Higher-order Displacement Field Research Articles

Generalized Finite-Volume Theory for Elastic Stress Analysis in Solid Mechanics—Part II: Results

In Part I, a generalized finite-volume theory was constructed for two-dimensional elasticity problems on rectangular domains based on a higher-order displacement field representation within individual subvolumes of a discretized analysis domain. The higher-order displacement field was expressed in terms of elasticity-based surface-averaged kinematic variables that were subsequently related to corresponding static variables through a local stiffness matrix derived in closed form. The theory was constructed in a manner that enables systematic specialization through reductions to lower-order versions, including the original theory based on a quadratic displacement field representation, herein called the zeroth-order theory. Comparison of predictions generated 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. While in certain instances the first-order theory produces acceptably smooth stress distributions, concentrated loadings require the second-order (generalized) theory to reproduce stress and displacement fields with fidelity comparable to analytical and finite-element results.

Thermally Induced Vibration Analysis of Laminated Plate Considering Radiation by Finite Element Method

ABSTRACTTransient temperature field of laminated plate which considering radiation is analyzed by laying temperature lines in every layer in this paper. Finite element equations are established by using high-order displacement field which considering shear effect based on the temperature analysis. Then thermally induced vibration of laminated plate is investigated by solving the finite element equation with numerical method. Compared with numerical results of ANSYS by using 3-D coupled element (SOLID5), the methods presented in this paper have a satisfied precision and better efficiency. It can give reference to the thermally induced vibration analysis of thin-large flexible laminated appendages of spacecraft.

Guidelines and Recommendations on the Use of Higher Order Finite Elements for Bending Analysis of Plates

This paper compares and evaluates various plate finite elements to analyse the static response of thick and thin plates subjected to different loading and boundary conditions. Plate elements are based on different assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner-Mindlin), refined (Reddy and Kant), and other higher-order displacement fields are implemented up to fourth-order expansion. The Unified Formulation UF by the first author is used to derive finite element matrices in terms of fundamental nuclei which consist of 3×3 arrays. The MITC4 shear-locking free type formulation is used for the FE approximation. Accuracy of a given plate element is established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates are implemented and compared using displacement and stress variables for various plate problems. Reduced models that are able to detect the 3D solution are built and a Best Plate Diagram (BPD) is introduced to give guidelines for the construction of plate theories based on a given accuracy and number of terms. It is concluded that the UF is a valuable tool to establish, for a given plate problem, the most accurate FE able to furnish results within a certain accuracy range. This allows us to obtain guidelines and recommendations in building refined elements in the bending analysis of plates for various geometries, loadings, and boundary conditions.

A Refined Shear Deformation Plate Theory

An improved higher-order shear deformation theory of plates is presented in this paper. The theory is developed from the transverse shear deformation theory presented by Ambartsumian [11]. The present plate theory contains kinematics of higher-order displacement field of plates, a system of higher-order differential equilibrium equations in terms of the three generalized displacements of bending plates, and a system of boundary conditions at each edge of plate boundaries. The present shear deformation theory of plates is validated by applying it to solve torsional plates and simply supported plates. The obtained solutions using the present theory are compared with the solutions of other shear-deformation theories. A good agreement is achieved through these comparisons and the advantages of the present theory are clearly verified. The shear deformation plate theory presented here can be applied to the analysis of laminated composite plates to better predict their dynamic and static behaviors. The proposed theory should also be supplemented to the theory of finite element analysis for developing new shell elements.

A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions

This paper presents the derivation of a new beam theory with the sixth-order differential equilibrium equations for the analysis of shear deformable beams. A sixth-order beam theory is desirable since the displacement constraints of some typical shear flexible beams clearly indicate that the boundary conditions corresponding to these constraints can be properly satisfied only by the boundary conditions associated with the sixth-order differential equilibrium equations as opposed to the fourth-order equilibrium equations in Timoshenko beam theory. The present beam theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the authors, a system of sixth-order differential equilibrium equations in terms of two generalized displacements w and ϕx of beam cross sections, and three boundary conditions at each end of shear deformable beams. A technique for the analytical solution of the new beam theory is also presented. To demonstrate the advantages and accuracy of the new sixth-order beam theory for the analysis of shear flexible beams, the proposed beam theory is applied to solve analytically three classical beam bending problems to which the fourth-order beam theory of Timoshenko has created some questions on the boundary conditions. The present solutions of these examples agree well with the elasticity solutions, and in particular they also show that the present sixth-order beam theory is capable of characterizing some boundary layer behavior near the beam ends or loading points.

Accuracy of refined finite elements for laminated plate analysis

This paper compares and evaluates various plate finite elements to analyze the static response of laminated plate when varying the thickness ratio, orthotropic ratio and the stacking sequence of the lay-out. Plate elements were created from different theoretical assumptions for the displacement distribution along the thickness direction. Classical (Kirchhoff and Reissner–Mindlin), known refined (Reddy, Pandya, and Kant), and other higher-order displacement fields were then implemented up-to fourth-order expansion. Following this the Carrera Unified Formulation was used to derive finite element matrices in terms of fundamental nuclei which consist of 3 × 3 arrays. The accuracy of a given plate element was established in terms of the error vs. thickness-to-length parameter. A significant number of finite elements for plates were implemented and compared using displacement and stress variables for various plate problems. In this paper concluding results are presented as number and type of required displacement variables vs. accuracy on a given stress/displacement parameter. These diagrams are labelled as Best Plate Theories BPT curves, which differ by changing geometry, material data and lay-out as well as loading and boundary conditions.

Optimal design of piezolaminated structures using B-spline strip finite element models

A package of B-spline finite strip models is developed for the linear analysis of piezolaminated plates and shells. This package is associated to a global optimization technique in order to enhance the performance of these types of structures, subjected to various types of objective functions and/or constraints, with discrete and continuous design variables. The models considered are based on a higher-order displacement field and one can apply them to the static, free vibration and buckling analyses of laminated adaptive structures with arbitrary lay-ups, loading and boundary conditions. Genetic algorithms, with either binary or floating point encoding of design variables, were considered to find optimal locations of piezoelectric actuators as well as to determine the best voltages applied to them in order to obtain a desired structure shape. These models provide an overall economy of computing effort for static and vibration problems.

Finite element formulations for thermopiezoelastic laminated composite plates

Finite element formulations are presented which account for the coupled mechanical,electrical, and thermal response of piezoelectric composite plate structures. Themathematical model is based upon a high-order displacement field, a quadratic electricpotential field and a linear temperature field. The dynamic governing equationsare derived using the virtual work principle. These formulations in general willenable the modeling of vibration and shape control applications in solving a widevariety of coupled thermal–piezoelectric problems. Three numerical examples,including a piezoelectric bimorph, a cantilevered aluminum plate with two bondedpiezoelectric patches and a clamped carbon/epoxy composite plate with an attached fullpiezoceramic layer, have been developed. The actual through-the-thickness electricpotential distribution is captured accurately with the present quadratic distributionassumption for each piezoelectric layer. However, the nonlinear induced electricpotentials of piezoelectric actuators are inferior to the directly applied linear electricpotentials. The higher induced electric potential of a piezoelectric sensor in opencircuit would lead to a higher induced stiffness and a higher global frequencyresponse. Numerical results for the thermopiezoelastic behavior of a piezoelectriclaminated plate have shown the significant impact of piezoelectric coupling andpyroelectric effects on the sensory response of thermal piezoelectric structures.

Damping and Viscoelastic Dynamic Response of RC Flexural Members Strengthened with Adhesively Bonded Composite Materials

A theoretical approach for the dynamic viscoelastic response of reinforced concrete (RC) beams and one-way slabs strengthened with adhesively bonded composite materials is developed. The analytical model is based on variational principles, dynamic equilibrium, and compatibility of deformations between the structural components (RC beam/slab, adhesive, composite material). The model accounts for the deformability of the adhesive layer and for its high order stress and displacement fields. The equations of motion and the boundary, continuity, and initial conditions are derived via the extended Hamilton’s principle. The Kelvin-Voigt approach is adopted for the consideration of the viscoelastic response of the adhesive material and the internal damping in the composite material and the RC member. The Rayleigh damping model is used for the external viscous damping of the RC member. The dynamic governing equations are solved using the Newmark time integration and a multiple shooting algorithm is used for the so...

A new finite element model for piezothermoelastic composite beam

Finite element modelling of piezothermoelastic composite beam with distributed piezoelectric sensor and actuator layers is considered in this paper. The mathematical model is based on a high-order displacement field, a new high-order electrical potential field and a linear temperature field. This model is developed for a composite beam structures using generalized virtual work principle and is facilitated by two nodes Hermitian beam element. Constant-gain negative velocity feedback control approach has been used for active vibration control with the structures subjected to impulse and thermal excitations. The influence of the pyroelectric effects on the vibration control performance is also investigated. Comparison of numerical results from this formulation with previous works, PVDF bimorph beam, shows that the present modelling method is very efficient. Additional numerical studies for piezothermoelastic composite beam demonstrate capabilities of the current formulation to predict the thermal deformation of composite beams, as well as the active compensation of these deformations using piezoelectric structures.

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.

Higher order mode-I elastodynamic stress and displacement fields of dynamic crack propagation in brittle materials

Higher order Mode-Ι elastodynamic fields at the tip of a crack are calculated by the perturbation method. We use a certain condition at the crack tip as one of the boundary conditions and resolve the difficulty encountered otherwise [1]. We find that the well-known hoop stress maximum at the crack velocity, 60 % of the transverse sound speed, disappears when the crack is accelerated.

A 20‐node hexahedron element with enhanced distortion tolerance

AbstractThis paper presents a distortion resistant 20‐node hexahedron element that employs two different sets of shape functions for the trial and test functions. The formulation seeks to satisfy the continuity and completeness requirements by exploiting the intrinsic properties of these two sets of shape functions. Several test problems are used to assess the performance of the element under various mesh distortions. The ability of the proposed as well as the classical 20‐node element to maintain solution accuracy under severe mesh distortions has been studied. The proposed element exhibits a very high tolerance to mesh distortions. In particular, for problems involving linear and quadratic displacement fields, the element is capable of reproducing exact solution under all admissible geometrical distortions of the mesh. For test problems involving higher‐order displacement fields, the performance of the present element is in general better than that of the classical element. Copyright © 2004 John Wiley & Sons, Ltd.

Local field effects in titanium matrix composites subject to fiber-matrix debonding

This paper addresses fiber-matrix debonding in titanium matrix composites (TMCs) using a recently developed micromechanics model known as the high-fidelity generalized method of cells (HFGMC). By employing a higher-order displacement field, this model supercedes its predecessor, the generalized method of cells (GMC), in terms of micro-scale field accuracy. The import of this micro-scale accuracy is amplified in the case of fiber-matrix debonding as the debonding phenomenon is dominated by local field effects. Via inclusion of appropriate constitutive relations for inelastic deformation and fiber-matrix debonding, both HFGMC and GMC have been applied herein to model the transverse deformation of titanium matrix composites, which exhibit obvious effects of interfacial debonding. Results indicate that HFGMC is considerably more quantitatively accurate than GMC for analysis of composites with debonding, enabling realistic predictions of the TMC transverse response. The improved accuracy of the HFGMC local fields also enables investigation of some qualitative aspects of the debonding phenomenon within TMCs.

A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

AbstractAn 8‐node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions, viz., the compatibility enforcing shape functions and completeness enforcing shape functions. The former are chosen to satisfy exactly the minimum inter‐ as well as intra‐element displacement continuity requirements, while the latter are chosen to satisfy all the (linear and higher order) completeness requirements so as to reproduce exactly a quadratic displacement field. Numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions. In this respect, the proposed 8‐node unsymmetric element emerges to be better than the existing symmetric QUAD8, QUAD8/9, QUAD9, QUAD12 and QUAD16 elements, and matches the performance of the quartic element, QUAD25. For test problems involving a cubic or higher order displacement field, the proposed element yields a solution accuracy that is comparable to or better than that of QUAD8, QUAD8/9 and QUAD9 elements. Furthermore, the element maintains a good accuracy even with the reduced 2× 2 numerical integration. Copyright © 2003 John Wiley & Sons, Ltd.

Active control of axisymmetric shells with piezoelectric layers: a mixed laminated theory with a high order displacement field

This paper presents the development of a semi-analytical axisymmetric shell finite element model with embedded and/or surface bonded piezoelectric ring actuators and/or sensors for active damping vibration control of the structure. A mixed finite element approach is used, which combines the equivalent single-layer higher order shear deformation theory, to represent the mechanical behavior with a layerwise discretization in the thickness direction to represent the distribution of the electric potential of each piezoelectric layer of the frusta conical finite element. To achieve a mechanism of active control of the dynamic response of the structure a feedback control algorithm is used coupling the sensor and active piezoelectric layers. Two illustrative examples are presented and discussed.

Comparative dynamic studies of thick laminated composite shells based on higher-order theories

Here, the dynamic response characteristics of thick cross-ply laminated composite cylindrical shells are studied using a higher-order displacement model. The formulation accounts for the nonlinear variation of the in-plane and transverse displacements through the thickness, and abrupt discontinuity in slope of the in-plane displacements at any interface. The effect of inplane and rotary inertia terms is included. The analysis is carried out using finite element approach. The influences of various terms in the higher-order displacement field on the free vibrations, and transient dynamic response characteristics of cylindrical composite shells subjected to thermal and mechanical loads are analyzed.

Modelling and design of adaptive structures using B-spline strip models

The aim of this paper is to present a family of laminated plate/shell B-spline finite strip models based on higher order displacement fields applied to the optimal design of laminated composite plate/shell structures with embedded and/or surface bonded piezoelectric actuators and sensors. Simulated annealing, as a stochastic global optimisation technique, is used to improve the performance of composite adaptive structures subjected to behavioural functions and/or constraints, with continuous and discrete design variables. To show the applicability of the proposed optimisation models, two illustrative examples are presented and discussed.

On a bending problem of thick laminated composite plates

A high-order displacement field in quadrilateral element with nine nodes and twelve-degrees-of-freedom per node is developed for bending analysis of thick arbitrary layered composite plates under transverse loads. Results for plate deformations, internal stress-resultants and stresses for selected examples are shown to compare well with the closed-form and other finite element solutions.

Doubly Curved Laminated Composite Shells with Hygrothermal Conditioning and Dynamic Loads, Part 1: A Theoretical Development and Semielastic Solution Using a Higher-Order Displacement Field

A hierarchical higher-order shear deformation theory for the analysis of composite shells is presented. The structural performance of doubly curved anisotropic laminated shells in adverse hygrothermal conditions is analyzed under static and dynamic loads. The development of such a theory is more appealing, as the thickness direction of the laminates plays a dominant role due to hygroscopic diffusion and heat conduction in that direction. This model further becomes important as much weaker material properties are experienced in the transverse direction compared to that in the fiber direction. The present theory can be applied with any desirable degrees of freedom in any coordinate direction. And as a result, remodeling is not necessary in the application to solve problems. The strain-displacement relations of a laminated shell are derived with equal emphasis paid to all components of the strain tensor. The equilibrium at ply interfaces is satisfied when the laminated shells are made of angle-ply setups or unidirectional plies. The variation principle is the basis of the formulation, and top and bottom surface stress boundary conditions are applied using Lagrange multipliers. These constraints result in an interdependency between the displacement components. The hygrothermal effects are introduced in terms of trigonometric functions as have been derived by classical moisture diffusion and heat conduction equations. The material properties are updated by a micromechanics model. The application of this theory to various static and dynamic problems is presented in Part 2 of this work.