A Coupled Zig-Zag Third-Order Theory for Piezoelectric Hybrid Cross-Ply Plates

2004 ◽  
Vol 71 (5) ◽  
pp. 604-614 ◽  
Author(s):  
S. Kapuria
Keyword(s):  
Electric Potential ◽  
Virtual Work ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Order Theory ◽  
Piecewise Linear Approximation ◽  
Shear Stresses ◽  
Normal Strain ◽  
Third Order ◽  
Hybrid Cross

A new zig-zag coupled theory is developed for hybrid cross-ply plates with some piezoelectric layers using third-order zig-zag approximation for the inplane displacements and sublayer wise piecewise linear approximation for the electric potential. The theory considers all electric field components and can model open and closed-circuit boundary conditions. The deflection field accounts for the transverse normal strain due to the piezoelectric d33 coefficient. The displacement field is expressed in terms of five displacement variables (which are the same as in FSDT) and electric potential variables by satisfying exactly the conditions of zero shear stresses at the top and bottom, and their continuity at layer interfaces. The governing equations are derived from the principle of virtual work. Comparison of the Navier solutions for the simply-supported plates with the analytical three-dimensional piezoelasticity solutions establishes that the present efficient zig-zag theory is quite accurate for moderately thick plates.

scholarly journals Bending of functionally graded plates via a refined quasi-3D shear and normal deformation theory

2018 ◽  
Vol 5 (1) ◽  
pp. 190-200 ◽  
Author(s):  
Asharf M. Zenkour ◽  
Rabab A. Alghanmi
Keyword(s):  
Transverse Shear ◽  
Deformation Theory ◽  
Virtual Work ◽  
Three Dimensional ◽  
Present Theory ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Normal Strain ◽  
Equilibrium Equations ◽  
Normal Deformation

Abstract Bending of functionally graded plate with two reverse simply supported edges is studied based upon a refined quasi three-dimensional (quasi-3D) shear and normal deformation theory using a third-order shape function. The present theory accounts for the distribution of transvers shear stresses that satisfies the free transverse shear stresses condition on the upper and lower surfaces of the plate. Therefore, the strain distribution does not include the unwanted influences of transverse shear correction factor. The effect of transverse normal strain is included. Unlike the traditional normal and shear deformation theories, the present theory have four unknowns only. The equilibrium equations are derived by using the principle of virtual work. The influence of material properties, aspect and side-to-thickness ratios, mechanical loads and inhomogeneity parameter are discussed. The efficiency and correctness of the present theory results are established by comparisons with available theories results.

Two-Dimensional Piezoelasticity and Zigzag Theory Solutions for Vibration of Initially Stressed Hybrid Beams

2005 ◽  
Vol 127 (2) ◽  
pp. 116-124 ◽  
Author(s):  
S. Kapuria ◽  
N. Alam ◽  
N. K. Jain
Keyword(s):  
Axial Strain ◽  
Piecewise Linear ◽  
Coefficient Matrix ◽  
Piecewise Linear Approximation ◽  
Two Dimensional ◽  
Governing Equations ◽  
Third Order ◽  
Hybrid Beams ◽  
Set Up ◽  
Zigzag Theory

Two-dimensional (2D) exact piezoelasticity and one-dimensional coupled zigzag theory solutions are presented for vibration of initially stressed simply-supported cross-ply symmetrically laminated hybrid piezoelectric beams under axial strain and actuation potentials. In the 2D exact solution, the coupled governing equations for the vibration mode are derived using Fourier series. Using transfer matrix approach and the boundary conditions, homogeneous equations are set up for the variables at the bottom. The determinant of their coefficient matrix is set to zero to obtain the natural frequency. An efficient coupled zigzag theory is developed for vibration of initially stressed hybrid beams. A piecewise linear approximation of the potential field, an approximation for the deflection to account for the piezoelectric strain and a combination of global third-order variation and layer-wise linear variation for the axial displacement are employed. The conditions of absence of shear tractions at the top and bottom and the conditions of continuity of transverse shear stress at the layer interfaces are exactly satisfied. The governing equations are derived from extended Hamilton’s principle. Comparison of natural frequencies of beams and panels of different configurations with the exact 2D piezoelasticity solution establish that the present zigzag theory is generally very accurate for moderately thick beams. The first-order and third-order shear deformable theories, which are also assessed, are found in some cases to yield poor results even for thin beams.

A procedure for the solution of general contact problems for the elastic half-space

2000 ◽  
Vol 35 (6) ◽  
pp. 559-565 ◽  
Author(s):  
C. E Truman ◽  
D A Hills ◽  
A Sackfield
Keyword(s):  
Pressure Distribution ◽  
Efficient Solution ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Piecewise Linear Approximation ◽  
Contact Pressure Distribution ◽  
True Contact

In this paper an efficient numerical procedure for the efficient solution of general three-dimensional elastic contacts where the bodies in the neighbourhood of the point of contact may be represented as half-spaces is described. The solution relies on a piecewise-linear approximation to the true contact pressure distribution.

Third-Order Polynomials Model for Analyzing Multilayer Hard/Soft Materials in Flexible Electronics

2016 ◽  
Vol 83 (8) ◽  
Author(s):  
Xianhong Meng ◽  
Boya Liu ◽  
Yu Wang ◽  
Taihua Zhang ◽  
Jianliang Xiao
Keyword(s):  
Mechanical Properties ◽  
Flexible Electronics ◽  
Virtual Work ◽  
Soft Materials ◽  
Normal Strain ◽  
Stress Distributions ◽  
Third Order ◽  
Inorganic Semiconductors ◽  
Design And Optimization ◽  
Good Agreement

In flexible electronics, multilayer hard/soft materials are widely used to utilize both the superior electrical properties of inorganic semiconductors and robust mechanical properties of polymers simultaneously. However, the huge mismatch in mechanical properties of the hard and soft materials makes mechanics analysis challenging. We here present an analytical model to study the mechanics of multilayer hard/soft materials in flexible electronics. Third-order polynomials are adopted to describe the displacement field, which can be used to easily derive both strain and stress fields. Then, the principle of virtual work was used to derive the governing equations and boundary conditions, which can be solved numerically. Two types of loadings, pure bending and transverse shear, are studied. The normal strain distributions along thickness direction in the bimaterial regions clearly show zigzag profiles, due to the huge mismatch in the mechanical properties of the hard and soft materials. The effect of very different mechanical properties of the hard and soft materials on shear stress distributions can also be predicted by this model. The results from this analytical mode show good agreement with finite-element modeling (FEM). This model can be useful in systems with multilayer hard/soft materials, to predict mechanical behavior and to guide design and optimization.

A Higher-Order Theory for Plane Stress Conditions of Laminates Consisting of Isotropic Layers

1999 ◽  
Vol 66 (1) ◽  
pp. 95-100 ◽  
Author(s):  
X. J. Wu ◽  
S. M. Cheng
Keyword(s):  
Three Dimensional ◽  
Order Theory ◽  
Higher Order ◽  
Shear Stresses ◽  
Stress Equilibrium ◽  
Material Discontinuities ◽  
Interlaminar Shear ◽  
Higher Order Theory ◽  
Three Dimensional Elasticity ◽  
Higher Order Functions

In this paper, a higher-order theory is derived for laminates consisting of isotropic layers, on the basis of three-dimensional elasticity with displacements as higher-order functions of z in the thickness direction. The theory employs three stress potentials, Ψ (an Airy function), p (a harmonic function), and its conjugate q, to satisfy all conditions of stress equilibrium and compatibility. Interlaminar shear stresses, i.e., antiplane stresses, are shown to be present at the interfaces, especially near material discontinuities where gradients of in-plane stresses are usually high. For illustrating its practical application, the problem of a plate containing a hole patched with an intact plate is solved.

The contact and friction problem in the rolling process

2000 ◽  
Vol 214 (1) ◽  
pp. 61-69
Author(s):  
J D Lee ◽  
S Shen
Keyword(s):  
Finite Element ◽  
Virtual Work ◽  
Large Strain ◽  
Three Dimensional ◽  
Rolling Process ◽  
Shear Stresses ◽  
Time Step ◽  
Element Analysis ◽  
Friction Law ◽  
Updated Lagrangian

In this work, a new generalized non-Euclidean friction law is proposed. This friction law allows the friction coefficients in the tangential and axial directions of the roll to be different. A three-dimensional, large-strain, non-steady state elastic-plastic finite element analysis has been performed for the flat rolling process. The contact and friction problem at the interface between the workpiece and the rolls is treated rigorously. The finite element procedures are based on the updated Lagrangian virtual work equations in incremental form. The solution at each time step is accepted only if the equilibrium of nodal forces, the calculation of which is an exact treatment, is reached pointwise. The numerical results, including the interfacial normal and shear stresses, are presented and discussed.

Theories for Elastic Plates Via Orthogonal Polynomials

1981 ◽  
Vol 48 (4) ◽  
pp. 900-904 ◽  
Author(s):  
S. Krenk
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Displacement Function ◽  
Shear Stresses ◽  
Transverse Displacement ◽  
Normal Strain ◽  
Two Dimensional ◽  
Elastic Plates

A complementary energy functional is used to derive an infinite system of two-dimensional differential equations and appropriate boundary conditions for stresses and displacements in homogeneous anisotropic elastic plates. Stress boundary conditions are imposed on the faces a priori, and this introduces a weight function in the variations of the transverse normal and shear stresses. As a result the coupling between the two-dimensional differential equations is described in terms of a single difference operator. Special attention is given to a truncated system of equations for bending of transversely isotropic plates. This theory has three boundary conditions, like Reissner’s, but includes the effect of transverse normal strain, essentially through a reinterpretation of the transverse displacement function. Full agreement with general integrals to the homogeneous three-dimensional equations is established to within polynomial approximation.

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