Biaxial Compressive Loads Research Articles

Normal form analysis of a forced aeroelastic plate

A nonlinear elastic plate in a supersonic unsteady flow forced by a dynamic excitation and a biaxial compressive load is studied. The physical behavior of the plate is modelized by the Von Kármán equations and the aerodynamic loads are modeled by using the piston theory including nonlinearities up to the third order. The space-continuum model is space-discretized by a Galerkin projection and then studied by a perturbation approach based on the Normal Form method in order to reduce the system to a simpler and essential form defined by its resonance conditions. A physical interpretation of the involved small divisors is given by analyzing how different equation parameters influence the reduced normal form model in the neighborhood of both static and dynamic bifurcation points.

Closed-form solution of the Ogden–Hill’s compressible hyperelastic model for ramp loading

This article deals with the visco-hyperelastic modelling approach for compressible polymer foam materials. Polymer foams can exhibit large elastic strains and displacements in case of volumetric compression. In addition, they often show significant rate-dependent properties. This material behaviour can be accurately modelled using the visco-hyperelastic approach, in which the large strain viscoelastic description is combined with the rate-independent hyperelastic material model. In case of polymer foams, the most widely used compressible hyperelastic material model, the so-called Ogden–Hill’s model, was applied, which is implemented in the commercial finite element (FE) software Abaqus. The visco-hyperelastic model is defined in hereditary integral form, therefore, obtaining a closed-form solution for the stress is not a trivial task. However, the parameter-fitting procedure could be much faster and accurate if closed-form solution exists. In this contribution, exact stress solutions are derived in case of uniaxial, biaxial and volumetric compression loading cases using ramp-loading history. The analytical stress solutions are compared with the stress results in Abaqus using FE analysis. In order to highlight the benefits of the analytical closed-form solution during the parameter-fitting process experimental work has been carried out on a particular open-cell memory foam material. The results of the material identification process shows significant accuracy improvement in the fitting procedure by applying the derived analytical solutions compared to the so-called separated approach applied in the engineering practice.

Fire Spalling Behaviour of Concrete: Role of Mechanical Loading (Uniaxial and Biaxial) and Cement Type

Fire poses one of the most severe environmental conditions that can act on concrete structures as an external load and can induce severe damages (cracks, spalling) or even lead to collapse. Fire spalling of concrete is a complex phenomenon, which might occur due to pressure build-up in the pores, thermal and load-induced stresses. In this context, ordinary concrete specimens (B40-II and B40-III: fc28days ≈ 40 MPa) were exposed to standard fire curve (ISO 834-1), while a constant uniaxial or biaxial compressive load was applied. Six different levels of uniaxial compressive stress on cubes and four different levels of biaxial compressive stress on slabs have been investigated. The test results showed that loaded specimens are more susceptible to spalling than unloaded specimens, with increasing amount of spalling for higher values of applied load. It has been found that biaxially loaded specimens are more prone to spalling than uniaxially loaded specimens. B40-II concrete (3% of slag) exhibited higher spalling than the B40-III concrete (43% of slag).

Initiation of unstable slips–microearthquakes by elastic impulses

A series of laboratory experiments have been carried out with a model of two granite blocks under biaxial compression loading. The experiments are mainly intended for assessing the possibilities of partially releasing the accumulated potential energy. The model was subjected to calibrated mechanical impacts (strokes) which induced elastic impulses. The mechanical stresses, strains, and acoustic emission were recorded. The strokes caused both large slips releasing the stresses down to their initial level and small slips which reduced the stresses by 5–8%. The small slips mostly occurred after the precursory emergence of the low frequency oscillations having low amplitudes. Before the large slips, the stages of speeding-up of the relative motion of the sides of the block contact was observed, similar to those emerging before the natural slips unrelated to the strokes. This feature was not universal: in some cases, the model recovered to the stationary state of the block contact without a large slip. All the slips occurred with a time delay after the stroke. The time delay was shorter when the energy of the blow was higher. With the shorter time delays, the small slip is more likely to occur. The energy of the impacts was by three orders of magnitude lower than the energy accumulated by the model, which points to the triggering mechanism of slip initiation. The series of strokes resulting in the small displacements partially reduced the accumulated energy and prevented the emergence of large motions such as the stick-slip events. If after a series of such blows a large sliding event still occurred, its energy was higher than in the slips unrelated to the impacts. The experiments revealed the difficulties in solving the problem of earthquake hazard reduction by elastic impacts.

양축 면내 압축하중 하의 이중판보강 선박판부재의 설계시스템 구축

Because of the importance of steel material saving and rational ship structural design due to the rapid increase in steel prices, a ship structural design system was developed for plate members reinforced by doubler plates subjected to biaxial in-plane compressive loads. This paper mainly emphasizes the design system improvement and upgrade according to the change in the in-plane loading condition of the doubler plate from the single load discussed in a previous paper to the biaxial in-plane compressive load discussed in this paper. A direct design process by a structural designer was added to this developed optimized system to increase the design efficiency and provide a way of directly inserting a designer's decisions into the design system process. As the second stage of preliminary steps of doubler design system development, design formulas subjected to these biaxial loads used in the doubler plate design system were suggested. Based on the introduction of influence coefficients <TEX>$K_t_c$</TEX>, <TEX>$K_t_d$</TEX>, <TEX>$K_b_d$</TEX> and <TEX>$K_a_d$</TEX> based on the variations in the doubler length, breadth, doubler thickness, and average corrosion thickness of the main plate reinforced by the doubler plate, respectively, the design formulas for the equivalent plate thickness of the main plate reinforced by the doubler plate were also developed, and a hybrid design system using these formulas was suggested for the doubler plate of a ship structure subjected to a biaxial in-plane compressive load. Using this developed design system for a main plate reinforced by a doubler plate was expected to result in a more rational reinforced doubler plate design considering the efficient reinforcement of ship plate members subjected to these biaxial loads. Additionally, a more detail structural analysis through local strength evaluations will be performed to verify the efficiency of the optimum structural design for a plate member reinforced by a doubler plate.

Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics

The maximum tensile stress criterion and the Mohr-Coulomb criterion are incorporated into the extended non-ordinary state-based peridynamics (NOSB-PD) to simulate the initiation, propagation and coalescence of the pre-existing flaws in rocks subjected to compressive loads. Wing cracks, oblique secondary cracks, quasi-coplanar secondary cracks and anti-wing cracks can be modeled and distinguished using the proposed numerical method. In the present study, a four-point beam in bending with two notches as a benchmark example is firstly conducted to verify the ability, accuracy and numerical convergence of the proposed numerical method. Then, the numerical samples of rock materials containing the one single pre-existing flaw with various lengths under uniaxial compression are modeled. Four significant factors, i.e. the axial stress versus axial strain curves, the peak strength, the ultimate failure mode and crack coalescence process, are obtained from the present numerical simulation. The effect of the flaw length on the propagation of cracks is investigated. Next, sandstone samples containing three pre-existing flaws with different ligament angles under uniaxial compression are also simulated. The effect of ligament angle on the propagation and coalescence of cracks is studied. Finally, rock-like samples containing two parallel pre-existing flaws subjected to biaxial compressive loads with confining stresses of 2.5, 5.0, 7.5 and 10.0MPa are simulated. The effect of the confining stresses on the initiation, propagation and coalescence of flaws is investigated. The present numerical results are in good agreement with the previous experimental ones.

Optimization design of unitized panels with stiffeners in different formats using the evolutionary strategy with covariance matrix adaptation

Curvilinear stiffener concept has been introduced to aircraft panel structures most recently for possible further weight reduction during optimization design. However, due to enlarged design space and high design complexity, more computational time is needed to optimize curvilinearly stiffened panels. Considering the requirement for both lighter structure and less design time, optimization designs of a unitized panel with stiffeners in three different formats, i.e. curvilinear, oblique, and evenly distributed straight, are conducted under various loading conditions to figure out which stiffener format should be selected in a real design environment. Four single loading cases with different compression-shear ratios and a set of multiple load cases are considered. Evolution strategy with covariance matrix adaption is combined with sequential quadratic programming to seek out the optimum structure with minimum mass taking into account buckling and strength constraints. Comparative analysis of the optimization results indicates that evenly distributed straight format is suitable for compression-dominant loading conditions whereas curvilinear format become superior in case that shear loading is considerable or multiple biaxial compression and shear loads are applied. Besides, curvilinear format adds more design flexibilities to the stiffeners, which enables them to play a more important role in the optimized structures. Furthermore, evolution strategy with covariance matrix adaption is found to be more efficient than particle swarm optimization for this optimization problem. This study can provide a useful guidance for future optimization design of aircraft structures.

Biaxial buckling analysis of functionally graded nanocomposite sandwich plates reinforced by aggregated carbon nanotube using improved high-order theory

In this work, biaxial buckling analysis of sandwich plates with symmetric composite laminated core and two functionally graded nanocomposite face sheets is carried out by a new improved high-order theory. The nanocomposite face sheets are carbon nanotube (CNT)-reinforced nanocomposites and the material properties of the nanocomposites plates are graded along the thickness and are estimated though the Mori–Tanaka approach. CNTs are assumed randomly oriented and aggregated into some clusters. The same third order theory is used for modeling of core and the faces sheets. The theory has third and second orders of z for in-plane and normal displacements, respectively. The principle of minimum potential energy is used to derive the equations of motion and boundary conditions. Analytical solution for static analysis of simply supported sandwich plates under biaxial in-plane compressive loads is presented using Navier’s solution. The effects of CNT volume fraction, CNT aggregation states, CNT distribution, biaxial loads ratio, and geometric dimensions of sandwich plate are investigated on the overall buckling of functionally graded carbon nanotube-reinforced nanocomposite sandwich plates.

Microscale failure mechanisms leading to internal short circuit in Li-ion batteries under complex loading scenarios

One of the least understood mechanisms of Li-ion batteries is the development of internal short circuits under mechanical loads. In this study, a micro mechanical model is developed and subjected to various loading scenarios to understand the sequence of failure in the multi-layer, multi-material structure of a Li-ion battery jellyroll. The constitutive response of each component of the electrode stack is obtained by comprehensive experimental tests using uniaxial and biaxial tensile and compressive loads. The homogenized response of the model is recovered through the computational homogenization theory. The model is validated by comparing the results of a macroscale simulation against experimental data. The study focuses next on the development of a failure criterion for the electrode stack based on the microstructural observations. Results show distinct failure mechanisms when the loading is predominantly tensile versus when it is compressive or combined tensile/compressive. A failure locus is plotted from the results of the simulations as a criterion to detect the onset of short circuit under complex multi-axial loading scenarios.

Using the modified strain gradient theory to investigate the size-dependent biaxial buckling analysis of an orthotropic multi-microplate system

On the basis of the modified strain gradient theory, this research presents an analytical approach to analyze elastic instability of an orthotropic multi-microplate system (OMMPS) embedded in a Pasternak elastic medium under biaxial compressive loads. Kirchhoff plate theory and the principle of total potential energy are applied to obtain the partial differential equations and corresponding boundary conditions. Various types of “chain” boundary conditions for the ends of the microplates system are assumed such as “Clamped-Chain,” “Free-Chain” and “Cantilever-Chain” systems. In order to analytically obtain the buckling load of the OMMPS, we use Navier’s approach which satisfies the simply supported boundary conditions and trigonometric method. In order to show the dependability of the presented formulation in this paper, several comparison studies are carried out to compare with existing data in the literature. Numerical results are presented to reveal variations of the buckling load of OMMPS corresponding to various values of the number of microplates, the length scale parameter \({\left({\frac{h}{l}}\right)}\), aspect ratio, Pasternak elastic medium parameters and the thickness of the microplate and the biaxial compression ratio. Some numerical results of this paper illustrate that when the number of microplates is small, especially becoming 2, there is an important difference between buckling loads obtained for “Clamped-Chain,” “Free-Chain” and “Cantilever-Chain” systems. Also, it is shown that by increasing the number of microplates in the system, the influence of the Pasternak elastic medium on the buckling load of system is reduced. It is anticipated that the results reported in this work are applied as a benchmark in future microstructure issues.

3D multiscale modeling of strain localization in granular media

A hierarchical multiscale modeling approach is used to investigate three-dimensional (3D) strain localization in granular media. Central to the multiscale approach is a hierarchical coupling of finite element method (FEM) and discrete element method (DEM), wherein the FEM is employed to treat a boundary value problem of a granular material and the required constitutive relation for FEM is derived directly from the DEM solution of a granular assembly embedded at each of the FEM Gauss integration points as the representative volume element (RVE). While being effective in reproducing the complex mechanical responses of granular media, the hierarchical approach helps to bypass the necessity of phenomenological constitutive models commonly needed by conventional FEM studies and meanwhile offers a viable way to link the macroscopic observations with their underlying microscopic mechanisms. To model the phenomenon of strain localization, key issues pertaining to the selection of proper RVE packings are first discussed. The multiscale approach is then applied to simulate the strain localization problem in a cubical specimen and a cylindrical specimen subjected to either conventional triaxial compression (CTC) or conventional triaxial extension (CTE) loading, which is further compared to a case under plane-strain biaxial compression (PBC) loading condition. Different failure patterns, including localized, bulging and diffuse failure modes, are observed and analyzed. Amongst all testing conditions, the PBC condition is found most favorable for the formation of localized failure. The CTC test on the cubical specimen leads to a 3D octopus-shaped localization zone, whereas the cylindrical specimen under CTC shows seemingly bulging failure from the outlook but rather more complex failure patterns within the specimen. The CTE test on a uniform specimen normally ends in a diffuse failure. Different micro mechanisms and controlling factors underlying the various interesting observations are examined and discussed.

The 3D Numerical Simulation for the Propagation Process of Multiple Pre-existing Flaws in Rock-Like Materials Subjected to Biaxial Compressive Loads

General particle dynamics (GPD), which is a novel meshless numerical method, is proposed to simulate the initiation, propagation and coalescence of 3D pre-existing penetrating and embedded flaws under biaxial compression. The failure process for rock-like materials subjected to biaxial compressive loads is investigated using the numerical code GPD3D. Moreover, internal crack evolution processes are successfully simulated using GPD3D. With increasing lateral stress, the secondary cracks keep growing in the samples, while the growth of the wing cracks is restrained. The samples are mainly split into fragments in a shear failure mode under biaxial compression, which is different from the splitting failure of the samples subjected to uniaxial compression. For specimens with macroscopic pre-existing flaws, the simulated types of cracks, the simulated coalescence types and the simulated failure modes are in good agreement with the experimental results.

Buckling analysis of thin rectangular plates under uniaxial or biaxial compressive point loads by the differential quadrature method

To obtain accurate buckling load for rectangular plates under compressive point loads, one of the important factor is that the in-plane stress distributions within the plate prior to buckling should be accurate enough. Although the differential quadrature method (DQM) has been successfully used in a variety of fields including the buckling analysis of thin rectangular plates under non-uniformly distributed edge compressions, however, the DQM has some difficulty in dealing with singular functions such as the Dirac-delta function appeared in the stress boundary condition. In this paper, novel formulations are presented to overcome the difficulty encountered in dealing with the Dirac-delta functions by using the DQM. The normal stress boundary condition is numerically integrated before being discretized in terms of the differential quadrature. Detailed formulations are given. Buckling of rectangular plates under either uniaxial or biaxial compressive point loads is successfully analyzed. It is demonstrated that accurate buckling loads can be obtained by the DQM for rectangular plates with nine combinations of boundary conditions and various aspect ratios. The compactness and computational efficiency of the DQM are retained in solving the partial differential equations with boundary conditions involving Dirac-delta functions. The accuracy of the differential quadrature (DQ) results is verified by comparing them with existing analytical solutions and finite element data. New results are tabulated which can be a reference for other researchers to develop new numerical methods.

Buckling and modal analysis of rotationally restrained orthotropic plates

This paper describes a theoretical approach for the static buckling, eigenfrequency and vibration buckling analysis of typical orthotropic plates used in aeronautical constructions. The plates are constrained with any combination of simply-supported and rotationally restrained edges, and are subjected to biaxial compressive loads. The derivation of closed-form solutions is discussed, and the results are presented in the form of design charts, where the effect of the restraint stiffness is highlighted. The proposed design charts allow a more accurate initial sizing of typical aerospace structural components, like stiffened panels, where both static and dynamic buckling loads are fundamental for the final design. As compared to classical solutions based on simply-supported and clamped edges, the modeling of the edge constraint as an elastic restraint provides a more refined description of the real stringer-stiffened panel. The results demonstrate that, in contrast to the commonly adopted simply-supported boundary condition, a reduction of the design conservativeness can be achieved if the stiffness of the restraint is accounted for. A novel set of results relative to the vibration buckling is derived, illustrating that the dynamic buckling tends to increase for load frequencies higher than the first natural frequency.

Working temperature variation effect on the failure envelope of continuous fiber-reinforced composites

In this study, a micromechanical method is presented to study working temperature variation effect on biaxial failure envelopes of continuous fiber-reinforced composites with imperfect interfacial bonding. Generalized viscoplastic potential structure model is used to describe nonlinear response of composites. The interfacial debonding model is incorporated into the micromechanical model for describing the interfacial damage evolution. Theoretical results show good consistency with experimental data. On this basis, a series of numerical examples are performed to investigate working temperature variation and interfacial debonding effect on macroscopic tensile response and biaxial loading failure, respectively. The results indicate that the stress–strain responses and failure strength are closely dependent on working temperature. And biaxial compressive loadings in axial-transverse and transverse–transverse, as well as axial tensile and compressive loadings, do not generate interfacial debonding.

In-plane biaxial compression and tension testing of thin sheet materials

Detailed experimental data on the behavior of textured sheet metals under compressive loading is important to describe their tension–compression asymmetry. This is particularly needed for materials that exhibit a strength-differential effect, or in cases where the Bauschinger effect occurs. So far, there is no systematic work describing the third quadrant in the 2D stress space under biaxial compressive loading. This paper presents a new device for biaxial, compressive in-plane testing of thin sheets. Biaxial and uniaxial compression experiments are carried out in the strain controlled device, analyzing the behavior of deep drawing steel sheets with and without skin-pass treatment. Moreover, in order to allow for the experimental description of the yield surfaces, biaxial tensile tests are performed. Detailed numerical validations and experimental strain analysis both for the new specimen for biaxial compressive testing and for the cruciform specimen for biaxial tensile testing show that reasonably homogeneous strain distributions can be achieved. The combined experimental and numerical method presented here allows to evaluate the tension–compression asymmetry of thin sheet materials. The results for the skin-passed condition clearly exhibit a tension–compression asymmetry, which highlights the necessity of biaxial compression tests already in the as-received material condition. The biaxial compression test opens a pathway to a more detailed analysis of the flow behavior of thin sheets under biaxial compression loading.

Material behavior modeling with multi-output support vector regression

Based on neural network material-modeling technologies, a new paradigm, called multi-output support vector regression, is developed to model complex stress/strain behavior of materials. The constitutive information generally implicitly contained in the results of experiments, i.e., the relationships between stresses and strains, can be captured by training a support vector regression model within a unified architecture from experimental data. This model, inheriting the merits of the neural network based models, can be employed to model the behavior of modern, complex materials such as composites. Moreover, the architectures of the support vector regression built in this research can be more easily determined than that of the neural network. Therefore, the proposed constitutive models can be more conveniently applied to finite element analysis and other application fields.As an illustration, the behaviors of concrete in the state of plane stress under monotonic biaxial loading and compressive uniaxial cycle loading are modeled with the multi-output and single-output support regression respectively. The excellent results show that the support vector regression provides another effective approach for material modeling.

Buckling failure analysis of cracked functionally graded plates by a stabilized discrete shear gap extended 3-node triangular plate element

We present new numerical results in buckling failure analysis of cracked composite functionally graded plates subjected to uniaxial and biaxial compression loads. An accurate extended 3-node triangular plate element in the context of the extended finite element method (XFEM) is developed, integrating the discrete shear gap method (DSG) to eliminate shear-locking. The plate kinematics is based on the Reissner–Mindlin theory, and material properties are assumed to vary through thickness direction, obeying a power law distribution. The developed DSG-XFEM is found to be effective and accurate as it owns many desirable advantages: conveniently representing crack geometry which is independent of the mesh; shear-locking effect is no longer valid; mesh distortion is insensitive and controllable; thin plates is possible; triangular elements are easily generated for problems even with complex geometries; and high accuracy. All these arisen features are demonstrated through numerical examples and the effects of crack-length, material gradation, mesh distortion, inclined angles of cracks, boundary conditions, width-to-thickness ratio, length ratio, etc. on the critical buckling coefficient (CBC) are analyzed. Numerical results reveal that the material gradation, crack-length, thickness, length ratios, etc. have a strong effect on the behavior of CBC. This phenomenon is mainly attributed to the plate stiffness degradation due to the presence of local defects and material composition. Also, the boundary conditions greatly alter the CBC whereas the inclination of cracked angle is found to be insignificant.

Experimental study on mechanical behavior of high performance concrete under multi-axial compressive stress

Experimental investigation was conducted to characterize the responses of high performance concrete (HPC) subjected to multiaxial compressive stresses. The HPC specimens were prepared with three different mix proportions, which corresponds to three different uniaxial compressive strengths. The cubic specimens with size of 100 mm for each edge were tested with servo-hydraulic actuators at different stress ratios. The principal stresses and strains of the specimens were recorded, and the failure of the cubic specimens under various stress states was examined. The experimental results indicated that the stress states and stress ratios had significant influence on the strength and deformation of HPC under biaxial and triaxial compression, especially under triaxial compression. Failure criteria were proposed for the HPC specimens under biaxial and triaxial compressive loading. The test results provided a valuable reference for obtaining multi-axial constitutive law for HPC.

Numerical Modelling of Cylindrical Hole Under Equal Biaxial and Uniaxial Tension Loadings at the Infinity

The determination of stresses and displacements is the vital preoccupation in the world of computing structural elements subjected to any loading. In this context, this work focuses on dealing with classical problem of elasticity which has a vested interest. The stress and displacements of a cylindrical cavity in an infinite elastic medium will be analyzed first on a theoretical level by highlighting the existing analytical solution. And then through the enormous development of computers and programming languages, this problem will be solved numerically where the results will be compared with the analytical solutions. Using the software FLAC2D, this work has a numerical analysis of two problems: the first one is to calculate the stresses and displacements around a cylindrical cavity subjected to a biaxial compression loading at the limits, this case is a verification problem in FLAC and a tool for validating the procedure for numeric simulation. The second one corresponds to the numerical analysis of the stress distribution and displacements around the cylindrical hole under uniaxial tension stress at the infinity. The results obtained by this model are in very good agreement with analytical results.