Thermo-electro-mechanical vibration of piezoelectric cylindrical nanoshells is studied using the nonlocal theory and Love’s thin shell theory. The governing equations and boundary conditions are ...derived using Hamilton’s principle. An analytical solution is first given for the simply supported piezoelectric nanoshell by representing displacement components in the double Fourier series. Then, the differential quadrature (DQ) method is employed to obtain numerical solutions of piezoelectric nanoshells under various boundary conditions. The influence of the nonlocal parameter, temperature rise, external electric voltage, radius-to-thickness ratio and length-to-radius ratio on natural frequencies of piezoelectric nanoshells are discussed in detail. It is found that the nonlocal effect and thermoelectric loading have a significant effect on natural frequencies of piezoelectric nanoshells.
The nonlocal elasticity theory of Eringen is used to study bending, buckling and free vibration of Timoshenko nanobeams. A meshless method is used to obtain numerical solutions. Results are compared ...with available analytical solutions. Two different collocation techniques, global (RBF) and local (RBF-FD), are used with multi-quadrics radial basis functions.
In this paper we apply high-polynomial order spectral/hp basis functions in the numerical implementation of a novel 7-parameter continuum shell finite element formulation using general quadrilateral ...finite elements. The implementation is applicable to the analysis of isotropic, functionally graded and laminated composite shells undergoing fully geometrically nonlinear mechanical response. The shell finite element formulation is constructed in a purely displacement-based setting such that no mixed variational procedures are employed and full numerical integration of all quantities appearing in the virtual work statement is performed using high-order Gauss–Legendre quadrature rules. An efficient procedure for numerically integrating the discrete weak formulation through the shell thickness is implemented; hence no thin- or shallow-shell type restrictions are imposed on the element. For the case of laminated composites, we introduce a discrete tangent vector field, defined on the approximate shell mid-surface, that permits the use of skewed and/or arbitrarily curved elements in the numerical simulation of complex shell structures. To reduce computer memory requirements in the numerical implementation we adopt element-level static condensation, wherein, the interior degrees of freedom of each element are implicitly eliminated prior to assembly of the global sparse finite element coefficient matrix; an approach that results in storage requirements that are on par with traditional low-order finite element implementations. The accuracy and overall robustness of the developed shell element are illustrated through the solution of several nontrivial benchmark problems taken from the literature. These numerical studies provide evidence that the proposed shell element may be used to obtain accurate locking-free results and that large load increments can be employed (in the numerical simulation of shell structures undergoing very large deformations).
A new beam element is developed to study the thermoelastic behavior of functionally graded beam structures. The element is based on the first-order shear deformation theory and it accounts for ...varying elastic and thermal properties along its thickness. The exact solution of static part of the governing differential equations is used to construct interpolating polynomials for the element formulation. Consequently, the stiffness matrix has super-convergent property and the element is free of shear locking. Both exponential and power-law variations of material property distribution are used to examine different stress variations. Static, free vibration and wave propagation problems are considered to highlight the behavioral difference of functionally graded material beam with pure metal or pure ceramic beams.
•2-D discrete lattices are modeled as classical and non-classical 1-D continuum beams.•Link is derived between the macrorotation and the micropolar antisymmetric shear.•Micropolar beam is reduced to ...a couple-stress and two classical lattice beam models.•Classical Timoshenko beam is an apt first choice for stretching-dominated cores.•The micropolar Timoshenko beam model works also for bending-dominated cores.
A discrete-to-continuum transformation to model 2-D discrete lattices as energetically equivalent 1-D continuum beams is developed. The study is initiated in a classical setting but results in a non-classical two-scale micropolar beam model via a novel link within a unit cell between the second-order macrorotation-gradient and the micropolar antisymmetric shear deformation. The shear deformable micropolar beam is reduced to a couple-stress and two classical lattice beam models by successive approximations. The stiffness parameters for all models are given by the micropolar constitutive matrix. The four models are compared by studying stretching- and bending-dominated lattice core sandwich beams under various loads and boundary conditions. A classical 4th-order Timoshenko beam is an apt first choice for stretching-dominated beams, whereas the 6th-order micropolar model works for bending-dominated beams as well. The 6th-order couple-stress beam is often too stiff near point loads and boundaries. It is shown that the 1-D micropolar model leads to the exact 2-D lattice response in the absence of boundary effects even when the length of the 1-D beam (macrostructure) equals that of the 2-D unit cell (microstructure), that is, when L=l.
Tailoring fiber orientation has been a very interesting approach to improve the efficiency of composite structures. For the discrete angle selection approach, previous methods use formulations that ...requires many variables, increasing the computational cost, and they cannot guarantee total fiber convergence (which is the selection of only one candidate angle). This paper proposes a novel fiber orientation optimization method based on the optimized selection of discrete angles, commonly used to avoid the multiple local minima problem found in fiber orientation optimization methods that consider the fiber angle as the design variable. The proposed method uses the normal distribution function as the angle selection function, which requires only one variable to select the optimized angle among any number of discrete candidate angles. By adjusting a parameter in the normal distribution function, total fiber convergence can be achieved. In addition, a usual problem in fiber angle optimization methods is that because fibers can be arbitrarily oriented, structural problems may exist at the intersection of discontinuous fiber paths. Besides, composite manufacturing technologies, such as Advanced Fiber Placement (AFP), produce better results when fiber paths are continuous. These problems can be avoided by considering continuously varying fiber paths. In the proposed method, fiber continuity is also achieved by using a spatial filter, which improves the fiber path and avoids structural problems. Numerical examples are presented to illustrate the proposed method.
We derive a thermodynamically-consistent, three-dimensional, rate form-based finite-deformation constitutive theory and computational approach for damage & fracture in nonlinear viscoelastic ...materials. The key ingredient that allows us to develop a physical criterion for fracture is the use of a Gibbs potential-based multinetwork formulation of viscoelasticity. An approach that is based on the use of the criticality of the averaged Gibbs potential over a fracture process zone is used for the initiation & propagation of cracks in a body. The rate form-based constitutive theory and fracture criterion are implemented into the Abaqus (2018) finite-element program through a user-material subroutine interface. Crack propagation is modeled as the failure of elements leading to loss of mechanical resistance to deformation. The tearing & fracture response of a viscoelastic material using a notch-in-plate sample deformed in simple tension is also simulated. By comparing the simulation results for a local damage criterion versus a truly nonlocal damage criterion, we show how the spread of Gibbs free energy around the crack tip influences the nature of the crack growth. As expected, the material fracture response described by a local damage criterion exhibits pathological mesh dependence whereas the response obtained using the nonlocal damage criterion is mesh objective regardless of mesh density, element type and orientation.
This work presents two major computational advances: (1) the conversion of the rate form-based finite-deformation constitutive theory into an objective time-integration procedure is straightforward because the material time derivative-based constitutive equations are frame-invariant, and (2) the physically-observed crack initiation & propagation process in solids can be accurately & robustly simulated using simple numerical techniques such as the element failure method if a truly nonlocal fracture criterion is utilized.
Finally, we benchmark results obtained from our computational framework and numerical simulations for modeling crack initiation & propagation to physical experimental data of mixed-mode crack propagation in a viscoelastic material.
•A new advance to modeling the fracture of viscoelastic solids is presented.•A novel rate form-based finitedeformation constitutive theory is derived.•A new computational framework for damage in viscoelastic solids is developed.•The crack propagation process in viscoelastic solids can be accurately simulated.
In this work, we develop a pseudoinverse-based static finite-element solver to model the elastic deformation and non-local brittle fracture of solids. The pseudoinverse of the finite-element ...stiffness matrix is calculated using a QR decomposition-based method which is faster than using the traditional approach based on the singular value decomposition method. This new finite-element framework has two advantages: (1) the finite-element equations can still be solved even when the finite-element stiffness matrix is singular, and (2) there is no need for introducing an artificial elastic rest energy or viscous regularization for the sole purpose of keeping the finite-element stiffness matrix non-singular in order to solve the finite-element equations. We also show that the proposed method is robust in solving chosen boundary value problems involving the elastic deformation and abrupt non-local mode I and mixed-mode brittle fracture of solids when compared to the traditional element kill method where a small but finite residual stiffness is maintained at a material in order to prevent the finite-element stiffness matrix from being singular. Hence, this proposed method allows the modeling of separation/fragmentation of solids when fracture occurs.
Finally, we use the new computational framework to model the fracture of PMMA beam samples at room temperature. By calibrating the material parameters in the constitutive theory using analytical methods and fitting to a Mode I fracture experiment force–displacement response, we show that our newly-proposed computational method is able to predict the experimental fracture loci and crack propagation characteristics in PMMA beam samples undergoing mixed-mode fracture conditions to good accord.
A consistent higher-order shear deformation non-linear theory is developed for shells of generic shape, taking geometric imperfections into account. The geometrically non-linear strain–displacement ...relationships are derived retaining full non-linear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only non-linear terms of the von Kármán type. Results show that inaccurate results are obtained by keeping only non-linear terms of the von Kármán type for vibration amplitudes of about two times the shell thickness for the studied case.
Postbuckling analysis of functionally graded ceramic–metal plates under edge compression and temperature field conditions is presented using the element-free
kp-Ritz method. The first-order shear ...deformation plate theory is employed to account for the transverse shear strains, and the von Kármán-type nonlinear strain–displacement relationship is adopted. The effective material properties of the functionally graded plates are assumed to vary through their thickness direction according to the power-law distribution of the volume fractions of the constituents. The displacement fields are approximated in terms of a set of mesh-free kernel particle functions. Bending stiffness is estimated using a stabilised conforming nodal integration approach, and, to eliminate the membrane and shear locking effects for thin plates, the shear and membrane terms are evaluated using a direct nodal integration technique. The solutions are obtained using the arc–length iterative algorithm in combination with the modified Newton–Raphson method. The effects of the volume fraction exponent, boundary conditions and temperature distribution on postbuckling behaviour are examined.
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