On the Mechanics of a Fiber Network-Reinforced Elastic Sheet Subjected to Uniaxial Extension and Bilateral Flexure

Abstract

The mechanics of an elastic sheet reinforced with fiber mesh is investigated when undergoing bilateral in-plane bending and stretching. The strain energy of FRC is formulated by accounting for the matrix strain energy contribution and the fiber network deformations of extension, flexure, and torsion, where the strain energy potential of the matrix material is characterized via the Mooney–Rivlin strain energy model and the fiber kinematics is computed via the first and second gradient of deformations. By applying the variational principle on the strain energy of FRC, the Euler–Lagrange equilibrium equations are derived and then solved numerically. The theoretical results highlight the matrix and meshwork deformations of FRC subjected to bilateral bending and stretching simultaneously, and it is found that the interaction between bilateral extension and bending manipulates the matrix and network deformation. It is theoretically observed that the transverse Lagrange strain peaks near the bilateral boundary while the longitudinal strain is intensified inside the FRC domain. The continuum model further demonstrates the bidirectional mesh network deformations in the case of plain woven, from which the extension and flexure kinematics of fiber units are illustrated to examine the effects of fiber unit deformations on the overall deformations of the fiber network. To reduce the observed matrix-network dislocation in the case of plain network reinforcement, the pantographic network reinforcement is investigated, suggesting that the bilateral stretch results in the reduced intersection angle at the mesh joints in the FRC domain. For validation of the continuum model, the obtained results are cross-examined with the existing experimental results depicting the failure mode of conventional fiber-reinforced composites to demonstrate the practical utility of the proposed model.

1. Introduction

Fiber-reinforced composites (FRCs), characterized by their distinguished long-term durability, high strength, light weight, fatigue resistance, and damage tolerance, are considered “promising” candidates for biological tissue manufacturing, aerospace, and construction sectors [1,2,3,4,5,6]. In general, the high-strength fibrous reinforcement and matrix are bonded to maintain the fiber location and orientation, while the matrix enclosure protects the FRC from surrounding physical and chemical damages, and the high performance of FRC can be achieved via bidirectional/multi-directional fiber distributions, structural strengthening E6, material selections, and manufacturing technique [7]. The extensive advantages of FRC includes the fact that FRC stiffness can be enhanced in a designated orientation and fabricated into flexural panels without further forming operation, and achieving FRC with zero coefficients of thermal expansion via structure design [8]. It is even believed that FRCs have started replacing conventional materials such as steel, aluminum, ceramics, and wood [7], and FRCs exhibit environmental and economic efficiency [9].

Nevertheless, it is still challenging to implement the wide use of FRCs in civil engineering, particularly in the case of seismic cases when FRCs become vulnerable under stretching, bending, and complicated loading conditions [1,10,11,12]. To address this, considerable research efforts have been dedicated to analyzing and describing the mechanical performance of FRC by evaluating the flexural and tensile characteristics of FRC. For instance, a sandwich structure for FRC is developed to improve the three-point bending performance by failure and statistics analysis, indicating its remarkable competency in the application of aeronautical and automotive industries [13]. The sandwich-structured FRC can be alternatively refined with corrugated steel faces and foamed concrete (FC) cores with 700 ${\mathrm{kg}/\mathrm{m}}^3$ density, under which the FRCs exhibit increased maximum load and energy absorption via four-point bending experiments [14]. To obtain a light weight and improve the performance of construction structures, theoretical and experimental works are implemented to substantiate the bending effectiveness of combined-reinforced glass fiber polymer composite concrete (GFPCC), showing that the efficiency of GFPCC bending elements is comparable and higher than the steel-reinforced concrete bending elements [15]. The authors of [16] investigated the flexural performance of FRC aggregate concrete beams by regulating fiber volume and the ingredients of large particle natural aggregate; the results show that the beam cracking resistance, the flexural capacity, and the flexural ductility are enhanced by a high fiber volume, and the new FRC ingredient formulas are developed to predict the cracking moment, crack width, and flexural stiffness. By admixing aluminum, mica, and ceramic particles, significant enhancements in tensile and bending properties of FRC are achieved [17]. Extensive studies on improving FRC mechanical performances include testing pantographic sheets via bias extension [18], examining the tensile and bending strength of doped-type FRC [19], and evaluating tensile–tensile cyclic performance to improve the extension performance differences for FRC with two types of structure [11].

The primary objective of the introduced studies is to investigate optimized manufacturing approaches, refined matrices, and reinforcement structures for FRC via pure bending or pure stretching tests, from which the flexure and stretch performance can be examined. Nevertheless, due to the intricate loading conditions and complex geometrical characteristics of FRC [11,12,20], FRCs are, in fact, subjected to combined loading conditions, including tension–twist, tension–tension, tension–bending, and bending–shear interactions. In this regard, massive studies emphasize investigating the mechanical performance of FRC under combined loading conditions: with a novel testing device, study [12] examines the coupling effects of tension and twist on anisotropic FRC, indicating the strong dependency of the tension–twist coupling effect on the orientation of the fibers and the structure of the reinforcement, for which reliable guidelines are provided to select adequate load-coupled material properties in terms of the matrix material and reinforcement. Intriguingly, it is observed that the hydrodynamic efficiency of FRC marine propellers is improved with bend–twist coupling effects that curved tows can reduce deflection or the stress and strain of the FRC blade [21]. The combined bending and shear strength are considered to be critical indicators for the experiments of evaluating ultra-high-performance FRC beams, providing that the medium-span cantilevers exhibit a lower probability of failure while the softening effects near the interface enhance the FRC deformation capacity [22]. Extensive studies for the combined loading of FRC encompass unveiling the mechanism for the failure patterns of FRC subjected to combined tension and bending [23], investigating the static behavior of laminated FRC panels in aircraft manufacturing via the joint work of transverse loading and in-plane loading conditions, where loading conditions of bending combined with various tensile and compression conditions are applied to examine the stretching and flexural strength of liquid silicon infiltrated (LSI) ceramic–matrix composite [24]. These studies highlight the important role and research merit of complex loading conditions in assessing the mechanical performance of FRC in a wide range of applications.

The intense research interests in experimental and theoretical studies for FRC mechanics performance improvement under complex loading conditions impel the advancement of continuum models for the strain–stiffness analysis of FRC undergoing flexural and tensile loading conditions, therefore providing theoretical guidance in manufacturing high-performance FRC and related synthetic composites. Within this context, the instability of FRC is theoretically investigated under uniaxial stretches, emphasizing that the instability of FRC significantly affects the critical strains and critical transition threshold [25]. The stress–stretch relations and failure evolution in FRC are characterized in [26] via tensile and cyclic loading tests for FRC improvements. These models theoretically analyze the mechanics of FRC by altering fiber arrangement and matrix while accounting for the incorporation of refined classic micromechanical theory and analysis methods, through which the micro deformation of FRC can be demonstrated. The ambition of this research is to correlate the microscopic and macroscopic behavior of FRC using microstructure analysis. Within this prescription, the authors of [27] developed a strain-gradient continuum model by accounting for the kinematics of embedded fibers such as extension and flexure, in which the fiber kinematics-associated strain energy is incorporated into the hyperelastic model of the matrix material through the calculation of the first and second gradients of continuum deformations. In there, the FRC “J-shape” stress–strain responses, shear strain intensity distributions, and the profile of FRC deformations are precisely predicted under the bilateral extension, yet, the FRC depicted by the continuum model is anisotropic in the bidirections, thereby falling short in investigating FRC mechanics in the transverse orientation. Proceeding work addressing this shortage has improved the continuum model by considering the fiber reinforcement in FRC as bidirectionally distributed, allowing isotropic FRC to be investigated [28]. Further, recent improvements in the strain-gradient model highlight describing the rapid strain-stiffening response of FRC and pursuing more accurate descriptions by considering fiber torsional deformation and highly non-linear strain potential. The obtained theoretical results are successfully cross-examined by in-house experiments in aspects of strain-stiffening responses, in-plane deformations, and shear strain distribution [29]. To capture the strain-stiffening responses locally (phenomena such as buckling and internal resonance), third-order strain-gradient theory was devised, where unique deformations are obtained under the Piola-type triple stress [30]. Extensive research interest in higher-order theories of FRC includes combining shear lag theory with strain gradient theory to analyze the mechanical response of the nanofiber-matrix system under a uniaxial extension test [31,32]; applying strain-gradient theory of FRC to analyze porous-ductile fracture [33]; examining the capability of the strain-gradient theory to depict the appearance of size effects in the uni-directionally reinforced FRC plates [34]; and developing nonlocal strain-gradient theory to analyze the dynamics of short-fiber-reinforced FRC beams [35]. The high-gradient performance of FRC has been experimentally evidenced in [36], showing that the strain-gradient theory can capture the shear in the matrix and the flexure in the fibers.

Nevertheless, the introduced strain-gradient continuum models mainly emphasize investigating the effects of tensile and flexural loading conditions separately, therefore neglecting the joint effects of combined stretch and flexure loadings on FRC. Moreover, the role of the microstructures in determining the overall FRC deformation remains to be quantitatively elucidated by the strain-gradient continuum models despite the impressive effects of embedded microstructures on the mechanical performance of FRC. Further, the introduced strain-gradient continuum models fall short in discussing the kinematic differences between matrix and fiber reinforcements. These underlying aspects impede the comprehensive understanding of the FRC mechanics subjected to the combined loading conditions in-plane, and critical indicators such as strain–stiffness relations, shear strain distributions, and network deformations remain to be investigated.

In this study, we investigate the mechanics of FRC subjected to bilateral tension and bending moment via a continuum model. The strain energy of FRC is described using the Mooney–Rivlin hyperelastic model for the matrix to investigate FRC subjected to complex loading scenarios (simultaneous bilateral bending and stretching) [37] and computing strain energy of bilateral fiber reinforcement via the first and second gradient of deformation. To accurately depict the strain–stiffness performance of fiber subjected to complex loading conditions, the Lagrange strains are introduced in the third-order polynomial form aimed at fitting the manner of stress–strain curves in the Mooney–Rivlin hyperelastic model [29]. Equilibrium equations and boundary conditions are formulated by applying the variation computation on the strain energy while considering that the FRC is incompressible. The equilibrium equations are solved numerically, and the results show that the matrix exhibited upward and transverse deformation invoked by the joint work of the bilateral bending and stretching. The case of plain woven reinforcement structures is conducted, illustrating the role of extension and flexure of fiber units in determining the overall deformation of the fiber network, while the matrix-reinforcement dislocation is observed; therefore, the case of pantographic reinforcement structure is investigated. In this case, the examination of the shear angles within the pantographic network demonstrates that bending moments increase the intersection angle within the domain and near the bilateral boundaries. In contrast, bilateral stretching exerts an opposing influence on the intersection angle. Further, the Lagrange strain reveals that transverse strain peaks near the bilateral boundaries, and transverse stretching effects are invoked near the bottom boundary. It is observed that the longitudinal strain peaks inside the domain, while the top and bottom boundaries of the FRC experience shrinkage in the longitudinal direction, resulting in negative Lagrange strain near the top and bottom boundaries.

For validation of the continuum model, the experimental and theoretical results from the published work are invoked with reasonable explanations.

2. Kinematics of Bidirectional Fibers

This section presents a theoretical framework for characterizing the strain energy of FRC. Initially, the strain energy is conceptually delineated by considering the individual contributions of the matrix material and the fiber reinforcement. Subsequently, the respective strain energy of matrix and fiber reinforcement is determined by the kinematics computations of matrix and fiber deformation, which encompasses the computations of the first-order and second-order deformation gradients. Particular attention is devoted to elucidating the kinematics of fiber extension, flexure, and twist within the matrix material.

To express the discussed strain energy of FRC, it is necessary to configure the kinematics of the matrix material of the FRC and the embedded fibers inside, via the computation of the gradient of deformation, the Green–Lagrange strain, and the geodesic curvature. Hence, we start by configuring the FRC in-plane coordinate components as $X_{\alpha }$ to parameterize the material position in the plane. The position of the material points of reference and current material configurations are respectively denoted as $\mathbf{X}$ and $\mathit{\chi }$ (vector). Their respective surface tangents are

$${\mathbf{X}}_{,\alpha }={\displaystyle \frac{\partial \mathbf{X}}{\partial X_{\alpha }}}\mathrm{and}{\mathit{\chi }}_{,\alpha }={\displaystyle \frac{\partial \mathit{\chi }}{\partial X_{\alpha }}}.$$

(1)

Within this description, the connections between the referential and current bases are established via the first-order gradient of deformation

$$\mathbf{F}={\mathit{\chi }}_{,\alpha }\otimes {\mathbf{X}}_{,\alpha }.$$

(2)

In Equation (2), the summation over $\alpha$ is required for the orthonormal coordinate system; for a non-orthogonal coordinate system, the surface metric should be calculated, yet the continuum model is projected and then simulated in the Cartesian coordinate system in the following sections; therefore, the surface metric is omitted for the sake of conciseness.

The extension of fiber units involves the derivative of the trajectories of the fibers in the reference configuration, computed as

$$\mathbf{L}={\displaystyle \frac{d\mathbf{X}}{d{X}_1}}\mathrm{and}\mathbf{M}={\displaystyle \frac{d\mathbf{X}}{\partial X_2}},$$

(3)

where $X_1$ and $X_2$ are the coordinate in the increasing directions of $\mathbf{L}$ and $\mathbf{M}$, respectively (shown in Figure 1). The initial FRC is characterized by uniformly and orthogonally oriented fibers, satisfying the condition $\mathbf{L}·\mathbf{M}=0$. Conversely, the initially non-orthogonal fibers exhibit a non-zero inner product, i.e., $\mathbf{L}·\mathbf{M}$$\ne 0$, as documented in [38]. This study adopts a configuration comprising uniform and orthogonal fibers to represent the initial fiber reinforcement. Subsequently, the deformed fiber unit vectors corresponding to the initial fibers are determined utilizing Equation (2); therefore,

$$\lambda \mathbf{l}=\mathbf{FL}\mathrm{and}\mu \mathbf{m}=\mathbf{FM},$$

(4)

where $\mathbf{l}$ and $\mathbf{m}$ are unit tangents to the trajectories of the fibers in the deformed configuration (Figure 1), and $\lambda$ and $\mu$ are the corresponding extension.

Then, to compute the flexure of fibers in FRC, the geodesic curvatures of the fiber units are given as

$$\begin{array}{ccc}\hfill {\mathbf{g}}_1& =& {\displaystyle \frac{d^2\mathit{\chi }}{d{X}_1^2}}={\displaystyle \frac{d\left(\mathbf{FL}\right)}{d\mathbf{X}}}{\displaystyle \frac{d\mathbf{X}}{d{X}_1}}=▿\left(\mathbf{FL}\right)\mathbf{L}=▿\mathbf{F}(\mathbf{L}\otimes \mathbf{L})\hfill \\ & & \mathrm{and}\hfill \\ \hfill {\mathbf{g}}_2& =& {\displaystyle \frac{d^2\mathit{\chi }}{d{X}_2^2}}={\displaystyle \frac{d\left(\mathbf{FM}\right)}{d\mathbf{X}}}{\displaystyle \frac{d\mathbf{X}}{d{X}_2}}=▿\left(\mathbf{FM}\right)\mathbf{M}=▿\mathbf{F}(\mathbf{M}\otimes \mathbf{M}),\hfill \end{array}$$

(5)

where $X_1$ and $X_2$ are the arc length parameters in the increasing directions of $\mathbf{L}$ and $\mathbf{M}$, $▿\mathbf{F}$ is the second-order gradient of deformation, and regarding no initial deformation for fiber units $▿\mathbf{L}=▿\mathbf{M}=\mathbf{0}$. By invoking Equation (2), we find

$$▿\mathbf{F}={Ø}_{,\alpha \beta }\otimes {\mathbf{X}}_{,\alpha }\otimes {\mathbf{X}}_{,\beta },{\mathbf{g}}_{\mathbf{1}}=({\mathit{\chi }}_{,\alpha \beta }\otimes {\mathbf{X}}_{,\alpha }\otimes {\mathbf{X}}_{,\beta })•(\mathbf{L}\otimes \mathbf{L})\mathrm{and}{\mathbf{g}}_{\mathbf{2}}=({\mathit{\chi }}_{,\alpha \beta }\otimes {\mathbf{X}}_{,\alpha }\otimes {\mathbf{X}}_{,\beta })•(\mathbf{M}\otimes \mathbf{M})$$

(6)

3. Strain Energy of FRC

By invoking the work of [30,39,40,41,42,43], the strain energy density for a single sheet of FRC might be demonstrated as

$$W(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)=(1-\Phi )W{\left(\mathbf{F}\right)}_{matrix}+\Phi [W{\left({\epsilon }_i\right)}_{fiberextension}+W{\left({\mathbf{g}}_i\right)}_{fiberbending/twist}],$$

(7)

in which $\Phi$ is the fiber volume fraction, and $W{\left(\mathbf{F}\right)}_{matrix}$, $W{\left({\epsilon }_i\right)}_{fiberextension}$, and $W{\left({\mathbf{g}}_i\right)}_{fiberbending/twist}$ are energy potentials deriving from the contributions of matrix materials, fiber extension, and fiber bending/twisting, respectively. In this study, we respectively investigate the bidirectional 50 × 50 fiber network and fiber network with a 20 × 20 pantographic structure, by which the fiber volume fraction can be computed by the representative volume element (RVE) method [44], which are merged into the material parameters determining the mechanical performance of FRC.

The primary purpose of this study is to investigate the FRC via continuum models. For clarification, $\mathbf{F}$ is the first-order gradient of deformation, ${\epsilon }_i$ is the Green–Lagrange strain of fiber extension, and ${\mathbf{g}}_i$ is the geodesic curvature of fiber. To present the energy potential explicitly, the strain energy potential of the matrix might be treated via the hyperelastic material model of strain energy, i.e., for the Mooney–Rivlin hyperelastic model describing the soft material [37], involving the computation of the first and second invariant, shown as

$$W{\left(\mathbf{F}\right)}_{matrix}={\displaystyle \frac{\kappa }{2}}(I_{\mathbf{1}}-3)+{\displaystyle \frac{\mu }{2}}(I_{\mathbf{2}}-3),$$

(8)

in which $\mu$ and $\kappa$ are material parameters, and $I_1=tr\left({\mathbf{F}}^T\mathbf{F}\right)$, $I_2={\displaystyle \frac{1}{2}}\left\{{\left[tr\left({\mathbf{F}}^T\mathbf{F}\right)\right]}^2-tr\left[{\left({\mathbf{F}}^T\mathbf{F}\right)}^2\right]\right\}.$ The Mooney–Rivlin hyperelastic model incorporates invariants $I_1$ and $I_2$. In particular, study [37] addresses “the inclusion of $I_2$ is crucial for a more comprehensive understanding of depicting the properties of biological soft tissues under complex loading scenarios where deformations cannot be accurately described by the first invariant alone”. In this regard, the Mooney–Rivlin hyperelastic model has been successfully applied to analyze and predict the matrix deformation of FRC [27,28,29,30] and showcase its capability of describing biological material (tissue, brain [45,46,47,48]). Therefore, the proposed model can be adapted to describe the FRC mechanical performance subjected to combined uniaxial stretch and bilateral flexure, while the application of the continuum model of this study might potentially extend to the field of depicting the mechanics of biological material.

Then, we consider how the FRC reinforcement potentially undergoes stretching/compressing-induced strain-stiffening behaviors contributing to the FRC strain energy, which might be illustrated by using the Green–Lagrange strain ${\epsilon }_i$ of the squared order [29], shown as

$$W{\left({\epsilon }_i\right)}_{fiberextension}={\displaystyle \frac{1}{2}}{E}_1{\epsilon }_1^2+{\displaystyle \frac{1}{2}}{E}_2{\epsilon }_2^2,$$

(9)

where $E_i$ are the stretch stiffness of fiber, and the Green–Lagrange strain ${\epsilon }_i$ can be computed under the assistance of Equation (4), shown as

$${\epsilon }_1={\displaystyle \frac{1}{2}}({\lambda }^2-1)={\displaystyle \frac{1}{2}}(\mathbf{FL}·\mathbf{FL}-\mathbf{1})\mathrm{and}{\epsilon }_2={\displaystyle \frac{1}{2}}({\mu }^2-1)={\displaystyle \frac{1}{2}}(\mathbf{FM}·\mathbf{FM}-\mathbf{1}).$$

(10)

Then, $W{({\mathbf{g}}_1,{\mathbf{g}}_2)}_{fiberbending/twist}$ demonstrates the bending and twisting contribution to the strain energy of FRC, which involves the dot product of geodesic curvature ${\mathbf{g}}_i$, yielding

$$W{\left({\mathbf{g}}_i\right)}_{fiberbending/twist}={\displaystyle \frac{1}{2}}{C}_1{\mathbf{g}}_1·{\mathbf{g}}_1+{\displaystyle \frac{1}{2}}{C}_2{\mathbf{g}}_2·{\mathbf{g}}_2+{\displaystyle \frac{1}{2}}T{\mathbf{g}}_1·{\mathbf{g}}_2,$$

(11)

in which $C_i$ is the bending stiffness of fiber and T is the torsional stiffness of fiber; $C_i$ and T are material parameters that are independent of the FRC deformation.

Notably, the study assumes no interaction between the matrix and fiber reinforcement, though it should address how the interaction plays a decisive role in the generation of FRC fracture [49], and a strong contact interface has the potential to achieve high load-bearing capacity of FRC [50]. The study highlights the kinematics of matrix and fiber reinforcement; therefore, the interaction-induced energy dissipation will be investigated in future studies and is not framed in the discussion of the work, and the fiber rotation-induced strain energy potential might be considered by invoking the couple stress theory [51]. The fiber bending/twist term represents the bending energy potential of the Spencer and Soldatos type, as elucidated in their seminal work [52]. Their formulation supports that the contribution of bending strain energy is solely governed by the geodesic curvature of the fibers, which is computed through the second-order gradient of the continuum deformation. This established concept of bending-induced strain energy has been widely adopted and integrated into numerous related investigations (cf. [38,53,54,55,56,57]). Furthermore, the computation of ${\mathbf{g}}_i$ necessitates the second-order gradient of deformation and must adhere to the principle of frame indifference. This requirement remains paramount not only within the framework of finite elastic deformations of general continuum bodies, as extensively discussed in [52,58,59], but also in the analysis of hyperelasticity in biological tissues, as highlighted in [60]. Within this context,

$$W(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)={\displaystyle \frac{\kappa }{2}}(I_{\mathbf{1}}-3)+{\displaystyle \frac{\mu }{2}}(I_{\mathbf{2}}-3)+{\displaystyle \frac{1}{2}}{E}_1{\epsilon }_1^2+{\displaystyle \frac{1}{2}}{E}_2{\epsilon }_2^2+{\displaystyle \frac{1}{2}}{C}_1{\mathbf{g}}_1·{\mathbf{g}}_1+{\displaystyle \frac{1}{2}}{C}_2{\mathbf{g}}_2·{\mathbf{g}}_2+{\displaystyle \frac{1}{2}}T{\mathbf{g}}_1·{\mathbf{g}}_2,$$

(12)

and by invoking the relations in Section 2, the strain energy potential of FRC is expressed in terms of the strain and strain gradient of FRC material, shown as

$$\begin{array}{ccc}\hfill W({\mathit{\chi }}_{,\alpha },{\mathit{\chi }}_{,\lambda \mu })& =& {\displaystyle \frac{\kappa }{2}}[tr\left({\mathbf{F}}^T\mathbf{F}\right)-3]+{\displaystyle \frac{\mu }{2}}\left\{{\left[tr\left({\mathbf{F}}^T\mathbf{F}\right)\right]}^2-tr\left[{\left({\mathbf{F}}^T\mathbf{F}\right)}^2\right]-3\right\}+{\displaystyle \frac{1}{2}}{E}_1{\epsilon }_1^2+{\displaystyle \frac{1}{2}}{E}_2{\epsilon }_2^2+\hfill \\ & & {\displaystyle \frac{1}{2}}{C}_1{\mathbf{g}}_1·{\mathbf{g}}_1+{\displaystyle \frac{1}{2}}{C}_2{\mathbf{g}}_2·{\mathbf{g}}_2+{\displaystyle \frac{1}{2}}T{\mathbf{g}}_1·{\mathbf{g}}_2\hfill \\ & =& {\displaystyle \frac{\kappa }{2}}({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\alpha }-3)+{\displaystyle \frac{\mu }{2}}\left\{{({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\alpha })}^2-{({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta })}^2-3\right\}+{\displaystyle \frac{1}{8}}{E}_1{({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta }{L}_{\alpha }{L}_{\beta }-1)}^2+\hfill \\ & & {\displaystyle \frac{1}{8}}{E}_2{({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta }{M}_{\alpha }{M}_{\beta }-1)}^2+{\displaystyle \frac{1}{2}}{C}_1({\mathit{\chi }}_{,\alpha \beta }·{\mathit{\chi }}_{,\lambda \mu }{L}_{\alpha }{L}_{\beta }{L}_{\lambda }{L}_{\mu })+\hfill \\ & & {\displaystyle \frac{1}{2}}{C}_2({\mathit{\chi }}_{,\alpha \beta }·{\mathit{\chi }}_{,\lambda \mu }{M}_{\alpha }{M}_{\beta }{M}_{\lambda }{M}_{\mu })+{\displaystyle \frac{1}{2}}T({\mathit{\chi }}_{,\alpha \beta }·{\mathit{\chi }}_{,\lambda \mu }{L}_{\alpha }{L}_{\beta }{M}_{\lambda }{M}_{\mu }),\hfill \end{array}$$

(13)

which is framed in the theory of the second gradient elasticity. The well-posedness of this class of continuum models has been thoroughly addressed in [61], with further extensive discussions available in [62,63] and the references therein. This study primarily focuses on the mechanical analysis of FRC through the continuum model and numerical simulations. Consequently, a detailed theoretical examination of well-posedness is omitted.

The above formulations (particularly Equations (8)–(11)) result in the Equations (7) and (12) being explicit in terms of the FRC in-plane configuration, and the equilibrium equations describing the mechanics of FRC sheets will be derived from the variation of the strain energy potential of FRC and the constitutive relations of continuum mechanics in the following sections.

4. Variational Framework and FRC In-Plane Equilibrium

Extensive literature, as documented in [64,65,66,67], has established the foundation of variational principles within the realm of second-gradient finite elasticity. Within this context, we proceed to derive the Euler–Lagrange equations via the variational framework. Hence, for the potential energy of FRC initially occupying domain $\Omega$, which then deforms to $\omega$, the primary work is to derive the Euler–Lagrange equation by minimizing the potential energy by computing the variation of E.

Before computing the variation of E, we consider that the FRC deformation is constrained under bulk incompressibility because for engineering materials, volume changes in material deformation can be costly [68,69], and by introducing the Lagrange multiplier p restricting intrinsic properties of FRC, we have [63]

$$\begin{array}{ccc}\hfill E& =& {\int }_{\omega }W(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)-pda\hfill \\ & =& {\int }_{\Omega }J[W(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)-p]dA,\hfill \end{array}$$

(14)

and

$$\begin{array}{ccc}\hfill \dot{E}& =& {\int }_{\Omega }\dot{J}[W(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)-p]+J\dot{W}(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)dA\hfill \\ & =& {\int }_{\omega }{\displaystyle \frac{\dot{J}}{J}}[W(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)-p]+\dot{W}(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)da,\hfill \end{array}$$

(15)

where the superposed dot means taking the derivative of the energy potential in respect to a parameter (e.g., $ϵ$) that identifies the configuration of the FRC surface.

4.1. Variational Formulation

The computation of the $\dot{E}$ in Equation (15) necessitates computing

$$\dot{W}(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)={\displaystyle \frac{\partial W{\left(\mathbf{F}\right)}_{matrix}}{\partial \mathbf{F}}}·\dot{\mathbf{F}}+{\displaystyle \frac{\partial W\left({\epsilon }_i\right)}{\partial {\epsilon }_i}}{\dot{\epsilon }}_i+{\displaystyle \frac{\partial W\left({\mathbf{g}}_i\right)}{\partial {\mathbf{g}}_i}}·{\dot{\mathbf{g}}}_i,$$

(16)

where

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial W{\left(\mathbf{F}\right)}_{matrix}}{\partial \mathbf{F}}}& =& {\displaystyle \frac{\partial W{\left(\mathbf{F}\right)}_{matrix}}{\partial I_{\mathbf{1}}}}{\displaystyle \frac{\partial I_{\mathbf{1}}}{\partial \mathbf{F}}}+{\displaystyle \frac{\partial W{\left(\mathbf{F}\right)}_{matrix}}{\partial I_{\mathbf{2}}}}{\displaystyle \frac{\partial I_{\mathbf{2}}}{\partial \mathbf{F}}}\hfill \\ & =& \mu \mathbf{F}+\kappa \mathbf{F}[(\mathbf{F}·\mathbf{F})\mathbf{I}-{\mathbf{F}}^T\mathbf{F}],\hfill \end{array}$$

(17)

and

$${\displaystyle \frac{\partial W(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)}{\partial {\epsilon }_1}}=E_1{\epsilon }_1\mathrm{and}{\displaystyle \frac{\partial W(\mathbf{F},{\epsilon }_i,{\mathbf{g}}_i)}{\partial {\epsilon }_2}}=E_2{\epsilon }_2,$$

(18)

in which

$${\epsilon }_1={\displaystyle \frac{1}{2}}({\lambda }^2-1)={\displaystyle \frac{1}{2}}({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta }{L}_{\alpha }{L}_{\beta }-1)\mathrm{and}{\epsilon }_2={\displaystyle \frac{1}{2}}({\mu }^2-1)={\displaystyle \frac{1}{2}}({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta }{M}_{\alpha }{M}_{\beta }-1).$$

(19)

For the partial derivative of the geodesic curvature,

$${\displaystyle \frac{\partial W\left({\mathbf{g}}_1\right)}{\partial {\mathbf{g}}_1}}=C_1{\mathbf{g}}_1+{\displaystyle \frac{1}{2}}T{\mathbf{g}}_2,{\displaystyle \frac{\partial W\left({\mathbf{g}}_2\right)}{\partial {\mathbf{g}}_2}}=C_2{\mathbf{g}}_2+{\displaystyle \frac{1}{2}}T{\mathbf{g}}_1$$

(20)

and $\dot{\mathbf{F}}$, ${\dot{\epsilon }}_i$ are computed via Equations (2) and (10), shown as

$$\begin{array}{ccc}\hfill \mathbf{F}& =& {\mathit{\chi }}_{,\alpha }\otimes {\mathbf{X}}_{,\alpha },\dot{\mathbf{F}}={\dot{\mathit{\chi }}}_{,\alpha }\otimes {\mathbf{X}}_{,\alpha }={\mathbf{u}}_{,\alpha }\otimes {\mathbf{X}}_{,\alpha }(\because {\dot{\mathbf{X}}}_{,\alpha }=0),\hfill \\ \hfill {\dot{\epsilon }}_1& =& {\displaystyle \frac{1}{2}}(\mathbf{FL}·\mathbf{FL}-1\dot{)}={\mathit{\chi }}_{,\alpha }·{\mathbf{u}}_{,\beta }{L}_{\alpha }{L}_{\beta }\mathrm{and}{\dot{\epsilon }}_2={\displaystyle \frac{1}{2}}(\mathbf{FM}·\mathbf{FM}-1\dot{)}={\mathit{\chi }}_{,\alpha }·{\mathbf{u}}_{,\beta }{M}_{\alpha }{M}_{\beta },\hfill \end{array}$$

(21)

By invoking Equation (5), ${\dot{\mathbf{g}}}_i$ proceeds to be

$$\begin{array}{ccc}\hfill {\dot{\mathbf{g}}}_1& =& {[▿\mathbf{F}(\mathbf{L}\otimes \mathbf{L})]}^{·}={(▿\mathbf{F})}^{·}(\mathbf{L}\otimes \mathbf{L})\hfill \\ & & \mathrm{and}\hfill \\ \hfill {\dot{\mathbf{g}}}_2& =& {[▿\mathbf{F}(\mathbf{M}\otimes \mathbf{M})]}^{·}={(▿\mathbf{F})}^{·}(\mathbf{M}\otimes \mathbf{M})\hfill \end{array}$$

(22)

which involves the computation of

$$\begin{array}{ccc}\hfill {(▿\mathbf{F})}^{·}& =& {[{\mathit{\chi }}_{,\alpha \beta }\otimes {\mathbf{X}}_{,\alpha }\otimes {\mathbf{X}}_{,\beta }]}^{·}\hfill \\ & =& {\mathbf{u}}_{,\alpha \beta }\otimes {\mathbf{X}}_{,\alpha }\otimes {\mathbf{X}}_{,\beta },\hfill \end{array}$$

(23)

by which the ${\dot{\mathbf{g}}}_i$ is transformed to

$$\begin{array}{ccc}\hfill {\dot{\mathbf{g}}}_1& =& ({\mathbf{u}}_{,\alpha \beta }\otimes {\mathbf{X}}_{,\alpha }\otimes {\mathbf{X}}_{,\beta })(\mathbf{L}\otimes \mathbf{L})\hfill \\ & =& {\mathbf{u}}_{,\alpha \beta }{L}_{\alpha }{L}_{\beta }\hfill \\ & & \mathrm{and}\hfill \\ \hfill {\dot{\mathbf{g}}}_2& =& ({\mathbf{u}}_{,\alpha \beta }\otimes {\mathbf{X}}_{,\alpha }\otimes {\mathbf{X}}_{,\beta })(\mathbf{M}\otimes \mathbf{M})\hfill \\ & =& {\mathbf{u}}_{,\alpha \beta }{M}_{\alpha }{M}_{\beta }.\hfill \end{array}$$

(24)

Accordingly, $W{\left(\mathbf{F}\right)}_{\mathbf{F}}·\dot{\mathbf{F}}$, $W{\left({\epsilon }_i\right)}_{{\epsilon }_i}{\dot{\epsilon }}_i$, and $W{\left({\mathbf{g}}_i\right)}_{{\mathbf{g}}_i}·{\dot{\mathbf{g}}}_i$ in Equation (16) are respectively obtained, yielding

$$W{\left(\mathbf{F}\right)}_{\mathbf{F}}·\dot{\mathbf{F}}=\left\{\kappa \mathbf{F}+\mu \mathbf{F}[(\mathbf{F}·\mathbf{F})\mathbf{I}-{\mathbf{F}}^T\mathbf{F}]\right\}·\dot{\mathbf{F}}=\left\{\kappa {\mathit{\chi }}_{,\beta }+\mu [{\mathit{\chi }}_{,\beta }({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\lambda })-{\mathit{\chi }}_{,\alpha }({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta })]\right\}·{\mathbf{u}}_{,\beta },$$

(25)

then

$$\begin{array}{ccc}\hfill \dot{W}{\left({\epsilon }_i\right)}_{fiberextension}& =& {\displaystyle \frac{E_1}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{L}_{\lambda }{L}_{\mu }-1){\mathit{\chi }}_{,\alpha }·{\mathbf{u}}_{,\beta }{L}_{\alpha }{L}_{\beta }+\hfill \\ & & {\displaystyle \frac{E_2}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{M}_{\lambda }{M}_{\mu }-1){\mathit{\chi }}_{,\alpha }·{\mathbf{u}}_{,\beta }{M}_{\alpha }{M}_{\beta }\hfill \\ & =& [{\displaystyle \frac{E_1}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{L}_{\lambda }{L}_{\mu }-1)L_{\alpha }{L}_{\beta }+{\displaystyle \frac{E_2}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{M}_{\lambda }{M}_{\mu }-1)M_{\alpha }{M}_{\beta }]{\mathit{\chi }}_{,\alpha }·{\mathbf{u}}_{,\beta },\hfill \end{array}$$

(26)

and

$$W_{{\mathbf{g}}_i}·{\dot{\mathbf{g}}}_i=C_1{\mathbf{g}}_1·{\dot{\mathbf{g}}}_1+C_2{\mathbf{g}}_2·{\dot{\mathbf{g}}}_2+{\displaystyle \frac{T}{2}}({\mathbf{g}}_1·{\dot{\mathbf{g}}}_2+{\mathbf{g}}_2·{\dot{\mathbf{g}}}_1),$$

(27)

by invoking Equations (20) and (24), Equation (27) explicitly includes the computation of

$$\begin{array}{ccc}\hfill W_{{\mathbf{g}}_1}·{\dot{\mathbf{g}}}_1& =& (C_1{\mathit{\chi }}_{,\mu \lambda }{L}_{\mu }{L}_{\lambda }+{\displaystyle \frac{1}{2}}T{\mathit{\chi }}_{,\mu \lambda }{M}_{\mu }{M}_{\lambda })·{\mathbf{u}}_{,\alpha \beta }{L}_{\alpha }{L}_{\beta }\mathrm{and}\hfill \\ \hfill W_{{\mathbf{g}}_2}·{\dot{\mathbf{g}}}_2& =& (C_2{\mathit{\chi }}_{,\mu \lambda }{M}_{\mu }{M}_{\lambda }+{\displaystyle \frac{1}{2}}T{\mathit{\chi }}_{,\mu \lambda }{L}_{\mu }{L}_{\lambda })·{\mathbf{u}}_{,\alpha \beta }{M}_{\alpha }{M}_{\beta }\hfill \end{array}$$

(28)

Lastly, we configure ${\displaystyle \frac{\dot{J}}{J}}$ via

$$\begin{array}{ccc}\hfill \dot{J}& =& {\mathbf{F}}^{*}·\dot{\mathbf{F}}=J{\mathbf{F}}^{-T}·\dot{\mathbf{F}}=J{\chi }_{\alpha ,\beta }{\mathbf{X}}_{,\alpha }·{\mathbf{u}}_{,\beta },\hfill \\ & \therefore & {\displaystyle \frac{\dot{J}}{J}}={\chi }_{\alpha ,\beta }{\mathbf{X}}_{,\alpha }·{\mathbf{u}}_{,\beta },\hfill \end{array}$$

(29)

where $J=det\left(\mathbf{F}\right)$ is the area dilatation and ${\mathbf{F}}^{*}$ is an adjugate of $\mathbf{F}$.

4.2. In-Plane Equilibrium of FRC

The substitution of Equations (25)–(29) into Equation (15) results in a bilinear form in terms of ${\mathbf{u}}_{,\beta }$ and ${\mathbf{u}}_{,\alpha \beta }$, shown as

$$\begin{array}{ccc}\hfill \dot{E}& =& {\int }_{\omega }\left\{\kappa {\mathit{\chi }}_{,\beta }+\mu [{\mathit{\chi }}_{,\beta }({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\lambda })-{\mathit{\chi }}_{,\alpha }({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta })]\right\}·{\mathbf{u}}_{,\beta }+(W-p){\mathit{\chi }}_{,\beta }·{\mathbf{u}}_{,\beta }\hfill \\ & & [{\displaystyle \frac{E_1}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{L}_{\lambda }{L}_{\mu }-1)L_{\alpha }{L}_{\beta }+{\displaystyle \frac{E_2}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{M}_{\lambda }{M}_{\mu }-1)M_{\alpha }{M}_{\beta }]{\mathit{\chi }}_{,\alpha }·{\mathbf{u}}_{,\beta }+\hfill \\ & & (C_1{\mathit{\chi }}_{,\mu \lambda }{L}_{\mu }{L}_{\lambda }{L}_{\alpha }{L}_{\beta }+C_2{\mathit{\chi }}_{,\mu \lambda }{M}_{\mu }{M}_{\lambda }{M}_{\alpha }{M}_{\beta }+{\displaystyle \frac{T}{2}}{\mathit{\chi }}_{,\mu \lambda }{M}_{\mu }{M}_{\lambda }{L}_{\alpha }{L}_{\beta }+{\displaystyle \frac{T}{2}}{\mathit{\chi }}_{,\mu \lambda }{L}_{\mu }{L}_{\lambda }{M}_{\alpha }{M}_{\beta })·{\mathbf{u}}_{,\alpha \beta }da,\hfill \end{array}$$

(30)

in which ${\mathbf{u}}_{,\beta }$ and ${\mathbf{u}}_{,\alpha \beta }$ respectively represent the strain and strain gradient of FRC, which involves the FRC material displacement $\mathbf{u}$. Then, for the sake of clarity, the treatments in [70,71,72,73,74] are invoked to clarify Equation (30), yielding

$$\dot{E}={\int }_{\omega }({\mathit{\phi }}_{\beta }·{\mathbf{u}}_{,\beta }+{\mathit{\psi }}_{\alpha \beta }·{\mathbf{u}}_{,\alpha \beta })da={\int }_{\omega }({\mathit{\phi }}_{\beta }-{\mathit{\psi }}_{\alpha \beta ,\alpha })·{\mathbf{u}}_{,\beta }da+{\int }_{\partial \omega }{\mathit{\psi }}_{\alpha \beta }·{\mathbf{u}}_{,\beta }{v}_{\alpha }ds,$$

(31)

where

$$\begin{array}{ccc}\hfill {\mathit{\phi }}_{\beta }& =& \kappa {\mathit{\chi }}_{,\beta }+\mu [{\mathit{\chi }}_{,\beta }({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\lambda })-{\mathit{\chi }}_{,\alpha }({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta })]+\hfill \\ & & [{\displaystyle \frac{E_1}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{L}_{\lambda }{L}_{\mu }-1)L_{\alpha }{L}_{\beta }+{\displaystyle \frac{E_2}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{M}_{\lambda }{M}_{\mu }-1)M_{\alpha }{M}_{\beta }]{\mathit{\chi }}_{,\alpha }+(W-p){\mathit{\chi }}_{,\beta }\mathrm{and}\hfill \\ \hfill {\mathit{\psi }}_{\alpha \beta }& =& (C_1{\mathit{\chi }}_{,\mu \lambda }{L}_{\mu }{L}_{\lambda }{L}_{\alpha }{L}_{\beta }+C_2{\mathit{\chi }}_{,\mu \lambda }{M}_{\mu }{M}_{\lambda }{M}_{\alpha }{M}_{\beta }+{\displaystyle \frac{T}{2}}{\mathit{\chi }}_{,\mu \lambda }{M}_{\mu }{M}_{\lambda }{L}_{\alpha }{L}_{\beta }+{\displaystyle \frac{T}{2}}{\mathit{\chi }}_{,\mu \lambda }{L}_{\mu }{L}_{\lambda }{M}_{\alpha }{M}_{\beta }).\hfill \end{array}$$

(32)

Hence, Equation (31) renders the Euler–Lagrange equation follows the relation in the FRC sheet domain $\omega$, i.e.,

$${\mathit{\phi }}_{\beta ,\beta }-{\mathit{\psi }}_{\alpha \beta ,\alpha \beta }=0.$$

(33)

To derive the governing equations of FRC, the substitution of Equation (32) into Equation (31) yields the equilibrium equations in-plane (which involves the deformed material position vector $\mathit{\chi }$), shown as

$$\begin{array}{ccc}\hfill 0& =& \kappa {\mathit{\chi }}_{,\beta \beta }+\mu {[{\mathit{\chi }}_{,\beta }({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\lambda })-{\mathit{\chi }}_{,\alpha }({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta })]}_{,\beta }+\hfill \\ & & {\left\{[{\displaystyle \frac{E_1}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{L}_{\lambda }{L}_{\mu }-1)L_{\alpha }{L}_{\beta }+{\displaystyle \frac{E_2}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{M}_{\lambda }{M}_{\mu }-1)M_{\alpha }{M}_{\beta }]{\mathit{\chi }}_{,\alpha }\right\}}_{,\beta }+\hfill \\ & & {\left[(W-p){\mathit{\chi }}_{,\beta }\right]}_{,\beta }-(C_1{L}_{\mu }{L}_{\lambda }{L}_{\alpha }{L}_{\beta }+C_2{M}_{\mu }{M}_{\lambda }{M}_{\alpha }{M}_{\beta }+{\displaystyle \frac{T}{2}}{M}_{\mu }{M}_{\lambda }{L}_{\alpha }{L}_{\beta }+{\displaystyle \frac{T}{2}}{L}_{\mu }{L}_{\lambda }{M}_{\alpha }{M}_{\beta }){\mathit{\chi }}_{,\mu \lambda \alpha \beta }\hfill \end{array}$$

(34)

on $\omega$. The last term in Equation (31) further indicates the loading conditions on the boundary $\partial \omega$; hence, we reformulate Equation (31) to analyze the boundary conditions for FRC through

$$\begin{array}{ccc}\hfill \dot{E}& =& {\int }_{\omega }({\mathit{\phi }}_{\beta }-{\mathit{\psi }}_{\alpha \beta ,\alpha })·{\mathbf{u}}_{,\beta }da+{\int }_{\partial \omega }{\mathit{\psi }}_{\alpha \beta }·{\mathbf{u}}_{,\beta }{v}_{\alpha }ds\hfill \\ & =& {\int }_{\partial \omega }({\mathit{\phi }}_{\beta }-{\mathit{\psi }}_{\alpha \beta ,\alpha })·\mathbf{u}{v}_{\beta }ds+{\int }_{\partial \omega }{\mathit{\psi }}_{\alpha \beta }·{\mathbf{u}}_{,\beta }{v}_{\alpha }ds-{\int }_{\omega }{({\mathit{\phi }}_{\beta }-{\mathit{\psi }}_{\alpha \beta ,\alpha })}_{,\beta }·\mathbf{u}da,\hfill \end{array}$$

(35)

in which ${\int }_{\partial \omega }({\mathit{\phi }}_{\beta }-{\mathit{\psi }}_{\alpha \beta ,\alpha })·\mathbf{u}{v}_{\beta }ds$ includes stretching effects on the boundaries, while ${\int }_{\partial \omega }{\mathit{\psi }}_{\alpha \beta }·{\mathbf{u}}_{,\beta }{v}_{\alpha }ds$ exhibits flexural effects on the boundaries, indicating the loading conditions of FRC for combined bilateral stretch and in-plane bending moment on boundaries simultaneously.

5. Governing Equations and Simulation

In this section, we derive the governing equations in the form of PDEs and configure the parameters of numerical experiments and simulation approaches, and the numerical simulation results will be illustrated in the following sections. To implement the proposed model, we express a material position in a deformed fiber-reinforced composite domain $\omega$ as

$$\mathit{\chi }={\chi }_{\alpha }{\mathbf{e}}_{\alpha },$$

(36)

where $\alpha$ takes the values in $\left\{1,2\right\}$ expressing the basis for 2-dimensional space in deformed configuration, and ${\chi }_k$ are corresponding components in the 2D coordinate system. For the material position of the initially undeformed fiber composite, $\mathbf{X}=X_{\alpha }{\mathbf{E}}_{\alpha }$, where $X_{\alpha }$ is the components of the referential coordinate, and $\left\{{\mathbf{E}}_{\alpha }\right\}$ is the orthogonal basis configured in the reference coordinate. The substitution of Equation (36) into Equation (34) renders a system of PDEs in terms of ${\chi }_1$, ${\chi }_2$ and p (p is the Lagrange multiplier), shown as

$$\begin{array}{ccc}\hfill 0& =& \kappa {\chi }_{1,\beta \beta }+\mu {[{\chi }_{1,\beta }({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\lambda })-{\chi }_{1,\alpha }({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta })]}_{,\beta }+\hfill \\ & & {\left\{[{\displaystyle \frac{E_1}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{L}_{\lambda }{L}_{\mu }-1)L_{\alpha }{L}_{\beta }+{\displaystyle \frac{E_2}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{M}_{\lambda }{M}_{\mu }-1)M_{\alpha }{M}_{\beta }]{\chi }_{1,\alpha }\right\}}_{,\beta }+\hfill \\ & & {\left[(W-p){\chi }_{1,\beta }\right]}_{,\beta }-(C_1{L}_{\mu }{L}_{\lambda }{L}_{\alpha }{L}_{\beta }+C_2{M}_{\mu }{M}_{\lambda }{M}_{\alpha }{M}_{\beta }+{\displaystyle \frac{T}{2}}{M}_{\mu }{M}_{\lambda }{L}_{\alpha }{L}_{\beta }+{\displaystyle \frac{T}{2}}{L}_{\mu }{L}_{\lambda }{M}_{\alpha }{M}_{\beta }){\chi }_{{}_1,\mu \lambda \alpha \beta }\hfill \end{array}$$

(37)

and

$$\begin{array}{ccc}\hfill 0& =& \kappa {\chi }_{2,\beta \beta }+\mu {[{\chi }_{2,\beta }({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\lambda })-{\chi }_{2,\alpha }({\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta })]}_{,\beta }+\hfill \\ & & {\left\{[{\displaystyle \frac{E_1}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{L}_{\lambda }{L}_{\mu }-1)L_{\alpha }{L}_{\beta }+{\displaystyle \frac{E_2}{2}}({\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\mu }{M}_{\lambda }{M}_{\mu }-1)M_{\alpha }{M}_{\beta }]{\chi }_{2,\alpha }\right\}}_{,\beta }+\hfill \\ & & {\left[(W-p){\chi }_{2,\beta }\right]}_{,\beta }-(C_1{L}_{\mu }{L}_{\lambda }{L}_{\alpha }{L}_{\beta }+C_2{M}_{\mu }{M}_{\lambda }{M}_{\alpha }{M}_{\beta }+{\displaystyle \frac{T}{2}}{M}_{\mu }{M}_{\lambda }{L}_{\alpha }{L}_{\beta }+{\displaystyle \frac{T}{2}}{L}_{\mu }{L}_{\lambda }{M}_{\alpha }{M}_{\beta }){\chi }_{{}_2,\mu \lambda \alpha \beta }\hfill \end{array}$$

(38)

on the $\omega$, where

$$\begin{array}{ccc}\hfill {\chi }_{1,\beta \beta }& =& {\chi }_{1,11}+{\chi }_{1,22},{\chi }_{2,\beta \beta }={\chi }_{2,11}+{\chi }_{2,22},\hfill \\ \hfill {\mathit{\chi }}_{,\lambda }·{\mathit{\chi }}_{,\lambda }& =& {\chi }_{1,\lambda }^2+{\chi }_{2,\lambda }^2\mathrm{and}{\mathit{\chi }}_{,\alpha }·{\mathit{\chi }}_{,\beta }={\chi }_{1,\alpha }{\chi }_{1,\beta }+{\chi }_{2,\alpha }{\chi }_{2,\beta }.\hfill \end{array}$$

(39)

The algebra procedures might not be complex but are tedious; for the sake of brevity, the procedures for explicitly expressing the PDEs are refrained. To demonstrate the continuum model, Equations (37) and (38) are solved numerically and post-processed via FEM packages FEniCS (version 2019.2.0.dey0 [75,76]) using Python language and its high-level Python and C++ interfaces, and the combined boundary conditions for the bilateral extension and the bending moment in Equation (35), as well as the solving method/procedures, can be found in [27,28,29,30] and the Appendix A. It is noteworthy that p is the Lagrange multiplier determining the intrinsic properties of the FRC. Here, we consider that the FRC is initially uniform for the strain energy distribution of FRC; therefore, p is treated as 0 in the simulation.

Specifically, we investigate the performance of fiber reinforcement subjected to a combination of bilateral stretching and bending moments applied to the bilateral edges. We present simulation results for a rectangular fiber-reinforced composite (FRC) with dimensions of 2 × 2 and the following material parameters: $\mu =4$ MPa, $\kappa =1$ MPa, $E_i=6$ MPa, $C_i=10$ MPa, $T_{torsion}=6$ MPa. Bilateral stretches are applied as ${\chi }_{1,11}=T$, and bending moments, denoted as ${\chi }_{2,11}=M$, are imposed on the left and right boundaries of the FRC, respectively.

In the accompanying figures, ${\chi }_1$, ${\chi }_2$ represent the deformed material positions with respect to the initial material positions $X_1$ and $X_2$, respectively. A detailed analysis and discussion of the obtained numerical simulation results are presented in the subsequent sections.

6. Results and Discussion

This section demonstrates the proposed model by illustrating the matrix and network deformations of FRCs. The analysis encompasses the matrix deformation, the Green–Lagrange strain, and fiber unit deformations in response to combined bending and stretching on the bilateral edges. To elucidate the effects of bilateral tension and bending moment, numerical experiments are conducted in the cases of fixing bilateral stretch and bending moment. Emphasis is placed on demonstrating the effects of bilateral stretch and bending moment on the matrix and network deformation of FRCs, and the role of embedded fiber microstructures in modulating the overall mechanical performance of the FRC network. Notably, the deformations of the fiber composite and the fiber meshwork are concurrently illustrated to address potential dislocation phenomena at the interface between the matrix material and the fiber meshwork.

6.1. FRC Deformation

6.1.1. Case 1: Increasing Bending Moments and Constant Bilateral Stretch

Figure 2 depicts the material migration of the FRC subjected to bilateral stretch and bending moments, indicating that the materials are simultaneously deformed in the bilateral and upward directions. Intensified deformations (1.2) are found near the bilateral boundaries due to the presence of bilateral stretches and bending moments applied directly there. Nevertheless, the middle FRC witnesses weak in-plane deformation (around 0) because of surface tension equilibrium.

Figure 3 compares the effects of increasing bending moment (1–2.5 MPa) on the FRC under fixed bilateral stretch (1.5 MPa). The FRC deformations, particularly for the bilateral edges, undergo significant increases due to the increased bilateral bending moment, i.e., the deformation on bilateral edges increases from 0.5 in Figure 3a to 3.06 in Figure 3d. The deformation distribution witnesses dramatic changes inside the domain because of the bending effects, leading to the material migrating upward while bilateral tension invokes deformation horizontally.

The inner FRC undergoes intensified deformation (0.2–0.5) in Figure 3a, resulting from the bilateral stretch dominating FRC deformation over the bending moment. Due to the enhanced bending moment in Figure 3b, the bending-induced compression balances the bilateral stretching, leading to weak deformation inside the FRC (around 0). As the bending moment continues to increase in Figure 3c,d, the FRC deformation increases in the upper area of FRC (from 0 in Figure 3b to 1 in Figure 3c and 1.7 in Figure 3d) due to the dominance of bending-induced deformation.

The theoretical results of Lagrange strain ${\epsilon }_1$ and ${\epsilon }_2$ are illustrated in Figure 4 under increasing bilateral bending moment (1 MPa in (a, b) to 1.5 MPa in (c, d)) and constant bilateral stretch (1.5 MPa). Both Figure 4a,c indicate ${\epsilon }_1$ peaking in the vicinity of bilateral boundaries, and inner FRC exhibits weak ${\epsilon }_1$ (0.625 to 0.765). The ${\epsilon }_1$ is weak (0.62 in (a) and 0.7 in (c)) inside the FRC because the bilateral bending moment induces compressive effects horizontally.

The intensified ${\epsilon }_1$ (1.1 in (a) and 1.7 in (c)) on the bilateral boundaries are invoked by the joint work of bending and stretching effects applied there, particularly for the bilateral bending moments contributing to local tensile effects in the horizontal directions (the bottom boundary of FRC exhibits large stretching strain area in (a) and (c)) which is induced by the bilateral tension and bending-induced tension effects. Further, the large ${\epsilon }_1$ near the bottom boundary may explain the densely distributed cracks near the FRC bottom boundaries when FRC is subjected to bending moment in Figure 4e [77]. The cracks near the bottom might be invoked by the large area of stretching strain near the bottom boundary (Figure 4c).

The longitudinal strain ${\epsilon }_2$ is shown in Figure 4b,d, showcasing the ${\epsilon }_2$ peaking inside the FRC (−0.2 in (b) and 0.255 in (d)), while the ${\epsilon }_2$ in Figure 4b are negative due to the shrinking effects (induced by the bilateral stretch) exceeding the stretching effects by the bilateral bending moments. In this regard, a larger bending moment is expected to result in a positive ${\epsilon }_2$ (0.178–0.255), as shown in Figure 4d, a result of stretching effects induced by bending moment exceeding the shrinking effects by bilateral stretch. It is noteworthy that the upper and bottom boundaries witness negative ${\epsilon }_2$ due to the shrinking effects in there. Such shrinking effects might explain the marked concrete crushing FRC beam crack generation in Figure 4e (top middle area in (e)), which is demonstrated in Figure 5a by the shrinking transverse strain ${\epsilon }_1$.

6.1.2. Case 2: Increasing Bilateral Stretching and Constant Bending

Figure 6 highlights the increasing tensile effects ($T=0.2$ in Figure 6a to $T=2$ MPa in Figure 6d) on the FRC. It is shown that the FRC shrinks to the width of 1.2 in Figure 6a but elongates to 3.9 transversely in Figure 6d. The bilateral boundary width increases from 1 in (a) to 1.5 in (d), mainly induced by the larger bilateral stretch. Further, the FRC deformation has been redistributed inside the domain subjected to increased bilateral tension from (a) to (d), where the material in the FRC undergoes increasing transverse deformation (from 0.12 in (a) to 0.51 in (d)), a result of tensile effects exceeding the bending-induced shrinking effects.

Additionally, the large matrix deformations near the bilateral boundaries (Figure 6) might induce the tensile failure of the matrix, as theoretically and experimentally validated by Figure 6e, where large tensile failures are observed near the bilateral boundaries.

Figure 5 further addresses the tensile effects of bilateral stretch on the Lagrange strain of FRC under increasing bilateral stretches (from $T=0.5$ in (a, b) to $T=0.8$ in (c, d)) and fixed bending moment $M=1.5$ MPa. It is apparent that a larger bilateral tension induces larger ${\epsilon }_1$ across the FRC, peaking from 0.684 in (a) to 0.864 in (c) in the vicinity of bilateral boundaries, and the increased bilateral tension transforms the compressing ${\epsilon }_1$ (negative strain in (a)) to be positive (around 0.1) in (c) in the middle of FRC. Meanwhile, the increased bilateral tension (from 0.5 in (b) to 0.8 in (d)) results in ${\epsilon }_2$ peaking from −0.04 in (b) to 0 in (d), indicating the tension reduces the bending-induced shrinking effects in the longitudinal direction. The large tensile strain area near the bottom boundary (Figure 5a,c) might explain the fiber tensile failure in the vicinity of the bottom boundary of FRC, shown in Figure 5e. Additionally, the shrinking strain near the top boundary in Figure 5a,b,d might further explain the compression failure in Figure 5e where the compression-induced fiber failure is marked near the top boundary.

6.2. Deformation of Plain Woven Network

The fiber network reinforcement, bonded with the matrix, is believed to determine the main strength of FRC; therefore, it is crucial to examine the fiber and its network deformation, particularly for the analysis of matrix-network dislocation. Hence, the fiber network deformation and kinematic differences between the matrix deformation and meshwork are investigated in this section. Further, regarding the microstructural features within the matrix material significantly affecting the overall mechanical responses of FRC [39,40,41,79], it is essential to clarify the role of fiber units in determining the overall deformation of the embedded meshwork. Therefore, particular attention should be emphasized on analyzing the kinematics of the fiber units and their impact on the overall deformation of the fiber meshwork. In this section, we theoretically investigate fiber unit extension and flexure; then, we discuss the effects of fiber unit deformations on the overall deformation of single fiber and fiber meshwork.

The extension of fiber units (i.e, $\mathbf{L}$ and $\mathbf{M}$) is illustrated in Figure 7 by computing $\mathbf{FL}$ and $\mathbf{FM}$, where the combination of bilateral stretch and bending moment tension leads to the shrinking (the deformed length of fiber unit is less than 1) and stretching effects (the deformed length of fiber unit is larger than 1) on the fiber units. Figure 7a indicates that the fiber units $\mathbf{L}$ are elongated across the domain (deformed $\mathbf{L}$ larger than 1), particularly near the bilateral boundaries and bottom boundary due to the bilateral stretch applied on the bilateral boundaries, and the bilateral bending moment has stretching effects on the bottom boundary. Further, the theoretical results of transverse fiber unit extension might be validated by Figure 7c, where the fibers are undergoing tensile failure near the bilateral boundaries as indicated by Figure 7c (top) theoretically and Figure 7 (bottom) experimentally.

It can be concluded that the bilateral tension elongates the FRC in the transverse direction, and the bilateral bending moment particularly exacerbates the stretching effects on the bottom boundary. Hence, in the horizontal direction, it is expected that the transverse fibers located near the bottom boundary exhibit a larger extension, while the fibers near the top boundary exhibit relatively less extension.

Besides the stretching effects, the bending moment induces the longitudinal extension of fiber as shown in Figure 7b, showing the $\mathbf{M}$ is stretched to 1.2 inside the FRC while the upper and bottom boundaries undergo shrinkage (deformed $\mathbf{M}$ length is less than 1) because the bending moment induces shrinking effects near the upper and bottom boundaries in the longitudinal direction while stretching effect is invoked inside the domain. Thus, due to more stretched fiber units located near the central area of FRC, the longitudinal fibers there are expected to exhibit intensified extension inside the FRC.

To verify the expectations of fiber deformation deduced from the fiber unit deformation in Figure 7, the elongations of transverse fibers are illustrated in Figure 8 by sampling five fibers with different locations (Figure 8a). Figure 8b describes the effects of increasing bilateral bending moment on the transverse fibers, where fiber 5 near the bottom boundary showcases the largest extension due to the largest bending-induced stretching effects there (as validated by the more stretched fiber units near the bottom boundary in Figure 7a), while the fiber extension decreases as the fibers are distanced from the bottom boundary (fibers 1–4) due to less stretched fiber units in there (Figure 7a). It is noteworthy that fiber 1 (near the top boundary) exhibits intensified extension in comparison to fiber 2, as a larger bending moment leads to a larger curvature near the top boundary that may exceed the fiber curvature inside the FRC. These results suggest that the fiber units’ extension determines the overall elongation of the single continuous fiber.

Additionally, Figure 8c shows the effects of increasing bilateral stretch on the transverse fibers, which exhibit consistent tensile effects on the sampled fibers (all the sampled transverse fibers exhibit uniform stretching effects). It should be noted that fiber 5 (near the bottom boundary) exhibits the largest extension due to the intensified bending-induced tensile effects therein.

To analyze the role of fiber unit deformation (Figure 7b) in determining the longitudinal fiber deformation of FRC, the elongation of the longitudinal fiber is investigated in Figure 9 by sampling fibers with different locations (Figure 9a). Figure 9b highlights the increasing bending moment leading to intensified extension of fibers 1 to 5, and fiber 5 (near the central area) exhibits the largest extension corresponding to the more stretched longitudinal fiber units (Figure 7b) and stretching strain there (Figure 4b,d and Figure 5b,d). It is observed that the longitudinal fibers 1–4 showcase gradually weakened extension due to the reduced amount of stretched fiber units in the longitudinal direction (see Figure 7b; the amount of the stretched longitudinal unit is reduced when it is approaching the bilateral boundaries) as they are distanced from the middle FRC.

Figure 9c exhibits the bilateral stretching on the longitudinal fibers in FRC; the increasing bilateral tensions consistently lead to larger extension (all the longitudinal fibers are stretched in the analogous manner/trend). Nevertheless, the bilateral tension might merely stretch the transverse fibers; therefore, it has no stretching effects on the longitudinal fibers. It is deduced that the larger bilateral tension indirectly enhances the bending effects due to the increased deformation in the transverse direction leading to the elongation of the bending arm; meanwhile, the bending moment stretches the longitudinal fiber units inside the domain (Figure 7b, particularly in the middle of FRC), hence leading to larger extension in the longitudinal direction of fibers.

To investigate the flexural properties of fibers, the curvature of the fiber unit is presented in Figure 10 and expressed by geodesic curvatures ${\mathbf{g}}_1$ and ${\mathbf{g}}_2$. In Figure 10a, the curved upper and bottom boundaries of FRC apparently indicate the presence of flexural fiber units ${\mathbf{g}}_1$, where the domain mainly performs ${\mathbf{g}}_1=2$, while ${\mathbf{g}}_1$ peaks around the middle of bottom boundary due to the partially fixed bottom boundary interacting with the ground, as illustrated by Figure 10c. The ${\mathbf{g}}_1$ is weak near the bilateral ends of the bottom boundary, a result induced by the bilateral extension effects and bending-induced stretching effects (indicated by the stretched $\mathbf{L}$ in Figure 7a). The curvature of $\mathbf{M}$(${\mathbf{g}}_2$) in Figure 10b indicates that the longitudinal fiber units mainly stay straight (observed by the weak ${\mathbf{g}}_2$) inside the FRC due to the stretching effects in the longitudinal direction in the central FRC (see, Figure 7b), while the $\mathbf{M}$ is locally curved in the vicinity of upper and bottom boundaries due to bending effects of bilateral bending moment near the top and bottom boundaries. It is noteworthy that ${\mathbf{g}}_2$ peaks in the middle area of the bottom edge (like ${\mathbf{g}}_1$) due to the bottom boundary interacting with the anchored ground, as shown in Figure 10c.

As illustrated by the transverse and longitudinal fiber extension and the fiber unit’s deformation, it is necessary to investigate the fiber mesh deformation and discuss how the extension/shrinkage (Figure 7) and flexural fiber units (Figure 8) determine the overall deformation of FRC. In this regard, the undeformed and deformed fiber networks are respectively depicted in Figure 11a,b.

In comparison to Figure 11a, Figure 11b shows that the mesh grids are bidirectionally elongated/shrunk across the FRC as indicated by the L and M deformation in Figure 7. It is observed that the meshwork shrinks near the upper and downward boundaries because of the bending effects induced by the bilateral bending moments, which has been validated by the shrunk fiber units in Figure 7b and negative strain in Figure 5b,d. Figure 11b further showcases that the transverse fibers are curved in FRC, as indicated by the curved fiber units ${\mathbf{g}}_1$ in Figure 10a. Meanwhile, the longitudinal fibers remain straight inside the domain as demonstrated by the weak ${\mathbf{g}}_2$ in Figure 10b. The densely distributed fibers near the top and bottom boundaries suggest the presence of intensified compressing effects near the top and bottom boundaries, leading to the curved and shrunk longitudinal fibers in the vicinity of the top and bottom boundaries (indicated by Figure 5b,d and Figure 7b,d). It is noteworthy that Figure 11b shows that the adjacent fibers in the single grid of meshwork are mutually tilted, suggesting the presence of fiber torsion as demonstrated in the FRC strain energy potential (Equation (12)). Further, the fiber grids are enlarged in the FRC, which can be validated by the experimental results of FRC subjected to in-plane bending in Figure 11c.

Additionally, Figure 11d indicates the underlying dislocation between the FRC matrix and network due to the kinematic difference between matrix and fiber reinforcement: the meshwork is significantly shrunk toward the upper and lower boundaries, while the matrix therein exhibits non-uniform deformations in the bilateral direction (see, Figure 2). The central FRC performs weak material displacements, yet the mesh grids there are dramatically elongated/tilted. Such dislocation might be alleviated by using the refined network structure to reduce the kinematic differences, as discussed in the following section.

6.3. The Pantographic Network Deformation of FRC

Due to the discussed numerical cases of FRC reinforced with plain network indicating the potential dislocation between matrix and plain network, it is necessary to consider alternative network structures for the reduction of matrix-network dislocation. Therefore, instead of using the plain network structure, we illustrate the pantographic structures for the network reinforcement in FRC. Pantographic structures are believed to be metamaterial structures that are capable of characterizing engineered materials for innovative material production [80] and have aroused considerable research attention in sections like solar roof panels, foldable stairs, civil engineering [81], “multiscale” mechanical systems [82], etc., due to the high flexibility and adaptivity, load-bearing capacity, toughness, light weight, and reliability ([81] and references therein), which particularly exhibits its iconic feature for constitutive modeling [82]. More importantly, the pantographic structures are expected to alleviate the potential matrix-reinforcement dislocation due to the flexible intersection joint and the kinematics coincidence between the matrix material and network intersection points. Hence, we apply the pantographic structures to demonstrate the mechanics of pantographic structures via bilateral extension and flexure testing.

Specifically, pantographic structures, known as scissor-like element structures, are typically composed of multiple parallel arrays of fibers that are orthogonally and mutually arranged with their intersections connected by pivots [82], as shown in Figure 12. In this regard, we investigate the case of a pantographic structure network (Figure 13a, see also [18]) subjected to the combination of bilateral stretch and bending moment.

Figure 13 compares the network deformation of the pantographic structure subjected to bilateral extension and bending, where the initial adjacent fibers are orthogonally distributed (Figure 13a), while the fiber mesh grids are sheared due to the presence of bilateral tension and bending moments (Figure 13b). Similar to the network deformation discussed in the previous section, the fiber network is shrunk near the upper and lower boundaries due to the flexure-induced shrinking effects, and the mesh grids are tilted or enlarged inside the domain because of the significant shearing/bending effects.

To investigate the bilateral bending and stretching effects on the featured shearing phenomenon illustrated in Figure 13b, the fiber intersection angles are investigated by respectively applying increased bilateral bending moment (Figure 14) and bilateral stretch (Figure 15) on the FRC pantographic network. Figure 14 indicates that the increased bending moment leads to a larger intersection angle peaking from 60 degrees in Figure 14a to 80 degrees in Figure 14b, while the central intersection angle remains at 60 degrees. Due to the width of the network being maintained at around 2 in Figure 14 (the initial width is 2; hence, the pantographic network is weakly stretched in the transverse direction), the changes of intersection angles near the bilateral boundaries are dominated by the bilateral bending effects, where the intersection angles therein are enhanced from 30 degrees in Figure 14a to 50 degrees in Figure 14b, as verified by the large ${\epsilon }_2$ near the bilateral boundaries (Figure 4b,d and Figure 5b,d).

It is noted that the larger intersection angle means weak shear strain, and it is indicated that the upper and downward boundaries suffer the largest shearing effects. Additionally, the large shear strain near the FRC boundaries might explain the FRC delamination subjected to bending and tensile tests [78].

Figure 15 further investigates the stretching effects on the intersection angle, and it is observed that a larger bilateral stretch leads to a larger intersection angle peaking from 50 degrees in Figure 15a to 60 degrees in Figure 15b near the bilateral boundaries. Due to the bilateral extension transversely stretching the network width from 2 (Figure 15a) to 3.4 (Figure 15b), the bending effects are enhanced by the larger width (the bending arm is enlarged), resulting in larger intersection angles near the bilateral boundaries (similar to Figure 14, near the bilateral boundaries). Therefore, bilateral stretch indirectly leads to increased intersection angles near the bilateral boundaries. Nevertheless, the bilateral stretching effects reduce the intersection angles in the middle of the FRC network (reducing from 60 degrees in Figure 15a to 50 degrees in Figure 15b) due to the large tension significantly shearing the mesh grids in the transverse direction. It can be deduced that a larger bilateral stretch induces decreased intersection angles with a larger shear strain in the middle of FRC. In addition, the large shear strain near the bilateral boundaries and in the central FRC might invoke matrix and fiber tensile failure, and the delamination of FRC [78].

7. Conclusions

This study demonstrates a continuum model to investigate the FRCs subjected to the combined loading of in-plane biaxial extension and flexure. The model incorporates a strain energy density function comprising the Mooney–Rivlin hyperelastic model for the matrix material and a bidirectional fiber meshwork for the reinforcement. The strain energy contribution of the fibers is computed by considering their kinematics, involving the first- and second-order deformation gradients.

Variational principles are utilized to derive the Euler–Lagrange equations and elucidate the boundary conditions associated with combined bilateral stretch and bending moments. Subsequently, the equilibrium equations are projected onto the Cartesian coordinate system, resulting in a system of PDEs. These PDEs are numerically solved using a custom-built finite element method (FEM) procedure. The FEM implementation enables the visualization and analysis of in-plane material displacement, Lagrange strain, shear strain, and network deformations for both plain bidirectional and pantographic reinforcement structures.

Theoretical results reveal that the matrix material undergoes simultaneous migration in both the biaxial and longitudinal directions, dictated by the interaction between bilateral stretching and bending. Notably, the Lagrange strain ${\epsilon }_1$ exhibits peaking values near the bilateral boundaries, primarily driven by the dominant stretching effects in those areas. Conversely, ${\epsilon }_2$ is negative near the upper and lower boundaries due to shrinking effects. Increasing the bending moment induces a transition from negative to positive strain ${\epsilon }_2$ inside the FRC.

For fiber network deformation, the transverse fiber units experience stretching across the entire domain, while the longitudinal fiber units primarily elongate within the interior of the domain, yet undergo shrinkage near the upper and lower boundaries. Additionally, bending effects induce fiber curvature in the transverse directions throughout the FRC, whereas the longitudinal fibers predominantly exhibit curvature near the top and bottom boundaries. An analysis of the kinematic discrepancies between the matrix and the fiber network suggests the presence of dislocations. More importantly, the illustrations of fiber network and individual fiber deformations are presented to understand how fiber unit deformation influences the overall deformation behavior of the FRC.

To reduce the dislocation in the case of plain network reinforcement, we investigate the mechanics of FRCs reinforced with pantographic structures, from which the bilateral extension and flexure tests are conducted. The results indicate that bending effects lead to an increase in intersection angles within the interior of the domain and near the bilateral boundaries with weak shearing strains. Further, the increased bilateral stretch indirectly induces a larger intersection angle near the bilateral boundaries by enhancing the bending arm, while the reduced intersection angle is invoked in the central region of the FRC.

Significantly, the validity of the continuum model is validated by invoking the experimental/theoretical results for comparison, where the observed high shear and bilateral strains near the bottom boundary and/or within the FRC’s interior provide a referential explanation for crack generation and delamination phenomena in practical applications.