Experimental Study on the Durability Performance of Sustainable Mortar with Partial Replacement of Natural Aggregates by Fiber-Reinforced Agricultural Waste Walnut Shells

Abstract

Through the recovery and reuse of agricultural waste, the extraction and consumption of natural aggregates can be reduced to realize the sustainable development of the construction industry. Therefore, this paper utilizes the inexpensive, surplus, clean, and environmentally friendly waste agricultural material walnut shell to partially replace the fine aggregates in mortar to prepare environmentally friendly mortar. Considering the decrease in mortar performance after mixing walnut shells, basalt fibers of different lengths (3 mm, 6 mm, and 9 mm) and different dosages (0.1%, 0.2%, and 0.3%) were mixed in the mortar. The reinforcing effect of basalt fibers on walnut shell mortar was investigated by mechanical property tests, impact resistance tests, and freeze–thaw cycle tests. The damage prediction model was established based on the Weibull model and gray model (GM (1,1) model), and the model accuracy was analyzed. The experimental results showed that after adding basalt fibers, the compressive strength, split tensile strength, and flexural strength of the specimens with a length of 6 mm and a doping amount of 0.2% increased by 13.98%, 48.15%, and 43.75%, respectively, and the fibers effectively improved the defects inside the walnut shell mortar. The R²s in the Weibull model were greater than 87.38%, and the average relative error between the predicted life of the impacts and the measured values was greater than 87.38%. The average relative errors in the GM (1,1) model ranged from 0.81% to 2.19%, and the accuracy analyses were all of the first order.

1. Introduction

Due to the rapid development of the construction industry, the exploitation of natural aggregates has accelerated, leading to the rapid depletion of natural resources worldwide [1]. In construction activities, the current trend is to utilize natural materials for sustainable development [2]. By substituting natural raw materials (aggregates), the overuse of natural resources is reduced [3,4]. Natural agricultural waste is waste residue generated from the processing of agricultural products [5], including oil palm shells [6], coconut shells [7], cashew nuts [8], peanuts [9], hazelnuts [10], pistachios [11], shea nuts [12] and walnuts. Due to the high cost of agricultural solid waste disposal and the reduction in the number and size of landfill sites [13], the most common disposal methods for agricultural wastes are in situ burial and open burning [14,15]. On the one hand, in situ landfilling takes up a large amount of space, and some agricultural wastes are dumped and exposed to the natural environment, polluting the air and water and spreading diseases [16,17]. On the other hand, open burning of agricultural waste has a negative impact on the environment [18,19]. Therefore, the use of agricultural wastes such as walnut shells is an inevitable trend in the production of cement mortar and concrete.

Walnut shells, as a renewable agricultural waste product with chemical stability and a large specific surface area [20], have a large number of benefits as a replacement for sand, both in terms of reducing the cost of manufacturing concrete or mortar and eliminating the problem of environmental pollution generated by these wastes [21], and these excellent properties give them great potential to become sustainable building materials [22].

Nahla Hilal et al. [19] used crushed walnut shells to replace coarse and fine aggregates in concrete to investigate the effect of changes in substitution rate on walnut shell concrete. The test results showed that when coarse and fine aggregates were replaced with walnut shells separately, the dry density and mechanical properties of the mortar decreased except for water absorption, and the overall performance was best when coarse and fine aggregates were replaced together. Alexey N. Beskopylny et al. [23] utilized walnut shells as equal volume replacements for coarse aggregates in concrete (5% to 30%), and the density, compressive, flexural, and tensile properties of the concrete were studied. The test results showed that at 5% replacement of walnut shells, the strength properties of the concrete increased by 3.5%, the strength-to-density ratio reached its maximum value, a 10% reduction in crushed stone consumption, and a 6% reduction in concrete mass. Mohanad Yaseen Abdulwahid et al. [20] utilized walnut shells as a replacement for sand in mortar and investigated the effects of these shells on dry density, thermal conductivity, compressive strength, and flexural strength. The results of the test showed that the density and thermal conductivity of the mortar with a 15% substitution amount treated with hot water decreased by 15% and 31%, respectively, and the compressive and flexural strengths decreased by 17% and 25%, respectively, compared to those of the control group. B. Venkatesan et al. [24] used proportions of steel slag fine aggregates at 0%, 10%, 20%, 30%, 40%, and 50%, and a 20% proportion of walnut shells. The mechanical properties of the concrete, such as compressive strength, split tensile strength, and flexural strength, were tested by replacing the coarse aggregate. The test results showed that the mechanical properties of the specimens with 20% substitution of walnut shells were all slightly lower than those of the normal specimens. Ayad S. Aadi et al. [21] used unsuperfine-treated walnut shells and superfine-treated walnut shells to replace cement. The test results showed that both substitutions decreased the flow and hardening properties of the cement mortar, with the negative effect being greater for the unsuperfine-treated mortar. Weimin Cheng et al. [25] used walnut shells as a substitute for natural coarse aggregate, and fibers made from discarded polyethylene terephthalate (PET) bottles were blended into lightweight wet-mixed concrete to improve the properties of the concrete. The test results showed that the compressive strength and splitting tensile strength of poured concrete decreased with increasing walnut shell admixture, and the compressive strength and splitting tensile strength decreased substantially at a replacement rate of 75%; moreover, the splitting tensile strength increased, and the compressive strength decreased after mixing with polypropylene fibers.

Traditional mortar is formulated by mixing cementitious materials and fine aggregates in a certain proportion [26], which is characterized by high compressive strength [27]. However, there are certain defects and microcracks on the internal interface of the mortar matrix [28], which are characterized by poor toughness, easy fracture, and low tensile strength [29]. The porous structure within the mortar affects its durability and reduces its lifespan. An effective means to enhance mortar properties involves mixing the mortar with fibers [30], enhancing the bonding between the matrix, and improving the toughness and ductility of the material [31,32], increasing the durability of the mortar. Basalt stone was melted at 1450 °C~1500 °C [33] and then was pulled and processed into a continuous fiber to form basalt fiber. As a new type of inorganic, environmentally friendly, green, high-performance material, basalt fiber is inexpensive, renewable, has excellent mechanical properties and durability [34,35], and is widely used in the testing of fiber-reinforced cementitious materials. Sateshkumar et al. [36] investigated the impact resistance of high-strength short-cut basalt fiber-reinforced concrete with the complete replacement of natural fine aggregates using manufactured sand. The results showed that as the number of strikes increased, the deflection decreased, and the impact resistance of the fiber-blended concrete improved. Chingxuan Wang et al. [37] investigated the damage modes and impact resistance of concrete blended with basalt fiber-reinforced polymers and polypropylene fibers under impact loading using a self-modified drop hammer impact test apparatus. The test results showed that the mixing of fibers changed the damage mode from brittle to ductile damage, and the two types of fibers produced a positive synergistic effect. John Branston et al. [38] investigated the relative merits of two different types of basalt fibers in enhancing the impact resistance of concrete. The test results showed that both fibers increased the strength of the concrete before cracking, as evidenced by an increase in the number of impacts, and that only the minibars fibers enhanced the performance after cracking, while the minibars fibers showed better ductility. Wenjun Li et al. [39] evaluated the mass loss rate and compressive strength of ordinary activated powder concrete and basalt fiber-activated powder concrete by comparing the mass loss rate and compressive strength loss rate of ordinary activated powder concrete and basalt fiber-activated powder concrete under freeze–thaw conditions. The loss rate was used to evaluate the effect of freeze–thaw conditions on the freeze–thaw durability of concrete. The test results showed that the mechanical properties and freeze–thaw resistance of the concrete were improved by mixing basalt fibers. Yang Li et al. [40] investigated the effect of the split tensile strength and flexural strength of basalt fiber-reinforced concrete after freeze–thaw cycles. The results showed that the compressive strength of the concrete significantly improved with increasing basalt fiber admixture density under an ultralow-temperature freeze–thaw environment, and the splitting tensile strength and flexural strength improved. Yan-Ru Zhao et al. [41] analyzed the flexural damage of basalt fiber-reinforced concrete after freeze–thaw cycling using the digital image correlation technique. Concrete under freeze–thaw cycling reduces the elastic deformation capacity of the specimen, while the fibers enhance the resistance to elastic–plastic deformation of the specimen, and the fiber incorporation changes the bending damage process from brittle to ductile damage and improves the flexural strength of the specimen.

The Swedish physicist Waloddi Weibull first proposed the Weibull distribution model in 1939, which is widely used in reliability assessment research on engineering materials as a data processing method for life tests [42,43,44,45]. Tingting Zhang et al. [46] used a two-parameter Weibull model to study the effect of high-performance polypropylene fibers on the impact resistance of recycled aggregate concrete. The test results show that the maximum impact energy consumption of recycled aggregate concrete with fibers increases significantly, and the incorporation of fibers greatly improves the impact resistance of concrete, which is in line with the two-parameter Weibull distribution. Qian Hui Xiao et al. [47] investigated the freeze–thaw resistance of recycled concrete in a sulfate environment and established damage equations and effective life expectancy values for recycled concrete using the two-factor Weibull model. The model accurately describes the damage changes in recycled concrete. Gray system prediction theory was first proposed by Prof. Deng Julong in 1981. As a kind of analytical tool based on partially known data, generating, developing, and extracting valuable information, determining the law describing the change in the system, and predicting the trend of the development of things, this theory has been widely used in scientific research [48,49,50,51]. The gray system GM (1,1) model is the core element in gray system theory, and life prediction is carried out by establishing a damage model. Yuan Qin et al. [52] introduced the GM (1,1) model to predict the change rule of the durability of fly ash fiber concrete in sulfate dry and wet cyclic tests. The results show that the addition of fly ash slightly improves the sulfate resistance of fiber concrete; the model prediction results of each group of specimens are reasonable, and the error is small. Yushi Yin et al. [53] studied the effect of the mechanical properties of concrete under the action of sulfate erosion and used the GM (1,1) model to establish the C20, C40, C60, and C80 models to predict the erosion resistance of concrete. The experimental results show that the sulfate erosion resistance of concrete increases with increasing concrete grade. The experimental data and the predictive model were well fitted with high accuracy.

According to the above literature, most of the studies on the incorporation of basalt fibers are limited to common construction aggregates, and studies on the testing of basalt fiber-reinforced walnut shell mortar with agricultural wastes are limited. In addition, most of the major studies on agricultural wastes have focused on mechanical properties, and studies on durability are rare. Considering the importance of cheap, renewable agricultural waste, walnut shells are an aggregate substitute in mortar production. In this paper, agricultural waste walnut shells were mixed with 20% sand, basalt fibers of different lengths (3 mm, 6 mm, and 9 mm), and different dosages (0.1%, 0.2%, and 0.3%) were mixed into the mortar, and 11 groups of specimens with different aggregate substitution rates, fiber types, and dosages were fabricated. The specimens were subjected to basic mechanical property tests, freeze–thaw cycle tests, and impact resistance tests to analyze the effects of basalt fibers on the mechanical properties, impact resistance, and freeze–thaw resistance of walnut shell mortar. Using the Weibull model and GM (1,1) model, a durability damage prediction model was established to compare and analyze the experimental and predicted results to assess the accuracy of the model and to analyze the enhancement effect of basalt fibers on the sustainability and durability of discarded walnut shells.

2. Materials and Methods

2.1. Test Materials and Mix Design

P.O42.5 ordinary silicate cement was used in this test. The burn loss of the cement was 3.13%, the main components were SO₃ 2.15%, MgO 3.52%, and Cl 0.051%, the fine aggregate used was high-purity quartz sand, the main component is SiO₂, the content was 99.8%, and the particle size is 1~2 mm. The fly ash used was of the first grade, and the fineness index of the 5 μm sieve residue (%) was not more than 18%. The silica fume was white silica fume, and the main component was SiO₂, accounting for 95.21%. The water-reducing agent used was a high-performance polycarboxylic acid water-reducing agent. The fibers were short-cut basalt fibers with lengths of 3 mm, 6 mm, and 9 mm. Waste walnut shells were recycled from the Xinjiang Aksu area, China. Different crushing stages of walnut shells are shown in Figure 1. Walnut shells were first crushed by a jaw crusher many times to reach a size of approximately 5–10 mm, subsequently crushed by a multifunctional pulverizer, and sifted by a sieve with a 1~2 mm fineness. Modification: Part of the initial crushed walnut shells were soaked in boiling water for half an hour after natural drying, air-drying, and drying in an oven at 50–55 °C for processing. The performance indexes of cement, fly ash, silica fume and basalt fiber are shown in

Table 1. Cement performance index.

Testing Program	Specific Surface Area (m2/kg)	Initial Condensation Time (min)	Final Coagulation Time (min)	Cubic Compressive Strength (MPa)		Stability
				3 d	28 d
Test results	341	258	328	27.4	43.5	Qualified

,

Table 2. Fly ash performance index.

Testing Program	Fineness (%)	Ablation (%)	Water Content (%)	Densities (g/cm3)	Packing Density (g/cm3)
Test results	16	2.62	0.84	2.23	1.12

,

Table 3. Silica fume performance index.

Testing Program	SiO2 (%)	Cauterization Reduction (%)	Water Demand Ratio (%)	28 d Activity Index (%)	Chloride Ion Content (%)
Test results	95.21	1.32	121	103	0.01

and

Table 4. Detailed parameters of performance indicators of basalt fiber.

Length (mm)	Monofilament Diameter (μm)	Densities (g/cm3)	Modulus of Elasticity (Gpa)	Tensile Strength (MPa)	Breaking Elongation (mm)
3, 6, 9	7~15	2.63~2.65	91~110	3000~4800	2.5~3.0

. The particle size distribution curves of cement, fly ash and silica fume are shown in Figure 2.

Basal fibers of 3 mm, 6 mm, and 9 mm were selected for use, with 0.1%, 0.2%, and 0.3% of the volume doped into the walnut shell mortar (doped into the group of fibers, walnut shell substitution rate of 20%). The specimen-specific parameters are shown in

Table 5. Mix design of mortar.

No.	Materials (kg/m3)						Water Reducing (%)	BF
	Cement	Fly Ash	Silica Fume	Quartz Sand	Walnut Shell	Water		Length (mm)	Dosage (%)
WS0	769	333	47	656	0	345	1	0	0
WS20	769	333	47	525	131	345	1	0	0
B3F1WS20	769	333	47	525	131	345	1	3	0.1
B3F2WS20	769	333	47	525	131	345	1	3	0.2
B3F3WS20	769	333	47	525	131	345	1	3	0.3
B6F1WS20	769	333	47	525	131	345	1	6	0.1
B6F2WS20	769	333	47	525	131	345	1	6	0.2
B6F3WS20	769	333	47	525	131	345	1	6	0.3
B9F1WS20	769	333	47	525	131	345	1	9	0.1
B9F2WS20	769	333	47	525	131	345	1	9	0.2
B9F3WS20	769	333	47	525	131	345	1	9	0.3

.

2.2. Specimen Production Method and Maintenance

The specimens were prepared according to the guidelines outlined in GB/T 50081-2019 [54] for testing the properties of ordinary concrete mixes and CECS 13:2009 [55] for fiber concrete tests. Cement, fly ash, silica fume, quartz sand, and walnut shells were added in a specific sequence and mixed for 1 min. Basalt fibers were introduced after the complete blending, followed by the addition of water and a water-reducing agent; the mixture was mixed thoroughly for approximately 90 s. The mixture was placed in a mill and subjected to vibration molding, after which the surface was leveled using a spatula to remove air pockets. Encapsulated specimens were stored in a chamber at 20 ± 5 °C for 24 h before demoulding and labeling. These were then transferred to a room maintained at a stable temperature of 20 ± 2 °C and a relative humidity of 95% ± 5% for a curing period of 28 days. Cubic compression test specimen size of 70.7 mm × 70.7 mm × 70.7 mm, each group using three blocks, a total of 33; split tensile test specimen size of 100 mm × 100 mm × 100 mm, each group using three blocks, a total of 33; flexural test specimen size of 100 mm × 100 mm × 400 mm, each group using three blocks. The size of the flexural test specimen is 100 mm × 100 mm × 400 mm. Each group adopts three specimens, for a total of 33. Cubic compression and flexural tests were performed using a microcomputer servo cement flexural testing machine, and the tensile strength was assessed using a microcomputer servo precompression testing machine. After the specimen was maintained for 28 d, the cement mortar specimen was removed from the standard maintenance room, the excess water on the surface of the specimen was wiped, and the appearance of the specimen was checked to determine whether it conformed to the test standard. The specimen was placed in the center of pressure, the power supply of the testing machine was turned on, the oil pump was loaded, the relevant parameters were set, and the compression test loading speed was 0.5 MPa/s. In the bending and splitting compression and flexural tests, the loading speed was 0.05 MPa/s. After the end of the test, the test was stopped, the destruction of the cement mortar was observed, and the results were recorded.

The drop hammer impact test employs the method recommended by the American Concrete Institute using a drop hammer, according to the American Institute of California, San Francisco [56]. The method is straightforward to use, and the test requirements are minimal. The chosen test equipment used was a CECS13-2009 concrete drop hammer impact tester (Figure 3). For the impact resistance test, cylindrical specimens of Φ150 mm × 63 mm were used, 6 in each group, for a total of 66 specimens. The test involved placing a Φ150 mm × 63 mm cylindrical specimen inside the device chassis, which was surrounded by four baffles. A 63 mm steel ball is positioned on top of the specimen, and a 4.5 kg hammer is dropped freely from a height of 500 mm onto the ball. This action transfers gravitational potential energy to the specimen through the steel ball. A counter was attached to the electromagnetic switch controlling the steel balls, and for each impact, the counter recorded the number of impacts. When the specimen was impacted until it cracked and contacted any of the three surrounding baffles, the specimen was considered to be ultimately destroyed, and the results were recorded. The reliability of the specimens against the number of impacts was analyzed based on the Weibull model, and the impact life of the mortar was predicted for different failure probabilities. Finally, impact damage evolution equations were established to derive the impact loss curve, and the impact damage process of the specimen was studied in depth.

According to the “long-term performance and durability of ordinary concrete test method standard” (GB/T 50082-2009) [57] for freeze–thaw cycle tests, this paper selected the freezing and thawing test method for the fast freezing method. The freeze-thaw cycle equipment is shown in Figure 4. This freeze–thaw test was performed according to the above specification requirements, using a concrete specimen size of 100 mm × 100 mm × 400 mm prismatic cylinder, with 3 specimens in each group of tests, for a total of 33 specimens. After 24 d of curing in the standard curing room, the specimens were removed in advance and soaked in (20 ± 2 °C) water for 4 days. Before the test started, the initial mass of the specimens was determined, and the original frequency was measured using a dynamic elastic modulus tester. The dynamic modulus of elasticity and quality were analyzed for different durations of freezing and thawing (0, 25, 50, 75, and 100 times). The relative error A, mean variance C, and small error probability P were used to check the error and accuracy of the GM (1,1) model and to analyze the applicability of the GM (1,1) model to the freeze–thaw damage model of the relative kinetic elastic modulus of the specimens.

3. Results

3.1. Basic Mechanical Properties Test Results

The test results for the 28 d compressive strength, split tensile strength, and flexural strength of the basalt fiber walnut shell mortar are shown in Table 3.

As shown in

Table 6. Mechanical properties test results.

No.	Compressive Strength (MPa)	Splitting Tensile Strength (MPa)	Flexural Strength (MPa)
WS0	47.8	3.1	4.8
WS20	32.9	2.7	3.2
B3F1WS20	34.9	2.9	3.3
B3F2WS20	39.2	3.4	4.1
B3F3WS20	36.3	2.7	2.3
B6F1WS20	36.0	3.2	3.6
B6F2WS20	37.5	4.0	4.6
B6F3WS20	33.5	2.6	2.8
B9F1WS20	35.0	3.0	3.4
B9F2WS20	36.7	3.8	3.9
B9F3WS20	31.9	2.2	2.1

, for walnut shells with an equal mass replacement of sand (20%), the mechanical properties of the mortar decreased significantly, and the amount of fibers on the walnut shell mortar increased and then decreased. Different lengths of fiber were used for a fiber volume doping of 0.2% to achieve the maximum value. When the fiber volume is 0.2% and the length is 6 mm, the enhancement effect is the greatest. Compared with those of the benchmark group, the compressive strength increased by 13.98%, the split tensile strength increased by 48.15%, and the flexural strength increased by 43.75%. According to the test data, the compressive strength of the specimen doped with fibers increased by a small amount, the splitting tensile strength and flexural strength increased by a large amount, and when the fiber doping continued to increase, the strength decreased. This is because there are more pores and microcracks inside the single mortar, and the addition of fibers can effectively improve the internal structure of the specimen. Fibers were randomly and uniformly distributed inside the matrix and intertwined with each other to form a fiber-mortar skeleton, which played a good supporting role, while the presence of fibers made internal holes in the specimen that were harmful and reduced the specimen size; the holes also played a certain role in resisting cracking during the specimen loading stage. When the length of the fiber was 6 mm and the dosage was 0.3%, the split tensile strength and flexural strength of B6F3WS20 slightly decreased because as the volume of the fiber increased, it was not easy to uniformly disperse the fibers during the mixing and vibration molding process; additionally, some of the fibers appeared to agglomerate, which affected the density of the mortar interior, and the specimen was likely to form internal defects, resulting in a reduction in strength. Moreover, compared with those of the B9F1WS20 and B9F2WS20 groups, the compressive strength, split tensile strength, and flexural strength of the B9F3WS20 group decreased by different degrees. With increasing fiber length and dosage, the area of cement mortar wrapped around the matrix fine aggregate decreased, resulting in an increase in the bonding of the fiber house. Because fibers are a kind of water-absorbing material, the longer the fiber length and the greater the dosage are, the stronger the water absorption, the less water there is inside the matrix, and the poorer the fluidity, which affects the development of strength in the later stage.

According to the masonry mortar proportion design regulations (JGJ/T98-2010) [58], the strength grade of the mortar is divided into M7.5, M10, M15, M20, M25, and M30. As shown in Table 1, the test results are in line with the relevant specification requirements.

3.2. Impact Resistance Test

3.2.1. Specimen Impact Damage Analysis

Figure 5 depicts the damage morphology of various specimens. Upon repeated hammering, Figure 5 shows the damage morphology of the various specimens. Upon repeated hammer impacts, the standard WSM exhibited a zigzag crack pattern, typically breaking into two sections due to its inherent brittleness. However, the BFWSM specimen underwent a notable change in damage morphology with the addition of fibers. The specimen exhibited three penetrating cracks and a ductile damage mode with the development of multiple cracks and was resilient to successive hammer impacts despite initial cracks.

The specimen surface was monitored to identify the first visible crack; at this point (initial crack), the number of impacts was recorded as N₁. When the specimen cracked and contacted any of the three surrounding baffles (indicating the end of the crack), the number of impacts was recorded as N₂. The number of impacts of the specimens is shown in

Table 7. Specimen impact test results.

No.	${\mathit{N}}_1$$/{\mathit{N}}_2$
	a	b	c	d	e	f
WS0	256/257	178/178	240/241	193/194	187/188	275/276
WS20	130/131	126/126	151/152	108/108	165/166	231/231
B3F1WS20	174/175	222/223	369/369	320/321	241/242	233/233
B3F2WS20	185/186	202/204	401/403	282/283	333/334	289/291
B3F3WS20	209/210	277/278	310/312	347/348	222/222	255/256
B6F1WS20	263/264	266/267	304/306	351/352	389/390	247/248
B6F2WS20	342/344	397/398	545/547	425/427	202/204	285/286
B6F3WS20	246/247	269/270	255/256	381/382	312/313	323/324
B9F1WS20	254/255	142/143	278/278	368/370	247/248	391/392
B9F2WS20	306/308	268/270	341/342	279/280	383/385	212/213
B9F3WS20	354/356	231/232	261/262	201/203	189/190	271/272

, the schematic is shown in Figure 6. The impact energy resistance is calculated as follows:

$$W=N_{2}mgh$$

(1)

In Equation (3):

• $W$—impact energy consumption (J);
• $N_2$—number of impacts at final destruction of the specimen;
• $m$—impact hammer quality (kg), 4.5 kg;
• g—gravitational acceleration (m/s²), 9.81 m/s²;
• h—drop height (m), 0.5 m.

Table 8. Specimen impact resistance index analysis results.

No.	Average Number of Impact Resistance
	${\mathit{N}}_1$	${\mathit{N}}_2$	${\mathit{N}}_2-{\mathit{N}}_1$	W (J)
WS0	260	261	1	4900.095
WS20	152	152	0	3355.02
B3F1WS20	282	284	2	5760.923
B3F2WS20	270	271	1	6268.59
B3F3WS20	303	305	2	5981.648
B6F1WS20	366	368	2	6732.113
B6F2WS20	298	299	1	8122.68
B6F3WS20	280	281	1	6599.678
B9F1WS20	298	300	2	6202.373
B9F2WS20	251	253	2	6621.75
B9F3WS20	222	222	0	5584.343

shows that fiber doping in the modified walnut shell mortar specimens enhances the toughness and ductility of the specimens. When the fibers were doped, the impact energy dissipation of all the groups of specimens increased significantly. At different fiber lengths, the impact energy consumption of the specimens tended to increase first and then decrease with increasing fiber doping. For fibers with a length of 6 mm and a doping amount of 0.2%, compared with that of the WS20 group, the impact energy consumption of this group of specimens increased by 142.11%, the best impact resistance performance. Before the specimen cracked, the basalt fibers and cement matrix shared an impact load. As the number of impacts increased, the cracking of the specimen included not only the fracture energy dissipation of the aggregate but also the pulling out and pulling off of the internal fibers from the matrix, which consumed a large amount of kinetic energy from the impact and effectively suppressed the development of cracks; thus, fiber doping improved the impact toughness of the mortar [59,60,61].

3.2.2. Impact Resistance Analysis Based on Weibull Modeling

Fatigue in mortar often originates from escalating cracks during impact resistance, and the number of impacts on mortar is a discrete random variable. Hence, this study incorporates the Weibull distribution theory model to treat the count of impacts (N) as a random research variable. This approach conducts probabilistic statistical analysis and formulates impact damage evolution equations to investigate the effect of basalt fiber on the impact resistance of modified walnut shell mortar. The probability density function f(N) concerning N can be represented as follows.

$$f\left(N\right)=\frac{\gamma }{N_x-N_0}{(\frac{N-N_0}{N_x-N_0})}^{\gamma -1}\exp [-(\frac{N-N_0}{N_x-N_0}{)}^{\gamma }]$$

(2)

In Equation (2):

• $\gamma$—shape parameters;
• $N_x$—characteristic life parameters.

The cumulative distribution function $F\left(N\right)$ is as follows:

$$F\left(N\right)=P_f\left(N\right)=1-\exp [-(\frac{N-N_0}{N_x-N_0}{)}^{\gamma }]$$

(3)

In Equation (3): $P_f\left(N\right)$ = failure probability function

To ensure test process safety, precision, and computational simplicity, setting the minimum impact life, $N_0$, to zero is viable, leading to the survival probability function of the two-parameter Weibull distribution.

$$P_s\left(N\right)=1-P_f\left(N\right)=\exp \left[-{\left(\frac{N}{N_x}\right)}^{\gamma }\right]$$

(4)

Taking the corresponding derivatives and natural logarithmic variations of the equation yields.

$$\ln \left[\ln \left(\frac{1}{P_s\left(N\right)}\right)\right]=\gamma \left[\ln N-\ln N_x\right]$$

(5)

Let $X=\ln N$; $Y=\ln [\ln \left(1/P_s\left(N\right)\right]; B=\gamma \ln N_x$. Then, the equation reduces to

$$Y=\gamma X-B$$

(6)

$$P_s\left(N\right)=1-\frac{m}{t+1},0<m⩽t$$

(7)

In Equation (7),

• t—is the total number of tests for each group of specimens, which is 6;
• m—ordinal number.

Equation (6) satisfies the linear relationship between $Y$ and $X$. The parameters $\gamma$ and $B$, along with the correlation coefficients $R^2$, are determined through Equations (2)–(7) using the linear fitting method.

Rahmani et al. [62] argue that a model is reasonable and credible when its regression coefficient R2 is greater than or equal to 0.7.

Table 9. Regression parameters and regression coefficients.

No.	Regression Parameters		Correlation Coefficient
	γ	B	R2
WS0	4.73433	−25.98736	0.89859
WS20	3.2595	−16.751	0.8863
B3F1WS20	3.29171	−18.67995	0.9232
B3F2WS20	3.06077	−17.64646	0.95392
B3F3WS20	4.62374	−26.29862	0.96021
B6F1WS20	4.80745	−27.89554	0.8811
B6F2WS20	2.66568	−16.09102	0.98962
B6F3WS20	5.21677	−30.13971	0.89344
B9F1WS20	2.41156	−13.94561	0.89962
B9F2WS20	4.1787	−24.2268	0.8738
B9F3WS20	3.86721	−21.77233	0.91125

indicates that for mortar, the regression coefficient $R^2$ ranges from 0.8738 to 0.98962. This signifies a notable linear correlation between $\ln [\ln \left(1/P_s\left(N\right)\right]$ and $\ln N$. In essence, the impact resistance test results for each specimen group adhere to the Weibull distribution function. The Weibull linear regression curve is shown in Figure 7.

The impact life of the specimen with different failure probabilities can be obtained by the calculation of Equation (8):

$$N=\exp \{\frac{\ln \left[\ln \left(1/1-P_{\mathrm{f}}\left(N\right)\right)\right]+B}{\gamma }\}$$

(8)

The impact life of mortar at different failure probabilities is shown in

Table 10. Impact life of mortar at different failure probabilities.

No.	0.1	0.3	0.5	0.7	0.9
WS0	289	252	224	195	150
WS20	220	181	152	124	153
B3F1WS20	375	308	262	213	147
B3F2WS20	419	339	283	228	153
B3F3WS20	354	307	273	236	181
B6F1WS20	394	344	307	267	207
B6F2WS20	572	449	365	284	180
B6F3WS20	379	335	301	265	210
B9F1WS20	459	351	279	212	128
B9F2WS20	402	345	302	257	192
B9F3WS20	346	292	254	213	156

. At various failure probabilities, the impact life of the mortar varies. However, the predicted impact life trend aligns with the data analysis from the tests. At a failure probability of 0.5, where the reliability rate of the mortar equals 0.5, the estimated impact life of the mortar better corresponds to real engineering scenarios.

The relative error between the predicted impact life of the mortar obtained through the Weibull probability density function and the actual test values, as depicted in

Table 11. The relative error between measured value and predicted value.

No.	Measured Value	Predicted Value	Relative Error
WS0	222	224	0.009009
WS20	152	153	0.006579
B3F1WS20	261	262	0.0038314
B3F2WS20	284	283	0.003521
B3F3WS20	271	273	0.0073801
B6F1WS20	305	307	0.0065574
B6F2WS20	368	365	0.008152
B6F3WS20	299	301	0.006689
B9F1WS20	281	279	0.007117
B9F2WS20	300	302	0.0066667
B9F3WS20	253	254	0.0039526

, is notably minimal. Ranging from 0.35% to 0.9%, with an average error of 0.63%, this indicates a high level of consistency between the predicted and actual values, affirming the model’s accuracy in prediction.

3.2.3. Impact Damage Analysis

The utilization of the two-parameter Weibull distribution model in the statistical analysis aimed to forecast the final count of cracks in cement mortar subjected to impact loads. This section systematically investigates the damage progression of cement mortar from the initial impact load until failure, elucidating the overarching evolution pattern of damage due to repeated drop hammer impacts.

Throughout the impact failure of cement mortar specimens, the extent and likelihood of damage occur simultaneously. Upon the specimen’s failure after “n” impacts, the failure probability $P_f\left(N\right)$ of the cement mortar equates to equals 1, aligning with the damage degree $D\left(N\right)$ being 1 as well. This equivalence signifies that the failure probability and damage degree of cement mortar are interchangeable and essentially represented as $P_f\left(N\right)=D\left(N\right)$. In summary, the random damage model of cement mortar based on the two-parameter Weibull probability distribution can be defined as follows:

$$D\left(N\right)=1-\exp \left[-{\left(\frac{n}{N_x}\right)}^{\gamma }\right]$$

(9)

$$N_x=\exp [-(\frac{B}{\gamma })]$$

(10)

By utilizing Formulas (5) and (6) in conjunction with Table 6, the values for $\gamma$ and $N_x$ are derived. These calculated results are subsequently applied to Formula (9), resulting in the formulation of the damage evolution equation for each cement mortar group under impact loading. A diagram depicting the damage evolution of the impact life for each group is provided in

Table 12. Two-parameter Weibull distribution damage evolution equation.

No.	Two-Parameter Weibull Distribution Damage Evolution Equation
WS0	$D\left(N\right)=1-\exp [-(\frac{n}{242.047}{)}^{4.734}]$
WS20	$D\left(N\right)=1-\exp [-(\frac{n}{170.568}{)}^{3.260}]$
B3F1WS20	$D\left(N\right)=1-\exp [-(\frac{n}{291.444}{)}^{3.292}]$
B3F2WS20	$D\left(N\right)=1-\exp [-(\frac{n}{319.056}{)}^{3.061}]$
B3F3WS20	$D\left(N\right)=1-\exp [-(\frac{n}{295.225}{)}^{4.624}]$
B6F1WS20	$D\left(N\right)=1-\exp [-(\frac{n}{331.148}{)}^{4.807}]$
B6F2WS20	$D\left(N\right)=1-\exp [-(\frac{n}{418.37}{)}^{2.666}]$
B6F3WS20	$D\left(N\right)=1-\exp [-(\frac{n}{322.94}{)}^{5.217}]$
B9F1WS20	$D\left(N\right)=1-\exp [-(\frac{n}{324.672}{)}^{2.412}]$
B9F2WS20	$D\left(N\right)=1-\exp [-(\frac{n}{329.537}{)}^{4.179}]$
B9F3WS20	$D\left(N\right)=1-\exp [-(\frac{n}{278.658}{)}^{4.734}]$

.

Damage evolution curve of mortar impact damage is shown in Figure 8. According to the damage change curve, during the initial stage of drop hammer impact, the cement mortar aggregates and fibers intertwine, creating a stable structure that results in minimal damage progression. However, as the number of drop hammer impacts increases, the cement matrix experiences continuous impacts, resulting in the proliferation of internal cracks and significant damage. This phase is characterized by a steeper damage curve, indicating a considerable increase in damage to the cement mortar. Eventually, as $D\left(N\right)$ reaches 1, the curve tends to smooth, indicating complete destruction of the cement mortar [45,62,63].

4. Freeze–Thaw Cycle Test Results

4.1. Freeze–Thaw Cycle Test Results

The mass loss rate of each specimen is calculated as follows:

$$\mathsf{\Delta }W=\frac{W_0-W_n}{W_0}\times 100$$

(11)

In the formula:

• the mass loss rate of concrete specimens after ΔW—N freeze–thaw cycles (%);
• $W_0$—the mass of concrete specimen before the freeze–thaw cycle test (g);
• $W_n$—the mass of concrete specimens after N freeze–thaw cycles (g).

There were three specimens in each group for the freeze–thaw cycle test, and the average mass loss rate of each group is the arithmetic mean of the three specimens. During the test, the quality of the specimen was measured every 25 freeze–thaw cycles, and the test was completed after 100 freeze–thaw cycles. The changes in the mass loss rate of the specimens in each group are shown in

Table 13. The change in mass loss rate of each group of specimens.

No.	Rate of Quality-Led Loss (%)
	0 Times	25 Times	50 Times	75 Times	100 Times
WS0	0	−0.65	0.36	0.95	1.51
WS20	0	−0.3	0.72	3.97	4.5
B3F1WS20	0	−0.62	0.43	1.74	2.56
B3F2WS20	0	−0.71	0.31	1.59	2.32
B3F3WS20	0	−0.83	0.48	1.65	2.61
B6F1WS20	0	−0.66	0.41	1.66	2.37
B6F2WS20	0	−0.75	0.22	1.48	2.21
B6F3WS20	0	−0.89	0.43	1.57	2.51
B9F1WS20	0	−0.64	0.5	1.86	2.65
B9F2WS20	0	−0.68	0.33	1.65	2.41
B9F3WS20	0	−0.85	0.41	1.69	2.58

.

In the freeze–thaw cycle test of mortar, the mass loss rate of the specimen is an important index for testing the frost resistance of mortar specimens. The mass loss rate reflects the degree of freeze–thaw damage to the specimen, and a lower mass loss rate after 100 freeze–thaw cycles indicate better frost resistance. As shown in Figure 9, the mass loss rate of each group of specimens first decreases and then increases because in the early stage of the freeze–thaw cycle, the influence of freeze–thaw damage results in an increase in the internal pores of the specimen, the cracks extend throughout the specimen, the specimen absorbs water, and the mass increases; in the late stage of the freeze–thaw cycle, the specimen aggregates and cementitious materials detach from the surface, which results in a decrease in mass. When the number of freeze–thaw cycles reached one hundred, the cumulative mass damage rate of WS20 reached 4.5%. After fiber doping, the cumulative mass damage rate of B6F2WS20 was 2.31%. Compared with that of the other groups, the specimen surface peeling was the best; the specimen was damaged by freezing and thawing damage to the lowest degree, and the mass loss rate was the smallest. The test results showed that the specimen with a fiber length of 6 mm and a doping rate of 0.2% had the best frost resistance. This is because the bridging effect of basalt fibers effectively inhibits the shedding of mortar and aggregates on the surface of mortar specimens, inhibits the expansion of cracks from the inside out, and effectively enhances the freeze–thaw resistance of specimens. Compared to the other fiber-blended groups, B9F1WS20 has a cumulative mass loss of 2.65% after one hundred freeze–thaw cycles, which means that, compared to the other groups, the specimen has the highest degree of freeze–thaw damage and the lowest resistance to freeze–thaw cycles.

4.2. Change in the Relative Dynamic Modulus of Elasticity of the Specimen

The relative dynamic elastic modulus of each specimen is calculated as follows:

$$P=\frac{f_n^2}{f_0^2}\times 100$$

(12)

In the formula: the relative dynamic elastic modulus (%) of concrete specimens after P-N freeze–thaw cycles;

The transverse fundamental frequency (Hz) of concrete specimens after $f_n$-N freeze–thaw cycles;

$f_0$—initial transverse fundamental frequency (Hz) of concrete specimens before freeze–thaw cycle test.

There are three specimens in each group for the freeze–thaw cycle test, and the relative dynamic elastic modulus of each group is the arithmetic mean of the three specimens. During the test, the transverse fundamental frequency and dynamic elastic modulus of the specimen were measured every 25 freeze–thaw cycles, and the test was completed after 100 freeze–thaw cycles. The relative dynamic modulus of elasticity for each group of specimens is shown in

Table 14. Changes in the relative dynamic modulus of elasticity for each group of specimens.

No.	Relative Dynamic Elastic Modulus (%)
	0	25	50	75	100
WS0	100	96.34	91.59	84.21	76.65
WS20	100	92.12	85.39	74.32	70.02
B3F1WS20	100	95.59	90.59	83.14	75.87
B3F2WS20	100	96.21	92.3	84.48	76.98
B3F3WS20	100	94.21	91.89	83.04	74.21
B6F1WS20	100	95.44	90.62	82.21	75.4
B6F2WS20	100	96.21	92.32	84.31	77.4
B6F3WS20	100	93.17	88.24	79.98	73.23
B9F1WS20	100	96.34	91.12	82.66	75.68
B9F2WS20	100	96.92	92.21	83.76	76.41
B9F3WS20	100	94.33	91.31	81.26	72.32

.

The freeze–thaw damage process of mortar is complex. Under freeze–thaw cycling, the mortar specimen is saturated with water, and the internal capillary water freezes and expands, triggering a variety of pressures; moreover, when the mortar cannot withstand pressure, new cracks are internally produced. When the temperature increases and the specimen thaws, the internal pore water of the mortar thaws, and the newly generated small voids and capillary cracks reabsorb water. When the freezing process occurs again, the internal pores and cracks gradually increase, expand and penetrate, and again produce new cracks. Under the freeze–thaw cycle, the mortar specimen will undergo freeze–thaw damage from the inside to the outside. In the freeze–thaw test process, after each freeze–thaw cycle, the larger the relative dynamic modulus of elasticity of the specimen is, the smaller the degree of freezing and thawing damage is, and the better the frost resistance is. As shown in Figure 10, the relative dynamic elastic modulus of each group of specimens decreased to different degrees after 100 freeze–thaw cycles. The relative kinetic elastic modulus of each group of specimens in the range of 0~50 cycles has a lower degree of change, and the relative elastic modulus of each specimen decreases more with the increase in the number of freeze–thaw cycles. The relative kinetic elastic modulus of WS20 is the smallest, at 70.02%, and the relative kinetic elastic modulus of BFWSB6F2 is the largest, at 77.4%. The results show that the BFWSB6F2 specimen has the best frost resistance. A freezing and thawing environment leads to the expansion of internal pores and microcracks in the mortar, and ultimately, the specimen surface cracks through and spalls. After adding basalt fibers, the fibers are irregularly distributed in the mortar, forming a three-dimensional spatial mesh structure, which can inhibit the development of internal cracks in the mortar and can offset a portion of the freezing and expansion force generated by water icing, effectively improving the frost resistance of walnut shell mortar [64,65,66].

4.3. Freeze–Thaw Performance of Specimens under the GM (1,1) Model

4.3.1. Gray System GM (1,1) Model

The freeze–thaw damage of mortar generally results from many factors. It is difficult to analyze the freeze–thaw damage of mortar from many factors in freeze–thaw tests. Intuitively, from the data analysis, it is difficult to determine which uncertain factors have the greatest impact on specimens. In this paper, the gray theory analysis method is used. The gray theory is applied to many professional fields and is considered to be a mature and reliable theory. The relative dynamic elastic modulus, an index of the freeze–thaw cycle test, conforms to the system characteristics of incomplete and uncertain information in gray theory. The gray theory data processing does not look for its statistical law and probability distribution but rather processes the original data to create regular time series data and establish a mathematical model on this basis. The GM series model is the basic model of gray prediction theory, especially the GM (1,1) model, which is widely used. The mean GM (1,1) model is widely used.

4.3.2. Gray System GM (1,1) Model Parameters

The relative dynamic elastic modulus of each specimen in each group under the freeze–thaw cycle test was used as the physical damage index of the freeze–thaw damage model. The measured value of the relative dynamic elastic modulus of the specimen under freeze–thaw cycling is taken as the original sequence $X^{\left(0\right)}$, and the predicted value is the simulated value $X^{\left(1\right)}$ to establish the prediction model of mortar.

$$X^{\left(0\right)}\left(X_1{}^{\left(0\right)},X_2{}^{\left(0\right)},\cdots X_n{}^{\left(0\right)}\right)$$

(13)

The first step is to determine the simulated value $X^{\left(1\right)}$, where $x_k^{\left(1\right)}$ can be obtained by performing a first-order cumulative sequence on $X^{\left(0\right)}$.

$$x_k^{\left(1\right)}={\sum }_{i=1}^k{x}_k^{\left(0\right)},x_k^{\left(0\right)}\ge 0,(k=1,2\cdots n)$$

(14)

Through the calculation of Equation (4), we can obtain $X^{\left(1\right)}$, where $X^{\left(1\right)}$ can also be called the first-order cumulative sequence 1-GAO and (1) can weaken the fluctuation of the $X^{\left(0\right)}$ data column and reduce the systematic error in the accumulation process to increase the accuracy of the prediction model.

$$X^{\left(1\right)}=\left(X_1^{\left(1\right)},X_2^{\left(1\right)},\cdots X_n^{\left(1\right)}\right)$$

(15)

Step 2: We determine the differential equation of the GM (1,1) model, which is expressed by Equation (8):

$$Z^{\left(1\right)}=\left(Z_2^{\left(1\right)},Z_3^{\left(1\right)},\cdots ,Z_n^{\left(1\right)}\right)$$

(16)

$$Z_k^{\left(1\right)}=\frac{1}{2}\left(x_k^{\left(1\right)}+x_{k-1}^{\left(1\right)}\right),k=2,3,\cdots n.$$

(17)

Step 3: We determine the whitening equation of the first-order linear differential equation of the GM (1,1) model. From Equation (9):

$$x_i^{\left(0\right)}+a{Z}_i^{\left(1\right)}=u$$

(18)

Step 4: We determine the whitening equation of the first-order linear differential equation of the GM (1,1) model. From Equation (9):

$$\frac{d{x}^{\left(1\right)}}{dt}+a{x}^{\left(1\right)}=u$$

(19)

In (15)–(19), $a$ is the development coefficient and $u$ is the gray work.

Step 5: We determine the parameter sequence $\widehat{\alpha },\widehat{\alpha }={\left[a,u\right]}^r$

$$\widehat{\alpha }={[A^{T}A]}^{-1}{A}^{T}Y$$

(20)

$$A=\left[\begin{array}{cc}-Z_2^{\left(1\right)}& 1\\ -Z_3^{\left(1\right)}& 1\\ ⋮& ⋮\\ -Z_n^{\left(0\right)}& 1\end{array}\right],Y=\left[\begin{array}{c}{x}_2^{\left(0\right)}\\ x_3^{\left(0\right)}\\ ⋮\\ x_n^{\left(0\right)}\end{array}\right]$$

(21)

Step 6: We determine the GM (1,1) time response:

$${\widehat{x}}_{k+1}^{\left(1\right)}=\left[x_1^{\left(0\right)}-\frac{u}{a}\right]e^{-ak}+\frac{u}{a},k=0,1,\cdots ,n-1.$$

(22)

4.3.3. Specimens Freeze–Thaw Damage Model

With the repeated cycles of freeze–thaw test, the background and one-time cumulative values of the relative dynamic elastic modulus of the specimen are shown in

Table 15. Relative dynamic modulus of elasticity background values for each group of specimens.

No.	Freeze–Thaw Cycles (Times)
	0	25	50	75	100
WS0	100	148.17	242.135	330.035	410.465
WS20	100	146.06	234.815	314.67	386.84
B3F1WS20	100	147.795	240.885	327.75	407.255
B3F2WS20	100	148.105	242.36	330.75	411.48
B3F3WS20	100	147.105	240.155	327.62	406.245
B6F1WS20	100	147.72	240.75	327.165	405.97
B6F2WS20	100	148.105	242.37	330.685	411.54
B6F3WS20	100	146.585	237.29	321.4	398.005
B9F1WS20	100	148.17	241.9	328.79	407.96
B9F2WS20	100	148.46	243.025	331.01	411.095
B9F3WS20	100	147.165	239.985	326.27	403.06

and

Table 16. Relative dynamic modulus of elasticity of each group of specimens.

No.	Freeze–Thaw Cycles (Times)
	0	25	50	75	100
WS0	100	196.34	287.93	372.14	448.79
WS20	100	192.12	277.51	351.83	421.85
B3F1WS20	100	195.59	286.18	369.32	445.19
B3F2WS20	100	196.21	288.51	372.99	449.97
B3F3WS20	100	194.21	286.1	369.14	443.35
B6F1WS20	100	195.44	286.06	368.27	443.67
B6F2WS20	100	196.21	288.53	372.84	450.24
B6F3WS20	100	193.17	281.41	361.39	434.62
B9F1WS20	100	196.34	287.46	370.12	445.8
B9F2WS20	100	196.92	289.13	372.89	449.3
B9F3WS20	100	194.33	285.64	366.9	439.22

, and the relative dynamic elastic modulus prediction model of the specimen and the parameters a, u can be calculated according to Equations (20)–(22).

Table 15, Table 16 and

Table 17. Relative dynamic modulus of elasticity prediction model for each group of specimens.

No.	a, u Parameter	Relative Dynamic Elastic Modulus
WS0	a = 0.076, u = 108.577	${\widehat{x}}_k=-1335.699e^{-0.076k}+1435.699$
WS20	a = 0.0965, u = 106.584	${\widehat{x}}_k=-1004.115e^{-0.0965k}+1104.115$
B3F1WS20	a = 0.0767, u = 107.841	${\widehat{x}}_k=-1306.237e^{-0.0767k}+1406.237$
B3F2WS20	a = 0.0742, u = 108.497	${\widehat{x}}_k=-1362.698e^{-0.0742k}+1462.698$
B3F3WS20	a = 0.0789, u = 107.951	${\widehat{x}}_k=-1268.265e^{-0.0789}+1368.265$
B6F1WS20	a = 0.0793, u = 108.149	${\widehat{x}}_k=-1364.04e^{-0.0793}+1264.04$
B6F2WS20	a = 0.073, u = 108.233	${\widehat{x}}_k=-1382.546e^{-0.073k}+1482.546$
B6F3WS20	a = 0.0809, u = 105.973	${\widehat{x}}_k=-1209.691e^{-0.0809k}+1309.691$
B9F1WS20	a = 0.081, u = 109.278	${\widehat{x}}_k=-1248.524e^{-0.081k}+1348.524$
B9F2WS20	a = 0.0795, u = 109.868	${\widehat{x}}_k=-1281.187e^{-0.0795k}+1381.187$
B9F3WS20	a = 0.0883, u = 109.456	${\widehat{x}}_k=-1139.37e^{-0.0883k}+1239.37$

can be used to calculate the predicted values of the specimens in 100 freeze–thaw cycles. The predicted values are shown in

Table 18. Relative dynamic modulus of elasticity predicted for each group of specimens.

No.	Relative Dynamic Elastic Modulus (%)
	0 (Times)	25 (Times)	50 (Times)	75 (Times)	100 (Times)
WS0	100	97.29	90.2	83.63	77.54
WS20	100	92.4	83.9	76.18	69.17
B3F1WS20	100	96.43	89.31	82.72	76.61
B3F2WS20	100	97.42	90.46	83.99	77.99
B3F3WS20	100	96.22	88.92	82.17	75.94
B6F1WS20	100	96.35	89.01	82.22	75.95
B6F2WS20	100	97.34	90.48	84.11	78.19
B6F3WS20	100	94.03	86.72	79.98	73.76
B9F1WS20	100	97.18	89.62	82.64	76.21
B9F2WS20	100	97.97	90.47	83.56	77.17
B9F3WS20	100	96.31	88.17	80.72	73.89

below.

4.3.4. Error and Precision Analysis of GM (1,1) Model

The relative error A, mean variance C, and small error probability P are quoted to test the error and accuracy of the GM (1,1) model and to analyze the applicability of the GM (1,1) model to the freeze–thaw damage model for the relative dynamic elastic modulus of the specimens. The true reliability of the GM (1,1) model is determined using the calculated mean variance, small probability error, and relative error. The GM (1,1) model has the advantage of requiring less data, does not require the data to satisfy a probability distribution, and is essentially a dynamic gray integral equation. The predicted and tested relative dynamic elastic moduli of each group of specimens under different numbers of freezing and thawing cycles are compared to determine whether the accuracy of the model is satisfactory. Test values, predicted values, and average relative errors for each group of specimens are shown in Figure 11. The relative error A between the experimental and predicted values of the relative dynamic elastic modulus of the specimens is given by Equation (23):

$$A=\mid \frac{x_k^{\left(1\right)}-x_k^{\left(1\right)}}{x_k^{\left(1\right)}}\mid \times 100\%$$

(23)

The mean variance C is calculated by Equation (15).

$$C=\frac{{\mu }_1}{{\mu }_2}$$

(24)

where μ₁ is the mean variance of the residual value and μ₂ is the mean variance of the test value calculated by Equations (16) and (17):

$${\mu }_1^2=\frac{1}{n}{\sum }_{k=1}^n{\left[x_k^{\left(0\right)}-\overline{x}\right]}^2\overline{x}$$

(25)

$${\mu }_2{}^2=\frac{1}{n}{\sum }_{k=1}^n{\left[{\lambda }_k-\overline{\lambda }\right]}^2$$

(26)

The small error probability P is calculated by Equation (18)

$$P=\{P\mid {\lambda }_k-\overline{\lambda }\mid <0.6745{\mu }_2\}$$

(27)

The residual between the initial test value sequence $X^{\left(0\right)}$ and the predicted value sequence $\widehat{X_k^{\left(1\right)}}$ is ${\mathsf{\lambda }}_k$, and the mean residual is $\overline{\lambda }$.

GM (1,1) model accuracy grade standard are shown in

Table 19. GM (1,1) model accuracy grade standard.

Precision Evaluation Coefficient	Accuracy Class
	Level 1	Level 2	Level 3	Level 4
C	<0.35	0.35~0.50	0.50~0.65	0.65~0.80
P	>0.95	0.95~0.80	0.80~0.70	0.70~0.60

. According to Equations (23)–(27), the relative error A, mean variance C, and small error probability P of the test and predicted values of the relative dynamic elastic modulus of the specimens can be calculated, and the results of the calculation of A, C, and P for each group of specimens are shown in

Table 20. GM (1,1) model accuracy test table.

Test index	WS0	WS20	B3F1WS20	B3F2WS20
C	0.133	0.145	0.117	0.167
P	1	1	1	1
Accuracy class	Level 1	Level 1	Level 1	Level 1
Test index	B3F3WS20	B6F1WS20	B6F2WS20	B6F3WS20
C	0.258	0.125	0.159	0.119
P	1	1	1	1
Accuracy class	Level 1	Level 1	Level 1	Level 1
Test index	B9F1WS20	B9F2WS20	B9F3WS20
C	0.114	0.138	0.234
P	1	1	1
Accuracy class	Level 1	Level 1	Level 1

. As shown in Table 20, under the effect of different fiber lengths and different fiber admixtures, the accuracy requirements of both the mean variance C and small error probability P of each group of specimens in the prediction model of GM (1,1) reached the first-class standard. The accuracy prediction requirement of the model is satisfied, which shows the real validity of the GM (1,1) model established with the relative dynamic elastic modulus as the index for further research on the freeze–thaw damage of mortar.

5. Conclusions

The following conclusions can be drawn from the basic mechanical property tests, freeze–thaw cycle tests and impact resistance tests of basalt fiber-reinforced walnut shell mortar:

• According to the mechanical property test, after the fine aggregates were replaced with 20% walnut shells of equal mass, the compressive strength, split tensile strength, and flexural strength of WS20 decreased by 31.17%, 12.09%, and 33.33%, respectively, compared with those of WS0. After mixing basalt fibers, the compressive strength of the B6F2WS20 group increased by 13.98%, the split tensile strength by 48.15%, and the flexural strength by 43.75% compared to those of the WS20 group. When the fiber length is 6 mm and the doping amount is 0.2%, the walnut shell mortar has optimal mechanical properties.
• According to the impact resistance test, compared with that of WS20, the impact energy consumption of B6F2WS20 increased by 142.11%. Similarly, the damage mode of the specimen changed from brittle damage to ductile damage, and the maximum number of impacts increased significantly. According to the freezing and thawing cycle test, after one hundred freezing and thawing cycles, the cumulative mass loss of B6F2WS20 was the smallest, at 2.21%, compared with that of WS20, at 50.89%.
• The impact damage prediction model was established based on the Weibull model, and the relative errors between the test values and the predicted values ranged from 0.35% to 0.9%. Through the establishment of the impact evolution equation, the life evolution trend of the specimen under repeated impact loading was obtained, and the damage process of the specimen was reasonably described.
• The GM (1,1) freeze–thaw damage prediction model was established with the mass damage rate and dynamic elastic modulus as the indices, and the average relative error between the experimental value and the predicted value was 2.19% at the maximum and 0.81% at the minimum. According to the accuracy analysis, the accuracy grade of each group of specimens is in the first tier, and the model accuracy is high.