimate load continues to rise. This confirms that high-strength, large-diameter reinforcement can reduce the concrete cracking load in prefabricated beams.

(b) Concrete strength grade (longitudinal reinforcement diameter = 16 mm; lap length = 10d; stirrup diameter = 8 mm; stirrup spacing = 100 mm)

Figure 24 and Figure 25 present the numerical simulations of precast beams with ordinary concrete of varying strength classes. As observed, modifying the compressive strength class of the ordinary concrete has no influence on the dominant failure mechanism of the precast beams. The UHPC shows almost no damage, whereas the ordinary concrete develops cracking—particularly diagonal cracks propagating into the compressive region. Concurrently, the longitudinal reinforcement bars on both sides of the interface have yielded, and the ordinary concrete fractures and collapses. The mises cloud diagram of the concrete also proves that the precast beams all present the reinforced beam failure mode. However, raising the strength class of the ordinary concrete leads to a decrease in both tensile and compressive damage within the concrete matrix. This is because the higher the strength grade of the concrete, the stronger its ability to resist compressive and tensile loads. Similar to reference [57], this has also been mentioned. Figure 26 shows the experimental and simulated load–displacement curves. These curves are nearly coincident across strength grades, indicating that while elevated concrete strength mitigates damage accumulation, it yields no substantial improvement in flexural stiffness or ultimate moment capacity.

Table 13. Comparison of initial cracking and ultimate loads of post-poured UHPC-connected prefabricated beams (HRB400) with different ordinary concrete strength grades.

Strength Grades of Ordinary Concrete	Initial Cracking Load/kN		Relative Error/%	Ultimate Load/kN		Relative Error/%
	Test Value	Simulated Value		Test Value	Simulated Value
C30	-	18.6	-	-	114.1	-
C35	-	19.2	-	-	113.9	-
C40	24	19.9	17.1	109.9	114.1	3.8
C45	-	20.5	-	-	114.1	-

Note:”-” indicates that this condition is not tested in reference [37], and it is an extended condition for the parameter analysis of this study.

and

Table 14. Comparison of initial cracking and ultimate loads of post-poured UHPC-connected prefabricated beams (HRB500) with different ordinary concrete strength grades.

Strength Grades of Ordinary Concrete	Initial Cracking Load/kN		Relative Error/%	Ultimate Load/kN		Relative Error/%
	Test Value	Simulated Value		Test Value	Simulated Value
C30	-	17.5	-	-	136.5	-
C35	-	18.1	-	-	136.5	-
C40	25	20.3	18.8	135.5	136.6	0.8
C45	-	19.5	-	-	137.5	-

Note:”-” indicates that this condition is not tested in reference [37], and it is an extended condition for the parameter analysis of this study.

summarize the initial cracking and ultimate loads. Both loads increase marginally with rising concrete strength grade—demonstrating that modest gains in bearing capacity can be achieved by upgrading ordinary concrete strength.

4. Limitations and Future Work

The limitations of this study are as follows: ① the validation of the constitutive model is limited to monotonic loading, without considering cyclic or seismic loads; ② the bonding slip parameters are obtained based on the literature of classical pull-out tests and may have slight differences from the actual bending stress conditions of the beam; ③ the parametric analysis is based on limited beam dimensions (150 mm × 250 mm), and the generalizability of the findings requires further verification; ④ only three lithium slag replacement ratios (10%, 20%, and 30%) are considered, without covering a broader range. Future work should include: ① conducting a direct shear test on the LS-UHPC-ordinary concrete interface; the bond–slip tests are carried out on the UHPC-rebar interface and the ordinary concrete-rebar interface; ② numerical simulation and experimental validation of LS-UHPC connection joints under cyclic loading; ③ extended parametric analysis for a wider range of beam cross-sectional dimensions, while also expanding the range of lithium slag content.

5. Conclusions

In this study, a stress–strain constitutive model of LS-UHPC is established through mechanical tests and embedded into a finite element framework. Numerical simulations and parametric analyses are conducted on the flexural behavior of post-poured LS-UHPC connections at the mid-span of precast concrete beams. The main conclusions are as follows:

(1)

The key mechanical parameters of LS-UHPC are: elastic modulus of 4.45 × 10⁴ MPa to 4.97 × 10⁴ MPa, compressive peak strain of 3150–3500 με, tensile peak strain of 2300–2450 με, and Poisson’s ratio of 0.17–0.19. Compared with existing UHPC studies, the elastic modulus is 5–8% lower, but the peak strain is comparable, indicating that replacing cement with lithium slag slightly reduces stiffness but has little effect on deformation capacity.

(2)

A piecewise constitutive model for LS-UHPC is established. The compressive model uses parameters a = 1.2 and α = 4, with goodness-of-fit R² = 0.962 (ascending branch) and R² = 0.886 (descending branch). The tensile model has a softening parameter β = 0.13, with R² = 0.879 (plastic ascending branch) and R² = 0.988 (softening descending branch). This model is the first complete constitutive expression specifically for LS-UHPC.

(3)

The failure mode of the post-poured LS-UHPC connection is as follows: cracks first appear at the interface between ordinary concrete and UHPC, then multiple short cracks form in the ordinary concrete zone, diagonal cracks in the flexural-shear zone propagate toward the compression zone, and finally, the ordinary concrete is crushed. Damage in the LS-UHPC zone is minor, indicating that LS-UHPC effectively transfers load in the connection region.

(4)

The simulated initial stiffness is 7–13% higher than the experimental value, mainly due to differences in bond–slip parameter sources, the assumption of perfect interface bonding, and shrinkage differences. However, predictions of ultimate load and failure mode agree well with experiments (error < 5%), so the model is suitable for ultimate limit state design. For serviceability limit states (e.g., deflection control), a stiffness reduction factor of 0.85 is recommended.

(5)

Increasing the longitudinal rebar diameter significantly enhances the flexural capacity: from 14 mm to 20 mm, the ultimate capacity increases from 90.0 kN to 172.2 kN for HRB400, and from 108.0 kN to 203.2 kN for HRB500. However, when the diameter exceeds 18 mm with HRB500 rebars, an over-reinforced failure mode occurs (concrete crushes before steel yielding). For connection nodes with similar cross-sectional dimensions (150 mm × 250 mm), it is recommended that the diameter of HRB500 longitudinal rebars should not exceed 18 mm.

(6)

Increasing the ordinary concrete strength grade from C30 to C45 improves the ultimate capacity of the connection by less than 5%, so this should not be used as a primary optimization measure in design.

6. Suggestions for Engineering Applications

(1)

It is recommended that the LS substitution rate be controlled within 20% to achieve the best mechanical properties;

(2)

It is recommended that in the connection nodes of precast beams with similar cross-sectional dimensions (150 mm × 250 mm), the diameter of the HRB500 longitudinal reinforcement should not exceed 18 mm to avoid excessive reinforcement failure;

(3)

Increasing the strength grade of ordinary concrete has a limited effect on the bending bearing capacity of the nodes; it is not recommended as the main optimization method;

(4)

During construction, it is necessary to ensure that the interface between LS-UHPC and ordinary concrete is roughly treated and key grooves are set.