This study presents a novel non-local model for the stress analysis of sandwich plates with a functionally graded core using Peridynamic Differential Operator (PDDO) and Refined Zigzag Theory (RZT). ...The through-thickness material properties of the functionally graded cores were tailored by means of mixing rules. The PDDO converts the equilibrium equations of the RZT from the differential form into the integral form. This makes the PDDO capable of solving the local differential equations accurately. The RZT is very suitable for the stress analysis, especially for thick and moderately thick plates. It contains only seven kinematic variables and eliminates the use of the shear correction factors. A typical sandwich structure consists of a soft core and stiff orthotropic face-sheets. The mismatch of the stiffness at the core and face sheet interfaces results in an increase in the interfacial shear stresses, leading to the core-face sheet delamination. The interfacial stresses can be mitigated by functionally grading the material properties of the core through the thickness. The PD-RZT stress and displacement predictions were compared with the analytical solutions by using the uniform and non-uniform mesh discretizations and good agreements were achieved. It was observed that the functionally graded cores offered some advantages with respect to the classical cores and minimized the stress concentrations at the interface of the core and the face sheets.
•A new nonlocal model is presented for the stress analysis of sandwich plates embedding functionally graded (FG) core.•Peridynamic Differential Operator (PDDO) is used to solve the equilibrium equations of the Refined Zigzag Theory (RZT).•The PDDO can determine any arbitrary order of derivatives accurately regardless of the presence of jump discontinuities or singularities.•The PDDO is free of the requirement of symmetric kernels, eliminating the necessity of ghost particles near the boundaries.•RZT contains only seven kinematic variables and eliminates the use of the shear correction factors.
In this study, stress analysis of laminated composite beams is carried out by using Refined Zigzag Theory (RZT) and Peridynamic Differential Operator (PDDO). The PDDO replaces local differentiation ...with nonlocal integration. This makes the PDDO capable of solving the local differential equations accurately. RZT is suitable for both thin and thick beams eliminating the use of the shear correction factors. Also, RZT ensures a constant number of kinematic variables regardless of the number of layers in the beam. The governing equations of the RZT beam and the boundary conditions were derived by employing the principle of virtual work. The capability of the present approach was assessed by considering various beams for different boundary conditions and aspect ratios. It provides robust and accurate predictions for the displacement and stress components in the analysis of highly heterogeneous laminates.
•A new nonlocal beam formulation is proposed for the stress analysis of adhesively bonded beams with modulus graded adhesives.•Peridynamic Least Square Minimization (PDLSM) approach is utilized for ...the approximation of the equilibrium equations of the Refined Zigzag Theory (RZT).•RZT is highly useful for the efficient and accurate stress analysis of thin and thick loadbearing structures.•The PDLSM introduces the local derivatives in terms of their nonlocal forms.•Various modulus graded adhesive layers are considered to investigate their effects on the stress minimization.
The present study provides a nonlocal beam model for the stress analysis of beams bonded with modulus adhesives using Peridynamic Least Square Minimization (PDLSM) and Refined Zigzag Theory (RZT). RZT is highly useful for the efficient and accurate stress analysis of thin and thick load-bearing structures. RZT avoids the use of shear correction factors to estimate the transverse shear stresses. PDLSM introduces the local derivatives in terms of their nonlocal forms. The PDLSM is applicable for the approximation of any order derivatives. In this study, the PDLSM was employed for the solution of the equilibrium equations of RZT. The robustness of the present approach was demonstrated by considering dissimilar bonded aluminum (Al)-carbon fiber-reinforced polymer composite (CFRP) beam. Modulus graded adhesives have been successfully implemented to minimize the stress concentrations occur in the bonded structures. In order to investigate the effects of the modulus graded adhesive layers on the stress minimization at the critical locations of the bonded beam, various adhesive models were investigated in detail. Each adhesive profile experienced different deformation and stress states. The peak stress levels near the adherend-adhesive interfaces were observed to be alleviated with the use of a modulus graded adhesive layer.
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•PeriDynamic Differential Operator is employed to solve the equations of the Refined Zigzag Theory (RZT).•The solution to the RZT equilibrium equations is achieved iteratively by using the ...Newton-Raphson method.•A cohesive zone model is used to model delamination initiation and evolvement inlaminated beams.•The present approach PD-RZT is computationally efficient and produces accurate stress and failure predictions for delaminated beams.
This study aims to model delaminations in laminated composite beams by using the PeriDynamic Differential Operator (PDDO) and Refined Zigzag Theory (RZT). A cohesive zone model is employed to monitor the delamination evolvement in laminated composite beams by embedding an interfacial resin layer between two potentially separable material layers. PDDO calculates the local derivatives in their nonlocal forms and produces highly accurate predictions for the solution of differential equations. RZT eliminates the consideration of shear correction factors and paves the way for modeling both thin and thick laminates. RZT enables computationally efficient analyses since it considers a constant number of kinematic variables regardless of the number of layers in the beam. The principle of virtual work is employed to derive the governing equations and boundary conditions of the RZT. The calculation of transverse normal and shear stresses at critical locations plays an important role in the delamination event. Therefore, this approach is promising for the delamination analysis of laminated composite beams. The present approach performs the nonlocal integration for the approximation of the local derivatives; hence, it reduces the undesirable localized stress peaks. It was demonstrated that the present approach successfully captured the deformation and stress fields as well as the delamination onset and evolvement of the laminated beams.
This study presents an ordinary state-based peridynamic (OSB PD) analysis within the finite-element framework while considering implicit/explicit solvers. The present PD formulation permits ...non-uniform discretization with a variable horizon and eliminates the use of external surface and volume correction factors. An implicit solver is employed until immediately before damage emerges, and then an adaptive time-stepping explicit solver for crack initiation and propagation. The major advantage of the present approach is the reduction in computational time. The PD interactions lead to a sparsely populated global stiffness matrix. The BiConjugate Gradient Stabilized (BICGSTAB) method is employed to determine the solution of the system equations. Damage onset and its evolution is investigated using the critical stretch criterion. The efficacy of the present approach is established by considering two different geometric configurations and loading/boundary conditions. The PD predictions for the crack patterns compare well with those of the analytical results and experimental observations.
Peridynamic least squares minimization Madenci, Erdogan; Dorduncu, Mehmet; Gu, Xin
Computer methods in applied mechanics and engineering,
05/2019, Letnik:
348
Journal Article
Recenzirano
Odprti dostop
This study presents the peridynamic approximation of a field variable and its temporal and spatial derivatives based on least squares minimization. Such capability permits the conversion of local ...form of differentiation to its nonlocal analytical integral form. It enables the analysis of discrete and scattered data for numerical differentiation and approximation of the field variable without employing any special techniques. Also, it enables the numerical solution of differential field equations with single and multiple variables in a computational domain of either uniform or nonuniform discretization. The implicit solution to the discrete form of the differential equations can be achieved by employing standard techniques for solving sparse non-symmetric systems. The accuracy of this approach is demonstrated by considering numerical differentiation of discrete data, and solving ordinary and partial differential equations with particular characteristics.
This study presents an accurate mixed variational formulation for the stress analysis of laminated composite plates based on Refined Zigzag Theory (RZT). A two-field variational concept based on the ...Hellinger-Reissner (HR) principle is employed associated with the kinematic assumptions of the RZT. The RZT provides a good mixture between the accuracy and computational efficiency for the thin and thick laminated composite structures without using the shear correction factors. A four-noded quadrilateral element and bi-linear shape functions are used for the discretization of the solution domain ensuring the C0-continuity. The main novelty of the present study is that the flexural behavior of the laminated composite plates is investigated based on RZT within the light of HR principle using monolithic approach for the first time. The proposed Mixed Finite Element (MFE) formulation assigns stress resultant type field variables in addition to the kinematic variables of the RZT. Therefore, the present approach, MRZT, paves the way of obtaining the stress resultants at each node directly from the solution of the system equations. Since the shear forces are obtained at each node, Equivalent (transformed) Section Principle (ESP) is utilized to achieve continuous through thickness transverse shear stress variations. In-plane strain components are calculated through the compliance matrix without resorting to the spatial derivatives of displacement components. The robustness and capability of the present approach are established through benchmark problems, and its applicability to challenging problems is demonstrated by modeling thick and highly heterogeneous plates, a delaminated plate and three-point bending tests.
This study investigates the crack initiation and its progression in two-dimensional functionally graded (FG) plates under dynamic and quasi-static loading conditions by using an ordinary state-based ...PeriDynamic (OSB PD) theory. Functionally gradient materials (FGMs) are a new class of advanced materials that provide a smooth transition among the layers of materials to meet the desired requirements in engineering structures. The effective material properties in the FG plate were functionally tailored in two directions by employing a rule of mixture. The present OSB PD theory is very suitable for the failure analysis of materials without the use of surface and volume correction factors. The robustness of the present approach on monitoring the mixed-mode fractures is established with the experimental results under various loading conditions. The PD predictions successfully captured the critical load levels and damage evolutions. Subsequently, numerical analyses were carried out to assess the influence of the one- and two-dimensional material variations, boundary, and loading conditions on the fracture behavior of the FG plates. The compositional gradient exponent played a major role on the reaction force levels and crack trajectories. It was observed that two-dimensional material variations paved the way for increasing the plate strength and fracture resistance.
This study presents the weak form of peridynamic (PD) governing equations which permit the direct imposition of nonlocal essential and natural boundary conditions. It also presents a variational ...approach to derive the PD form of first- and second-order derivatives of a field variable at a point which is not symmetrically located in its domain of interaction. This capability enables the nonlocal PD representation of the internal force vector and the stress components without any calibration procedure. Furthermore, it removes the concern of truncated domain of interaction for a point near the surface. Thus, the solution is free of nonlocal boundary forces and surface effects. The numerical solution of the resulting equations can be achieved by considering an unstructured nonuniform discretization. The implicit solution to the discrete form of the equations is achieved by employing BiConjugate Gradient Stabilized (BICGSTAB) method which is an iterative technique for solving sparse non-symmetric linear systems. The explicit analysis is performed by constructing a global diagonal mass matrix, and using a hybrid implicit/explicit time integration scheme. The accuracy of this approach is demonstrated by considering an elastic isotropic plate with or without a cutout subjected to a combination of different types of boundary conditions under plane stress conditions.
•Weak form of bond-associated non-ordinary state based peridynamics.•Bond-associated deformation gradient tensor and force density vectors.•Use of peridynamic differential operator.•Implicit and ...hybrid implicit/explicit solution of resulting governing equations.•Combination of nonlocal essential and natural boundary conditions.
The non-ordinary state-based peridynamics (NOSB PD) is attractive because of its ability to employ existing constitutive relations for material models. The deformation gradient tensor and the force density vector appearing in the equilibrium equations are expressed in terms of nonlocal integrals. The definitions of these nonlocal integrals affect the accuracy and stability of PD predictions. Therefore, this study introduces a more accurate representation of the deformation gradient and the bond associated (BA) force density vector by using the peridynamic differential operator (PDDO). Also, it presents the weak form of BA-NOSB PD governing equations in order to impose natural and essential boundary conditions without the use of Lagrange multipliers for implicit and explicit analysis. By considering a two-dimensional rectangular plate with and without a hole under tension, the numerical results demonstrate the accuracy of BA-NOSB PD with no oscillations and zero energy modes.
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