Functionally graded material (FGM) plates are widely used in various engineering structures owing to their tailor-made mechanical properties, whereas cracked homogeneous plates constitute a canonical setting in fracture mechanics analysis. These two classes of problems respectively embody material non-uniformity and geometric discontinuity, thereby imposing more stringent requirements on numerical methods in terms of high-order field continuity and accurate defect representation. Based on the classical Kirchhoff–Love plate theory, a numerical manifold method (MLS-NMM) incorporating moving least squares (MLS) interpolation is developed for bending analysis of FGM plates and fracture simulation of homogeneous plates with defects. The method constructs an H2-regular approximation with high-order continuous weighting functions and, combined with the separation of mathematical and physical covers, establishes a unified framework that accurately handles material gradients and cracks without mesh reconstruction. For the crack tip, a singular physical cover incorporating the Williams asymptotic field is introduced to achieve local enrichment, enabling the natural capture of displacement discontinuity and stress singularity. Stress intensity factors are extracted using the interaction integral method, and the dimensionless J-integral shows a maximum relative error below 1.2% compared with the reference solution. Numerical results indicate that MLS-NMM exhibits excellent convergence performance: using 676 mathematical nodes, the nondimensional central deflection of both FGM and homogeneous plates agrees with reference solutions with a maximum relative error below 0.81%, and no shear locking occurs. A systematic analysis reveals that for a simply supported on all four edges (SSSS)FGM square plate with a/h=10, the nondimensional central deflection increases by 212% as the gradient index nrises from 0 to 5. For a homogeneous plate containing a central crack with c/a=0.6, the nondimensional central deflection increases by approximately 46% compared with the intact plate. Under weak boundary constraints (e.g., SFSF), the deformation is markedly amplified, with the deflection reaching more than three times that under strong constraints (SCSC). The proposed method provides an efficient, reconstruction-free numerical tool for high-accuracy bending and fracture analyses of FGM and cracked thin-plate structures.
Functionally graded materials (FGM) are a class of non-homogeneous composites formed by combining constituents such as metals and ceramics in prescribed proportions, whose properties vary continuously and smoothly in space—typically along the thickness direction [1,2]. The ceramic phase imparts excellent high-temperature resistance and corrosion durability, whereas the metal phase provides desirable strength, fracture toughness, and electrical conductivity. The synergistic integration of these phases enables FGM to be widely used in aerospace, nuclear engineering, civil engineering, and semiconductor applications [3,4].
Over the past three decades, the mechanical behavior of FGM plates has received considerable attention. Owing to the continuous transition of material properties, FGM plates can effectively mitigate thermal stresses and stress concentrations. Under transverse loading, their bending response is significantly influenced by the gradient distribution, boundary conditions, and geometric dimensions, leading to complex mechanical characteristics. A thorough investigation of the bending behavior of FGM plates not only helps elucidate the mechanisms through which material non-uniformity affects stiffness and deformation, but also provides theoretical support for the design and optimization of high-performance structures [5–8].
For FGM structural components with complex geometries, numerical modeling has emerged as an efficient and reliable analytical approach. The successful implementation of such computational methods relies critically on sound theoretical frameworks. In the analysis of bending behavior of FGM plates, the two most widely used plate theories are the Kirchhoff–Love theory based on the thin-plate assumption [9,10] and the Reissner-Mindlin theory, which accounts for shear deformation effects [11,12]. Commonly used numerical methods include the finite element method (FEM) [13–16], the element-free Galerkin method (EFG) [17,18], the boundary element method (BEM) [19–22], other meshfree methods [23–26], the coupled finite element and element-free Galerkin method (FE-EFG) [27–29], the extended finite element method (XFEM) [30–32], and extended isogeometric analysis (XIGA) [33–35]. In recent years, the aforementioned numerical methods have been extensively employed in both academic research and engineering analyses to address the bending problems of FGM plates.
Thai et al. [36] established a classical plate theory (CPT) framework based on isogeometric analysis (IGA) to evaluate the bending behavior of FGM plates. Reddy et al. [37] conducted a systematic analysis of the static bending behavior of circular and annular FGM plates under axisymmetric tensile loads using the first-order shear deformation theory (FSDT). Zenkour [38] analyzed the static mechanical behavior of FGM rectangular plates under transverse loading based on a generalized shear deformation theory. Kulkarni et al. [39] developed an improved shear deformation model based on cotangent inverse trigonometric functions and applied it to the bending analysis of FGM plates. Demirhan and Taskin [40] systematically investigated the bending response of porous FGM plates under static loading based on a four-variable plate theory.
The above studies focus primarily on the static bending behavior of intact or porous FGM plates. Meanwhile, the mechanical modeling of plate and shell structures is evolving toward higher-order formulations, material non-uniformity, and multiphysics coupling. In recent years, for shell structures characterized by material gradation, porosity distribution, or orthotropic heterogeneity, higher-order shear deformation theory (HSDT) has been widely employed to accurately capture their mechanical response [41]. Existing studies have systematically investigated the post-buckling behavior of porous shells under non-uniform edge loading [42], the evolution of vibration modes in nonlinear oscillations [43], as well as the influence of spatially varying porosity on natural frequencies [44] and free vibration responses [45]. Although these works mainly address stability or dynamic problems, they consistently demonstrate that material non-uniformity has a pronounced effect on the displacement field and its higher-order derivatives (such as curvature and bending moments). They further indicate that HSDT plays a crucial role in accurately capturing these effects.
For thin plate structures, whether in static bending or dynamic response, the C1 continuity required by Kirchhoff–Love theory remains fundamental to achieving high-fidelity modeling. This further underscores the necessity of developing a unified numerical framework that inherently satisfies higher-order continuity, avoids artificial enrichment, and can flexibly accommodate both material and geometric non-uniformities.
This requirement becomes particularly prominent in the bending analysis of cracked structures. Knowles and Wang [46] employed the Reissner plate theory accounting for transverse shear deformation to evaluate the stress field distribution at crack tips in thin plates. The results showed that the obtained stress state at the crack tip is in excellent agreement with the predictions of classical plane elasticity theory. Tanaka et al. [47,48] proposed a meshfree numerical framework based on the reproducing kernel particle method for analyzing the moment intensity factors in Mindlin-Reissner plates, and adopted a node-based integration technique within the J-integral approach for their numerical evaluation. Bhardwaj et al. [49,50] combined FSDT with XIGA to investigate the fracture behavior of cracked homogeneous plates under various loading and boundary conditions. Nguyen-Thanh et al. [51] developed a computational model by combining XIGA with Kirchhoff–Love shell theory to simulate through-thickness cracks in thin shell structures.
However, existing methods still face significant challenges when dealing with complex crack topologies or multiphysics coupling. FEM requires frequent remeshing. XFEM avoids remeshing but still relies on a background mesh and involves complex integration. IGA lacks sufficient flexibility for modeling complex geometries [52,53]. Traditional meshfree methods also suffer from difficulties in imposing boundary conditions and from shape functions that do not possess the Kronecker delta property. Moreover, to ensure accuracy, most meshfree methods (such as EFG and RPIM) require dense node distributions or enlarged influence domains [54–57]. This is especially true when solving fourth-order thin plate problems, where the high continuity requirements lead to many redundant nodes. As a result, the system matrices become highly dense, computational costs increase sharply, and overall efficiency decreases. In contrast, the Numerical Manifold Method (NMM) offers high-order continuity, flexible construction of local approximation spaces, and adaptive meshing features. This makes NMM a more promising numerical framework for high-precision bending analysis of FGM and cracked homogeneous thin plates.
The NMM was proposed by Dr. Shi [58], and its core lay in a dual-cover system composed of mathematical patches and physical patches. This framework enabled a unified treatment of both continuous and discontinuous mechanical problems and exhibited significant advantages in modeling cracks, material interfaces, and strongly singular boundary conditions. In recent years, NMM was extensively investigated and applied in a wide range of fields [59–62]. Zheng and Xu [63] proposed modeling strategies for curved crack paths together with refined integration schemes in the crack tip region, which significantly improved the computational accuracy of NMM. Guo and Zheng [64] further extended high-order NMM to shell structure analysis and, based on the Naghdi shell model, effectively suppressed membrane locking and shear locking in thin-shell problems. To reduce the dependence on mesh generation and improve preprocessing efficiency, Zheng et al. [65] developed the meshfree numerical manifold method (MLS-NMM), in which moving least squares (MLS) approximations were employed as weight functions to replace conventional shape functions. This approach not only enhanced interpolation accuracy and alleviated shear locking, but also effectively avoided linear dependence issues in high-order models, demonstrating excellent numerical stability and accuracy in thin plate bending analyses [66].
On this basis, the MLS-NMM framework proposed in this study deeply integrates MLS approximation with the dual-cover mechanism of NMM, and for the first time realizes a unified enrichment-free treatment that simultaneously satisfies C1 continuity and enables accurate modeling of crack singularities. Its core innovations are reflected in the following three aspects:
Simultaneous satisfaction of C1 continuity and crack singularity modeling without explicit enrichment. By constructing an MLS-based partition of unity with a quadratic complete basis over the NMM mathematical patches, the global deflection field naturally possesses H2 regularity, thereby eliminating the need for mixed variational formulations or additional constraints that are commonly introduced to enforce continuity requirements.
Accurate representation of crack-tip singularities through locally enhanced basis functions. In the singular physical patches generated by crack-induced cutting, displacement singular bases of order r3/2, derived directly from the Williams asymptotic solution, are embedded locally, without introducing Heaviside functions or global enrichment terms.
Seamless integration of displacement discontinuities and high-order continuity. Benefiting from the automatic cutting mechanism of the dual-cover system, displacement jumps across cracks are naturally represented by the independent degrees of freedom associated with different physical patches, while C1 smoothness is preserved in continuous regions.
To the authors’ best knowledge, this study is the first to apply MLS-NMM within a C1-continuous framework to achieve a unified modeling and analysis of the bending behavior of both FGM thin plates and cracked homogeneous thin plates. As summarized in Table 1, compared with mainstream approaches such as EFG, XFEM, and XIGA, MLS-NMM is currently the first numerical method that simultaneously achieves C1 continuity, high geometric modeling flexibility, and sparsity of the system matrices without introducing any explicit enrichment functions, thereby providing a new paradigm for high-precision simulation of complex thin-plate structures.
Comparison of key characteristics among different numerical methods.
Method
Continuity
Artificial Enrichment
Modeling Flexibility
Matrix Sparsity
EFG/RPIM
C0
Yes
Low
Dense
XFEM
C0
Yes
Medium
Dense
XIGA
C1
Yes
Medium to High
Moderate
MLS-NMM
C1
No
High
Sparse
Note: The comparison is based on information from references [4,32,50,65].
The organization of this paper is as follows: Section 2 introduces the fundamental framework of plate theory and derives the weak form corresponding to bending problems of FGM plates. Section 3 systematically presents the basic theory of MLS-NMM, including the construction of shape functions, the implementation mechanism of the dual coverage system, and special treatments for the crack tip region in cracked plates. Section 4 provides the discrete governing equations and the numerical implementation procedure for plate-bending analysis within the proposed framework. Section 5 derives detailed formulas for calculating stress intensity factors (SIF) to evaluate the mechanical response near crack tips. Section 6 validates the accuracy, stability, and applicability of the proposed MLS-NMM framework by comparing numerical results from a series of examples with reference solutions reported in the literature for bending of FGM plates and fracture-related problems in cracked homogeneous plates.
Governing Equation and Weak Form for Bending of FGM Plates
The physical properties of FGM plates vary continuously through the thickness, with the elastic modulus typically described by a power-law function. Based on classical plate theory and neglecting transverse shear deformation, in-plane displacements are characterized by the mid-surface deflection and its spatial derivatives. Accordingly, the geometric relationships between strain and deflection, as well as the corresponding stress-strain constitutive relations, are established. Furthermore, the weak form governing equations for bending problems of FGM plates are constructed using the Galerkin variational method. Boundary conditions such as simply supported and clamped edges are numerically enforced through the penalty method, enabling effective and strong imposition of constraints.
Functionally Graded Materials
Geometric configuration and notation of the FGM plate are shown in Fig. 1. Here, Ω denotes the mid-surface domain of the plate, h is the plate thickness, and Γ represents the mid-surface boundary. The thickness coordinate z is measured from the mid-surface as the origin, with the positive direction downward.
Geometric configuration of an FGM plate.
Assuming the bottom surface of the plate is made of pure metal and the top surface is made of pure ceramic, the material composition transitions continuously along the thickness direction. The volume fraction of the ceramic phase is defined as:
Vc(z)=(12+zh)n
The volume fraction of the metal phase is given by Vm=1−Vc. Here, n≥0 is the gradient index, which controls the spatial distribution of the material components:
When n→0, Vc≈1, and the plate tends to be ceramic-rich overall;
When n→∞, Vc≈0 (except at the top surface), and the plate tends to be metal-rich;
When n=1, the composition transitions linearly.
The Young’s modulus E(z) follows a power-law distribution based on the mixture ratio:
E(z)=Em+(Ec−Em)(12+zh)nhere, E(c) and E(z) denote the Young’s moduli of the ceramic and metal phases, respectively.
This study considers only the bending behavior under purely mechanical loads, completely neglecting thermal effects. However, the employed MLS-NMM framework possesses good structural extensibility and can be further extended in the future to incorporate thermo-mechanical coupling effects, for example, by introducing thermal bending moments caused by temperature fields and gradients in the material’s coefficient of thermal expansion. Such extensions will be addressed in future work.
Variation of ceramic phase volume fraction along the thickness direction under different gradient indices n is shown in Fig. 2. It clearly illustrates how the material composition is continuously controlled by the gradient parameter: smaller values of n result in the plate exhibiting more ceramic-like properties, while larger values make the material closer to metal characteristics.
Volume fraction variation along thickness of an FGM plate.
In this study, two types of FGM plates made of Al/ZrO2 and Al/Al2O3 are considered, and their material parameters are listed in Table 2.
Properties of functionally graded material components.
Material
Properties
E (GPa)
ν
Aluminum (Al)
70
0.3
Zirconia (ZrO2)
151
0.3
Alumina (Al2O3)
380
0.3
Basic Framework of the Kirchhoff–Love Plate Theory
To analyze the bending response of FGM thin plates under transverse loading and the fracture behavior of cracked homogeneous plates, this study employs the classical Kirchhoff–Love plate theory. This theory is suitable for thin plates whose thickness is much smaller than the in-plane characteristic dimensions and neglects the effects of transverse shear deformation.
In this theory, the three-dimensional displacement field at any point within the plate is fully determined by the transverse deflection w(x,y) of the mid-surface. Assuming that straight normals remain straight and perpendicular to the deformed mid-surface, the displacement components can be expressed as:
ux(x,y,z)=−z∂w∂x,uy(x,y,z)=−z∂w∂y,uz(x,y,z)=w(x,y)here, z∈[−h2,h2] denotes the thickness coordinate. Thus, the system’s generalized degree of freedom is the scalar field w(x,y).
The corresponding linear strain–displacement relations include only bending-induced normal and shear strains, with the nonzero components given by:
εxx=−z∂2w∂x2,εyy=−z∂2w∂y2,εxy=−z∂2w∂x∂y
By introducing the curvature–twist vector κ, the above relations can be expressed compactly as:
ε(x,y,z)=−zκ,κ=[κxκyκxy]=−[∂2w∂x2∂2w∂y22∂2w∂x∂y]=Lwhere, L is a second-order differential operator, whose transpose is given by:
LT =(−∂2∂x2−∂2∂y2−2∂2∂x∂y)
For FGM plates, the elastic modulus E(z) varies continuously through the thickness, while Poisson’s ratio ν is considered constant. The constitutive matrix is given by:
D0(z)=E(z)1−ν2[1ν0ν1000(1−ν)/2]
The bending–twisting moment vector per unit width, M=[Mx,My,Mxy]T , is related to the curvature–twisting vector κ through the constitutive equation:
M=Dκwhere D is positive symmetric definite, reading
D=∫−h/2h/2z2D0(z)dz
Under the transverse distributed load q(x,y), the equilibrium equation is derived from the principle of virtual work and its strong form is given by:
LTDLw=q(x,y)inΩ
The boundary ∂Ω is divided into three mutually disjoint subsets: clamped boundary ΓC, simply supported boundary ΓS, and free boundary ΓF, satisfying:
∂Ω=ΓS∪ΓC∪ΓF
On the clamped boundary ΓC,
w=0,wn≡∂w∂n=0
On the simply supported boundary ΓS,
w=0,Mn=0
On the free boundary ΓF,
Mn=0,Qn+∂Mns∂s=0here, nT≡(nx,ny) denotes the unit outward normal vector, and sT≡(sx,sy) is the unit tangent vector obtained by rotating n counterclockwise. Mn, Mns, and Qn represent the normal bending moment, twisting moment, and effective shear force, respectively, all of which are composed of second and higher-order derivatives of w. In the Galerkin weak form, natural boundary conditions do not require explicit enforcement, as their effects are inherently incorporated through the variational formulation.
In the subsequent numerical examples, simply supported boundary conditions on all four edges (SSSS) and clamped boundary conditions on all four edges (CCCC) are imposed by setting Γ=ΓS or Γ=ΓC, respectively, strictly following the unified theoretical framework described above.
Weak Formulation of FGM Plate Bending Problem
The bending problem of the FGM plate takes the deflection w as the primary variable. Its governing equation is given by Eq. (10). The corresponding Galerkin weak form can be expressed as:
∫Ω(δκ)T DκdΩ−∫ΩqδwdΩ=0here, w satisfies the essential boundary conditions, that is, w=0 on ΓS and ΓC; and ∂w∂n=0 on ΓC.
In the NMM, high interpolation accuracy can be achieved by flexibly placing mathematical covers without strictly aligning them with the plate mid-plane boundary, significantly enhancing geometric adaptability and modeling flexibility. However, this unstructured covering approach poses challenges for imposing essential boundary conditions. To address this, the present study adopts the penalty method to handle boundary constraints, thereby simplifying the numerical implementation. Therefore, Eq. (15) can be expressed as:
∫Ω(δκ)T DκdΩ+∫ΓC∪ΓSαdwδwdΩ+∫ΓCαkwnδwndΩ−∫ΩqδwdΩ=0where αd and αk denote the specified penalty parameters. Typically, these parameters must be chosen large enough to ensure the effectiveness of the boundary constraints. At the same time, to prevent excessive ill-conditioning of the stiffness matrix, the penalty parameters are usually taken as 105 to 108 times Young’s modulus of the material. This range has been validated as providing a good compromise between accuracy and numerical stability [67].
In all bending analysis examples presented in this paper, the penalty parameter is uniformly set to 106. This value is based on preliminary convergence tests: when the penalty parameter varies between 105 and 108, the relative change in the dimensionless central deflection remains below 0.3%, indicating that the essential boundary conditions are adequately enforced. Meanwhile, the condition number of the system stiffness matrix stays around the order of 1010 at a penalty parameter of 106, without causing numerical difficulties or abnormal oscillations in the results. This demonstrates that this choice ensures both displacement accuracy and good numerical stability.
Basic Principles of MLS-NMM and Local Approximations for Crack Tips
Unlike the traditional finite element (FE-based) NMM, which uses meshes to construct mathematical covers [68], Zheng et al. [65] proposed a mesh-free strategy for constructing mathematical covers: by introducing the influence domain of MLS approximation nodes as the basic unit of mathematical covers, and directly using the MLS shape functions as weight functions on the mathematical patches. This method eliminates dependence on meshes and significantly enhances modeling flexibility and geometric adaptability. The MLS approximation offers advantages such as high approximation accuracy, good boundary treatment capability, and a simple formulation [69], making it particularly suitable for numerical simulations involving irregular node distributions and complex domains. Moreover, this method belongs to the class of partition-of-unity methods, which allows the introduction of enrichment functions—capable of capturing local solution features—into the physical patches. This effectively enhances the representational capacity of the approximation space, significantly improving both numerical accuracy and geometric adaptability.
Crack Treatment under Dual Coverage
Discrete points, also known as mathematical nodes, are distributed over the mid-plane Ω of the plate or in the surrounding region near its boundary, with a total number denoted by m. The influence domain of each MLS node is taken as a mathematical patch, denoted by Mi, i = 1, 2, …, m, where m is the total number of mathematical patches. Each mathematical patch may be circular or rectangular in shape. In this study, rectangular influence domains are adopted for all mathematical patches to ensure computational stability and applicability. Different mathematical patches may partially overlap, but gaps or voids between them are not allowed. When placing mathematical patches, it is not necessary to pay excessive attention to the structural details of the problem domain. For example, a crack tip can terminate anywhere within a mathematical patch. As observed in Fig. 3, mathematical patches M1, M5 and M6 are entirely inside the problem domain, while patches M2 through M4 partially overlap with the domain. All mathematical patches must collectively cover the entire problem domain Ω, meaning that every point in Ω must belong to at least one mathematical patch. In set-theoretic terms, this is expressed as Ω⊂⋃iMi.
MLS node distribution for a cracked plate.
To accurately capture geometric and material discontinuities in the problem domain Ω (such as boundaries and cracks), physical boundaries are used to cut each mathematical patch Mi in the mathematical covering. The portion inside Ω is retained, and the outside portion is discarded. The resulting subregions are called physical patches P(i−j) with j=1,2,…,nip, where nip is the number of physical patches generated from Mi. Since the cutting strictly follows the physical interfaces, each physical patch contains only a single material and may include three types of boundaries: the original mathematical patch boundary (mathematical boundary), physical boundaries, and crack segments.
This process automatically adapts to various geometric scenarios (see Fig. 3):
(i) If Mi lies entirely within Ω without being cut, it directly becomes a single physical patch; (ii) If it is truncated by the external boundary, one physical patch is generated; (iii) If it is intersected by a crack, multiple physical patches are generated. If a crack tip lies inside a patch, that patch is marked as a singular physical patch, while the others are non-singular physical patches.
All physical patches together form the physical covering set that precisely covers Ω: Ω=⋃i=1m⋃j=1nipPi−j.
In summary, the MLS-NMM combined with the cutting strategy offers the following advantages over standard XFEM or XIGA methods when simulating plates with crack-induced discontinuities:
Simplified modeling: Based on meshfree mathematical covering, it eliminates the need for geometry-conforming mesh generation. Physical patches can directly span discontinuities such as cracks, naturally capturing displacement jumps and singularities.
High efficiency and accuracy: It achieves high-precision approximation of defects like cracks without local mesh refinement or remeshing, balancing computational efficiency and numerical accuracy.
Strong adaptability: It can uniformly and efficiently handle various geometric discontinuities without algorithm adjustments or additional preprocessing for different defect types [70,71].
MLS-NMM Space
The MLS shape function ωi(x), which constitutes a partition of unity, is used as the weight function associated with the mathematical patch. For each mathematical patch Mi in the set {Mi}, there exists a smooth weighting function ωi(x) that satisfies the following conditions:
ωi(x)=0,x∉Miωi(x)≤1,x∈Mi∑i=1mωi(x)=1,x∈Ωwhere x denotes the position vector on the mathematical patch, used to define the spatial coordinates of the local approximation function. The set of weight functions {ωi(x)} forms a smooth and compactly supported function family, which constitutes a partition of unity over the solution domain subordinate to the mathematical cover {Mi}.
For solving the fourth-order partial differential equations of plates, the H2 regularity requirement must be satisfied. However, it is difficult for the conventional finite element method to construct partition of unity functions with sufficient smoothness [72]. In contrast, the MLS approximation can effectively generate higher-order continuous weighting functions by selecting appropriate intrinsic weight and basis functions [66]. Therefore, within the MLS framework, the weight function ωi(x) can be expressed as:
ωi(x)=pT(x)A−1(x)p(xi)ψi(x)here, p(x) is a polynomial basis function, which in the two-dimensional case is generally taken as:
The linear basis
pT(x)=(1,x1,x2)
The secondary basis
pT(x)=(1,x1,x2,x12,x1x2,x22)
And the cubic basis
pT(x)=(1,x1,x2,x12,x1x2,x22,x13,x12x2,x1x22,x23)here, to satisfy the H2 regularity requirement of MLS interpolation, this study employs second-order complete polynomials as basis functions. The resulting weight functions possess C1 continuity, which is sufficient to accurately reproduce the constant curvature condition.
Although cubic bases theoretically offer higher smoothness, preliminary tests showed that under the same node distribution, they significantly increase the condition number of matrix A(x) (typically by 1–2 orders of magnitude), which can cause interpolation oscillations or ill-conditioning of the stiffness matrix. In contrast, quadratic bases provide a better balance between accuracy and numerical stability and are therefore chosen as the standard configuration in this study.
The function A(x) in Eq. (18) takes the following form:
A(x)=∑i=1mψi(x)p(xi)pT(xi)here, ψi(x) represents the MLS weight function. In addition to being smooth, the most important requirement for ψi(x) is that it possesses compact support. For two-dimensional problems, the support of ψi(x) is generally chosen to be either rectangular or circular.
In general, all MLS weight functions ψi(x) are generated from the same univariate generating function ϕ(z), whose support is [0, 1].
In this study, the support of ψi(x) is taken as a rectangle centered at x =(xi,yi), which can be expressed as:
ψi(x)=ϕ(|x−xi|Dxi)ϕ(|y−yi|Dyi)here, Dxi and Dyi denote the half-lengths of the support of ψi(x) (which is a rectangle) along the x-axis and y-axis, respectively. The generating function ϕ(z) is chosen as a cubic spline function and can be expressed as:
ϕ(z)={2/3−4z2+4z3,z∈(0,0.5]4/3−4z+4z2−4z3/3,z∈(0.5,1)0,z∉(0,1)where z represents either |x−xi|Dxi or |y−yi|Dyi.
All mathematical patches are cut by the problem domain boundaries, holes, and cracks to form physical patches. The physical patch Pi−j inherits the weight function of the mathematical patch Mi, and its weight function ωi−j(x) is expressed as:
ωi−j(x)={ωi(x),x∈Pi−j0,x∉Pi−j
After assigning a unified numbering to all physical patches, a local approximation of the plate deflection w can be constructed on each physical patch. Let the local approximation on the i-th physical patch Pi be expressed as:
wih(x)=bi(x)aihere, bi(x) denotes the basis function vector defined on the i-th physical patch Pi, used to construct the local approximation of the deflection field, which can be expressed as:
bi(x)=(1,bi1,⋯,biDi)and a coefficient vector ai∈RDi+1, where Di denotes the dimension of the basis functions on the i-th physical patch.
For a non-singular physical patch Pi that does not contain a crack tip, its local approximation adopts a constant function form. In this case, the basis function vector degenerates to bi(x)=(1), and the corresponding unknown coefficient ai is a constant to be determined.
In contrast, for singular physical patches containing crack tips, enhanced basis functions incorporating singular terms are introduced to more accurately capture stress concentration behavior. This can be expressed as:
bi(x)=(1,r3/2sin3θ2,r3/2cos3θ2,r3/2sinθ2,r3/2cosθ2)
Specifically, a local r−θ coordinate system is established with the crack tip as the origin, where r represents the radial distance from the field point to the crack tip, and θ is the polar angle measured counterclockwise from the crack extension line to the field point (see Fig. 4), with θ∈(−π,π].
Crack-tip coordinates in local and global coordinate systems.
In the Kirchhoff–Love thin-plate theory, the displacement field (i.e., the deflection) near a crack tip exhibits the characteristic r3/2 asymptotic behavior, which is a fundamental feature of the Williams crack-tip solution. The enriched basis functions in Eq. (28) are constructed directly from this r3/2 displacement singularity, allowing the method to accurately capture the mechanical response in the vicinity of the crack tip.
By performing a weighted summation of the local approximation functions over all physical patches and utilizing the partition of unity property, a globally continuous approximate displacement field can be constructed as follows:
wh(x)=∑iωi(x)wih(x)
Substituting Eq. (26) into Eq. (29) yields the global approximate displacement field as follows:
wh(x)=N(x)awhere a denotes the vector of global degrees of freedom, whose transpose is expressed as:
aT=[a1T,⋯,anpT]while each subvector aiT corresponds to the local unknown coefficients on the i-th physical patch Pi, with dimension of Di + 1, expressed as:
aiT =[ai0,⋯,aiDi]
The shape function matrix N(x) is composed of the local shape function vectors corresponding to each physical patch Pi, and is defined as:
N(x)=[N1(x),⋯,Nnp(x)]where Ni(x) is the local shape function vector associated with physical patch Pi, having a dimension of 1 + Di, and is explicitly expressed as:
Ni(x)=(ωi,ωibi1,⋯,ωibiDi)
Therefore, in the NMM, the mathematical cover is automatically cut by discontinuous interfaces (such as cracks or boundaries), generating multiple physical patches. Each physical patch independently carries a set of nodal degrees of freedom and serves as a computational unit for local approximation. This patch-based structure allows discontinuities in the displacement field across interfaces to be naturally represented, thereby efficiently capturing and simulating various discontinuous features in materials or structures through Eq. (30).
Thanks to its intrinsic piecewise approximation mechanism, the NMM can accurately model displacement jumps and singular behaviors without introducing additional enrichment functions (such as the Heaviside function used in XFEM to describe strong discontinuities) [73]. This significantly simplifies the modeling process of discontinuous problems while enhancing computational robustness and implementation efficiency.
MLS-NMM Discretization Formulation for Plate Bending
This section performs bending analyses of FGM plates and cracked homogeneous plates within the MLS-NMM framework, based on classical thin-plate theory. The objective is to validate the effectiveness and applicability of the proposed method for these two categories of bending problems.
According to thin plate theory, neglecting transverse shear deformation, the displacement field is described by the mid-surface deflection w. Substituting the approximate displacement wh from Eq. (30) into w in Eq. (16), the following equilibrium equation is obtained:
Ka=phere, K denotes the global stiffness matrix, a is the vector of unknown degrees of freedom, and p represents the equivalent nodal load vector.
The submatrix Kij of the stiffness matrix K is determined jointly by the physical patches Pi and Pj, and is explicitly expressed as:
Kij=∫ΩBiTDBjdΩ+∫ΓC∪ΓSαdNiT NjdΩ+∫ΓCαk∂NiT ∂n∂Nj∂ndΩwhere the strain-displacement matrix Bi is defined as:
Bi=LNiwhere L is the differential operator defined in Eq. (6).
The i-th component of the equivalent nodal load vector pi is given by:
pi=∫ΩNTqdΩhere, q denotes the transverse distributed load acting on the plate surface.
In particular, within MLS-NMM, when a physical patch is intersected by a crack, its integration domain Ω is automatically partitioned into several continuous subdomains free of discontinuities. A differentiated Gauss integration strategy is then employed: standard Dunavant rules are used for regular subdomains; crack-cut subdomains without a tip use higher-order quadrature; and subdomains containing the crack tip are remapped into quadrilateral elements and integrated using Gauss–Legendre rules. The load vector pi is assembled by independently integrating over all subdomains and summing their contributions, ensuring accurate computation even in the presence of strong discontinuities.
Calculation of Stress Intensity Factor (SIF)
In this section, the stress intensity factor is calculated within the MLS-NMM framework based on thin plate theory. The objective is to validate the applicability of the stress intensity factor in the bending analysis of cracked plates.
Asymptotic Displacement near Crack Tip and Fracture Modes
In thin plate theory, two fracture modes exist depending on the type of loading: symmetric bending mode k1, and antisymmetric bending mode k2. Williams [74] derived the stress field near the tip of a through-thickness crack in the plate. The out-of-plane displacement w is given in Ref. [75] as follows:
w=(2r)32(1−ν2)2Eh(3+ν){k1[13(7+ν1−ν)cos3θ2−cosθ2]+k2[13(5+3ν1−ν)sin3θ2−sinθ2]}where k1 and k2 are defined as:
k1=limr→02rσθθ(r,0,h2);k2=limr→03+ν1+ν2rσrθ(r,0,h2)
In polar coordinates (r,θ), the stress field near the crack tip induced by bending in thin-plate theory can be expressed as [51,76]:
{σrrσrθσθθ}=k12rz2h13+ν{(3+5ν)cosθ2−(7+ν)cos3θ2−(1−ν)sinθ2+(7+ν)sin3θ2(5+3ν)cosθ2+(7+ν)cos3θ2}+k22rz2h13+ν{−(3+5ν)sinθ2+(5+3ν)sin3θ2(−1+ν)cosθ2+(5+3ν)cos3θ2−(5+3ν)(sinθ2+sin3θ2)}where z is the out-of-plane component of the current coordinate vector.
J-Integral and Interaction Integral
The classical J-integral is widely used in fracture mechanics to determine stress intensity factors. In this study, its domain-integral form is adopted. For a through-thickness crack that is perpendicular to the midplane (with the crack front also perpendicular to the midplane) [76], the J-integral is defined as:
J=1h∫V(σij∂wi∂x1−12σijεijδ1j)∂q∂xjdVwhere σij denotes the components of the stress tensor; εij represents the components of the strain tensor; wi is the displacement component; and δ1j is the Kronecker delta. The function q represents a sufficiently smooth weight function that takes the value 0 on the outer boundary of the domain V and 1 on the inner boundary. For detailed procedures, refer to the study by Moës et al. [77].
In the present MLS-NMM implementation, we tested multiple integration domains of different shapes and sizes. The results show that the variation of the computed J-integral values is less than 0.5%, indicating that the method maintains the path-independence of the J-integral very well. This not only verifies the numerical satisfaction of energy conservation, but also demonstrates that the asymptotic crack-tip field is accurately captured.
Based on thin plate theory, under mixed-mode loading conditions, the relationship between the J-integral and stress intensity factors is given by [75]:
J=G=π3E(1+ν3+ν)(k12+k22)where k1 and k2 are the symmetric and antisymmetric bending stress intensity factors [75], respectively.
To separate k1(1) and k2(1) under mixed-mode conditions, the interaction integral method is introduced. Let the field variables of the actual state be (σij(1),εij(1),wi(1)), and those of the auxiliary state be (σij(2),εij(2),wi(2)). According to Eq. (42), the J-integral of the combined state is expressed as:
J(1+2)=1h∫V[(σij(1)+σij(2))(∂wi(1)∂x1+∂wi(2)∂x1)−12(σij(1)+σij(2))(εij(1)+εij(2))δ1j]∂q∂xjdV
After expansion and rearrangement, it can be expressed as:
J(1+2)=J(1)+J(2)+I(1,2)where:
J(α)=1h∫V(σij(α)∂wi(α)∂x1−12σij(α)εij(α)δ1j)∂q∂xjdV,α=1,2and the interaction integral I(1,2) is defined as:
I(1,2)=1h∫V(σij(1)∂wi(2)∂x1+σij(2)∂wi(1)∂x1−σij(1)εij(2)δ1j)∂q∂xjdV
Combining with Eq. (43), the J-integral for the superposed states can also be expressed as:
J(1+2)=J(1)+J(2)+2π3E(1+ν3+ν)(k1(1)k1(2)+k2(1)k2(2))
From this, the relationship between the interaction integral I(1,2) and the stress intensity factors can be obtained as:
I(1,2)=2π3E(1+ν3+ν)(k1(1)k1(2)+k2(1)k2(2))
The auxiliary fields serve to provide analytical reference solutions with known modal characteristics. By coupling these with the actual fields, the intensity factors of specific fracture modes can be extracted. In the numerical implementation, these auxiliary states are directly constructed using Eqs. (39)–(41), where the asymptotic displacement and stress fields corresponding to a unit symmetric mode (k1(2)=1,k2(2)=0) or a unit antisymmetric mode (k1(2)=0,k2(2)=1) are substituted into Eq. (47) as the auxiliary solutions.
Finally, the stress intensity factors in the actual state can be obtained by the following expressions:
k1(1)=3E2π(3+ν1+ν)I(1,2)with k1(2)=1,k2(2)=0and
k2(1)=3E2π(3+ν1+ν)I(1,2)with k1(2)=0,k2(2)=1
Numerical Examples
To systematically verify the accuracy and applicability of MLS-NMM in simulating the bending behavior of intact FGM thin plates and cracked homogeneous thin plates, this section presents four typical numerical examples: (i) convergence analysis of homogeneous square plates; (ii) bending response of Al/Al2O3 FGM square plates; (iii) bending behavior of Al/ZrO2 FGM circular plates; (iv) fracture mechanics analysis of homogeneous square plates with central cracks. All examples explicitly specify geometric dimensions, material properties, boundary conditions, loading types, and nondimensionalization schemes.
In particular, Example 6.4 employs two independent yet geometrically similar parameter systems: one defined in physical units for the analysis of the actual bending stress field, and the other formulated as a dimensionless reference model dedicated to the validation of the SIF and the J-integral. For clarity and ease of reference, the key modeling parameters for all examples are summarized in Tables 3 and 4.
Summary of geometric and material parameters for bending analysis examples.
Example
Geometry
Dimensions (m)
Material system
Material Properties
6.1
Square plate
a = b = 1, h = 0.01
Homogeneous
E = 1.092 MPa, ν = 0.3
6.2
Square plate
a = b = 1, h = 0.05 or 0.1
FGM(Al/Al2O3)
See Table 2; power-law indexn = 0–5
6.3
Circular plate
r = 1, h = 0.1
FGM(Al/ZrO2)
See Table 2; power-law indexn = 0, 0.5, 2
6.4
Square plate with crack
2a = 2b = 1, h = 0.01, crack length = 2c
Homogeneous
E = 1.092 MPa, ν = 0.3
Summary of boundary conditions, loading, and non-dimensionalization for bending analysis examples.
Example
Boundary Conditions
Load Type
Load Definition and Purpose
Non-DimensionAlization
6.1
SSSS, CCCC
Uniform
q = 1 KN/m2
w¯=100wDqa4, D=Eh312(1−ν2)
6.2
SCSC, SSSC, SSSS, SFSC, SFSS, SFSF
Sinusoidal
q=psin(πxa)sin(πya),p = 1 KN/m2
w¯=10h3Ecwpa4
6.3
C, S
Uniform
q=pEmh4r4 (with p as a dimensionless load parameter)
Note: The boundary condition abbreviations indicate the support type on the four edges of the plate in counter-clockwise order: edge 1 (bottom), edge 2 (right), edge 3 (top), and edge 4 (left). Here, S denotes simply supported, C denotes clamped (fully fixed), and F denotes free.
For crack-free FGM plates, the material properties vary continuously through the thickness direction, ensuring good structural integrity. Conducting bending analysis helps elucidate the influence of material heterogeneity, geometric dimensions, and boundary conditions on deflection distributions and stiffness characteristics, while also validating the accuracy of the MLS-NMM in capturing smooth graded fields and continuous deformation responses.
When cracks or other defects are present in the plate, local stiffness degradation occurs, leading to stress field redistribution. This alters the deformation pattern, increases deflection, and reduces load-carrying capacity. Furthermore, the location and length of the crack significantly influence the complexity of the bending response, potentially triggering local instability or structural failure.
Therefore, comparing the bending behavior of intact and cracked plates not only helps assess the extent to which defects influence structural performance but also provides a basis for validating the applicability of the MLS-NMM in modeling both undamaged and damaged structures. In this section, several numerical examples are presented to demonstrate the bending response of isotropic and FGM plates. Model validation is performed through representative case studies to evaluate the convergence and computational reliability of the MLS-NMM.
To evaluate convergence and accuracy, the relative error of the nondimensional central deflection is defined as:
εiRel=|ϖinum−ϖirefϖiref|where ϖinum and ϖiref denote the numerical solution and the reference solution, respectively.
Convergence Analysis
To verify the convergence and accuracy of MLS-NMM in classical thin plate bending problems, numerical simulations are first conducted on homogeneous square plates. Two typical boundary conditions—simply supported (SSSS) and clamped (CCCC)—are considered. The central deflection under uniformly distributed load is used as the evaluation metric. Relevant geometric, material, and nondimensionalization parameters are detailed in Tables 3 and 4.
Under uniformly distributed load, for a homogeneous square plate with an aspect ratio of a/h = 100, the nondimensional central deflection computed by MLS-NMM using different numbers of mathematical nodes (100, 144, 196, 324, and 676) is listed in Table 5. The results from Ref. [78] are adopted as the reference exact solution for comparative analysis. It is observed that the MLS-NMM solutions converge as the number of nodes increases, and when the node count reaches 676, the numerical results achieve excellent agreement with the reference solutions.
Nondimensional central deflection <inline-formula id="ieqn-212"><mml:math id="mml-ieqn-212"><mml:mover><mml:mi>w</mml:mi><mml:mo accent="false">¯</mml:mo></mml:mover></mml:math></inline-formula> of a homogeneous square plate under various boundary conditions.
Methods
Mathematical Nodes
Boundary Condition
Clamped Support
Simply Supported
Exact [78]
0.1265
0.4064
MLS-NMM
100
0.1193
0.3881
144
0.1232
0.3985
196
0.1251
0.4036
324
0.1260
0.4060
676
0.1264
0.4064
Fig. 5 shows the relative error of the dimensionless central deflection for a homogeneous square plate as a function of the number of mathematical nodes under different boundary conditions. It can be observed that, as the number of mathematical nodes increases, the numerical solution obtained by the MLS-NMM gradually stabilizes and converges toward the exact solution. When the number of nodes reaches 676, the relative error is minimized, and the numerical results are in excellent agreement with the exact solution. This demonstrates that the MLS-NMM exhibits high numerical accuracy and good convergence performance in solving thin plate bending problems, and is capable of effectively capturing the structural mechanical responses under various boundary conditions, such as simply supported and clamped edges.
Convergence analysis of a homogeneous square plate under various boundary conditions using MLS-NMM.
Figs. 6 and 7 illustrate the dimensionless deflection distributions of a homogeneous square plate calculated using the MLS-NMM under different boundary conditions. It can be observed that the deformation magnitude under simply supported (SSSS) conditions is significantly larger than that under clamped (CCCC) conditions, indicating that stronger boundary constraints effectively suppress the deformation, highlighting the significant influence of boundary stiffness on the overall bending response. The maximum deflection occurs at the center of the plate, with the dimensionless deflection showing a symmetric parabolic distribution about the center, decreasing gradually from the center toward the edges and approaching zero at the boundaries, consistent with the imposed boundary constraints.
Nondimensional deflection contours of a homogeneous square plate: (<bold>a</bold>) simply supported (SSSS); (<bold>b</bold>) clamped (CCCC).
Nondimensional deflection <inline-formula id="ieqn-213"><mml:math id="mml-ieqn-213"><mml:mover><mml:mi>w</mml:mi><mml:mo accent="false">¯</mml:mo></mml:mover></mml:math></inline-formula> of a homogeneous square plate under uniform load along the line <italic>x</italic> = 0.5 m.
Bending Analysis of FGM Square Plate
To further assess the capability of MLS-NMM in handling nonhomogeneous material fields, this section conducts bending analysis of an Al/Al2O3 FGM square plate. The plate is subjected to sinusoidally distributed load and six different combinations of boundary conditions are considered to comprehensively evaluate the method’s applicability under complex support scenarios. Detailed modeling parameters are listed in Tables 3 and 4.
Within the MLS-NMM framework, mathematical nodes are uniformly distributed for the FGM square plate, with the node distribution shown in Fig. 8a and the Gauss integration mesh depicted in Fig. 8b.
FGM square plate: (<bold>a</bold>) distribution of mathematical nodes, (<bold>b</bold>) Gauss integration mesh.
Fig. 9 illustrates the three-dimensional distribution of dimensionless deflection for an FGM square plate under sinusoidal loading across six typical boundary conditions. It can be observed that, regardless of the loading scenario, the deflection patterns conform to the fundamental mechanical behavior of thin plate bending. Additionally, different boundary constraints significantly influence both the deformation mode and its magnitude, clearly demonstrating the critical role of boundary conditions in governing the bending response of FGM plates.
Nondimensional deflection contours of a square FGM plate under various boundary conditions. (<bold>a</bold>) SCSC. (<bold>b</bold>) SSSC. (<bold>c</bold>) SSSS. (<bold>d</bold>) SFSC. (<bold>e</bold>) SFSS. (<bold>f</bold>) SFSF.
The bending analyses for different gradient indices n and boundary conditions all use the same mathematical coverage arrangement with 676 nodes. Thanks to the separation mechanism of MLS-NMM coverage, the functionally graded characteristics are defined only through material parameters in the physical coverage without modifying the mathematical coverage. As a result, the total number of degrees of freedom remains unchanged. The CPU time results in Table 6 show that, under the same mesh and solver settings, the computational time remains basically stable across different n values, indicating that the method demonstrates good computational efficiency and robustness when handling material heterogeneity.
Dimensionless central deflection <inline-formula id="ieqn-216"><mml:math id="mml-ieqn-216"><mml:mover><mml:mi>w</mml:mi><mml:mo accent="false">¯</mml:mo></mml:mover></mml:math></inline-formula> and computational cost of square FGM plates under sinusoidal loading with various boundary conditions.
a/h
Boundary Condition
Methods/ Computational Cost
Mathematical Nodes
Gradient Index n
0
0.5
1
2
5
10
SCSC
Thai and Choi [79]
0.2424
0.3676
0.4732
0.6061
0.7406
Demirhan and Taskin [80]
0.2321
0.3537
0.4584
0.5910
0.7215
Ye et al. [20]
0.2399
0.3696
0.4695
0.6074
0.7393
MLS-NMM
676
0.2403
0.3546
0.4708
0.6046
0.7382
CPU time(s)
17.063
16.117
15.362
16.521
16.854
SSSC
Thai and Choi [79]
0.3343
0.5099
0.6591
0.8454
1.0235
Demirhan and Taskin [80]
0.3270
0.5001
0.6488
0.8351
1.0104
Ye et al. [20]
0.3292
0.5108
0.6559
0.8463
1.0206
MLS-NMM
676
0.3300
0.5164
0.6480
0.8374
1.0109
CPU time(s)
15.682
15.413
15.244
16.653
17.256
SSSS
Thai and Choi [79]
0.4693
0.7190
0.9324
1.1973
1.4393
Demirhan and Taskin [80]
0.4666
0.7154
0.9288
1.1940
1.4349
Ye et al. [20]
0.4662
0.7218
0.9212
1.1956
1.4210
MLS-NMM
676
0.4613
0.7166
0.9142
1.1903
1.4372
CPU time(s)
15.865
15.494
15.279
15.814
16.546
SFSC
Thai and Choi [79]
0.6684
1.0221
1.3217
1.6922
2.0335
Demirhan and Taskin [80]
0.6490
0.9954
1.2925
1.6611
1.9945
Ye et al. [20]
0.6602
1.0234
1.3059
1.6817
2.0269
MLS-NMM
676
0.6636
1.0225
1.3190
1.6873
2.0389
CPU time(s)
15.147
15.495
15.289
16.368
17.159
SFSS
Thai and Choi [79]
0.9079
1.3925
1.8054
2.3143
2.7687
Demirhan and Taskin [80]
0.8963
1.3770
1.7887
2.2971
2.7467
Ye et al. [20]
0.8961
1.3987
1.7814
2.2901
2.7654
MLS-NMM
676
0.9040
1.3966
1.7973
2.2945
2.7503
CPU time(s)
15.236
14.875
15.426
15.707
16.465
SFSF
Thai and Choi [79]
1.4985
2.2990
2.9786
3.8133
4.5524
Demirhan and Taskin [80]
1.4688
2.2584
2.9345
3.7668
4.4934
Ye et al. [20]
1.4653
2.3079
2.9155
3.7454
4.5120
MLS-NMM
676
1.4551
2.3294
2.9952
3.7898
4.4919
CPU time(s)
15.258
15.409
15.926
16.297
17.139
20
SCSC
Thai and Choi [79]
0.2256
0.3449
0.4448
0.5670
0.6776
Demirhan and Taskin [80]
0.2151
0.3307
0.4297
0.5515
0.6579
Ye et al. [20]
0.2151
0.3363
0.4265
0.5482
0.6618
MLS-NMM
676
0.2275
0.3245
0.4270
0.5421
0.6777
CPU time(s)
15.160
15.203
15.304
15.902
16.526
SSSC
Thai and Choi [79]
0.3173
0.4869
0.6304
0.8058
0.9597
Demirhan and Taskin [80]
0.3099
0.4770
0.6199
0.7954
0.9463
Ye et al. [20]
0.3100
0.4869
0.6162
0.7913
0.9519
MLS-NMM
676
0.3122
0.4745
0.6110
0.7987
0.9471
CPU time(s)
17.149
16.901
15.815
16.473
16.941
SSSS
Thai and Choi [79]
0.4521
0.6956
0.9033
1.1572
1.3747
Demirhan and Taskin [80]
0.4494
0.6921
0.8997
1.1539
1.3703
Ye et al. [20]
0.4490
0.7059
0.8946
1.1483
1.3714
MLS-NMM
676
0.4567
0.6952
0.9044
1.1505
1.3775
CPU time(s)
16.799
15.608
15.731
16.067
17.118
SFSC
Thai and Choi [79]
0.6460
0.9917
1.2838
1.6400
1.9493
Demirhan and Taskin [80]
0.6264
0.9649
1.2543
1.6087
1.9098
Ye et al. [20]
0.6273
0.9864
1.2480
1.6017
1.9232
MLS-NMM
676
0.6351
0.9665
1.2426
1.6045
1.9233
CPU time(s)
15.245
15.436
15.746
16.251
16.815
SFSS
Thai and Choi [79]
0.8851
1.3618
1.7670
2.2615
2.6835
Demirhan and Taskin [80]
0.8736
1.3463
1.7503
2.2443
2.6615
Ye et al. [20]
0.8739
1.3787
1.7415
2.2337
2.6761
MLS-NMM
676
0.8825
1.3613
1.7610
2.2504
2.6749
CPU time(s)
15.606
16.054
15.948
16.693
17.214
SFSF
Thai and Choi [79]
1.4692
2.2594
2.9293
3.7454
4.4427
Demirhan and Taskin [80]
1.4396
2.2189
2.8851
3.6989
4.3837
Ye et al. [20]
1.4389
2.2710
2.8682
3.6781
4.4035
MLS-NMM
676
1.4327
2.2884
2.9094
3.7064
4.4026
CPU time(s)
15.971
15.625
15.961
16.663
17.035
Under sinusoidal loading, Table 6 presents the dimensionless central deflections of an FGM square plate under various boundary conditions, computed using the MLS-NMM. A comparative analysis demonstrates good agreement between the present numerical results and the reference solutions, verifying the accuracy and reliability of the MLS-NMM in solving bending problems of FGM plates.
Further analysis shows that the nondimensionalized center deflection significantly depends on both material and geometric parameters: for a fixed gradient index n, the nondimensional center deflection decreases as the plate thickness decreases (i.e., as a/h increases); conversely, for a constant plate thickness, the center deflection increases monotonically with n, since a higher n reduces the ceramic phase fraction, leading to an overall decrease in stiffness.
To quantify the effect of the gradient index n, the relative increase in nondimensional center deflection was calculated as n varied from 0 to 5. At a/h=10, under SCSC boundary conditions, the nondimensional center deflection rose from 0.2403 to 0.7382 (+207%), while under SFSF conditions it increased from 1.4551 to 4.4919 (+209%). For a/h=20, the increases were 198% (SCSC) and 207% (SFSF), respectively. Although the percentage increases are similar, the absolute deflection increment under SFSF is much larger than that under SCSC, indicating that structures with weaker constraints are more sensitive to material softening, and the nondimensional center deflection varies more significantly with n.
As shown in Fig. 10, the nondimensional center deflection w¯ exhibits a nonlinear increasing trend with respect to the gradient index n. The increase is pronounced for n<2, after which it gradually levels off. This indicates that when the gradient index is small, changes in material composition have the most significant impact on structural stiffness. As n increases, the ceramic phase proportion gradually decreases, enhancing the overall material softening effect; however, further increases in n lead to a saturation in the stiffness reduction effect.
Dimensionless central deflection <inline-formula id="ieqn-231"><mml:math id="mml-ieqn-231"><mml:mrow><mml:mrow><mml:mover><mml:mi>w</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:math></inline-formula> of FGM square plates for different gradient indices <italic>n</italic>: (<bold>a</bold>) <italic>a/h</italic> = 10, (<bold>b</bold>) <italic>a/h</italic> = 20.
Moreover, under identical geometric and material parameters, the boundary constraints have a significant influence on the deflection response. Among the cases considered, the SCSC condition (simply supported on two opposite edges and clamped on the other two) yields the smallest dimensionless central deflection, indicating the highest constraint stiffness and the most effective deformation suppression. In contrast, the SFSF condition (simply supported on two opposite edges and free on the other two) results in the largest deflection, reflecting the lowest support stiffness and a greater susceptibility to bending deformation.
Fig. 11 illustrates the distribution of dimensionless deflection along the two central cross-sectional lines (x = 0.5 m and y = 0.5 m) for an FGM square plate under sinusoidal loading, with a side-to-thickness ratio a/h = 20 and a gradient index n = 0.5. The figure clearly reveals the influence of different boundary conditions on the spatial distribution of the deformation field within the plate.
Nondimensional deflection <inline-formula id="ieqn-232"><mml:math id="mml-ieqn-232"><mml:mrow><mml:mrow><mml:mover><mml:mi>w</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:math></inline-formula> of a square FGM plate (<italic>a/h</italic> = 20, <italic>n</italic> = 0.5) under sinusoidal load along the lines: (<bold>a</bold>) <italic>x</italic> = 0.5 m; (<bold>b</bold>) <italic>y</italic> = 0.5 m.
As shown in Fig. 11a, the dimensionless deflection curves along the cross-sectional line at x = 0.5 m (i.e., the transverse section) exhibit excellent symmetry under all boundary conditions, with the maximum deflection occurring at the geometric center of the plate (y = 0.5 m). This indicates that the deformation in this direction is less influenced by the combined effects of symmetric loading and boundary constraints, leading to a structural response that is concentrated around the central region.
In contrast, the deflection distribution along the y = 0.5 m section (longitudinal section) in Fig. 11b exhibits significant asymmetry, reflecting the asymmetric constraint effects of boundary conditions in different directions. Specifically, under SCSC and SSSS boundary conditions, the deflection curves remain symmetric with peaks at the center; under SSSC condition, due to one clamped edge and one simply supported edge, the maximum deflection shifts toward the simply supported side and occurs near it; for SFSC and SFSS conditions, with one free edge, the maximum deflection clearly appears at the midpoint of that free edge, reflecting intensified local deformation caused by released constraints; and for the SFSF condition with two pairs of free edges, the maximum deflections occur simultaneously near the midpoints of both free edges, forming a dual-peak distribution, further highlighting the weakening effect of free edges on structural stiffness.
Bending Analysis of FGM Circular Plate
To extend the application of the method to non-rectangular domains and different FGM systems, this section investigates the bending response of an Al/ZrO2 FGM circular plate under uniform loading. Both simply supported (S) and clamped (C) boundary conditions are considered. Results are presented using a nondimensional load parameter based on the metal’s modulus. Detailed parameter settings are listed in Tables 3 and 4.
In the MLS-NMM framework, mathematical nodes are uniformly distributed for the FGM circular plate. The node distribution is illustrated in Fig. 12a, and the corresponding Gauss integration mesh is depicted in Fig. 12b.
FGM circular plate: (<bold>a</bold>) distribution of mathematical nodes, (<bold>b</bold>) Gauss integration mesh.
Fig. 13 presents the nondimensional deflection profiles of the FGM circular plate under different boundary conditions, computed using MLS-NMM. The results show that, under both boundary conditions, the plate deformation exhibits an axisymmetric, bowl-shaped pattern. The maximum deflection occurs at the plate center and gradually decreases toward the edges. Compared to the clamped boundary, the plate under simply supported boundary exhibits larger overall deformation.
Nondimensional deflection contours of an FGM circular plate: (<bold>a</bold>) clamped boundary; (<bold>b</bold>) simply supported boundary.
Under uniform distributed loading, Tables 7 and 8 present the nondimensional center deflections of clamped and simply supported FGM circular plates, respectively, for different gradient indices and applied loads p. The MLS-NMM results show excellent agreement with references [81,82], fully validating the reliability and accuracy of the method in handling varying material gradient distributions and load intensities.
Nondimensional central deflection of clamped FGM circular plate (<italic>r/h</italic> = 10).
p
n = 0
n = 2
[81]
[82]
Mathematical Nodes
MLS-NMM
[81]
[82]
Mathematical Nodes
MLS-NMM
−1.4286
−0.1167
−0.1171
1017
−0.1121
−0.1804
−0.1808
1017
−0.1854
−2.8571
−0.2285
−0.2292
1017
−0.2243
−0.3446
−0.3450
1017
−0.3407
−4.2857
−0.3323
−0.3331
1017
−0.3364
−0.4868
−0.4868
1017
−0.4861
−7.1429
−0.5128
−0.5132
1017
−0.5107
−0.7143
−0.7131
1017
−0.7168
−11.4290
−0.7264
−0.7255
1017
−0.7271
−0.9639
−0.9604
1017
−0.9662
−14.2857
−0.8416
−0.8397
1017
−0.8368
−1.0940
−1.0891
1017
−1.0903
−17.1429
−0.9415
−0.9387
1017
−0.9422
−1.2055
−1.1994
1017
−1.2043
−21.4286
−1.0705
−1.0662
1017
−1.0635
−1.3484
−1.3407
1017
−1.3404
Nondimensional central deflection of simply supported FGM circular plate (<italic>r/h</italic> = 10).
p
n = 0.5
n = 2
[82]
Mathematical Nodes
MLS-NMM
[82]
Mathematical Nodes
MLS-NMM
−1.4286
−0.4675
1017
−0.4655
−0.5553
1017
−0.5539
−2.8571
−0.6949
1017
−0.6940
−0.8030
1017
−0.8017
−4.2857
−0.8422
1017
−0.8419
−0.9608
1017
−0.9626
−7.1429
−1.0446
1017
−1.0452
−1.1771
1017
−1.1739
−11.4290
−1.2513
1017
−1.2526
−1.3979
1017
−1.3962
−14.2857
−1.3577
1017
−1.3593
−1.5117
1017
−1.5103
−17.1429
−1.4491
1017
−1.4496
−1.6096
1017
−1.6076
−21.4286
−1.5670
1017
−1.5692
−1.7361
1017
−1.7359
Further analysis indicates that boundary constraints play a decisive role in the deformation behavior of the plate: under the same loading conditions, the dimensionless central deflection of simply supported plates is consistently and significantly larger than that of clamped plates. For instance, at p=−21.4286, when the gradient index increases from n=0 (fully ceramic) to n=2, the dimensionless central deflection of the clamped plate increases from −1.0635 to −1.3404, corresponding to a relative increase of 26.0%. In contrast, for the simply supported plate, as n varies from 0.5 to 2, the dimensionless central deflection increases from −1.5692 to −1.7359, with a relative increase of only 10.6%. Although the simply supported plate exhibits a larger absolute deformation, the clamped plate, owing to its higher initial stiffness, shows a greater relative sensitivity of the dimensionless central deflection to material softening. This observation demonstrates that stronger boundary constraints lead to a more pronounced structural response to variations in material properties.
Overall, as the gradient index n increases (from n=0 to 2 for clamped plates and from n=0.5 to 2 for simply supported plates), the reduction in ceramic volume fraction leads to a decrease in the effective elastic modulus, resulting in weakened structural stiffness and a corresponding increase in the dimensionless central deflection. This trend is observed under both boundary conditions. Moreover, the relative increase in the dimensionless central deflection becomes more pronounced with stronger boundary constraints, highlighting the coupled effect of material gradation and boundary conditions on the deformation response.
As shown in Fig. 14, under uniform distributed load, the nondimensional center deflection w¯ of the FGM circular plate increases approximately linearly with the load p, indicating that the structure exhibits good linear elastic behavior within the small deformation range. Under the same load, the deflection with simply supported boundary conditions is significantly larger than that with clamped boundaries, further confirming the dominant role of boundary constraint stiffness on structural deformation: the weaker the constraint, the greater the deformation.
Dimensionless central deflection <inline-formula id="ieqn-242"><mml:math id="mml-ieqn-242"><mml:mrow><mml:mrow><mml:mover><mml:mi>w</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:math></inline-formula> of an FGM circular plate under loading <italic>p</italic>.
In addition, under the same boundary conditions, as the gradient index n increases from 0 to 2 for clamped plates or from 0.5 to 2 for simply supported plates, the material stiffness decreases and the dimensionless central deflection increases accordingly. This effect is more pronounced under simply supported boundaries, indicating that weakly constrained structures are more sensitive to stiffness degradation induced by material nonhomogeneity. This observation demonstrates that the influence of the gradient index on deformation is regulated by boundary conditions: the weaker the constraint, the more readily the contribution of material gradation to the overall structural flexibility manifests.
Bending Analysis of Square Plate with Center Crack
To validate the effectiveness of MLS-NMM in handling geometric discontinuities such as cracks, this section conducts bending simulations on a homogeneous square plate with a central crack. The plate is simply supported and subjected to a uniformly distributed load. Besides the deflection response, the fracture mechanics behavior near the crack tip is also examined.
It should be emphasized that this example employs two distinct parameter systems to serve different analysis objectives. Specifically, the bending stress field analysis adopts parameters with physical dimensions in order to obtain realistic stress distributions, whereas the calculations of the SIF and the J-integral are carried out based on a dimensionless reference model (SIF¯=h2qa2cSIF, J¯=Eh4q2a4cJ), enabling direct comparison with standard fracture mechanics solutions. The two parameter systems share the same relative crack length c/a, thereby ensuring identical geometric configurations. The detailed parameter settings are summarized in Tables 3 and 4.
Fig. 15a illustrates the distribution of mathematical nodes and the configuration of the horizontal central crack (c/a = 0.6) in the homogeneous square plate, while Fig. 15b displays the corresponding Gauss integration mesh.
Square plate with a horizontal center crack: (<bold>a</bold>) mathematical nodes and crack configuration; (<bold>b</bold>) Gauss integration mesh.
To quantify the impact of crack defects on structural stiffness, the maximum (center) deflection of the cracked plate is compared with that of the intact plate under the same load and boundary conditions. As shown in Table 9, the nondimensional center deflection increases significantly with the crack length ratio c/a. For example, when c/a=0.6, the center deflection rises from 0.4064 for the intact plate to 0.5924 for the cracked plate, an increase of 45.8%. This marked growth indicates that even cracks of moderate length can substantially weaken the overall deformation performance of bending-dominated thin plate structures, highlighting the sensitivity of structural stiffness to cracks.
Comparison of dimensionless central deflection <inline-formula id="ieqn-250"><mml:math id="mml-ieqn-250"><mml:mover><mml:mi>w</mml:mi><mml:mo accent="false">¯</mml:mo></mml:mover></mml:math></inline-formula> between simply supported square plates with varying crack length ratios and the intact plate.
Crack Ratio c/a
Nondimensional Central Deflection w¯ (Intact Plate)
Nondimensional Central Deflection w¯ (Cracked Plate, MLS-NMM)
Deflection Increase (%)
0.0
0.4064
0.4064
0.0
0.1
0.4064
0.4110
1.1
0.2
0.4064
0.4708
15.8
0.3
0.4064
0.4833
18.9
0.4
0.4064
0.4977
22.5
0.5
0.4064
0.5449
34.1
0.6
0.4064
0.5924
45.8
0.7
0.4064
0.6032
48.4
0.8
0.4064
0.6375
56.9
0.9
0.4064
0.6700
64.9
Under the same simply supported boundary conditions (SSSS) and geometric settings, a comparative analysis between Tables 6 and 9 show that the nondimensional center deflection is significantly more sensitive to the gradient index n than to the crack length ratio c/a. Specifically, for an FGM square plate with an aspect ratio a/h=20, when n increases from 0 to 5, the nondimensional center deflection rises from 0.4567 to 1.3775, an increase of 202%. In contrast, for a homogeneous cracked square plate, when c/a increases from 0 to 0.9, the nondimensional center deflection only increases from 0.4064 to 0.6700, a rise of 64.9%. This difference arises because an increase in the gradient index leads to a systematic reduction in the overall material stiffness, while cracks mainly cause local stiffness degradation and stress redistribution, resulting in a comparatively limited impact on the global deformation.
The convergence of the dimensionless stress intensity factor (SIF¯) with respect to the number of mathematical nodes for different crack lengths is shown in Fig. 16. The results indicate that as the number of nodes increases, the SIF¯ values rapidly stabilize and remain nearly constant once the node count exceeds 200, demonstrating good convergence of the current numerical model. For the three crack length ratios ca=0.4,0.6,0.8, the SIF¯ converge around 1.0 with a maximum deviation within ±5%, indicating that the method is insensitive to crack geometric parameters and yields stable and reliable results. Furthermore, the obtained SIF¯ values are in good agreement with reference solutions reported in [83], validating the accuracy of the proposed method in fracture mechanics analysis.
Convergence of the dimensionless stress intensity factor (<inline-formula id="ieqn-264"><mml:math id="mml-ieqn-264"><mml:mover><mml:mrow><mml:mi>S</mml:mi><mml:mi>I</mml:mi><mml:mi>F</mml:mi></mml:mrow><mml:mo accent="false">¯</mml:mo></mml:mover></mml:math></inline-formula>) with respect to the number of mathematical nodes.
Under uniform loading, Table 10 presents the dimensionless J-integral values at the crack tip computed by MLS-NMM for different crack lengths. The results are in good agreement with the reference solutions [76]. This indicates that MLS-NMM achieves high numerical accuracy in fracture analysis of cracked structures and can accurately capture the effect of crack length variations on the energy release rate.
Dimensionless <inline-formula id="ieqn-265"><mml:math id="mml-ieqn-265"><mml:mi>J</mml:mi></mml:math></inline-formula>-integral values for different crack lengths.
Methods
Mathematical Nodes
c/a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Chau-Dinh et al. [76]
0.738
1.265
1.672
1.938
2.036
1.947
1.668
1.216
0.640
MLS-NMM
676
0.735
1.271
1.670
1.952
2.023
1.931
1.659
1.221
0.637
Fig. 17 presents the bending response of a homogeneous square plate with a central crack under a uniformly distributed load, as computed by MLS-NMM. Fig. 17a shows the deflection contour, indicating that the maximum downward deflection occurs in the central region and forms a symmetric bowl-shaped profile; the deflection gradually decreases from the center toward the edges and approaches zero along the simply supported boundaries. Due to the presence of the crack, the deformation is significantly amplified in the vicinity of the crack tips, exhibiting a pronounced feature of locally intensified indentation. Fig. 17b–d depicts the distributions of the normal stress σxx in the x direction, the normal stress σyy in the y direction, and the shear stress τxy, respectively. The σxx field shows clear tensile stress concentration zones on both sides of the crack, while the stress along the crack faces tends to zero, consistent with the traction-free boundary condition. The σyy distribution exhibits steep gradients near the crack tips, indicating pronounced stress concentration effects. The τxy contours reveal a clear redistribution of shear stresses around the crack region, with particularly complex stress patterns forming near the crack tips.
Bending response of a homogeneous square plate with a central crack (bottom surface): (<bold>a</bold>) Deflection distribution, (<bold>b</bold>) normal stress in the x direction <inline-formula id="ieqn-275"><mml:math id="mml-ieqn-275"><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, (<bold>c</bold>) normal stress in the y direction <inline-formula id="ieqn-276"><mml:math id="mml-ieqn-276"><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, (<bold>d</bold>) shear stress <inline-formula id="ieqn-277"><mml:math id="mml-ieqn-277"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></inline-formula>
Overall, these results demonstrate that the presence of the crack not only alters the global deformation mode of the plate but also induces strong localized stress concentrations, thereby confirming the capability of the proposed numerical method to accurately capture key features of fracture mechanics problems.
As shown in Fig. 18, under uniform loading, the dimensionless J-integral (J¯) of a centrally cracked square plate initially increases and then decreases with the increase in crack ratio c/a, reaching a peak value at approximately c/a = 0.5. This indicates that stress concentration effects are significant in the short crack regime, while the reduction in structural stiffness for longer cracks leads to a decrease in the energy release rate. Furthermore, the trend of J¯ variation exhibits a symmetric parabolic shape, reflecting the combined influence of crack geometry and loading conditions. The MLS-NMM accurately captures these variations, making it suitable for fracture mechanics analysis of cracked structures.
Variation of the dimensionless <inline-formula id="ieqn-278"><mml:math id="mml-ieqn-278"><mml:mi>J</mml:mi></mml:math></inline-formula>-integral <inline-formula id="ieqn-279"><mml:math id="mml-ieqn-279"><mml:mo stretchy="false">(</mml:mo><mml:mover><mml:mi>J</mml:mi><mml:mo accent="false">¯</mml:mo></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> with crack length under uniform pressure.
Conclusions
This study proposes an MLS-NMM with H2-regular approximation, integrating the high-order smoothness of MLS shape functions and the flexible geometric discontinuity modeling capability of NMM. The developed framework successfully performs unified static analysis of bending behavior in FGM thin plates and fracture responses in cracked homogeneous plates. The main conclusions are summarized as follows:
High accuracy and convergence: MLS-NMM naturally satisfies the high-order continuity requirements of the Kirchhoff–Love theory. Key quantities such as the non-dimensional deflection and dimensionless J-integral show excellent agreement with reference solutions, demonstrating stable convergence and superior accuracy.
Gradient index influences stiffness: Increasing the gradient index leads to a significant rise in the center deflection of the FGM plate, reflecting structural softening due to the reduced ceramic phase fraction.
Boundary conditions dominate deformation: Strong constraints suppress deformation, while weak constraints significantly amplify deflection; the type of boundary condition decisively affects the response magnitude and distribution.
Defects weaken bending performance: Cracks cause local stiffness degradation, markedly increasing overall deflection, highlighting the adverse impact of geometric defects on mechanical response.
Unified treatment of complex defects: By integrating Williams’ asymptotic singular physical patches with the NMM double coverage mechanism, displacement discontinuities and stress singularities of cracks are accurately captured without the need for additional enrichment.
This framework focuses on the static bending and fracture analysis of FGM and defective plates, featuring strong extensibility. It can be further extended to sandwich plates, fiber-reinforced composite plates, and dynamic response problems, providing an effective numerical tool for the multi-scale, multi-physics coupled analysis of complex structures.
It should be noted that the current model is based on the classical thin-plate theory and neglects transverse shear deformation, making it mainly suitable for thin plates with large aspect ratios. When extended to three-dimensional problems, geometric cutting of physical patches and high-accuracy numerical integration still pose challenges. Nevertheless, the method shows promising potential in thermo-mechanical coupling and dynamic fracture problems, and related research is actively underway.
None.
Funding Statement
This study is supported by Beijing Natural Science Foundation(L233025).
Author Contributions
Shouyang Huang: Conceptualization, Investigation, Software, Writing-original draft, Formal analysis, Writing–review & editing, Visualization, Data curation, Validation. Hong Zheng: Conceptualization, Funding acquisition, Methodology, Writing–review & editing, Project administration. Xuguang Yu: Writing–review & editing, Validation. Ziheng Li: Data curation. Zhiwei Pan: Supervision. All authors reviewed and approved the final version of the manuscript.
Availability of Data and Materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Ethics Approval
Not applicable.
Conflicts of Interest
The authors declare no conflicts of interest.
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