Mindlin Theory Research Articles

The Immersed Boundary Conformal Method for Kirchhoff–Love and Reissner–Mindlin shells

This work utilizes the Immersed Boundary Conformal Method (IBCM) to analyze linear elastic Kirchhoff–Love and Reissner–Mindlin shell structures within an immersed domain framework. Immersed boundary methods involve embedding complex geometries within a background grid, which allows for great flexibility in modeling intricate shapes and features despite the simplicity of the approach. The IBCM method introduces additional layers conformal to the boundaries, allowing for the strong imposition of Dirichlet boundary conditions and facilitating local refinement. In this study, the construction of boundary layers is combined with high-degree spline-based approximation spaces to further increase efficiency. The Nitsche method, employing non-symmetric average operators, is used to couple the boundary layers with the inner patch, while stabilizing the formulation with minimal penalty parameters. High-order quadrature rules are applied for integration over cut elements and patch interfaces. Numerical experiments demonstrate the efficiency and accuracy of the proposed formulation, highlighting its potential for complex shell structures modeled through Kirchhoff–Love and Reissner–Mindlin theories. These tests include the generation of conformal interfaces, the coupling of Kirchhoff–Love and Reissner–Mindlin theories, and the simulation of a cylindrical shell with a through-the-thickness crack.

Density-Based Isogeometric Topology Optimization of Shell Structures

Shell structures with high stiffness-to-weight ratios are desirable in various engineering applications. Topology optimization serves as a popular and effective tool for generating optimal shell structures. The solid isotropic material with penalization (SIMP) method is often chosen because of its simplicity and convenience. However, SIMP method is typically integrated with conventional Finite Element Analysis (FEA) which has limitations in computational accuracy. Achieving high accuracy with FEA necessitates a substantial number of elements, leading to computational burdens. In addition, the discrete representation of the material distribution function may result in rough boundaries. Owing to these limitations, this paper proposes an Isogeometric Analysis (IGA) based SIMP method for optimizing the topology of shell structures based on Reissner–Mindlin theory. This method uses Non-Uniform Rational B-Splines (NURBS) to represent both the shell structure and the material distribution function with the same basis functions, allowing for higher accuracy and smoother boundaries. The optimization model takes compliance as the objective function with a volume fraction constraint and the coefficients of the density function as design variables, resulting in an optimized shell structure defined by the material distribution function. To obtain fairing boundaries of the holes in the optimized shell structure, further process is conducted by fitting the boundaries with fair B-spline curves automatically. Furthermore, the proposed IGA-SIMP framework is applied to generate porous shell structures by imposing different local volume fraction constraints. Numerical examples are provided to demonstrate the feasibility and efficiency of the IGA-SIMP method, showing that it outperforms the FEA-SIMP method and produces smoother boundaries.

Thermomechanical bending of functionally graded carbon nanotubes reinforced composite plate by meshless method

AbstractFunctionally graded (FG) carbon nanotubes (CNTs) reinforced composite possessing continuously varied material properties have received extensive concern in the area of aviation, automotive, military, and spaceflight. The nonlinear bending of FG CNT composite is performed by developing a novel meshless model on the base of Smoothed Particle Hydrodynamics (SPH) method. The equivalent temperature relative material properties of composite is established in the light of the extended rule of mixture model. Different gradient dispersions of CNTs are taken into account and modeled. The governing equations are explored using the Mindlin theory and discretized though a symmetric SPH method with high prediction ability. The developed model is verified by analyzed several examples and compared to the solution obtained in literature. Influences of CNT dispersion, content, plate aspect ratio, boundary conditions and lamination angle on the bending response are elaborated. The present study provides an alternative potential method to solve the mechanical performances of FG CNT composites based on the meshless SPH formulation.Highlights Thermo‐mechanics of FG‐CNTRC is firstly solved by SPH method. The model is verified by solving the benchmarks in literature. Nonlinear bending behaviors of FG‐CNTRC are demonstrated. The minimum deflection take place in FG‐X type composite. For laminate plate, the smallest deformation occurs in 0° CNTRC.

Isogeometric Topology Optimization of Multi-patch Shell Structures

Shell structures refer to structural elements that derive strength and load-bearing capacity from their thin and curved geometry. In practical applications, shell structures are commonly composed of multiple patches to represent intricate and diverse architectural configurations faithfully. Nevertheless, the design of multi-patch shell structures holds considerable promise. However, most of the previous work is devoted to the numerical analysis of multi-patch shell structures without further optimization design. The work proposes an inverse design framework, specifically focusing on multi-patch configurations based on Reissner–Mindlin theory. First, reparameterization and global refinement operations are employed on the provided multi-patch shell structures. Renumbering the indices of control points with shared degrees of freedom at the interface naturally ensures C0-continuity between patches. Subsequently, this study investigates the amalgamation of Isogeometric Analysis (IGA) and the Solid Isotropic Material with Penalization (SIMP) method for topology optimization of shell structures. The proposed approach is validated through numerical examples, emphasizing its capacity to enhance multi-patch shell structure design, showcasing robustness and efficiency.

Equivalent single‐layer Mindlin theory of laminated piezoelectric plates and application

AbstractBased on the Mindlin first‐order shear deformation theory, this paper proposes an equivalent single layer (ESL) plate theory to analyze the electro‐mechanical coupling problem of laminated piezoelectric plates (LPPs). The main features of the proposed approach are: (i) It assumes that the electric potential across the thickness is a polynomial function, ensuring its continuity at the interface. (ii) The electric displacements are continuous at the interface, in line with the interface continuity condition between laminated plates. The theoretical solutions for the deformation and electric potential of LPPs are obtained. The validity and accuracy of the theoretical solutions are confirmed through comparison with results of two‐ and four‐layer LPPs obtained from the three‐dimensional finite element method (FEM). The numerical results discuss the influence of different series expansions and emphasize the necessity of high‐order expansion. Meanwhile, the range of application of three‐dimensional FEM is discussed. It is expected that such a new analytical method can be instructive to the optimal design of piezoelectric device.

Rotating Sonotrode Design for Ultrasonic-Assisted Arc Welding of Metal Materials.

In the process of the ultrasonic-assisted arc welding of metal materials, traditional ultrasonic application methods, such as the low-frequency impact of ultrasonic horns on a base material, can easily cause the non-fusion defect. In order to solve this problem, a rotating sonotrode with a groove and double thin ends was designed in this study. The ultrasonic vibration is transmitted into the weld pool by the rolling of the sonotrode on both sides of the weld. The resonant frequency was set at 50 kHz. Firstly, based on the Mindlin theory, a rotating sonotrode without a groove was designed by solving the frequency equation and by conducting a finite element simulation. Secondly, the effects of the groove, perforation, and transition mode on the resonant frequency, stress distribution, and amplification factor were investigated by finite element simulation. Finally, the optimum rotating sonotrode with a groove was obtained. The results show that the size of a rotating sonotrode that has a small frequency error can be obtained by using the discrete interval solver method combined with finite element simulation. The groove can significantly reduce the resonant frequency. The stress concentration can be effectively reduced by using the elliptical transition mode. The resonant frequency and amplification factor of a rotating sonotrode with a groove could be effectively adjusted by a method of double-position joint perforation. The final resonant frequency was 49.721 kHz and the amplification factor was 3.02. This study provides an effective design method for a sonotrode with double thin ends and a groove structure.

On mechanics of piezocomposite shell structures

This study presents an original and novel investigation into the mechanics of piezo-flexo-magneto-elastic nanocomposite doubly-curved shells (PFMDCSs) and the ability to detect the lower and higher levels of electro-magnetic fields. In this context, by utilizing the first-order shear deformation shell model, stresses and strains are acquired. By imposing Hamilton's principle and the von Kármán approach, the governing equations have been obtained. The intelligent shell model consists of size-dependent influences, viz., strain gradients. This will take place via Mindlin's strain gradient elasticity theory and the subsequent re-establishing of the mathematical framework by incorporating this concept. The strain gradient results in a flexoelectric/flexomagnetic effect. The converse effect of the magnetic field on the basis of a close circuit has been assumed. The developed bending equations have been transferred into the algebraic ones by substituting an analytical technique based on homogeneous immovable simple support for the four edges. The problem has been solved according to the Newton-Raphson iteration scheme, and transverse deflections have been computed. For researching the rightness and precision of the shell models together with the solution process, a comparison is prepared by the finite element method (FEM) results for simplified shells, and a good correlation has been observed. At last, by examining several factors governing the problem, the conditions under which the magnetic effects can be noticeable and dominant in doubly-curved shells have been sought. This study could serve as a benchmark reference for piezoceramic-DCSs, as the presented governing equations are original. The most interesting outcome of this research is that the electro-magnetic response of intelligent structures can be entirely geometry-dependent.

Free-Vibration Analysis for Truncated Uflyand–Mindlin Plate Models: An Alternative Theoretical Formulation

Plates are flat structural elements whose thickness is small in relation to the size of the surface. Their use may include engine foundations, reinforced concrete bridge elements or parts of various floating structures. Consequently, knowledge of their mechanical behavior under static and dynamic loads is of primary importance in engineering applications and of interest from a structural point of view. As a result, numerous works existing in the literature have investigated the mechanical properties of plates using various plate models, such as Reissner’s theory, Levinson’s theory, Kirchhoff’s theory and Mindlin’s theory, and their static and dynamic behavior has been examined. In the present paper the truncated Uflyand–Mindlin plate equation is proposed. According to Uflyand–Mindlin theory, an alternative theoretical formulation is presented for the free-vibration analysis of plates, and the equations of motion and the general corresponding boundary conditions are derived. This paper develops the truncated Uflyand–Mindlin plate equation, i.e., without the fourth-order derivative, by means of the direct method and variational formulation. The first-order shear deformable plate theory developed by Elishakoff, which takes into account rotational inertia and shear deformation and does not include a fourth-order time derivative, is variationally derived here. This derivation complements that performed by Mindlin some 70 years ago. The innovative aspect of the suggested strategy is that variational and direct methods for studying plate dynamics are analogous. Finding the third equation of the reduced Uflyand–Mindlin equations, the accompanying boundary conditions and their mathematical resemblance are the goals of the presented formulations. In order to solve the dynamic equilibrium problem of a truncated Uflyand–Mindlin equation via a variational formulation, it is demonstrated that the differential equations and the corresponding boundary conditions have the same form as those found using the direct technique. This paper successfully completes this task. Finally, in order to validate the effectiveness and correctness of the proposed procedure, a numerical example of the case of a plate simply supported at all four ends is proposed.

Ispitivanje vršeno savijanjem napredne generacije kompozitnih struktura sa specifičnim svojstvima izloženih različitim opterećenjima

Introduction/purpose: This article presents the bending examination of advanced-generation composite structures with specific properties exposed to different loads. Methods: This paper thus proposes and introduces a new generalized five-variable shear strain theory for calculating the static response of functionally graded rectangular plates made of ceramic and metal. Notably, our theory eliminates the need for a shear correction factor and ensures zero-shear stress conditions on both the upper and lower surfaces. Numerical investigations are introduced to interpret the influences of loading conditions and variations of power of functionally graded material, modulus ratio, aspect ratio, and thickness ratio on the bending behavior of FGPs. These analyzes are then compared to the results available in the literature. Results: Preliminary results include a comparative analysis with standard higher-order shear deformation theories (PSDPT, ESDPT, SSDPT), as well as Mindlin and Kirchhoff theories (FSDPT and CPT). Conclusion: Our theory contributes alongside established theories in the field, providing valuable insights into the static thermomechanical response of functionally graded rectangular plates. This encompasses the influence of volume fraction exponent values on non-dimensional displacements and stresses, the impact of aspect ratios on deflection, and the effects of the thermal field on deflection and stresses. Numerical examples of the bending examination of advanced-generation composite structures with specific properties exposed to different loads demonstrate the accuracy of the present theory.

Coupled five-parameter dynamics of Mindlin and third-order shear deformable FG graphene-platelets reinforced viscoelastic plates with geometric and material imperfections

The present work investigates the coupled five-parameter dynamics, in particular, the coupled five-parameter structural resonance frequency, of functionally graded (FG) graphene-platelets reinforced viscoelastic plates in the presence of geometric and material (porosity) imperfections, based on the Mindlin theory and the third-order shear deformation plate theory (TSDPT). Various imperfections are often found in structures as a result of the challenges in manufacturing and operations over time. Geometric imperfection is considered in the FG reinforced plate structure by modelling an initial curvature in the out-of-plane direction. Material imperfections, in the form of closed-cell voids, are formulated as uniform and non-uniform FG porosity distributions. Uniform and thickness-wise FG gradation effects of graphene-platelets are accounted for in the determination of the effective material properties using a two-dimensional Halpin-Tsai micromechanics model, in conjunction with the rule of mixtures. Viscoelasticity of the FG imperfect plate structure is modelled by employing the Kelvin-Voigt model to consider energy dissipation. Using the Mindlin plate theory and the TSDPT, five equations of motion, which are coupled through the five parameters u, v, w, ϕ1, and ϕ2, are derived using Hamilton's variational principle. An assumed mode method is then employed to obtain the real and the imaginary components of resonance frequencies. The combined influences of the different effects on the structural resonance are examined. The results obtained offer a comprehensive understanding of how the FG graphene-platelets, FG porosity, plate thickness, geometric imperfection, and the utilisation of two different shear deformation plate theories collectively influence the coupled five-parameter dynamics of the FG graphene-platelets reinforced viscoelastic plate structure with imperfections. It was found that the difference in frequency resulted from the use of the two plate theories is most dominant with the FG GX and/or FG PX graphene-platelets reinforced porous plates, and that geometric imperfection was found to alter a graphene-platelets reinforced plate's sensitivity towards material imperfection.

Pullback exponential attractors in nonlocal Mindlin's strain gradient porous elasticity

ABSTRACT In this paper, we study the long-time dynamics of a pullback exponential attractors for non-autonomous one dimensional system in nonlocal Mindlin's strain gradient porous elastic theory recently developed by Aouadi [Well-posedness, lack of analyticity and exponential stability in nonlocal Mindlin's strain gradient porous elasticity. Z Angew Math Phys. 2022;73:111]. In this theory, the second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. Then we establish a Lipschitz stability result. We also prove the existence of pullback exponential attractors which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. Finally, we prove the upper-semicontinuity of minimal pullback attractors with respect to the perturbations parameter in natural energy space.

Wave reflection and Rayleigh waves in the context of the complete Toupin–Mindlin theory of strain gradient elasticity

Wave reflection and Rayleigh waves in the context of the complete Toupin–Mindlin theory of strain gradient elasticity

Numerical Simulations in Ultrasonic Guided Wave Analysis for the Design of SHM Systems—Benchmark Study Based on the Open Guided Waves Online Platform Dataset

Structural health monitoring (SHM) strategies based on ultrasonic guided waves are very promising regarding thin-walled lightweight structures. To study the performance of such systems, validated numerical analysis tools have to be used. For that procedure, a benchmark between two numerical methods will be presented. The first promising approach is the elastodynamic finite integration technique (EFIT). Miscellaneous research shows that its capability of capturing wave characteristics and interactions is advanced in various media and structures, including thin-walled composites. The second approach employs conventional shell-type finite elements following the Reissner–Mindlin theory for modelling layered composite structures. The advantage of using such finite element methods (FEM) is their high availability in general purpose simulation tools. As a reference model, the measurement data coming from the Open Guided Waves Project (OGW) was taken into account. The OGW dataset provides the experimental data of ultrasonic guided wave propagation in carbon fiber composite plates with an additional omega stringer. By using this contribution, this experiment was reproduced by simulation. The paper presents the results of a validation and motivates further research, such as in research related to the probability of detection analysis and numerical performance.

Study on Compaction Characteristics and Compaction Process of an Unsaturated Silt Based on PFC3D

In order to reduce the construction cost of excessively wet silt roadbeds and improve the compaction quality, this paper studies the compaction characteristics and compaction technology of excessively wet silts through theoretical derivation, indoor experiments, numerical simulation, engineering verification, and other research methods. Based on the pendulum-type liquid bridge structure, this paper uses the Hertz–Mindlin theory, particle motion equation of state, and other theories to modify the force between particles of an unsaturated soil. Through the contact angle, which is a medium, the matrix suction is linked to the water content and is used to approximate the water content state of soil samples. Using PFC3D (particle flow code 3D) numerical simulation software, an unsaturated silt model was established based on Hill contact, and the model parameters were calibrated through basic geotechnical tests and triaxial compression tests. The correctness of the theory is verified by comparing the simulated triaxial compression test of wet silts with the indoor test. By simulating subgrade rolling, considering factors such as the distribution of contact force chain and the variation of compaction degree, the effects of rolling times, strong vibration, and weak vibration on the compaction effect were compared and analyzed. Finally, the optimal rolling and compaction process under the optimal water content is obtained through on-site engineering tests. The results show that: (1) The Hill contact model is reasonable for simulating the wet silt. (2) During the simulation of the roadbed compaction process using PFC3D software, it was found that the compaction degree change during the entire compaction process can be roughly divided into the initial compaction stage and the re-compaction stage. The reasonable number of compaction times was determined to be five through discrete element simulation. (3) It is found from the numerical simulation results that compared to static compaction, both strong and weak vibration compaction can effectively improve the compaction effect of subgrade compaction. The larger the vibration amplitude, the more obvious the improvement of the compaction effect. The compaction effect of strong vibration followed by weak vibration is stronger than that of weak vibration followed by strong vibration. (4) The optimal compaction process obtained under the optimal moisture content of 15% is static pressure once + strong vibration twice + weak vibration twice.

Determination of eigen vibration modes for three-layer plates using the example of solar panels

Solar panels are considered as three-layer plates with a thick hard outer layer and a thin soft inner layer. To describe the mechanical behavior of the plates on the example of a solar panel, a model for anti-sandwich plates was used. The literature review includes scientific articles describing models for analytical and numerical calculations of three-layer plates. During the scientific study of the mechanical behavior of the solar plate under the influence of external factors, the method of finite element analysis using the element of the spatial shell was used. This type of elements is used for theories of single and multilayer plates. Shell elements were used for calculations and modeling of the natural forms of vibrations of three-layer plates. The paper presents scientific studies under static loading under different exposure conditions, as well as an analysis of self-oscillations of a three-layer plate using the Kirchhoff and Mindlin theories as an example. As part of the scientific work, a study of the mechanical model of a thin solar panel was carried out using finite element analysis in the ABAQUS program, taking into account different temperature conditions. The article provides analytical calculations of the application of various theories to determine the natural forms of plate vibrations.

High-fidelity analytical modeling of asymmetric CFRP composites using Reissner–Mindlin theory and hygroscopic degradation

Multi-stable carbon fiber-reinforced composites with asymmetric ply layouts have shown promising potential for creating multi-functional structural systems. To design and analyze these complex composites, high-fidelity and computationally efficient analytical models are crucial. To this end, Hyer and Dano formulated the first model four decades ago based on classical lamination theory (CLT). Surprisingly, despite many follow-up studies to refine this analytical approach, current models still follow the same fundamental ingredients used in Hyer and Dano’s original work. As a result, they all inherited the same limitations. For example, they ignore the interlaminar stresses and lack a rigorously calibrated hygroscopic term. To overcome these limitations, we propose a new analytical modeling approach by adopting the Reissner–Mindlin theory and hygroscopic degradation. This new model introduces in-plane rotations as two additional degrees of freedom and a combined expansion coefficient for the epoxy matrix materials. Compared to finite element simulations and experiment data, the new model can estimate the interlaminar stress reasonably well (especially regarding its concentration near the edges) and predict the laminates’ external shape more accurately than an equivalent Hyer’s model. The new model can also accurately predict the reduction in the laminate’s curvature and snap-through force in different design configurations, as well as their linear correlation to the moisture content levels. Overall, the results of this study present a fundamental advancement in the analytical modeling of asymmetric multi-stable composite laminates, offering a broad and versatile foundation for designing and analyzing the next generation of multi-functional composite structures.

A force controlled tribometer for pre-sliding measurements at the nanometer scale

In the pre-sliding friction regime, interfaces partially stick and partially slip. The pre-slip is thought to be locally initiated at regions of the interface where the ratio of shear stress to normal stress exceeds a critical value. The displacements involved in pre-slip can be limited to the nanoscale, especially for stiff interfaces. Furthermore, little is known experimentally about the interplay between surface topography, pre-sliding behavior and wear. In this work, we introduce a pre-sliding tribometer that enables the study of how the pre-sliding friction at various types of ball-on-flat interfaces evolves as a function of wear. Polytetrafluoroethylene-on-silicon (PTFE-on-Si) pre-sliding measurements covering interfacial displacements up to 50 nm, conducted with the new instrument, show good agreement with Mindlin theory predictions, without adjustable parameters.

ON THE INFLUENCE OF THE MECHANICAL CHARACTERISTICS OF A THIN ADHESION LAYER ON THE COMPOSITE STRENGTH. PART 1. ELASTIC DEFORMATION

The problem of deformation of a DCB sample, which is a composition of bodies bound by an adhesive layer of finite thickness, is considered. Based on the variational equilibrium equation containing the layer thickness as a linear parameter, a finite element solution of the problem of loading the layer with a normal discontinuity in the plane strain state is constructed. The stresses averaged over the layer thickness are related to the stresses along the layer boundary by the equilibrium equations. The boundary stresses of the layer form the boundary conditions for the mating bodies. In the layer, along with shear stresses, stresses orthogonal to shear are also taken into account. The constitutive relations in the layer are represented in terms of average stresses. With a significant difference in the Young's moduli of the adhesive and mating bodies, the convergence of the value of the J-integral with a decrease in the layer thickness is shown. To find the J-integral, its representation is used as a product of the specific free energy at the end of the layer and its thickness. It has been established that the Poisson's ratio of the bodies affects the value of the J-integral, and the Poisson's ratio of the adhesive layer has almost no effect on the value of the J-integral. Using the theory of plates Mindlin - Reisner at zero Poisson's ratio of the adhesive, an analytical representation of the J-integral is obtained. The representation includes energy terms related to the pull-off stress and the axial stress in the layer. In this case, the term associated with the axial stress in the layer is proportional to the square of the ratio of the Young's moduli of the adhesive layer and the bodies mating with it. From the solution obtained, it follows that the mechanical properties of the adhesive layer with a small thickness compared to bodies do not affect the value of the J-integral if the elastic modulus of the adhesive layer is significantly less than the elastic modulus of the mating bodies. Thus, the use of replacing the adhesive layer with a layer of zero thickness is correct under these restrictions.

Mesh-free discretization of peridynamic shell structures and coupling model with isogeometric analysis

Peridynamic (PD) theory applies nonlocal framework to reform the equilibrium equation in fracture analysis. The mesh-free formulation of the PD model performs well in two- and three-dimensional formulations. A mesh-free discretization of the PD shell using Reissner–Mindlin theory is proposed to meet the demands of crack simulation in shell structures. A micro-beam bond with three translatory and three rotational degrees of freedom is applied to describe the in-shell planar, shearing, and bending deformations. A particle-based method for the coupling of isogeometric analysis (IGA) and PD shells is proposed to improve the computational efficiency. The smooth geometry of the shell structure is precisely described by nonuniform rational B-splines (NURBS). The zone around the crack is transformed from IGA mesh into PD particles by applying mesh-free PD discretization to account for the crack propagation. A particle equilibrium-based method is used for the coupling of IGA and PD zones. The proposed coupling model combines the advantages of both methods and successfully simulates crack growth in shell structures with better computational efficiency compared with the pure PD model. Numerical examples are conducted wherein the complex propagation and intersecting of cracks can be captured, thus proving the accuracy and effectiveness of the proposed method.

A theoretical justification of the slip index concept in fretting analysis

Fretting in the partial-slip and gross-slip regimes under a constant normal load is considered. The tangential force—displacement relations for the forward and backward motions are described based the generalized Cattaneo—Mindlin theory of tangential contact and Masing’s hypothesis on modelling the force—displacement hysteretic loop. Besides the critical force and displacement parameters (characterizing the triggering of sliding), the model includes one dimensionless fitting parameter that tunes the tangential contact stiffness of the friction—contact interface. Explicit expressions are derived for the main tribological parameters of the fretting loop, including the slip index and the signal index. The presented phenomenological modelling approach has been applied to the analysis of two sets of experimental data taken from the literature. It has been shown that the experimentally observed simple relation of a rational type between the slip index and the slip ratio corresponds to the gross-slip asymptotics of the corresponding model-based predicted relation. The known quantitative criteria for the transition from the partial slip regime to the gross slip regime are expressed in terms of the stiffness parameter, and a novel geometric transition criterion is formulated.