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Natural frequencies of cracked functionally graded material plates by the extended finite element method
S Natarajan, PM Baiz, S Bordas, T Rabczuk, P Kerfriden
TL;DR
Cracks alter the dynamic response of functionally graded material plates, motivating analysis of their free flexural vibration. The paper uses an XFEM-based field-consistent plate element to compute natural frequencies and examines how crack and plate parameters affect them. The reported trends show lower frequencies with longer cracks, higher gradient indices, and central crack locations, alongside additional effects from thickness, aspect ratio, orientation, and multiple cracks.
Problem
Cracks introduce local flexibility and anisotropy in FGM plates, so their dynamic response requires analysis across crack and plate parameters.
Method
The study uses XFEM with a field-consistent four-noded plate element and enrichments for crack discontinuities and near-tip behavior.
Results
Increasing crack length and gradient index decreases natural frequency, while frequency is lowest when the crack is centered; thickness, aspect ratio, orientation, and multiple cracks also affect frequency.
Takeaways & Limitations
Crack length, location, orientation, gradient index, thickness, aspect ratio, and crack multiplicity are relevant parameters for the computed vibration characteristics of FGM plates.
Abstract
from arXiv · showhide
In this paper, the linear free flexural vibration of cracked functionally graded material plates is studied using the extended finite element method. A 4-noded quadrilateral plate bending element based on field and edge consistency requirement with 20 degrees of freedom per element is used for this study. The natural frequencies and mode shapes of simply supported and clamped square and rectangular plates are computed as a function of gradient index, crack length, crack orientation and crack location. The effect of thickness and influence of multiple cracks is also studied.
1. Introduction
Engineered materials can suffer degradation and premature failure from material discontinuities, motivating functionally graded materials and study of cracked plates’ dynamic response.
- Laminated composites offer high strength- and stiffness-to-weight ratios but can experience premature failure or stiffness loss from delaminations and unstable interfaces.
- Functionally graded materials combine metals and ceramics and were initially developed as thermal-barrier materials for aerospace and fusion-reactor applications.
- Cracks and local defects affect structural dynamics by introducing local flexibility and anisotropy, while crack opening and closure depend on vibration amplitude.
- The paper addresses the need to understand the dynamic response of FGM plates containing internal flaws.
2. Theoretical Formulation
The plate model represents a metal–ceramic FGM whose properties vary through thickness according to a power-law distribution and Mori–Tanaka homogenization.
- 2.1. Functionally Graded Material: The modeled rectangular plate has length a, width b, and thickness h, with x and y in-plane coordinates and z through the thickness.
- 2.1. Functionally Graded Material: The FGM is formed by mixing metal and ceramic phases, with ceramic at z = h/2 graded to metal at z = −h/2.
- 2.1. Functionally Graded Material: The material variation through thickness follows a power-law distribution for the phase composition.
- 2.1. Functionally Graded Material: Effective homogenized material properties are computed using the Mori-Tanaka scheme.
Estimation of mechanical and thermal properties
The formulation defines temperature-dependent, thickness-graded FGM properties and uses Mindlin plate kinematics, constitutive relations, and finite-element equations to obtain natural frequencies.
- Mori-Tanaka homogenization evaluates the FGM’s effective bulk and shear moduli from ceramic and metal phase properties.
- The ceramic and metal volume fractions satisfy Vc + Vm = 1, with n serving as the volume-fraction exponent or gradient index.
- The ceramic fraction varies through thickness, producing a ceramic-rich top surface and metal-rich bottom surface.
- Effective Young’s modulus, Poisson’s ratio, and mass density are computed from the graded constituent properties.
- Temperature-dependent constituent properties are represented using coefficients multiplying T^-1, T, T^2, and T^3.
- Mindlin kinematics express u, v, and w through mid-plane displacements and independent rotations θx and θy.
- The formulation relates membrane, bending, and shear strains to stress resultants through extensional, coupling, bending, and transverse-shear stiffness coefficients.
- Lagrange’s equations produce stiffness and consistent mass matrices, and harmonic motion reduces the system to an eigenvalue problem with natural frequency ω.
3. Overview of the extended finite element method
XFEM enriches a field-consistent Q4 plate element with discontinuity and crack-tip functions, allowing cracks to be modeled independently of the background mesh.
- XFEM augments polynomial finite-element bases with localized functions to represent steep gradients and material discontinuities.
- 3.1. Element Description: The formulation uses a field-consistent, shear-flexible QUAD-4 element with five degrees of freedom per node and redistributed shape functions to avoid locking.
- 3.2. Enriched Q4 element: The enriched approximation represents an independent crack geometry through standard displacement fields, Heaviside enrichment, and crack-tip functions.
- 3.2. Enriched Q4 element: Nodes cut by the crack interior receive discontinuous enrichment, while near-tip nodes receive elastic asymptotic crack-tip functions.
- 3.2. Enriched Q4 element: The enrichment functions represent the discontinuous surface independently of the mesh, using polar coordinates centered at the crack tip.
- 3.2. Enriched Q4 element: Nodes are removed from the crack-interior enrichment set when a small cut-support region would make the matrix ill-conditioned; the computation uses a tolerance of 10^-4.
- 3.3. Discretized equations for enriched Q4 plate element: Substituting the enriched displacement field into the energy equations yields discretized equations with standard and enriched strain-displacement contributions to stiffness and mass matrices.
- 3.3. Discretized equations for enriched Q4 plate element: The eigenvalues are solved using a QR algorithm based on QR decomposition.
4. Numerical results
The numerical study computes nondimensional free-flexural frequencies for cracked FGM plates across geometries, boundary conditions, and crack configurations. The formulation is evaluated using material data and mesh refinement, with validation against existing cracked-plate results.
- The study considers square and rectangular plates with simply supported, cantilevered, and clamped boundary conditions.
- Natural frequencies are examined against thickness, aspect ratio, crack length, orientation, location, multiple cracks, and boundary condition.
- A 34 × 34 structured mesh is found adequate for modeling the full plate after progressive mesh refinement.
- The formulation is validated against available linear-frequency results for cracked isotropic and functionally graded material plates.
- The FGM component properties and temperature-dependent coefficients are documented in Tables 1 and 2.
Effect of crack length, crack orientation and gradient index
For a simply supported square FGM plate, longer cracks and higher gradient indices reduce the fundamental frequency, while crack orientation produces its lowest value at 45°.
- Increasing crack length decreases the fundamental frequency by increasing local flexibility.
- Increasing gradient index n decreases frequency because the metallic volume fraction increases and stiffness degrades.
- The combined increase in crack length and gradient index lowers the fundamental frequency.
- The frequency is lowest at crack orientation θ = 45° and tends to be symmetric about that orientation.
Effect of crack location
Crack location strongly affects the natural frequency of the simply supported square plate. The frequency is highest near a corner and lowest when the crack is centered.
- As the crack moves along an edge toward its center, the natural frequency monotonically decreases.
- The natural frequency is maximum when the crack is located at a corner.
- Along the plate center lines, frequency first increases from the edge toward an intermediate distance and then decreases.
- The natural frequency is minimum when the crack is situated at the plate center.
Plate with multiple cracks
For plates with two cracks, the study varies crack orientations under fixed crack spacing and lengths. Frequency increases with orientation and reaches its maximum when both cracks are at 90°.
- The two-crack configuration uses fixed horizontal separation H = 0.2 and vertical separation V = 0.1 between crack tips.
- For gradient index n = 5 and equal crack lengths a1 = a2 = 0.2, fundamental frequency is studied as a function of crack orientations.
- Increasing crack orientation raises the frequency, which reaches its maximum at θ1 = θ2 = 90°.
- The frequencies at θ = 0° and θ = 90° differ because the 90° cracks lie away from the plate center and disturb the mode shape slightly.
Effect of aspect ratio, thickness and boundary conditions
The study examines how aspect ratio, thickness, and boundary conditions affect the fundamental frequency of cracked FGM plates, including simply supported and clamped cases. For fixed crack length and location, frequency increases as thickness decreases, aspect ratio increases, or support stiffness changes from simply supported to clamped.
- Decreasing plate thickness a/h and increasing aspect ratio b/a increase the fundamental frequency for fixed crack length and location.
- Changing the boundary condition from simply supported to clamped increases frequency because the plate stiffness increases.
- The analyzed configuration is a cracked Si3N4/SUS304 FGM plate with a horizontal center crack, using simply supported and clamped supports.
- The cantilevered side-crack geometry is parameterized by plate dimensions, crack length d, crack position cy, and orientation θ.
5. Conclusion
The paper uses an extended finite element formulation to study natural frequencies of temperature-dependent FGM plates with cracks. Numerical studies vary crack, material-gradient, geometric, boundary, and multiple-crack parameters to identify their effects on frequency.
- The formulation uses first-order shear deformation theory and a four-noded field-consistent enriched element for cracked FGM plates.
- Increasing crack length decreases natural frequency, and frequency is lowest when the crack is located at the plate center.
- Increasing gradient index n decreases natural frequency because the metallic volume fraction increases.
- Decreasing plate thickness a/h and increasing aspect ratio b/a increase the frequency.
- For a cantilevered plate with a side crack, the horizontal crack has maximum frequency, with the trend changing at θ = ±40o.
- Crack orientation θ = 45o is identified as a critical angle where the frequency trend changes for a square plate.
- Increasing the number of cracks decreases overall plate stiffness and frequency, with the lowest frequency occurring when both cracks are oriented at θ = 50o.