https://orcid.org/0009-0003-0773-8972
Figures
Abstract
Based on the Weibull random distribution of the micro-unit strength of fiber reinforced concrete, the parameters are introduced to describe the micro-unit strength of concrete reasonably, and it is used to establish a damage-softening constitutive model that can reflect the whole process of fiber reinforced concrete damage with the Drucker-Prager yield criterion. On this basis, according to the stress-strain curves of concrete with different palm fiber content and cement content, the functional relationship is discussed among the parameters with palm fiber content and cement content of concrete damage-softening model based on Weibull distribution, then a more realistic concrete damage- softening constitutive model is established with the modified model parameters. To consider the dynamic characteristics of micro-elements in palm fiber concrete during loading-unloading, a modified linear regression model parameter solving method is proposed. By comparing the experimental results with the calculated values, the rationality of the constitutive model and the parameter solution method is verified. The constitutive model proposed in this paper can well reflect the stress-strain relationship and damage evolution process of plam fiber reinforced concrete under different mix ratios, which can provide reference for the engineering application of brown fiber concrete.
Citation: Zhan Q, Huang X, Huo Z, Yi F, Bo H, Jing J, et al. (2025) Damage-softening constitutive model of palm fiber reinforced concrete based on weibull distribution and its correction method. PLoS One 20(6): e0325602. https://doi.org/10.1371/journal.pone.0325602
Editor: Roohollah Kalatehjari, Auckland University of Technology, NEW ZEALAND
Received: January 12, 2025; Accepted: May 14, 2025; Published: June 13, 2025
Copyright: © 2025 Zhan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting Information files.
Funding: This research was financially supported by the High-Level Talent Research Startup Fund of Xiangyang Polytechnic in 2024 (Number: XYZYZZ202405), and Hubei Provincial Department of Education (Number: B2023518); Zhitao Huo and Feiting Yi are the funding recipients. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The authors have declared that no competing interests exist.
Competing interests: NO authors have competing interests.
Introduction
In the process of treating the bare rock slope formed by excavation, it is not only necessary to meet the requirements of slope strength and stability, but also to use plam fiber reinforced concrete for ecological protection [1–5]. Plam fiber reinforced concrete is a composite material composed of brown fiber, cement, soil, special additives and organic materials, the slope spray layer formed by it has a certain strength and is not easy to produce cracks, which has strong anti-erosion ability, especially suitable for steep rock slope protection and ecological restoration projects. At present, a lot of research has been carried out on the mechanical properties and constitutive model of plam fiber reinforced concrete. For example, some researchers have carried out experimental research on the performance of concrete under special conditions, and studied the influence of CFRP, steel tube, polypropylene and polyvinyl alcohol fibers on the performance of concrete [6–10]. Other scholars have described the macroscopic mechanical properties of plam fiber reinforced concrete through tests, and analyzed the influence of fiber content, length and cement content on the properties of concrete, which showed that plam fiber reinforced concrete is strain softening, and the strength is significantly different with different fiber content, length and cement content are significantly different [11–14]. Therefore, it is necessary to explore the mechanical properties and damage constitutive model of plam fiber reinforced concrete under different fiber content and cement content.
The constitutive relationship of fiber-reinforced concrete has always been a key and difficult problem in academic research, and it is particularly necessary to establish a reasonable constitutive model of fiber reinforced concrete with the popularization and application of plam fiber reinforced concrete improvement technology in engineering. At present, there are many constitutive models of fiber reinforced concrete. For example, Babu and Chouksey used data regression analysis method to fit the stress-strain curve obtained from the test, and established the constitutive model of fiber reinforced cohesive soil based on Cambridge model [15]. Notta-Cuvier considered the distribution and orientation of fibers, and established the anisotropic damage model of plam fiber reinforced concrete by homogenization method and stress equilibrium equation [16]. EI-Helou established a UHPC strength model and a multi-axial constitutive model by simplifying the internal fiber orientation of the material and considering the influence of the slip energy consumption of the fiber on the material damage [17]. Koci and Foster established a new elasto-plastic constitutive model of fiber-reinforced material, which considered the effect of distributed fibers embedded in a matrix [18]. The above model promotes the development and application of damage constitutive model to a certain extent. However, to establish a scientific and reasonable damage constitutive model of fiber reinforced concrete, it is necessary to understand not only the macroscopic mechanical behavior of materials, but also the microscopic damage evolution mechanism and law of materials. Based on the randomness of the distribution of defects in fiber reinforced concrete, some scholars combine the continuous damage theory with the statistical strength theory, and establish the damage constitutive model of plam fiber reinforced concrete according to the concept that the strength of the material element obeys the Weibull distribution, which can not only reflect the softening characteristics of the whole process of deformation and failure of plam fiber reinforced concrete, but also reflect the characteristics of the strength changing with the pressure [19–23]. However, because the model does not consider the difference of material strength or model parameters under different fiber content and cement content, it cannot reflect the characteristics of softening and strength of plam fiber reinforced concrete with the change of material content, which limits the rationality and applicability of the model.
In this paper, focusing on the functional relationship between the parameters based on Weibull distribution between the fiber content and cement content, establishing a more realistic damage softening constitutive model of plam fiber concrete by modifying the model parameters. Analyzing the structure and deformation failure characteristics of plam fiber concrete, a modified parameter solution method is proposed. Finally, the experimental results are compared with the calculated values to verify the rationality of the model and parameter solution method. The research results make a breakthrough in the study of the constitutive model of the whole process of deformation and failure of fiber reinforced concrete.
Damage-softening model of plam fiber reinforced concrete based on Weibull distribution
Traditional damage-softening constitutive model on Weibull distribution
There are two methods to establish the constitutive relation of plam fiber reinforced concrete by using damage theory, the former is based on the principle of energy equivalence of materials before and after damage, and the latter is based on the principle of strain equivalence of materials before and after deformation [24–27]. Due to the mechanical concept of the second method is more clear, so this paper established the constitutive relationship of plam fiber reinforced concrete damage with J.Lemaitre strain equivalence principle, as follows [28],
where, is the effective stress matrix,
is the total stress matrix,
is the stiffness matrix,
is the strain matrix and D is the damage variable. It can be seen that the key to establish the damage structure relationship of plam fiber reinforced concrete is to solve the damage variable D from Eq. (1). It is assumed that the plam fiber reinforced concrete is composed of damaged and undamaged elements, and the failure criteria of the elements satisfy,
where, is the constant related to the cohesion and internal friction angle, and
represents the micro-element strength. Assuming that
is the probability of damage, then the damage variable is,
It is known that Obtaining the damage variable with Eq. (3), it is necessary to solve the strength and damage probability
of the plam fiber reinforced concrete micro-element.
Micro-element strength of plam fiber reinforced concrete F.
The form of the microelement strength is directly determined by the failure mechanism of the plam fiber reinforced concrete and the form of the failure criterion. At present, the failure criteria have many forms of expression, which used to study mechanics of plam fiber reinforced concrete. Considering that the parameters of the Drucker-Prager failure criterion are simple and applicable, the micro-element strength of plam fiber reinforced concrete is derived based on the Drucker-Prager failure criterion as follows [29],
where, ,
is the internal friction angle,
and
are the first invariant of stress tensor and the second invariant of stress deviator, respectively, and having,
For the triaxial test, there are and
, which are obtained by Hooke’s law,
Substituting Eqs. (7), (8), and (9) to Eqs. (5) and (6), we obtain,
Then substituting Eqs. (10) and (11) to Eq. (4), the micro-element strength of plam fiber reinforced concrete can be obtained as follows,
It can be seen from Eq. (12), the parameters to be solved are the internal friction angle , the Poisson’s ratio
, and the elastic modulus E.
Damage constitutive model based on Weibull distribution.
Assuming that the strength or damage of the plam fiber reinforced concrete element is subject to the Weibull distribution, the probability density function of the element can be expressed,
where, m and F0 are the Weibull distribution parameters. Substituting Eq. (13) to Eq. (3), obtaining,
The damage model based on Weibull distribution is,
The model parameters are the Weibull distribution parameters m and F0 in Eq. (15). It can be divided into two stages in the loading process with plam fiber reinforced concrete, the first stage will not be damaged and the performance of linear elastic deformation, that is, D = 0 in Eq. (15), and the second stage is be damaged and the performance of nonlinear law.
Modified damage constitutive model based on Weibull distribution
The author carried out unconsolidated undrained triaxial compression tests on plam fiber reinforced concrete with different cement content (xc) and palm fiber content (wf), the test scheme is shown in Table 1, and the test results are shown in Fig 1 [30].
(a) xc = 0%, (B) xc = 4%, (c) xc = 6%, (a) xc = 8%.
E50 represents the ratio of stress to strain at 50% of the stress peak strength, and the peak strength under different cement content and palm fiber content is shown in the literature [30], where the elastic modulus E50 is determined according to the stress and strain at 50% of the peak strength, and it is found that the elastic modulus E50 is not a constant under different mix ratios (Fig 2), indicating that the elastic modulus E50 is related to the cement content and palm fiber content.
By constructing the relationship between the elastic modulus E50 and the cement content xc and the palm fiber content wf, the damage constitutive model of plam fiber reinforced concrete can be effectively modified. If the cement content xc and palm fiber content wf are taken as x and y coordinates respectively, and the elastic modulus E50 is taken as z coordinate, then the (x,y,z) scatter distribution map can be obtained. Matlab is used for exponential surface fitting (Fig 3), the fitting degree is R2 = 0.9567, and the fitting function is obtained as follows,
Eq. (16) represents the relationship between the elastic modulus E50 and the cement content xc and the palm fiber content wf. The modified damage model of plam fiber reinforced concrete can be obtained by substituting Eq. (16) to Eq. (15) as follows,
The parameter solution method of the correction
Linear regression method
The solution is usually linear regression to obtain the Weibull distribution parameters m and F0, for which Eq. (15) is linearized to get,
Further simplification,
Assumptions,
Then Eq. (19) is,
Assuming that I1 and I2 are the slope and intercept of linear in Eq. (22), respectively, the Weibull distribution parameters m and F0 can be expressed,
Modified linear regression parameter method
The rock has the typical characteristics of brittle failure, that is the failure of the performance of local rock cracking and rock fragmentation shedding, which generally will not affect the change of the internal micro-element structure of the rock, so it is usually considered to be constant m and F0 in the calculation of the Weibull distribution parameters of the rock. Different from rock, plam fiber reinforced concrete belongs to the semi-rigid material with a porosity of 40%, the cohesion between the particles is insufficient, and the rheological properties of the concrete will be changed due to the incorporation of brown fibers, which caused that its internal micro structure are prone to change under the action of external force (Fig 4), thus the Weibull distribution parameters are also changed before and after the destruction of plam fiber reinforced concrete [31–32]. Therefore, when the linear regression method is used to solve the Weibull distribution parameters, it is necessary to consider the dynamic change characteristics of micro-elements and improve the regular linear regression method.
(a) before destruction, (B) after destruction, (c) columnar cross section,(d) physical sample.
Considering the dynamic change characteristics of micro-elements in the loading process of plam fiber reinforced concrete, this paper uses the segmented method to solve the slope I1 and intercept I2 of each segment, and then, it calculates the parameters m and F0 according to the slope and intercept of each segment. The specific method is as follows, take Xj (j = 1,2,3,......,j is an integer) as the center, take n/2 groups of (X,Y) data (excluding X j) to the left, and take n/2 groups of (X,Y) data (including X j) to the right. It is calculated according to the actual number of groups when the data is less than n/2 groups (Fig 5), the Weibull distribution function corresponding to each Xj point is denoted as mj, F0j.
How to determine the value of n? In this paper, it is taking plam fiber reinforced concrete with cement content of 8% and palm fiber content of 0.0 as an example. Considering that when linear regression method is used for calculation, the value of n is at least 10 times the number of independent variables (xi), then n = 10,15,20,......,60 is taken to analyze the relationship between the model calculation value and the test value under different values of n, as shown in Fig 6.
= 8%, wf = 0.0%).
As can be seen from Fig 6, with the decrease of the n value, the peak value of the model calculation curve gradually increases, and the smaller the error between the calculated value and the test value, that is, when n = 10, the calculated value is in good agreement with the test value. The value of n is best taken from 10 times the number of variables when solving the parameters using linear regression segments. Therefore, the paper selects the segmentation method of n = 10 to calculate the parameters of plam fiber reinforced concrete.
Validation
In order to verify the modified damage model and parameter solution method, this paper uses the test data under the condition of cement content of 6% for analysis. It is also divided into two stages in determining the calculated value, the first is the initial stage of loading and the material is not damaged, which taking D = 0 to calculate the stress in Eq.(15), where elastic modulus E according to Eq. (16). In the second stage, the average slope I1 and intercept I2 corresponding to each point Xj are solved according to the modified parameter solution method, that is, n = 10, and then the parameters mj and F0j are solved corresponding to point xj with Eqs. (23) and (24), respectively. Finally, mj and F0j are substituted into Eq.(17) to obtain the calculated stress value.
The modified damage model can well describe the stress-strain behavior and fully reflect the softening characteristics of plam fiber reinforced concrete (Fig 7), so the model can be used to analyze and predict the mechanical properties of plam fiber reinforced concrete with different cement content and palm fiber content.
= 6%).
Conclusions
In this paper, the micro-element strength and micro-element damage of plam fiber reinforced concrete obey the Weibull distribution, introducing the random distribution variable F of the micro-element strength of the material, and it deduces the damage evolution equation of the pressure damage of plam fiber reinforced concrete.
- (1). The relationship between the elastic modulus E50 with cement content xc and palm fiber content wf is discussed basing on the unconsolidated and undrained triaxial test results, and then the damage model is revised in combination with E50, where a damage model is established that it is more in line with the actual deformation. The model fully reflects that the damage in the destruction process not only changes with the strength, but also is affected by the cement content and palm fiber content in the plam fiber reinforced concrete, the modified damage model is more reasonable.
- (2). Plam fiber content and cement content could improve the compressive strength of concrete significantly, plam fiber reinforced concrete showed strain softening with the increase of plam fiber content and cement content. The established model can reflect the whole failure process of plam fiber reinforced concrete under complex stress state, especially the softening characteristics of plam fiber reinforced concrete.
- (3). Plam fiber formed a woven network structure in concrete, showing the characteristics of cracks but not broken, which made the process of damage slowly and ductility. Therefore, plam fiber has a certain crack resistance effect on the deformation of concrete.
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