Higher-order Displacement Field Research Articles

Modal and aeroelastic analysis of trapezoidal corrugated-core sandwich panels in supersonic flow

Corrugated-core sandwich panels have a great potential in the application to acreage thermal protection system of aerospace vehicles. However, due to their structural complexity, it takes high computational cost to conduct modal analysis with detailed 3D finite element models and the aeroelastic behaviors of these sandwich structures have not been fully investigated. In this paper, an analytical model of a trapezoidal corrugated-core sandwich panel using homogenization technique and layerwise theory is presented for modal and aeroelastic analysis. The trapezoidal corrugated core is homogenized as an equivalent orthotropic layer based on an energy approach and the overall sandwich panel is treated as a three-layer continuum. The proposed layerwise theory which is developed for a sandwich plate, assumes higher-order displacement field for the core layer and first-order displacement field for the top and bottom layers. The unsteady aerodynamic pressure is evaluated by the supersonic Piston theory. An eight-noded C0 isoparametric element with 13 degrees of freedom per node is used for finite element method. The accuracy and reliability of the present method are verified by comparing natural frequencies and mode shapes as well as flutter speeds with those obtained from commercial software. Parametric studies concerning different boundary conditions and parametric variables are also conducted. It is shown that the proposed method has sufficient accuracy and requires less computational effort, providing a theoretical basis for the utilization of the trapezoidal corrugated-core sandwich panel in aircraft designing.

Thermally induced large deflection analysis of shear deformable FGM shallow curved tubes using perturbation method

AbstractIn the present investigation, geometrically nonlinear response of a shallow curved tube subjected to thermal loading is analyzed. The tube is made of a functionally graded material (FGM), where properties are distributed across the tube radius. Thermomechancal properties of the constituents are assumed to be temperature dependent. The simple case of uniform temperature rise loading is considered. A higher order displacement field is considered for the tube which satisfies the conditions of zero tangential tractions on the inner and outer surfaces of the tube. With the aid of von Kármán strain‐displacement relations, linear thermoelastic constitutive law, and Hamilton's principle the governing equations of the tube are obtained. These equations are reduced to new two coupled equations for the case of axially immovable tubes. These new equations are solved using the two step perturbation technique for pinned and clamped tubes. Closed form expressions are provided to estimate the deflections in the FGM temperature dependent curved tubes under uniform thermal load. The effect of boundary conditions, temperature dependency of material properties, and geometrical properties are discussed via numerical examples.

Static analysis of a stepped main shaft in a mine hoist by means of the modified 1D higher-order theory

The analysis of a stepped main shaft by 1D refined beam theories in cylindrical coordinate system is presented. High-order displacement fields are achieved by employing the Carrera unified formulation (CUF), which takes direct implementation of any-order theory without the requirement of considering special formulations. The classical beam theories can be derived from the formulation as particular cases. The principle of minimum potential energy is used to obtain the governing differential equations and the related boundary conditions in a cylindrical coordinate system. These explicit terms of the stiffness matrices are exhibited and a global stiffness matrix is then obtained by matrix transformation. For the special working condition in a mining hoist and stepped shaft, the resulting global stiffness matrix and the loading vector are modified and applied with the boundary conditions in the static analysis of shaft parts. The accuracy of static analysis based on the refined beam theory is confirmed by comparing ANSYS solid theory and classical beam theories. An experiment for verifying the availability of the modified 1D refined beam model on the surface strain of segment 9 of the main shaft is conducted in a field experiment at Zhaojiazhai Coal Mine, China. Experimental results demonstrate the practicability of the present theory in predicting the strain field on the surface of the stepped main shaft of a mining hoist.

Analysis of functionally graded sandwich plates using a higher-order layerwise theory

A higher-order layerwise finite element formulation is presented for static and dynamic analyses of functionally graded material (FGM) sandwich plates. A higher-order displacement field is assumed for core and first-order displacement field is assumed for top and bottom facesheets maintaining a continuity of displacement at layer interface. An eight noded isoparametric element using a C0 based finite element formulation with thirteen degrees of freedom per node has been considered in the present work. Two configurations of FGM sandwich plates, one with FGM core and homogenous facesheets and second having top and bottom layers made of FGM and homogenous core are considered. Effective material properties of the FGM are computed using rule of mixture (ROM). In order to establish the correctness of the present finite element formulation for wide range of problems for two configurations of FGM sandwich plates, comparison studies are presented. Next, parametric studies are taken up to investigate the effects of volume fraction index, span to thickness ratio and boundary conditions on static and dynamic behavior of FGM sandwich plate. It is shown here that present formulation is simple, straightforward and accurate for static and dynamic analyses of functionally graded material (FGM) sandwich plates.

Case report of infrequent variation of third head of biceps brachii muscle

The numerical analysis of laminated composite plates and shells resting on nonlinear elastic foundation is the main topic of the paper. The generalized differential quadrature (GDQ) technique and the Newton-Raphson iteration are employed to obtain the solution of the static problems under consideration. The nonlinear elastic foundation is modeled using the Winkler-Pasternak model embedded with quadratic and cubic nonlinearities in order to a have a more complete description of the interaction. The structural behavior is modeled by means of higher-order displacement fields developed in the framework of a unified formulation. Several lamination schemes are studied. The class of sandwich structures with an inner soft-core is also taken into account with the help of the Murakami’s function, which correctly captures the so-called zig-zag effect. The presented approach can deal with doubly-curved surfaces characterized by two radii of curvature that can vary in each point of the reference domain, whereas most of the examples available in the literature considers only shells with constant curvature, such as spherical and cylindrical shells. Solutions are presented in terms of through-the-thickness variations of strains, stresses, and displacements. For these purposes, a posteriori recovery procedure based on the GDQ method is introduced. The accuracy and effectiveness of the proposed approach are proven by means of the comparison with the numerical results obtained by a three-dimensional finite element model.

A posteriori stress and strain recovery procedure for the static analysis of laminated shells resting on nonlinear elastic foundation

Abstract The numerical analysis of laminated composite plates and shells resting on nonlinear elastic foundation is the main topic of the paper. The generalized differential quadrature (GDQ) technique and the Newton-Raphson iteration are employed to obtain the solution of the static problems under consideration. The nonlinear elastic foundation is modeled using the Winkler-Pasternak model embedded with quadratic and cubic nonlinearities in order to a have a more complete description of the interaction. The structural behavior is modeled by means of higher-order displacement fields developed in the framework of a unified formulation. Several lamination schemes are studied. The class of sandwich structures with an inner soft-core is also taken into account with the help of the Murakami’s function, which correctly captures the so-called zig-zag effect. The presented approach can deal with doubly-curved surfaces characterized by two radii of curvature that can vary in each point of the reference domain, whereas most of the examples available in the literature considers only shells with constant curvature, such as spherical and cylindrical shells. Solutions are presented in terms of through-the-thickness variations of strains, stresses, and displacements. For these purposes, a posteriori recovery procedure based on the GDQ method is introduced. The accuracy and effectiveness of the proposed approach are proven by means of the comparison with the numerical results obtained by a three-dimensional finite element model.

A finite element formulation for thermally induced vibrations of functionally graded material sandwich plates and shell panels

A finite element formulation based on a higher-order layerwise theory is presented for the first time to investigate thermally induced vibrations of functionally graded material (FGM) sandwich plates and shell panels. The properties of FGM sandwich are assumed to be position and temperature dependent. The upper and lower layers of the sandwich panel are considered to be made of pure ceramic and metal, respectively and the elastic properties of FGM core are varied according to a power-law function. The top surface is exposed to a thermal shock and the bottom surface of the panel is either kept at a reference temperature or thermally insulated. The one-dimensional transient heat conduction equation is solved using a central difference scheme in conjunction with the Crank-Nicolson method. A higher-order layerwise theory is used for FGM sandwich panels, in which a higher-order displacement field for the FGM core and a first-order displacement field for the facesheets are assumed. The governing equations are solved using Newmark average acceleration method. It is shown that the proposed layerwise finite element formulation is simple and can easily be applied to investigate FGM sandwich plates and shell panels subjected to rapid heating.

A new efficient higher-order shear deformation theory for a doubly curved laminated composite shell

Appropriate representation of the displacement field is important to establish proper stress distribution including shear-free condition at free surfaces for a laminated composite shell used in real engineering applications. The present study attempted to develop a more accurate higher-order displacement field for the analysis of a doubly curved laminated composite shell with \(\hbox {C}^{0 }\) finite element model based on higher-order shear deformation theory. A new displacement function is proposed for static and free vibration analysis of such a shell. The accurate strain displacement relationship is applied in the analysis of a shell structure with exactly zero shear-free condition at top and bottom surfaces. The proposed model is capable of determining the accurate shear stress distribution across the thickness of the laminate. Moderately deep and thick shells can be analyzed very accurately as the ratio of thickness coordinate to radius of curvature is incorporated in the formulation. An eight-noded isoparametric shell element with seven degrees of freedom at each node is used to formulate the finite element model. The numerical results in terms of deflection, stresses and natural frequencies obtained by the present formulations are compared with those available in the published literature to validate the accuracy of the proposed model.

Free vibration analysis of simply supported beams with solid and thin-walled cross-sections using higher-order theories based on displacement variables

Solutions for undamped free vibration of beams with solid and thin-walled cross-sections are provided by using refined theories based on displacement variables. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. Subsequently, the differential equations of motion and the natural boundary conditions are formulated in terms of fundamental nuclei by using CUF and the strong form of the principle of virtual displacements. The second-order system of ordinary differential equations is then reduced into a classical eigenvalue problem by assuming simply supported boundary conditions. The proposed methodology is extensively assessed for different solid and thin-walled metallic beam structures and the results are compared with those appeared in published literature and also checked by finite element solutions. The research demonstrates that: (i) the innovative 1D closed form CUF represents a reliable and compact method to develop refined beam models with solely displacement variables; (ii) 3D-like numerically exact solutions of complex structures can be obtained with ease; and (iii) the numerical efficiency of the present method is uniquely robust when compared to other methods that provide similar accuracies.

Generalized FVDAM theory for elastic–plastic periodic materials

Generalized FVDAM theory for the analysis of periodic heterogeneous materials with elastic–plastic phases undergoing infinitesimal deformation is constructed to overcome the limitations of the original or 0th-order version of the theory, and to concomitantly extend the theory's range of modeling capabilities. The 0th-order theory suffers from intrinsic constraints stemming from limited displacement field representation at the local level, which results in the deterioration of pointwise continuity of interfacial tractions and displacements with increasing plasticity, requiring greater unit cell discretization. The generalization is based on a higher-order displacement field representation within individual subvolumes of a discretized analysis domain, in contrast with the second-order expansion employed in the 0th-order theory. The higher-order displacement field is expressed in terms of elasticity-motivated surface-averaged kinematic variables which are subsequently related to corresponding static variables through a generalized local stiffness matrix. Comparison of local fields in metal matrix composites with large moduli contrasts obtained using the generalized FVDAM theory, its predecessor and finite-element method illustrates substantial improvement in the pointwise satisfaction of interfacial continuity conditions at adjacent subvolume faces in the presence of plasticity, producing smoother stress and plastic strain distributions and excellent interfacial conformability with smaller unit cell discretizations. The generalized theory offers several advantages relative to the Q9-based finite-element method, including direct relationship between static and kinematic variables across subvolume faces and superior satisfaction of pointwise traction continuity which facilitate modeling of various interfacial phenomena, such as fiber/matrix cracks illustrated herein.

Effects of geometric ratios and fibre orientation on the natural frequencies of laminated composite plates

The present investigation aims to examine the influence of geometric ratios and fibre orientation on the natural frequencies of fibre-reinforced laminated composite plates using finite element method based on Yang’s theory and his collaborators. The transverse shear and rotatory inertia effects were taken into consideration in the developed Fortran computer program. It has been shown that the use of first-order displacement field provides the same accuracy as higher-order displacement field when the number of elements representing the plate structure is increased (refined mesh). However, poor precision may appear for plates with high thickness-to-side ratio h/a (thickness/side length). This discrepancy limits the application of the developed theory to thick plates (h/a<0.5). The various curves show the evolution of the dimensionless frequency (w*) versus fibre orientation angle (q) and illustrate the apparition of a “triple-point” phenomenon engendered by the increase of the plate aspect ratio a/b (length/width) for a specific value of h/a. This point defines the maximum natural frequency and the associated fibre orientation. Also, results show that for high and/or low aspect ratios, the triple-point phenomenon does not occur. This latter is rapidly reached for thick plates than thin plates when the plate aspect ratio a/b is progressively increased. Keywords : composite plates, dimensionless frequency, fibre orientation, finite element method, geometric ratios, triple-point.

On the effect of inter-laminar contact on the dynamics of locally delaminated FRP strengthened walls

The paper studies the effect of the inter-laminar contact and its temporal variation on the behavior of FRP strengthened walls with delaminated regions. The paper adopts an analytical approach that unifies the displacement and stress fields of sub-regions where the delaminated faces are in contact with ones where the layers are free of contact. This is achieved by means of a Robin type of condition at the delaminated interface and a 2D time dependent Heaviside function that automatically switches between contact states. The model is converted into a finite element form that utilizes the high order displacement fields, overcomes the obstacles due to the scatter of geometric scales, avoids the need to mesh through the thickness of the layers, and simplifies the implementation in a broader finite element framework. The role of the inter-laminar contact non-linearity is demonstrated through the study of the dynamic behavior triggered by the geometrically non-linear snap-out of a delaminated layer in a shear wall strengthened with FRP. The analysis quantifies the impact of the 2D time dependent contact map on the snap-out load, the dynamic post-buckling behavior, and the interfacial stresses. The paper closes with conclusion regarding the delaminated wall as well as many other laminated, adhesively bonded, or sandwich structural forms that are prone to delamination, dynamic instabilities, and the combination of the two.

A new C0 higher-order layerwise finite element formulation for the analysis of laminated and sandwich plates

In this paper, a new layerwise plate formulation based on a C0 higher-order finite element model is presented for static and free vibration analyses of laminated composite and sandwich plates. The proposed layerwise theory which is developed for a three layered composite plates, assumes higher-order displacement field for middle layer and first-order displacement field for top and bottom layers. Compatibility conditions are imposed at the layer interface to satisfy the interlaminar displacement continuity. An eight-noded isoparametric element is used to model the plate. The accuracy of the proposed formulation is assessed for linear static and free vibration analyses by comparing the authors’ results with available 3D elasticity, finite element and analytical solutions. It has been shown here that the present finite element formulation is much simpler, straightforward and accurate for static and free vibration analyses of laminated composite and sandwich plates.

Dynamic edge debonding in FRP strengthened beams

Abstract The debonding failure in beams strengthened with externally bonded composite materials, as well as in many other layered and adhesively bonded structural elements, is a dynamic process. This paper studies the dynamics of the edge debonding process in beams strengthened with Fiber Reinforced Polymer (FRP) composite materials and derives an analytical model for the dynamic debonding process. The model adopts an extended high order approach, incorporates the dynamic inertial effects, and considers the high order displacement, acceleration, and stress fields in the adhesive layer. The model uses two cohesive interfaces for the consideration of independent debonding mechanisms at the two physical interfaces of the adhesive layer. It also accounts for the inelastic accumulation of damage at the interfaces and for their degradation under the effect of the dynamic and cyclic response. Along with the derivation of the model and its validation through comparison with 2D dynamic finite element analysis, the paper studies the dynamics of the edge debonding mechanism under two loading conditions. In the first case, the load is applied statically but the debonding mechanism drives the structure into the dynamic regime. In the second case, the dynamic debonding mechanism is triggered by an impulse load. In both cases, the results reveal and quantify the critical dynamic features of the edge debonding mechanism, its time and velocity scales, and its impact on the performance and resilience of the strengthened beam.

Homogenization of periodic materials with viscoelastic phases using the generalized FVDAM theory

In the last decades, new generations of advanced materials have been designed and manufactured for specific applications. The micromechanics plays an important role in the development of heterogeneous materials, enabling efficient analyses of composite materials with complex geometries, circumventing the traditional trial-and-error approach, producing substantial cost savings. The unit cell problem to the analysis of periodic heterogeneous media can be solved by the well-established 0th order version of the finite-volume theory, named finite-volume direct averaging micromechanics (FVDAM) theory. This standard version of the FVDAM theory employs an incomplete second-order displacement field within individual subvolumes of a discretized analysis domain together with a surface-averaging framework, which does not enforce displacement or traction continuity in a point-wise manner. This, in turn, produces interfacial interpenetrations and non-traction stress discontinuities, thereby demanding very refined meshes in order to produce good interfacial conformability and pointwise stress continuity between adjacent subvolumes. To overcome these shortcomings, a generalized FVDAM theory has been proposed to enable analysis of periodic heterogeneous materials in the finite-deformation domain. The generalization is based on a higher-order displacement field representation and on the definition of elasticity-based surface-averaged kinematic and static variables related through a local stiffness matrix. Herein, we specialize the generalized FVDAM theory to the infinitesimal analysis of periodic materials with viscoelastic phases, where a total or secant formulation is employed, with the viscoelastic strains evaluated incrementally using an algorithm based on the concept of state variables. The generalized or 2nd order version considerably improves interfacial conformability and pointwise traction and non-traction stress continuity between adjacent subvolumes in comparison with the 0th order version, but with a higher computational cost. Furthermore, for the same mesh discretization, these two versions provide comparable macroscopic response. Considering these features, the 0th order version is recommended to evaluate the effective elastic properties and the homogenized creep and relaxation functions, while the 2nd order version is more efficient in the evaluation of the microscopic displacement and stress fields.

Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part I: Framework

The recently constructed generalized finite-volume theory for two-dimensional linear elasticity problems on rectangular domains is further extended to make possible simulation of periodic materials with complex microstructures undergoing finite deformations. This is accomplished by embedding the generalized finite-volume theory with newly incorporated finite-deformation features into the 0th order homogenization framework, and introducing parametric mapping to enable efficient mimicking of complex microstructural details without artificial stress concentrations by stepwise approximation of curved surfaces separating adjacent phases. The higher-order displacement field representation within subvolumes of the discretized unit cell microstructure, expressed in terms of elasticity-based surface-averaged kinematic variables, substantially improves interfacial conformability and pointwise traction and nontraction stress continuity between adjacent subvolumes. These features enable application of much larger deformations in comparison with the standard finite-volume direct averaging micromechanics (FVDAM) theory developed for finite-deformation applications by minimizing interfacial interpenetrations through additional kinematic constraints. The theory is constructed in a manner which facilitates systematic specialization through reductions to lower-order versions with the 0th order corresponding to the standard FVDAM theory. Part I presents the theoretical framework. Comparison of predictions by the generalized FVDAM theory with its predecessor, analytical and finite-element results in Part II illustrates the proposed theory's superiority in applications involving very large deformations.

Generalized FVDAM Theory for Periodic Materials Undergoing Finite Deformations—Part II: Results

In Part I, a generalized finite-volume direct averaging micromechanics (FVDAM) theory was constructed for periodic materials with complex microstructures undergoing finite deformations. The generalization involves the use of a higher-order displacement field representation within individual subvolumes of a discretized analysis domain whose coefficients were expressed in terms of surface-averaged kinematic variables required to be continuous across adjacent subvolume faces. In Part II of this contribution we demonstrate that the higher-order displacement representation leads to a substantial improvement in subvolume interfacial conformability and smoother stress distributions relative to the original theory based on a quadratic displacement field representation, herein called the 0th order theory. This improvement is particularly important in the finite-deformation domain wherein large differences in adjacent subvolume face rotations may lead to the loss of mesh integrity. The advantages of the generalized theory are illustrated through examples based on a known analytical solution and finite-element results generated with a computer code that mimics the generalized theory's framework. An application of the generalized FVDAM theory involving the response of wavy multilayers confirms previously generated results with the 0th order theory that revealed microstructural effects in this class of materials which are important in bio-inspired material architectures that mimic certain biological tissues.

Non-linear oscillations of a thin-walled composite beam with shear deformation

A geometrically non-linear theory is used to study the dynamic behavior of a thin-walled composite beam. The model is based on a small strain and large rotation and displacements theory, which is formulated through the adoption of a higher-order displacement field and takes into account shear flexibility (bending and warping shear). In the analysis of a weakly nonlinear continuous system, the Ritz’s method is employed to express the problem in terms of generalized coordinates. Then, perturbation method of multiple scales is applied to the reduced system in order to obtain the equations of amplitude and modulation. In this paper, the non-linear 3D oscillations of a simply-supported beam are examined, considering a cross-section having one symmetry axis. Composite is assumed to be made of symmetric balanced laminates and especially orthotropic laminates. The model, which contains both quadratic and cubic non-linearities, is assumed to be in internal resonance condition. Steady-state solution and their stability are investigated by means of the eigenvalues of the Jacobian matrix. The equilibrium solution is governed by the modal coupling and experience a complex behavior composed by saddle noddle, Hopf and double period bifurcations.

Shear-deformable thin-walled composite Beams in internal and external resonance

The non-linear dynamic response of thin-walled composite beams is analyzed considering the effect of shear deformation. The model is based on a small strain and large rotation and displacements theory, which is formulated through the adoption of a higher-order displacement field and takes into account shear flexibility (bending and warping shear). The beam is assumed to be in internal resonance conditions of the kind 2:3:1, so that quadratic, cubic and combination resonances occur. In the analysis of a weakly nonlinear continuous system, the Galerkin’s method is employed to express the problem in terms of generalized coordinates. Then, the perturbation method of multiple scales is applied to the reduced system in order to obtain the equations of amplitude and modulation. The equilibrium solution is governed by the modal coupling and experience a complex behavior composed by saddle–noddle and Hopf bifurcations. The results of the analysis show that the equilibrium solutions are influenced by the shear effect, when this effect is ignored the amplitude of vibration is reduced significantly, thus altering the dynamic response of the beam. This alteration can infer in an incorrect stability prediction of the periodic solutions.

Generalized Finite-Volume Theory for Elastic Stress Analysis in Solid Mechanics—Part I: Framework

A generalized finite-volume theory is proposed for two-dimensional elasticity problems on rectangular domains. The generalization is based on a higher-order displacement field representation within individual subvolumes of a discretized analysis domain, in contrast with the second-order expansion employed in our standard theory. The higher-order displacement field is expressed in terms of elasticity-based surface-averaged kinematic variables, which are subsequently related to corresponding static variables through a local stiffness matrix derived in closed form. The novel manner of defining the surface-averaged kinematic and static variables is a key feature of the generalized finite-volume theory, which provides opportunities for further exploration. Satisfaction of subvolume equilibrium equations in an integral sense, a defining feature of finite-volume theories, provides the required additional equations for the local stiffness matrix construction. The theory is constructed in a manner which enables systematic specialization through reductions to lower-order versions. Part I presents the theoretical framework. Comparison of predictions by the generalized theory with its predecessor, analytical and finite-element results in Part II illustrates substantial improvement in the satisfaction of interfacial continuity conditions at adjacent subvolume faces, producing smoother stress distributions and good interfacial conformability.