In-plane Biaxial Loading Research Articles

Vibration analysis of biaxially compressed double-layered graphene sheets based on nonlocal strain gradient theory

ABSTRACTThis paper develops a nonlocal strain gradient plate model for vibration analysis of double-layered graphene sheets under biaxial in-plane mechanical loads. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. In fact, frequency increment due to the strain gradient effect is neglected in all previous papers related to double-layer graphene sheets. Governing equations of a nonlocal strain gradient double-layer graphene sheet on elastic substrate are derived via Hamilton's principle. Galerkin's method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as in-plane loading, nonlocal parameter, length scale parameter, elastic foundation, interlayer stiffness and boundary conditions on vibration characteristics a double-layer graphene sheet are examined.

Fatigue crack propagation under biaxial fatigue loading with single overloads

The crack propagation behavior and the governing crack growth micromechanisms in aluminum alloy under in-plane biaxial fatigue loading with single overloads, of different magnitudes and occurring at different fatigue crack lengths, is investigated. The microscale fracture mechanisms governing crack growth behavior under these conditions are identified through detailed fractography. Crack growth retardation behavior observed due to the occurrence of single overloads is correlated with overload magnitude, instantaneous fatigue crack length, crack-tip plasticity and fracture surface morphology. The results obtained provide insight into the relationship between macroscale crack growth behavior to microstructural mechanisms, which is essential to understanding the fatigue behavior of metallic materials under variable amplitude biaxial loading scenarios.

Wrinkling of Functionally Graded Sandwich Structures Subject to Biaxial and In-Plane Shear Loads

Benefits of a functionally graded core increasing wrinkling stability of sandwich panels have been demonstrated in a recent paper (Birman, V., and Vo, N., 2017, “Wrinkling in Sandwich Structures With a Functionally Graded Core,” ASME J. Appl. Mech., 84(2), p. 021002), where a several-fold increase in the wrinkling stress was achieved, without a significant weight penalty, using a stiffer core adjacent to the facings. In this paper, wrinkling is analyzed in case where the facings are subject to biaxial compression and/or in-plane shear loading, and the core is arbitrary graded through the thickness. Two issues addressed are the effect of biaxial or in-plane shear loads on wrinkling stability of panels with both graded and ungraded core, and the verification that functional grading of the core remains an effective tool increasing wrinkling stability under such two-dimensional (2D) loads. As follows from the study, biaxial compression and in-plane shear cause a reduction in the wrinkling stress compared to the case of a uniaxial compression in all grading scenarios. Accordingly, even sandwich panels whose mode of failure under uniaxial compression was global buckling, the loss of strength in the facings or core crimpling may become vulnerable to wrinkling under 2D in-plane loading. It is demonstrated that a functionally graded core with the material distributed to increase the local stiffness in the interface region with the facings is effective in preventing wrinkling under arbitrary in-plane loads compared to the equal weight homogeneous core.

The effect of bi-axial in-plane loads on the natural frequency of nano-plates

In this paper, natural frequencies of nano-plates subjected to two-sided in-plane tension or compressive loads, based on Eringen nonlocal elasticity theory and displacement field of first-order shear deformation plate theory (FSDT), are investigated. By considering total rotational variables as the two rotations due to bending and shear, another formulation form of FSDT nano-plate is achieved, that can simultaneously consider classical plate theory (CLPT) and FSDT. In a comprehensive study, the effects of different parameters such as a nonlocal parameter, aspect ratio, thickness to length ratio, mode number, boundary conditions and also length of nano-plate are examined on the dimensionless natural frequency. The results show that simultaneously applying two-sided tension and compressive in-plane loads changes frequency in a manner which is different to one-directional loading.

Experimental response of historic brick masonry under biaxial loading

Unreinforced masonry (URM) structures in seismic areas are usually considered weak in terms of supporting dynamic action. Sufficient regularity of structural organisms plays an important part in fostering adequate behaviour of masonry building although the response of historic-URM (HURM) is greatly influenced by the strength of brick and mortar materials. The analysis of damaged masonry structures following an earthquake puts emphasis on knowing the proper values of strength under biaxial in-plane loading to estimate the response of HURM. On the other hand, damage to masonry buildings subjected to the last earthquakes in Italy (2016–17) which invested mainly its central regions, highlighted the need to further investigate HURM present in many small Italian towns to define the actual behaviour of masonry walls. This paper describes and discusses the results of an extensive experimental research campaign, developed over several years, on wallets, triplets, and walls built with solid historic bricks in 1/3rd scale. Additional experimental tests reported in this paper provide data useful to calibrate failure criteria available for assessing HURM buildings by numerical modelling using FE codes.

SIMULATION OF FRICTIONAL DISSIPATION UNDER BIAXIAL TANGENTIAL LOADING WITH THE METHOD OF DIMENSIONALITY REDUCTION

The paper is concerned with the contact between the elastic bodies subjected to a constant normal load and a varying tangential loading in two directions of the contact plane. For uni-axial in-plane loading, the Cattaneo-Mindlin superposition principle can be applied even if the normal load is not constant but varies as well. However, this is generally not the case if the contact is periodically loaded in two perpendicular in-plane directions. The applicability of the Cattaneo-Mindlin superposition principle guarantees the applicability of the method of dimensionality reduction (MDR) which in the case of a uni-axial in-plane loading has the same accuracy as the Cattaneo-Mindlin theory. In the present paper we investigate whether it is possible to generalize the procedure used in the MDR for bi-axial in-plane loading. By comparison of the MDR-results with a complete three-dimensional numeric solution, we arrive at the conclusion that the exact mapping is not possible. However, the inaccuracy of the MDR solution is on the same order of magnitude as the inaccuracy of the Cattaneo-Mindlin theory itself. This means that the MDR can be also used as a good approximation for bi-axial in-plane loading.

Strength and plastic deformation behavior of nanolaminate composites with pre-existing dislocations

Pre-existing dislocations (PED) are ubiquitous inside crystalline lattices which in turn affect the yield stress and the process of plastic deformation. Hence, understanding the onset of dislocations motion and interaction is critical in modifying or designing new materials with advanced properties so that their mechanical behavior approach realistic conditions. One such family of new materials is the ceramic/metallic nanolaminates. In this work, we have investigated the effect of pre-existing dislocations on the mechanical behavior of NbC/Nb nanolaminates using molecular dynamics simulations. Upon unloading at different strains from stress-strain curve of 3nm NbC/7nm Nb sample, we were able to generate structures with various pre-existing dislocation densities inside the layers. Uniaxial loadings parallel to the interface at two different temperatures (10K and 300K) were performed on each structure. Also, the yield locus was determined at 300K by applying biaxial in-plane loading and fitted with a general flow potential to be used in macroscale analysis. Finally, the tension-compression asymmetry (TCA) was investigated for the structures with pre-existing dislocations along two different in-plane loading directions.

면내 전단하중과 양축압축하중을 받는 선박 판부재의 이중판 설계시스템 개발

A design system for doubler reinforcement of the ship plate members subjected to in-plane shear and biaxial compressive loads was developed. This design system of doubler reinforcement on ship plate members established by design supporting system and this system was based on the buckling evaluation process of ship plate members for these in-plane loads. Each design parameters were suggested by equations as the form of influence coefficients for the doubler reinforcement subjected to the various in-plane loads including shear load. Strength of doubler plate member reinforced on the plate member could be suggested by the equivalent flat plate thickness after the consideration of corelation equations in the design system of doubler reinforcement. Level of strength recovery of ship plate members for these in-plane loads according to the local reinforcement by doubler could be suggested by use of this design system in the initial repair design stage of shipyards.

Characterization of forming limits at fracture with an optimized cruciform specimen: Application to DP600 steel sheets

In order to identify experimental limit strains at fracture for metallic sheets under in-plane biaxial loadings, a specific specimen shape has to be defined. Ideally, the proposed geometry must lead to fracture at the specimen center whatever the strain path imposed by the experimental device. If such condition is ensured, the whole strain paths ranging from equi-biaxial to uniaxial could be experimentally investigated. Based on the literature review, four in-plane cross specimen shapes have been selected owing to strong potential to reach large strain and fracture in their central point. From finite element simulations, the behavior of these different shapes is evaluated. Based on these numerical results, a new shape is proposed and optimized for determining the forming limits at fracture of DP600 sheet metal with a thickness of 2mm. The forming limit strains at fracture are identified by a time-dependent method and the forming limit strains at necking are determined by a position-dependent method. The experimental forming limit strains at fracture of DP600 sheet metal are fitted by a forming line. Three ductile fracture criteria are then calibrated with experimental points. The Oyane criterion with two parameters gives the best prediction.

Optimal Cruciform Specimen Design Using the Direct Multi-search Method and Design Variable Influence Study

Nowadays the development of new testing machines and the optimization of new specimen geometries are two very demanding activities. In order to study complex material stress and strain distributions, as in-plane biaxial loading, one must develop new technical solutions. A new type of testing machine has been developed by the present authors, for the fatigue testing of cruciform specimens, but the low capacity of the testing machine requires the optimization of the specimen in order to achieve higher but uniform stress and strain distributions on the specimen center. In this paper, the authors describe the procedure to optimize one possible geometry for cruciform specimens, able to determine the fatigue initiation life of material subjected to out of phase in-plane biaxial fatigue loadings. The high number of design variables were optimized using the direct multi-search method, considering two objective functions, the stress level on the specimen center and the uniformity of the strain distribution on a 1.0 mm radius of the specimen center. Several Pareto Fronts were obtained for different material thickness, considering the commercially available sheet metal thickness. With the optimal solution, the influence of every design variable was studied in order to provide others with a powerful tool that allows selecting the optimal geometry for the desired application. The results are presented in the form of design equations considering that the main design variable, the material thickness, was chosen from a Renard series of preferred numbers. The end user is then able to configure the optimal specimen for the required fatigue test.

An original single variable shear deformation theory for buckling analysis of thick isotropic plates

This work proposes an original single variable shear deformation theory to study the buckling analysis of thick isotropic plates subjected to uniaxial and biaxial in-plane loads. This theory is built upon the classical plate theory (CPT) including the exponential function in terms of thickness coordinate to represent shear deformation effect and it involves only one governing differential equation. Efficacy of the present theory is confirmed through illustrative numerical examples. The obtained results are compared with those of other higher-order shear deformation plate theory results.

Identification of strain rate-dependent mechanical behaviour of DP600 under in-plane biaxial loadings

The rate-dependent hardening behaviour of dual phase DP600 steel sheet is investigated by means of in-plane biaxial tensile tests, in an intermediate strain rate range (up to 20s−1) and for large strains (up to 30% of equivalent plastic strain) at room temperature. The dynamic biaxial in-plane tensile tests are performed on a dedicated cruciform specimen shape previously developed and validated for large strains at quasi-static conditions. To perform dynamic biaxial tensile tests, a specific device is used to reduce oscillations on measure of forces. Based on these experiments, parameters of two phenomenological strain-rate hardening models (saturated and power law), are identified. Material parameters are obtained from an optimisation process, based on the finite element modelling of the biaxial test, by minimising differences between experimental and simulated principal strains at the specimen centre. A comparison of the strain hardening functions calibrated from both biaxial and uniaxial tensile tests under static and dynamic conditions are then proposed. Results show that the biaxial procedure developed in this work, associated with a specific shape of the cross specimen, are more efficient to determine the rate-dependent hardening behaviour for large strains than the conventional uniaxial tensile test.

Positive and negative surface effects on the buckling and vibration of rectangular nanoplates under biaxial and shear in-plane loadings based on nonlocal elasticity theory

In this paper, positive and negative surface effects on the buckling and vibration of rectangular nanoplates under biaxial and shear in-plane loadings are investigated based on the nonlocal elasticity theory. Aluminum and silicon materials, respectively, with positive and negative surface properties are considered. The surface effect is considered using Gurtin–Murdoch’s theory. Using Hamilton’s principle, the governing equations considering small scale for both bulk and surface properties of nanoplate are derived. Generalized differential quadrature method (GDQM) is used. GDQM results are validated by comparing with the Navier’s method. The influence of nonlocal parameter, boundary conditions, in-plane biaxial and shear loads, and width-to-length aspect ratio, on the positive and negative surface effects influencing buckling and vibration are investigated. Numerical results show that by increasing the nonlocal parameter, the effect of positive surface property increases on the buckling and vibration behavior of nanoplate, while this conclusion is reversed for negative surface property.

An exact solution for stability analysis of orthotropic rectangular thin plate under biaxial nonlinear in-plane loading resting on Pasternak foundation

In this research, the buckling analysis of orthotropic rectangular plate resting on Pasternak elastic foundation was studied, using Frobenius exact solution method. The plate is subjected to biaxial in-plane loading with non-uniform distribution. It is assumed that it is simply supported by two opposite sides, and the remaining two edges can have any arbitrary conditions. To extract the governing equations on the buckling of the plate, the classical plate theory based on Kirchhoff hypothesis is employed. According to Levy solution, the buckling equation is reduced to an ordinary differential equation. Frobenius method is exploited in the governing equation, and the eigenvalue equation is obtained, imposing the boundary conditions on the other two sides. By solving the eigenvalue equation, the dimensionless critical buckling loads are determined. The accuracy of presented results is validated by comparing with available results in previous studies and also finite element method. Furthermore, the influences of some parameters such as aspect ratio, the ratio of elasticity modulus of the plate in two in-plane directions, the type of non-uniform loading in two states of uniaxial and biaxial loadings, various combinations of boundary conditions, lateral and shear stiffness coefficients of elastic foundation are examined on critical buckling.

Characterization of the nonlinear behavior of nodular graphite cast iron via inverse identification: Analysis of biaxial tests

In this paper the nonlinear behavior of nodular graphite cast iron is investigated under in-plane biaxial loading regimes. Multiaxial experiments enable a wide variety of loading paths to be investigated. In the presented study two loading histories are proposed, namely, equibiaxial and “snail” loading paths corresponding to proportional and nonproportional regimes. The center gauge zone of the testing cruciform specimen is observed. Mean strain fluctuations are measured and presented in order to determine the material response for the prescribed biaxial loading paths. For the equibiaxial and snail loading histories finite element model updating will couple 2D-DIC measurements with 3D-FE analyses to calibrate material parameters describing elasticity, isotropic or kinematic hardening.

Buckling analysis of orthotropic rectangular plate resting on Pasternak elastic foundation under biaxial in-plane loading

ABSTRACTIn this study, buckling of rectangular orthotropic plates resting on a Pasternak elastic foundation under biaxial in-plane loading by the power series method (the method of Frobenius) was analyzed. Similar to many studies, two opposite edges of loading are simply supported and two other edges are assumed clamped. In order to extract the characteristic equations of orthotropic rectangular plate under in-plane loading resting on a Pasternak elastic foundation, the classical plate theory, by considering the interaction between plate and foundation, is used. The results showed that in the aspect ratio of less than 2, the existing Pasternak foundation caused the buckling load to increase severely, but by increasing the aspect ratio, the effect of the foundation is negligible. Applying the in-plane load in the y-direction caused the buckling load to decrease, but by increasing the aspect ratios the effect of the load is negligible.

Stochastic progressive failure analysis of laminated composite plates using Puck's failure criteria

ABSTRACTThe present work deals with second-order statistics of the progressive failure response of laminated composite plates subjected to in-plane uniaxial and bi-axial loadings with random system properties. A stochastic finite element method based on higher-order shear deformation theory combined with first- and second-order perturbation technique is used for solution of random progressive failure equation. A Puck failure criterion is used for the evaluation of first ply and last-ply failure load. The results obtained using the present solution approach are validated with the results available in the literature and Monte Carlo simulation.

3-D elasticity buckling solution for simply supported thick rectangular plates using displacement potential functions

This paper concentrates on the elastic buckling of thick plates. The shear strains in thick structures such as thick plates are important and thus, cannot be neglected. On the other hand, shear strains in the analysis of thick plates leads to complicity of governing equations. In this paper, with the use of displacement potential functions, the governing equations are simplified to two partial differential equations for the potential functions, one of which is second order and the other is forth order. These PDEs are established for a rectangular isotropic body, so that they are applicable to any arbitrary thickness of plate with no limitation on its thickness ratio. By solving the governing differential equations using separation of variables method and satisfying the exact boundary conditions, an analytical solution is obtained for linear elastic buckling of simply supported rectangular thick plates, subjected to in-plane either uniaxial or biaxial static loads, one of which could also be tensile force. Then critical buckling load is expressed in terms of non-dimensional buckling factor. The results of this paper are compared with other analytical and numerical works for thin and moderately thick plates, and also with numerical works for thick plates, which proves an excellent agreement between the results of this paper and others.

Shear vibration and buckling of double-layer orthotropic nanoplates based on RPT resting on elastic foundations by DQM including surface effects

In this article, shear vibration and buckling of double-layer orthotropic nanoplates resting on elastic foundations are analyzed subjected to in-plane loadings including surface and nonlocal effects. These effects are considered by Gurtin---Murdoch's theory. Using the principle of virtual work, the governing equations for bulk and surface of double-layer orthotropic nanoplate are derived using refined plate theory. Differential quadrature method (DQM) is implemented. DQM solutions are validated by Navier's method and journal references. The influences of nonlocal parameter, van der Waals, Winkler, shear modulus, orthotropic material properties, boundary conditions, and in-plane biaxial, uniaxial, and shear loadings, are investigated on the surface effects of buckling and vibration modes of out-of-phase and in-phase. Results demonstrate that by augmenting nonlocal parameter, the surface effects on the vibration and buckling modes of both out-of-phase and in-phase increase. This result is in contrast with the works of other researchers in the field. Moreover, by enhancing in-plane loadings, the degree of surface effects on the vibration increase. On the other hand, the effects of nonlocal parameter on the vibration and buckling under in-plane shear load are more influential than those of biaxial and uniaxial, while the surface effects on the biaxial vibration and buckling are more remarkable than those of shear vibration and buckling.

DYNAMIC INSTABILITY OF COMPOSITE SKEW PLATES USING BOUNDARY CHARACTERISTIC ORTHOGONAL POLYNOMIALS

Dynamic instability analysis of laminated composite skew plate for different skew angles subjected to different type of linearly varying in-plane loadings is investigated. The analysis also includes the instability of skew plate under uniform bi-axial in-plane loading. The skew plate structural model is based on higher order shear deformation theory (HSDT), which accurately predicts the numerical results for thick skew plate. The total energy functional is derived for the skew plates from total potential energy and kinetic energy of the plate. The strain energy which is the part of total potential energy contains membrane energy, bending energy, additional bending energy due to additional change in curvature and shear energy due to shear deformation, respectively. The total energy functional is mapped into a square plate over which a set of orthonormal polynomials satisfying the essential boundary conditions is generated by Gram–Schmidt orthogonalization process. Different boundary conditions of skew plate have been correctly incorporated by using Rayleigh–Ritz method in conjunction with Boundary Characteristics Orthonormal Polynomials (BCOPs). The boundaries of dynamic instability regions are traced by the periodic solution of governing differential equations (Mathieu type equations) with period T and 2T. The width of instability region for uniform loading is higher than various types of linearly varying loadings (keeping the same peak intensity). Effect of various parameters like skew angle, aspect ratio, span-to-thickness ratio, boundary conditions and static load factor on dynamic instability has been investigated.