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Printed Edition of the Special Issue Published in Mathematical and Computational Applications

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Nguyễn Gia Hào

Academic year: 2023

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Porous functionally graded panels: Evaluation of the effect of shear correction factor on static behavior. Porous functionally graded plates: Evaluation of the effect of the shear correction factor on the static behavior.Math.

Porous Functionally Graded Plates: An Assessment of the Influence of Shear Correction Factor on

Static Behavior

Introduction

Functionally graded porous materials (FGPs) combine both porosity and functional gradient characteristics, where the porosity can have a graded evolution across the volume, providing desirable properties for some applications (as in biomedical implants), and undesirable in others, where voids can cause serious problems (as in the aerospace sector). To the best of our knowledge, there are no previously published works that focus on the assessment of the influence of the shear correction factor in the static bending behavior of porous plates.

Materials and Methods

In this model, the material properties, namely the Young's modulus and Poisson's ratio, are estimated by equations (8) and (9), and the graph of the evolution of the normalized Young's modulus through the thickness is presented in Figure 4. 53] calculated the shear correction factor as a function of the exponent of the power law, the ratio of thickness to length (a/h) and some constant coefficients that depend on the material phases involved.

Figure 1. Example of dual-phase functional gradient material (FGM) mixture, with variation in one direction.
Figure 1. Example of dual-phase functional gradient material (FGM) mixture, with variation in one direction.

Results

The case studies show the influence of the different porosity distribution models on the neutral surface displacement, shear correction factor and maximum transverse displacement for a range of power law exponents. For other power law exponents, as the maximum porosity values ​​increased, the neutral surface shift decreased and the shear correction factor increased.

Table 2. Results of the porosity distribution model 1 (M1) verification study.
Table 2. Results of the porosity distribution model 1 (M1) verification study.

Discussion and Conclusions

Mechanical analysis of functionally graded composite beams reinforced with graphene oxide based on first-order shear deformation theory.Mech. Free vibrations of functionally graded rectangular plates using first-order shear deformation plate theory. Appl. A simple first-order shear deformation theory for bending and free vibration analysis of functionally graded plates. Compos.

Higher-order displacement theory for static analysis of functionally graded plates with porosity. Comptes Rendus Mécanique. Bending, free vibration and buckling analysis of functionally graded porous microplates using a general third order plate theory.J.

A Continuation Procedure for the Quasi-Static Analysis of Materially and Geometrically

Preliminaries

The system of Equation (1) can be seen in terms of continuation techniques after noting that the set of equations implicitly defines a curve of solution points that can be continuously parameterized by means of the parameter ˜λ. One possibility is given by the adoption of the load factorλ, which leads to a power control approach. In fact, despite the similarity between the two strategies, the correction process in the continuation methods thrives on the powerful contractive properties of the solution set H−1(0) for iterative methods such as Newton's method[21].

It can be observed that the system of equation (3) involves the presence of derivatives with respect to the parameters, which is one of the main differences between arc length and continuation methods. The performance of the method can be modified by applying different strategies to make the prediction or the correction.

Critical Points

The critical points following the first one are captured by monitoring the sign of the lowest positive eigenvalue. Npos(si)

Once critical points are identified and classified, an approximation of the critical state is needed. It is based on the polynomial approximation of the critical point ˜x(s∗), and seeks the zero of ωk(s).

Branch-Switching Method

Δλ(λ∗−λ(si−1)) (16) This strategy is fast and tends to guarantee satisfactory accuracy for obtaining an estimate of the limit points. Regarding the approximation of the bifurcation points, an iterative procedure is implemented to obtain improved accuracy. It relies on the idea of ​​perturbing the configuration at the critical state using the eigenvector vk associated with the eigenvalueωk(s∗) =0 [27].

It is emphasized that a proper choice of ˜τ is needed to guarantee convergence on the desired branch. To address this problem, it is necessary that the branch-switching algorithm allows restarting the analysis of the bifurcation branch with a different value of ˜τ.

Path-Following Continuation Technique

However, the two equations differ in the term1 of equation (34)b, namely in the derivative of the displacement vector in the last converged equilibrium point, which is replaced by the incremental displacement of the predictor step Δa1 in the Riks formulation. Moreover, equation (36) can be thought of as a projection of the Riks constraint equation onto a subspace of displacements. Based on the sign assumed by β, the orientation is defined by flipping ˜x when β<0.

As observed in the previous section, the orientation of the tangent vector is not available from the solution of equation (49), and the evaluation of the parameter β of equation (38) is thus performed. It is noted that the adoption of the two mentioned strategies determines the presence of a double parametrization, since the modified Riks continuation method is parametrized with respect to the arc lengths and the dissipated energy limitation equation is dependent on the parameterτ.

Implementation

The successive step consists in the evaluation of the dissipated energy, which is subsequently used to define the continuation algorithm. The evaluation of the tangent vector requires the solution of the linear systems according to equation (37) or (49), while the parameter β of equation (38) is used to define the orientation. The successive phase consists in detecting the critical points, which is performed by evaluating the eigenvalues ​​of the stiffness matrix K.

Depending on the number of positive eigenvectors and the value of the stiffness parameter, it can be determined whether a critical point is reached and its nature, i.e. bifurcation or boundary. The method to be used in the subsequent step is defined by comparing the deposited energy Δτ with the threshold values ​​defined by the user.

Results

The procedure is then switched to the continuation procedure and the equilibrium path tracing is resumed from the last available equilibrium point. In the second stage of the solution process, the equilibrium branches starting from the bifurcation points are calculated. In this test, the advantages offered by using the hybrid constraint are more clear.

In contrast, the hybrid implementation is able to cover the following parts of the equilibrium path. The results give a clear picture of what was highlighted by examining the force-displacement response.

Figure 2. Sketch of the truss-spring system.
Figure 2. Sketch of the truss-spring system.

Conclusions

A semi-analytical model for local post-buckling analysis of string-frame braced cylindrical panels. Thin-walled structures. An adaptive model reduction strategy for postbuckling analysis of braced structures. Thin-walled structure. Experimental and numerical post-buckling analysis of thin aluminum aeronautical panels under shear loading.

A nonlinear finite element approach for the analysis of mode-I free edge delamination in composites. Int. Hybrid geometric-dissipative arc length methods for the quasi-static analysis of delamination problems. Comput.

Hydrodynamic and Acoustic Performance Analysis of Marine Propellers by Combination of Panel Method

  • Formulation of the Panel Method
  • Validation of Results of Panel Method
  • Acoustic Formulations
  • Noise of DTMB 4119 Propeller
  • Conclusions

Also, they used the combination of time-domain acoustic analogy and the panel method for the numerical investigation of propeller noise. On the right-hand side of equation (1), the integration is done on the surface S, which includes the body surface, the Kutta strip, and the wake surfaces of the object (Figure 1). In this research, the combination of the panel method and the FW-H equations is used to calculate the non-cavitating noise of marine propellers.

As the distance from the propeller increases, propeller noise is significantly reduced. According to the above results, the combination of the panel method and the FW-H equations can be used for the hydrodynamic and acoustic analysis of marine propellers.

Figure 1. Body surface, Kutta strip and wake surface.
Figure 1. Body surface, Kutta strip and wake surface.

Natural Frequency Analysis of Functionally Graded Orthotropic Cross-Ply Plates Based on the Finite

Finite Element (FE) Formulation for Laminated Thick and Thin Plates

The degrees of freedom (2) are approximated in each element by quadratic Lagrange interpolation functions. Once the orientation of the fibers has been defined, the following relations are used to calculate the coefficients Q(k)ij. In the current study, a non-uniform distribution of the fibers along the sheet thickness is defined.

The values ​​of the roots of Legendre polynomials and the corresponding weight coefficients used in the numerical integration are listed in Table 1. Finally, u¨ is the vector collecting the second-order time derivatives of the nodal displacements.

Figure 1. Nine-node quadratic Lagrange rectangular element.
Figure 1. Nine-node quadratic Lagrange rectangular element.

Numerical Applications

Convergence features of the numerical approach and comparison of the first ten natural frequencies (Hz) with the semi-analytical solutions provided by Reddy [23] for a single supported thin plate with a uniform distribution of the reinforcing fibers through the thickness. Convergence features of the numerical approach and comparison of the first ten natural frequencies (Hz) with the semi-analytical solutions provided by Reddy [23] for a single supported thick plate with a uniform distribution of the reinforcing fibers through the thickness. The first fourteen natural frequencies of a single supported thin plate for the different thickness distributions of the reinforcing fibers specified in Table 5 are presented in Table 6, whereas Table 7 collects the same results for a single supported thick plate.

Note that the mode shapes have taken on different aspects by changing the distributions through the thickness of the fibers in the four layers, keeping their orientation constant. First fourteen natural frequencies (Hz) of a simply supported thin plate for various through-thickness distributions of the reinforcing fibers.

Table 3. Convergence features of the numerical approach and comparison of the first ten natural frequencies (Hz) with the semi-analytical solutions provided by Reddy [23] for a simply-supported thin plate with a through-the-thickness uniform distribution of
Table 3. Convergence features of the numerical approach and comparison of the first ten natural frequencies (Hz) with the semi-analytical solutions provided by Reddy [23] for a simply-supported thin plate with a through-the-thickness uniform distribution of

Conclusions

Bending, free vibration and buckling of modified pairs of stress-based functionally graded porous microplates.Compos. Free Vibration Analysis of Functionally Graded Conical, Cylindrical Shell and Annular Plate Structures with a Four-Parameter Power-Law Distribution.Comput. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells.Compos.

Elasticity solution of functionally graded carbon nanotube reinforced cylindrical composite panel subjected to thermomechanical loading. Free vibration of reinforced carbon nanotubes (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method.Compos.

Hình ảnh

Figure 2. Evolution of the volume fraction through the thickness direction.
Figure 4. Variation of the normalized Young’s modulus (E(z)) through the thickness for porosity model M2.
Figure 3. Normalized: (a) porosity model M1 and (b) Young’s modulus (E(z)) through the thickness direction.
Figure 5. Porosity model M3: normalized Young’s modulus (E(z)) through the thickness.
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