Displacement Derivatives Research Articles

Transformation of surface [formula omitted]-forces in high grade elasticity

The expression of the elastic energy, postulated for a given material, specifies through the principle of virtual work which external loadings turn out to be admissible. For the elastic bodies, the energy of which depends on the gradients of the placement higher than (H+1) (or, equivalently, on the derivatives of the deformation gradient higher than H), external force densities can be prescribed over the boundary surface, which do work on the H-th derivative of the virtual displacements along the normal direction. Such generalized forces with H=1,2 have been dealt with in the literature, referred to as double and triple forces: when H is increased, however, the analytical evaluation of these work terms may become prohibitive. Each Eulerian H-force density of this kind, acting on the current boundary surface, turns out to be equivalent to a set of diverse actions defined in the Lagrangian configuration. These actions include: (i) surface terms doing work on the virtual displacement vector; (ii) surface terms doing work on the higher order derivatives of the virtual displacement along the normal direction; (iii) edge terms, doing work on the virtual displacement or on its higher order gradients, which may be further reduced. In this study, a recursive algorithm is utilized to generate all these Lagrangian contributions for any H. Such an approach rests on the binomial expansion of the tensor products of complementary surface projectors, symmetrized in the work duality, and on the integration by parts combined with the generalized divergence theorem.

Application of Slip Interface Conditions in the Elastic Parabolic Equation

The parabolic equation method is the most effective approach for solving wave propagation problems in environments with strong vertical variations and gradual horizontal variations. This approach efficiently provides accurate solutions for problems involving sloping fluid-fluid interfaces, sloping solid-solid interfaces, and sloping solid boundaries. There remains a need for improvement for the case of sloping fluid-solid interfaces. The most promising approach to date is the combination of a single-scattering approximation and the treatment of part of the fluid layer as a solid layer with low shear speed. This is a singular perturbation, which results in rapid oscillations near the interface between the low-speed layer and the true solid layer. It is demonstrated here that the rapid oscillations may be eliminated by using slip conditions at the interface between the low-speed layer and the true solid layer. Stability problems were encountered when attempting to directly implement the slip conditions, but this problem was resolved by using an equivalent condition involving the normal derivative of the tangential displacement. This progress does not fully resolve the case of the sloping fluid-solid interface. There remains a need to find a vertical interface condition that is compatible with the slip conditions and that provides accurate results for problems involving sloping fluid-solid interfaces.

Efficient equilibrium-based stress recovery for isogeometric laminated Euler–Bernoulli curved beams

Laminated curved composite parts, used, e.g., in the spar and ribs in aircraft and wind turbine blades, are typically subjected to high interlaminar stresses. This work focuses on a two-step procedure to study laminated Euler–Bernoulli curved beams discretized via Isogeometric Analysis (IGA). First, we solve a (planar) Euler–Bernoulli curved beam formulation in primal form to obtain the tangential and transverse displacements. This formulation features high-order PDEs, which we can straightforwardly approximate using either an IGA-Galerkin or an IGA-collocation approach. Starting from the obtained displacement solution, which accounts for bending-stretching coupling, we can directly compute the normal stress only, while we do not have information concerning the transverse shear stress state, typically responsible for delamination. However, by imposing equilibrium in strong form in a curvilinear framework which eases the post-processing, eliminating the need for coordinate changes, we can easily recover interlaminar transverse shear stresses at locations of interest. Such a posteriori step requires calculating the high-order displacement derivatives in the equilibrium equations and, therefore, demands once again higher-order regularity that can be easily fulfilled by exploiting the high-continuity properties of IGA. Extensive numerical tests prove the effectiveness of the proposed approach, which is also aided by the IGA’s superior geometric approximation.

Advanced Thick FGM Plates-Cylindrical Shells in Linear Unsteady Supersonic Flow

Introduction:: Thick functionally graded (FG) plates-cylindrical shells with third-order shear deformation theory (TSDT) of displacement models under thermal vibration in linear unsteady supersonic flow are investigated by using the generalized differential quadrature (GDQ) method. Method:: The nonlinear coefficient term of TSDT models and the advanced nonlinear shear correction coefficient expression are used to study the thermal vibration of thick FG plates and cylindrical shells in linear unsteady supersonic flow. At the intersection of FG plates-cylindrical shells, the displacements, stress resultants, and moment resultants could be reasonably assumed in the continuity condition for the differential volume in linear unsteady supersonic flow, respectively. Result:: The dynamic equilibrium differential equations of thick FG plates and cylindrical shells in linear unsteady supersonic flow, respectively in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads consist of the linear supersonic aerodynamic pressure and inertia terms can be presented. Two parametric effects including environment temperature and functionally graded material (FGM) power law index on the thermal stress and center displacement of thick FG plates-cylindrical shells in linear unsteady supersonic flow are investigated. Conclusion:: The linear unsteady supersonic flow over the outer surface of thick FG platescylindrical shells provides the additional aerodynamic effect on the values of displacements and stresses.

Measurement of out-of-plane first-order displacement derivatives in orthogonal shear directions using dichroic mirrors

Measurement of out-of-plane first-order displacement derivatives in orthogonal shear directions using dichroic mirrors

Data fusion based on short-term memory Kalman filtering using intermittent-displacement and acceleration signal with a time-varying bias

Accurate displacement information plays a vital role in various applications from structural health monitoring to damage detection in physical systems. Due to practical reasons, displacement sensing is often performed using sensors that provide less accuracy or a low sampling rate. This study proposes an algorithm for displacement signal estimation that fuses high sampling rate acceleration data (which consists of unknown time-varying bias) with low sampling rate displacement data. In contrast to the present literature, the state vector in the proposed state-space formulation consists of short-term memory of the past realizations of displacement and velocity, and bias terms which allow (i) online estimation of the displacements by avoiding multi-rate Kalman filtering, and (ii) estimate time-varying bias in the measured acceleration signal. The state equations comprise the finite difference approximations of the time derivative of velocity and displacement which act as a moving average filter to provide smooth estimates, thus preventing the need for offline smoothing procedures. By inclusion of bias terms in the state vector, the drifts in displacement and velocity caused due to time-varying bias in measured acceleration are easily eliminated. The proposed method was found to provide accurate estimates of displacement, velocity, and time-varying bias in measured acceleration signals when tested on different types of bias, including but not limited to constant, linear, and parabolic variations. Additional tests on three oscillators subjected to three earthquake ground motions showed significantly better state estimates for stiff oscillators and improved estimates for flexible oscillators. Experimental validation on a base-excited two-story frame further affirms the capability of the method to estimate the hidden bias in the measured acceleration and provide an accurate estimate of displacement.

A Sparsely-Measured Fitting Method for Identifying the Coefficients in the Governing Equation of a Beam

This study proposes a method for identifying the coefficients in the governing equation of a beam from the sparsely measured response of bridge displacement. The following three steps constitute the procedure: First, the full-field displacement is reconstructed by the compressive sensing theory, which is innovatively applied in the spatial domain with a proposed general basis matrix. Then, the reconstructed displacement is fitted by the B-spline surface basis functions, and the derived control points are further adopted to compute the derivatives of displacement. The genetic algorithm is finally applied to seek the coefficients of the governing equation by substituting the fitted response into the equation. The proposed method is validated numerically and experimentally, including a parametric study to evaluate the performance subjected to various practical conditions. The salient advantages of the proposed method lie in that it requires few sensors for measurement, and the derivatives of displacement are computed robustly.

MixTURE: L1-Norm-Based Mixed Second-Order Continuity in Strain Tensor Ultrasound Elastography.

Energy-based displacement tracking of ultrasound images can be implemented by optimizing a cost function consisting of a data term, a mechanical congruency term, and first- and second-order continuity terms. This approach recently provided a promising solution to two-dimensional axial and lateral displacement tracking in ultrasound strain elastography. However, the associated second-order regularizer only considers the unmixed second derivatives and disregards the mixed derivatives, thereby providing suboptimal noise suppression and limiting possibilities for total strain tensor imaging. We propose to improve axial, lateral, axial shear, and lateral shear strain estimation quality by formulating and optimizing a novel L1-norm-based second-order regularizer that penalizes both mixed and unmixed displacement derivatives. We name the proposed technique L1-MixTURE, which stands for L1-norm Mixed derivative for Total UltRasound Elastography. When compared to simulated ground truth results, the mean structural similarity (MSSIM) obtained with L1-MixTURE ranged 0.53 to 0.86 and the mean absolute error (MAE) ranged 0.00053 to 0.005. In addition, the mean elastographic signal-to-noise ratio (SNR) achieved with simulated, experimental phantom, and in vivo breast datasets, ranged 1.87 to 52.98, and the mean elastographic contrast-to-noise ratio (CNR) ranged 7.40 to 24.53. When compared to a closely related existing technique that does not consider the mixed derivatives, L1-MixTURE generally outperformed the MSSIM, MAE, SNR, and CNR by up to 37.96%, 67.82%, and 25.53% in the simulated, experimental phantom, and in vivo datasets, respectively. These results collectively highlight the ability of L1-MixTURE to deliver highly accurate axial, lateral, axial shear, and lateral shear strain estimates and advance the state-of-the-art in elastography-guided diagnostic and interventional decisions.

Alternating direction method of multipliers for displacement estimation in ultrasound strain elastography.

Ultrasound strain imaging, which delineates mechanical properties to detect tissue abnormalities, involves estimating the time delay between two radio-frequency (RF) frames collected before and after tissue deformation. The existing regularized optimization-based time-delay estimation (TDE) techniques suffer from at least one of the following drawbacks: (1) The regularizer is not aligned with the tissue deformation physics due to taking only the first-order displacement derivative into account; (2) The -norm of the displacement derivatives, which oversmooths the estimated time-delay, is utilized as the regularizer; (3) The modulus function defined mathematically should be approximated by a smooth function to facilitate the optimization of -norm. Our purpose is to develop a novel TDE technique that resolves the aforementioned shortcomings of the existing algorithms. Herein, we propose employing the alternating direction method of multipliers (ADMM) for optimizing a novel cost function consisting of -norm data fidelity term and -norm first- and second-order spatial continuity terms. ADMM empowers the proposed algorithm to use different techniques for optimizing different parts of the cost function and obtain high-contrast strain images with smooth backgrounds and sharp boundaries. We name our technique ADMM for totaL variaTion RegUlarIzation in ultrasound STrain imaging (ALTRUIST). ALTRUIST's efficacy is quantified using absolute error (AE), Structural SIMilarity (SSIM), signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and strain ratio (SR) with respect to GLUE, OVERWIND, and -SOUL, three recently published energy-based techniques, and UMEN-Net, a state-of-the-art deep learning-based algorithm. Analysis of variance (ANOVA)-led multiple comparison tests and paired -tests at overall significance level were conducted to assess the statistical significance of our findings. The Bonferroni correction was taken into account in all statistical tests. Two simulated layer phantoms, three simulated resolution phantoms, one hard-inclusion simulated phantom, one multi-inclusion simulated phantom, one experimental breast phantom, and three in vivo liver cancer datasets have been used for validation experiments. We have published the ALTRUIST code at http://code.sonography.ai. ALTRUIST substantially outperforms the four state-of-the-art benchmarks in all validation experiments, both qualitatively and quantitatively. ALTRUIST yields up to , , and SNR improvements and , , and CNR improvements over -SOUL, its closest competitor, for simulated, phantom, and in vivo liver cancer datasets, respectively, where the asterisk (*) indicates statistical significance. In addition, ANOVA-led multiple comparison tests and paired -tests indicate that ALTRUIST generally achieves statistically significant improvements over GLUE, UMEN-Net, OVERWIND, and -SOUL in terms of AE, SSIM map, SNR, andCNR. A novel ultrasonic displacement tracking algorithm named ALTRUIST has been developed. The principal novelty of ALTRUIST is incorporating ADMM for optimizing an -norm regularization-based cost function. ALTRUIST exhibits promising performance in simulation, phantom, and in vivoexperiments.

Integration of digital shearography and FEM for non-destructive evaluation of internal cracks in GRE pipes

Pipes made of glass fiber-reinforced epoxy (GRE) are often exposed to complex environmental conditions, which can make them prone to internal cracking. Quantitative, non-destructive evaluation of cracks can reduce unintended failures and improve system reliability. The use of digital shearography, a modern non-destructive method based on laser interferometry, in the diagnosis of composite materials has drawn attention in recent years. In this paper, a model for the quantitative evaluation of subsurface cracks in various locations on GRE pipes was developed by integrating the finite element method and shearography with thermal stimulation. The response surface method was used to design the simulations, and the effect of crack dimensions and location on the phase map and out-of-plane surface strain component was investigated. The results demonstrated that the crack placement angle relative to the shearography device is effective on crack detectability due to the change in the actual shearing size. A model was then presented to estimate the length and depth of the crack based on the displacement derivative graph by using analysis of variance and regression techniques. By determining the out-of-plane displacement derivative, the proposed model enables precise estimation of the dimensions and location of subsurface cracks.

УТОЧНЁННАЯ ЛИНЕЙНАЯ ТЕОРИЯ УПРУГОСТИ С УЧЁТОМ КВАДРАТИЧНЫХ СЛАГАЕМЫХ

A variant of the linear theory is proposed, based on taking into account in Taylor expansions for stresses and in power series and the expansion of the radical into a binomial series for deformations of not only linear, but also quadratic terms. In this case, static equilibrium equations in stresses are written in the form of second-order partial differential equations. The law of pairing of tangential stresses is observed. The relationship between deformations and displacements is described by relations similar to Cauchy relations, but taking into account the second derivatives of displacements along spatial coordinates. The equations of continuity of deformations for this version of the theory of elasticity are written in the form of two groups of relations. The first group of relations establishes the continuity equations for the linear terms of the deformation components; The second group of relations establishes the continuity equations for the quadratic terms of the deformation components. The physical equations for this variant of the linear theory of elasticity are written in the same way as for the classical linear theory of elasticity. Equilibrium equations in stresses are represented by second-order partial differential equations. The equilibrium equations in displacements are written in the form of fourth-order partial differential equations. The equations of equilibrium, as well as the equations of continuity of deformations, along with other parameters – the physical constants of the medium – contain small parameters dx, dy, dz, the magnitude of which, as shown by numerical modeling, has little effect on the nature of the stress-strain state. To determine them, it is necessary, apparently, to use experimental data. The construction of resolving equations in both stresses and displacements for special cases of the stress-strain state of an elastic continuous medium is considered: uniaxial dressed-up state; uniaxial deformed state; plane strain; generalized plane stress state. The problem of the uniaxial deformed state of a continuous medium (solution in stresses) is numerically solved. The proposed version of the linear theory of elasticity extends the classical linear theory of elasticity and, with appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

Minor and Major Strain: Equations of Equilibrium of a Plane Domain with an Angular Cutout in the Boundary

Large values and gradients of stress and strain, triggering concentrated stress and strain, arise in angular areas of a structure. The strain action, leading to the finite loss of contact between structural elements, also triggers concentrated stress. The loss of contact reaches an irregular point and a line on the boundary. The theoretical analysis of the stress–strain state (SSS) of areas with angular cutouts in the boundary under the action of discontinuous strain is reduced to the study of singular solutions to the homogeneous problem of elasticity theory with power-related features. The calculation of stress concentration coefficients in the domain of a singular solution to the elastic problem makes no sense. It is experimentally proven that the area located near the vertex of an angular cutout in the boundary features substantial strain and rotations, and it corresponds to higher values of the first and second derivatives of displacements along the radius in cases of sufficiently small radii in the neighborhood of an irregular boundary point. As far as these areas are concerned, it is necessary to consider the plane problem of the elasticity theory, taking into account the geometric nonlinearity under the action of strain, to analyze the effect of relationships between strain orders, rotations, and strain on the form of the equation of equilibrium. The purpose of this work is to analyze the effect of relationships between strain orders, rotations, and strain on the form of the equilibrium equation in the polar system of coordinates for a V-shaped area under the action of temperature-induced strain, taking into account geometric non-linearity and physical linearity.

KINEMATIC ANALYSIS OF CONSTANT BREADTH CAM DRIVEN LINKAGES

Several constant breadth curves are defined that can be used as cam profiles in constant breadth cam mechanisms that are closed cam mechanisms. There are two objectives for this study. One of them is to study the kinematic analysis of different type of constant breadth cam mechanisms. The other objective is to obtain a dwell period for constant breadth cam driven linkages that is impossible for a standard cam mechanism. A general kinematic analysis of a constant breadth cam mechanism with translating flat-faced follower was carried out with the principle of kinematic inversion. With the results, the kinematic analyses of the constant breadth cam driven inverted slider crank mechanism and four bar mechanism were examined in detail and a general method is given for all constant breadth cam profiles and cam driven linkages. It has been seen that a dwell period of 45° (with the fixed joint coordinates as x_n = 18 mm and y_n= 8.5 mm) and 40° (with the fixed joint coordinates as x_n = 18.5 mm and y_n= 8.5 mm) can be obtained in designed cam driven four bar and inverted slider crank mechanism respectively. After the displacement analysis, some velocity and acceleration analysis examples are given by taking the derivative of displacement. Similar kinematic analyses are possible for cam-driven mechanisms with more links. Also, it has been seen that changing the location of fixed joint of the cam profile can affect the displacement, velocity and acceleration graphics of the mechanism. With this, the dwell period can be changed too.

GDQ computation for thermal vibration of thick FGM plates by using third-order shear deformation theory

The third-order shear deformation theory (TSDT) effects on the thick functionally graded material (FGM) plates under thermal vibration are investigated by using the generalized differential quadrature (GDQ) method. The nonlinear coefficient term of displacement field of TSDT is used to derive the equations of motion for thermal vibration of thick FGM plates. The simpler form stiffness of FGM plates under linear temperature rise is considered in the thermo-elastic stress–strain relations. The dynamic equilibrium differential equations of FGM plates in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads and inertia terms are derived. Two parametric: environment temperature and FGM power law index effects on the thermal stress and center deflection of FGM thick plates are investigated.

Irreducible representation of surface distributions and Piola transformation of external loads sustainable by third gradient continua

In this paper Piola transformations are found that relate the Eulerian and Lagrangian external loads which third gradient continua can sustain. As shown by Gabrio Piola and Paul Germain, the most effective postulation scheme in mechanics is based on the principle of virtual work and therefore continuum mechanics must be mathematically founded based on the theory of distributions and on differential geometry. Using the principle of virtual work, the set of admissible external loads sustainable by third gradient continua is seen to include: i) volume force density, ii) surface density of contact force, iii) surface density of contact double force, iv) surface density of contact triple force, v) line density of edge contact forces, vi) line density of contact edge double forces and vii) contact forces concentrated on wedge points. Following the nomenclature introduced by Paul Germain, forces are dual in virtual work of virtual displacements, surface and line double forces are dual of the derivatives of virtual displacements in the normal direction(s) of the surfaces and edges constituting the boundary of the continuum, and surface triple forces are dual of the second normal derivatives of virtual displacements. Volume and surface forces transform as in first gradient Cauchy continua. Moreover we find that: a) the virtual work expended by Eulerian surface triple force, when transformed into the Lagrangian description, must be represented as the work expended by all the kinds of external Lagrangian loads listed in i)-vii); b) Eulerian surface double force transforms into Lagrangian surface double force, surface contact force and edge contact line force; c) Eulerian edge contact line double force transforms into Lagrangian edge contact line double forces, edge line forces and point concentrated wedge forces; d) Eulerian edge and wedge contact line forces transforms into their Lagrangian counterpart only. The Piola transformation formulas deduced in this paper depend on the first, second and third gradients of placement. The presented results allow for the formulation of well-posed boundary condition problems for third gradient continua in the Lagrangian description, and are relevant in computational mechanics. In view of the obtained Piola transformation formulas, the concept of dead loads needs to be modified. We believe to have given an example of how the Mechanics in the French Style, as developed on the ideas by D’Alembert and Lagrange, is still a fertile tool of invention.

A bond-based peridynamics model based on variable material properties for modeling elastoplastic behavior

Crack tip singularities and material discontinuities are taken to be the main challenges for analyzing cracks in fracture mechanics problems. Peridynamic (PD) theory has been recently introduced as a nonlocal extension of continuum mechanics capable of improved modeling of progressive damage and rupture in cracked structures. In this theory, the internal forces have been characterized by nonlocal interaction between the pairs of the material points whose displacements or displacement derivatives are discontinuous. Given that the bond-based PD (BB-PD) has not been satisfactorily employed for predicting ductile fracture, the main purpose of this work is to present a new PD plasticity model in conjunction with the Variable Material Property (VMP) method, aiming to provide the possibility of elastoplastic modeling in BB-PD. In the present work, the quasi-static loading has been selected for simulation and the von-Mises yield criterion has been utilized for describing the plastic deformation. The loadings are applied as the displacement is controlled and outputs from the PD model are compared with the results achieved from ABAQUS software based on the continuum mechanic assumptions. To validate the accuracy of the present formulation, two example problems of the plate with a central crack and a central hole subjected to tensile load are examined. The results of von-Mises stress, the plastic zone size, horizontal and vertical displacement, and equivalent plastic strain have been verified in comparison with the outputs calculated by the finite element method (FEM), evidencing good accuracy of the VMP-based PD elastoplastic model in predicting the desired parameters. To predict the formation of spontaneous cracks and crack growth trajectory, two more examples that include a plate with a central hole, as well as, a plate with two edge cracks and a double hole are discussed. The obtained results are in good agreement with those from the experiment and ABAQUS software.

Free vibration response of bidirectional functionally graded rotating micro-disk under mechanical and thermal loading

The present work studies the free vibration response of functionally graded rotating micro-disks subjected to transverse pressure and thermal loading based on the modified couple stress theory. The disk material is considered to be functionally graded along the radial and thickness directions, and its properties are assumed to be temperature-dependent following the Touloukian model. The mathematical formulation is based on an energy functional involving the von Kármán type non-linearity, in which appropriate displacement derivatives and its conjugate stress measures are used to define the strain energy of the micro-disk. The minimum potential energy principle is employed to develop the governing equations for determining the deformed configuration of the micro-disk under combined centrifugal, pressure and thermal loading. Further, the governing equations for free vibratory motion of the micro-disk are derived following Hamilton’s principle and incorporating the tangent stiffness of the deformed micro-disk. The governing equations are discretized and solved employing the Ritz method. The mathematical model is successfully validated with different reduced problems available in the literature. The influence of rotational speed, transverse pressure, thermal loading, size-dependent thickness and volume fraction indices are investigated for a wide range of parametric values. Some illustrative mode shapes along with the contour have also been presented. The present study is first of its kind and the presented results would definitely serve as benchmarks for any further study in this field.

Nonlinear aerodynamic analysis of functional graded plates using an HSDT-based isogeometric approach

In this paper, a nonlinear FGM plate analysis model is presented based on IGA, which suppresses shear locking and is applicable to plates with a wide range of thickness-to-length ratios. Furthermore, the nonlinear aerodynamic characteristics of FGM plate subjected to the combined effect of transverse load, in-plane load, and supersonic flow are investigated. HSDT is used to construct a displacement field suitable for nonlinear vibration analysis of FGM plates with a wide range of thickness-to-length ratios. For the problem that shear locking cannot be avoided entirely in the nonlinear analysis of plates, the first derivative of transverse displacement is adopted as the first term of HSDT, which weakens the effect of shear strain. Therefore, the shear locking is effectively restrained. The nonlinear FGM plate analysis model is derived using NURBS, which meets the C1 continuity requirement of HSDT. The computational efficiency, accuracy and the suppressing capacity to shear locking are verified in three aspects: static displacement, eigenfrequency, and nonlinear frequency ratio. The Bathe method is used to obtain the displacement response of an FGM plate subjected to supersonic flow, in-plane load, and transverse load. The dynamic characteristics of the FGM plate are studied using time history, bifurcation diagrams, PSD curves, phase diagrams, and Poincaré maps.

Analysis of Plane Displacement of Pressure Vessels with Defects

Aiming at the problem that the internal defects of the pressure vessel are not easy to detect, the out-of-plane displacement method is used to study it. The out-of-plane displacement field of circular flat plate heads with defects is obtained by the method of elastic theory analysis and compared with the finite element method to verify the accuracy of the finite element method. Based on the finite element method, the effects of vessel material, pressure load, defect shape, and defect size on the first-order out-of-plane displacement derivative are studied. The results show that the material of the container has little effect on the out-of-plane displacement. The first derivative of out-of-plane displacement increases linearly with the increase of pressure, and this relationship will not change with the increase of pressure. With the increase of defect aspect ratio, the maximum value of the first derivative of out-of-plane displacement increases gradually. When δ ˃ 9, the defect is detectable. The research results lay a theoretical foundation for the internal defect detection of pressure vessels based on digital speckle shearing interferometry.

Dynamic analysis of graphene-reinforced sandwich plates with piezoelectric macro fiber composite face sheets

Graphene is considered as an ideal candidate for composite materials owing to the outstanding physical characteristics. However, existing higher-order shear deformation theories in the literature may meet severe challenges for the accurate prediction of natural frequencies of graphene-reinforced sandwich plates with piezoelectric macro fiber composite (MFC) face sheets. If transverse shear deformations cannot be described accurately, the dynamic behaviors of piezoelectric sandwich plates with graphene reinforcements will be significantly impacted by the large differences of material properties at interfaces of adjacent layers and electro-mechanical coupling characteristic. Thereby, a novel electro-mechanical coupling theory is to be proposed for the free vibration of graphene-reinforced sandwich plates with MFC face sheets. Based on the Reissner mixed variational theorem (RMVT), the precision of the interlaminar shear stresses including the electro-mechanical coupling effect can be improved. The modified interlaminar shear stress field is absorbed in the strain energy, which can enhance the accuracy of the free vibration response of piezoelectric sandwich plates. Additionally, the second-order derivatives of in-plane displacements have been eliminated from the interlaminar shear stress components, which can substantially simplify the finite element implementation. Thus, a simple C0-type finite element formulation is developed for the free vibration of piezoelectric sandwich plates with graphene reinforcements. The 3D elasticity solutions and the results obtained from other theories are used to evaluate the performance of the proposed theory. In comparison with the existing higher-order theories, the proposed theory can produce more accurate results. In addition, a comprehensive parametric study is conducted to explore the impacts of significant parameters on the free vibration behaviors of piezoelectric graphene-reinforced sandwich plates.