Tangent Stiffness Research Articles

Tangential bond stiffness evaluation of adhesive lap joints by spectral interference of the low-frequency A0 lamb wave

Transmission spectra of the lowest-order antisymmetric (A0) Lamb wave for adhesively bonded single lap joints are experimentally investigated in a frequency range below 0.4 MHz. Based on the theoretical results of the wave interference, a tangential bond stiffness evaluation method is proposed for the lap joints. Aluminum alloy plates bonded with epoxy adhesive on different conditions were prepared as bonded specimens. An air-coupled ultrasonic transducer was used to generate the A0 mode in the specimens, and the signals of the transmitted waves across the joints were measured with a pin-type piezoelectric transducer. Spectral analysis was performed for the measured waveforms, and the transmission coefficient was calculated for each specimen as a function of frequency. The experimental results showed that the transmission coefficient has local maxima and minima at multiple frequencies, called peak and notch frequencies, respectively. The sets of the peak and notch frequencies were different among the bonded specimens. Based on the peak and notch frequencies, the tangential bond stiffness was estimated for each bonded specimen. As a result, the bond stiffnesses obtained for the specimens with the same nominal bond thickness and adherend pretreatment were almost equal even if the bond lengths were different. Furthermore, the evaluation results showed that pretreatments such as sanding and contamination of adherend surfaces affect the bond stiffness due to the change in the interfacial stiffness.

Geometrically nonlinear formulation of Generalized Beam Theory for the analysis of thin-walled circular pipes

In this paper, the formulation of Generalized Beam Theory (GBT) for the geometrically nonlinear analysis of thin-walled circular pipes is presented. GBT is a computationally efficient numerical method which is especially formulated for thin-walled members with a capacity of determining the cross-sectional deformation through a combination of a set of pre-determined cross-sectional deformation modes. In this study, the current GBT analysis of circular pipes which is limited to buckling analysis is enhanced to a full geometrically nonlinear analysis. This new formulation considers the nonlinear membrane kinematic description based on the Green–Lagrange strain definition. The nonlinear tangent stiffness matrices and the internal force vectors are derived from the variation of the internal energy which results in third and fourth-order GBT deformation mode coupling tensors. These tensors can predetermine the type of GBT deformation modes needed for the nonlinear analysis based on the applied loading conditions. In addition to the classical GBT deformation modes, the non-conventional GBT deformations modes have a vital role since without these modes the coupling tensors and the nonlinear stiffness matrix related to the transverse and the shear membrane energy will be lost. Here, to illustrate the application and capabilities of the developed GBT formulation, two numerical examples involving transverse and longitudinal bending are presented to show the nonlinear relationship between bending, cross-sectional ovalization, and higher local deformation modes. For the purpose of validation, these examples are compared with refined shell finite element models in both displacement and stress fields.

Model order reduction of thermo-mechanical coupling flexible multibody dynamics via free-interface component mode synthesis method

A systematical procedure of model order reduction for the thermo-mechanical coupling flexible multibody system (FMS) is proposed. The displacement and temperature field are discretized by unified element mesh based on the absolute nodal coordinate formulation (ANCF). Since the expression of the elastic forces is highly nonlinear and the tangential stiffness matrix of ANCF is time varying, the Taylor expansion is used to linearized the dynamic and heat transfer equations to obtain constant tangential stiffness matrix. The thermo-mechanical coupling system is divided into several substructures and the free-interface component mode synthesis method is used at the interface. The modal truncation for the dynamic and thermal coordinates are performed to reduce the system degrees of freedom (DOF). Therefore, the computational efficiency is greatly improved. The system thermo-mechanical equations are reunited and the solving scheme is given. Numerical results show that the proposed method can significantly improve the simulation efficiency without losing accuracy.

Improved Finite Element Method for Inflated Beams with Local Wrinkles

An improved finite element method is proposed for inflatable beams with a global finite deformation and a local wrinkling behavior, based on the corotational approach and tension field theory. The finite deformation is decomposed into a rigid-body displacement in the global coordinate system and a small strain measured in the local coordinate system, by using the corotational approach. The tangent stiffness matrix of a three-node triangular membrane element is presented. With the modified bimodulus constitutive law, a wrinkling model is constructed for determination of the status of element (tension, wrinkled, or slack). To improve the rate of convergence, a consistent tangent material stiffness matrix is embedded into the local coordinate system. Numerical results are in good agreement with the existing experimental data. In some case, the proposed method can obtain more accurate deflection of inflatable beams than the postbuckling analysis of thin shells. A quadratic rate of convergence is observed for the proposed method, and thus the computational efficiency is improved greatly. The limitation of the method is also discussed.

Influence of strain-hardening on the load-carrying behaviour of bearing type bolted connections

The investigations presented in this paper focus on the hole bearing resistance of symmetric single-bolt connections, taking into account the distinct hardening behaviour of various steel grades and quantifying its impact on the resulting load-carrying behaviour. The main focus of the study was put on the limitation of the hole bearing deformations by defining a corresponding “resistance”, which is thus a quantity with the background of an implicit deformation limitation. This represents a further expansion of new regulations, which have currently found their way into the draft version of the part of the Eurocode 3 standard dedicated to the resistance of connections prEN1993-1-8. In contrast to this draft standard, the specific differences between steels with very pronounced and less pronounced hardening behaviour are dealt with. Experimental investigations were carried out on four different steel grades, each with eight different geometries, in order to investigate the influence of this hardening behaviour on the hole bearing deformations.The paper presents a new approach to define the point on the load-deformation curve between acceptable and excessive deformations. This is defined as a point with a certain slope of the normalised and standardised load-deformation curve, namely the point where the tangent stiffness corresponds to the secant stiffness defined by connecting the origin of the load-deformation point to the point with the peak load.Based on the results of the hole bearing tests in combination with the new approach to define the transition point between acceptable and excessive deformations, an adapted resistance model is presented. In this model, the hardening behaviour of the steel is taken into account, whereby here specifically the ratio fu/fy of tensile strength to yield strength influences the load-bearing behaviour.In addition to laboratory tests, numerical simulations were carried out with an extended range of geometries. This made it possible to create a representative data set with which the proposed model could be verified. It was shown that with the developed proposal a significant improvement of the prediction accuracy can be achieved compared to the resistance model in prEN1993-1-8.The proposed adaptation of the hole bearing verification which takes the fu/fy ratio into account facilitates the optimal use of material and can thus save resources. The format of the hole bearing verification remains almost unchanged, which should simplify its applicability in practice.

Experimental investigation on the tribological behavior and surface damage due to embedment of ceramic proppant – analog mudrock composite systems

The analysis of interface problems involving proppants and rocks is of major interest in unconventional reservoir exploitation and modeling as well as in geophysics, mining engineering and rock mechanics. Despite the significant progresses that have been reported in hydraulic fracturing, there has been given less attention in the fundamental understanding of the interactions between proppants and rocks, which can provide enhanced input parameters in computational-based analyses involving complex multi-phase processes. In this study, the tribological behavior of three types of contacts including ceramic-ceramic (analog) proppants, kaolinite-kaolinite blocks (analog mudrock) and ceramic-kaolinite composite interfaces was investigated performing laboratory tests using a custom-developed loading apparatus and the experiments involved force and displacement measurements by the application of high-precision transducers. Based on a series of normal loading tests, it was observed that the ceramic beads exhibited a completely elastic behavior, whereas the kaolinite blocks showed a softer response with, predominantly, plastic deformations. The results of the shearing tests indicated that for the ceramic-kaolinite composite contacts, the arrangement of the clay microfabric significantly affected the shearing load-displacement response in terms of the post-peak softening behavior, the development of the microslip regime as well as the dynamic friction. It was also shown that even though the interface friction was controlled, predominantly, by the softer kaolinite block, the interface tangential stiffness had different involved mechanisms, in which case the kaolinite blocks showed the highest stiffness specifically at greater magnitudes of normal load, whereas the ceramic-kaolinite composite interfaces had the lowest values of stiffness.

Contact stiffness of jointed interfaces: A comparison of dynamic substructuring techniques with frictional hysteresis measurements

The tangential contact stiffness is an important parameter used in non-linear dynamic analyses of jointed structures since it can strongly affect the prediction of resonance frequencies. Many experimental techniques are available for contact stiffness estimations, but the reliability of such estimations remains unknown due to a lack of comparative studies. This paper proposes a comparative study of contact stiffness measurements obtained with two experimental techniques: hysteresis loop measurements and Frequency Based Substructuring (FBS). Hysteresis loops are traditionally measured with dedicated friction test rigs to provide, amongst others, contact stiffness estimations through local interface measurements. The assumption with hysteresis measurements is that the measured parameters are independent of the dynamics of the test rig and can therefore be used as input for analyses of other structures, as long as loading conditions and contact interfaces are comparable. An alternative approach to identify the contact stiffness is FBS, which uses information from the overall system dynamics. FBS has the advantage that it can be applied to any structure, without the need of building ad-hoc test rigs, consequently giving a structure-specific information. Despite this advantage over hysteresis measurements, it is as of yet not well understood how accurately FBS can extract contact stiffness values. This paper presents FBS measurements and hysteresis loop measurements performed simultaneously on the same contact interface of a traditional high-frequency friction rig during vibration, thus enabling a cross-validation of the results of both techniques. This novel comparison validates FBS approaches against local hysteresis measurements and shows the strengths and limitations of both experimental methods, making it possible to improve the current understanding of the contact stiffness of jointed structures.

Analysis of tangential characteristics of the paraboloidal contact model

A paraboloidal contact model with different axial crossing angles is established. The contact model is subjected to a combined action of normal and tangential loads. Based on the Hertzian theory, the contact states of the model under the action of the normal load are obtained by deriving the equations. The contact states include critical interference values of the elastic phase, elastoplastic phase, and plastic phase. Under the action of the normal preload, the tangential stiffness of the contact model is obtained and verified by finite element simulation. Simulation results indicate that tangential displacements of the contact model in the deformation phase correspond well with analytical solution equations. Stresses and strains on the contact surface are also investigated. By employing the proposed contact model, the effects of axial crossing angle, friction factor, focal length, and normal load on mechanical properties are investigated. The paraboloidal contact model is advantageous to contact analysis for paraboloidal gears and cams.

Probabilistic-based analysis and model selection of the tangential stiffness reduction: displacement curves of nonconforming contacts

Probabilistic-based analysis and model selection of the tangential stiffness reduction: displacement curves of nonconforming contacts

A symmetric tangent stiffness approach to cohesive mechanical interfaces in large displacements

The present article proposes a formulation for a cohesive interface element in large displacement conditions. Theoretical and computational aspects, useful for an effective and efficient finite element implementation, are examined in details. A six-node (or higher) isoparametric interface element for two dimensional cohesive fracture propagation problems is developed. The element operators are consistently derived by a variational approach enforced in the current configuration, where a current frame is defined with axes tangential and normal to the middle line of the interface opening displacement gap. Under the constitutive assumption of small value of the modulus of the vector product between the Cauchy traction and the displacement jump vector, explicit expression of the interface nodal force vector and of the consistent tangent stiffness matrix are derived in a closed form. At difference with most of the available formulations, the proposed approach allows to naturally achieve a symmetric tangent stiffness matrix, provided that all the involved terms are maintained in the development, namely without neglecting second-order partial derivatives involved in the exact linearization procedure.

Nonlinear multiscale modeling of thin composite shells at finite deformations

In this work a formulation for non linear multi-scale analysis of thin shells is presented and a modeling scheme is proposed in a FE2 method’s context. The shells may undergo large deformations and exhibit heterogeneous micro-structures consisting of nonlinear materials and cohesive interfaces. Appropriate use of an attached coordinate system, for the projection of the strain measures, enables the elimination of large rotations from the kinematic constraints that are imposed at the Representative Volume Element (RVE), simplifying this way the boundary value problem (BVP) to be solved at the micro-structural level. This way, the BVP formulation refers to the local coordinate system of the RVE in which the imposition of the plane stress condition becomes totally independent from the current orientation of the midsurface of the shell. Emanating from Hill’s averaging theorem, which is satisfied by use of appropriated averaging relations, the principle of virtual work is formulated in the attached auxiliary system basis, where a consistent tangent stiffness matrix is analytically derived. The resulting methodology is assessed against popular benchmarks for thin shells and allows for a straight forward integration of Kirchhoff–Love shells under large deformations in existing FE2 codes. The simulation possibilities originating from this formulation are countless and are related to the wide applicability of the Kirchhoff Love theory in many fields of engineering.

Compressive loading of structural insulated panels and conventional stud walls: a study

Structural insulated panels (SIPs) are structural elements comprising a core of expanded polystyrene insulation sandwiched between two oriented strand boards. They can replace conventional joist floors and stud walls in low-rise residential construction. Three identical samples of full-scale SIP walls and stud walls were tested according to ASTM standards and experimental results for the specimens were obtained in terms of their ultimate load capacity and failure mechanism. Comparison of the load–deformation diagrams of the SIPs and stud walls showed that tangential stiffness variation of the former occurred in the hardening mode, whereas the variation for the latter was in two stages, hardening and softening. Regarding compressive loading, the results showed that SIPs are as good as conventional wood framing of the same size. Therefore, SIP walls are very efficient in the case of axial compressive loading.

Bistable inclined beam connected in series for quasi-zero stiffness

This paper investigates a structure system that expresses stable and controllable quasi-zero stiffness in an integrated form, which is based on inclined bistable beams connected in series. This paper shows that interaction between the connected beams and their nonlinear axial stiffness always leads to a plateau trajectory in its force-displacement curve with very low tangential stiffness. The plateau part of the force-displacement occurs while each of the connected beam snaps through sequentially and ends when all of the beams are going though hardening behavior. A series of tests were conducted to validate this behavior. A numerical model capable of predicting the behavior is developed. Studies were performed based on the model for this kind of structure with different geometric configuration and details of coupling links. The results show that in addition to being able to achieve quasi-zero stiffness of an integrated structures, this kind of structure has increased initial stiffness and loading capacity compared with traditional bistable structures subjected to similar limitations of extreme strains.

Accurate determination of the stiffness properties for an elastic interface under peeling conditions between homogeneous materials

In this paper, several models for analysing peeling tests are considered, including numerical models based on FEM, semi-analytical approaches, and beam models including an elastic interface. Results show that none of the beam models including an elastic interface represents correctly the mixity at the crack tip for small thickness ratios. To avoid this inconsistency, the problem is studied in the framework of Fracture Mechanics, concluding that the stiffness in both directions must fulfil a material-dependent relationship (besides, the tangential stiffness is independent of the shear modulus). The new proposed interface stiffnesses depict accurately both the mixed-mode conditions at the crack tip and the load versus displacement evolution for the full range of thickness ratios.

Implicit implementation of the nonlocal operator method: an open source code

In this paper, we present an open-source code for the first-order and higher-order nonlocal operator method (NOM) including a detailed description of the implementation. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combined with the method of weighed residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. The implementation in this paper is focused on linear elastic solids for sake of conciseness through the NOM can handle more complex nonlinear problems. The NOM can be very flexible and efficient to solve partial differential equations (PDEs), it’s also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Finally, we present some classical benchmark problems including the classical cantilever beam and plate-with-a-hole problem, and we also make an extension of this method to solve complicated problems including phase-field fracture modeling and gradient elasticity material.

Second-order arbitrarily-located-refined plastic hinge model for high-strength steel frame design

A second-order inelastic analysis method is developed for welded high-strength steel (i.e., Q460GJ steel in this study) frames with box-sections and H-sections. Due to the great differences in the distribution of the residual stresses between welded Q460GJ steel and mild steel, traditional tangent modulus and stiffness degradation function are unsuitable for the design of welded 460GJ steel members and structures. This paper proposes a new beam–column element using the stability functions which allows refined plastic hinges to form in an arbitrary location along the member. With the considering of the effects of residual stresses in Q460GJ steel sections, new tangent modulus formulae and stiffness degradation functions are proposed for the welded Q460GJ steel members and structures. Existing experimental results of welded Q460GJ steel columns under axial and eccentric compression are introduced to verify the reasonability and accuracy of the proposed method. Based on the theoretical model, a computation program is developed for the analysis of steel frames with welded Q460GJ steel section and the analyzed results agree well with the finite element results. The proposed analysis is shown to be an efficient tool ready to be implemented into the design practice.

Elastic properties and anharmonicity of solids

The squares of the velocities of the longitudinal and transverse acoustic waves separately are practically not associated with anharmonicity, and their ratio (νL2/νS2) turns out to be a linear function of the Gruneisen parameter γ --- the measure of anharmonicity. The obtained dependence of (νL2/νS2) on γ is in satisfactory agreement with the experimental data. The relationship between the quantity (νL2/νS2) and anharmonicity is explained through its dependence on the ratio of the tangential and normal stiffness of the interatomic bond λ, which is a single-valued function of the Gruneisen parameter λ(γ). The relationship between Poisson's ratio μ and Gruneisen parameter γ, established by Belomestnykh and Tesleva, can be substantiated within the framework of Pineda's theory. Attention is drawn to the nature of the Leont'ev formula, derived directly from the definition of the Gruneisen parameter by averaging the frequency of normal lattice vibration modes. The connection between Gruneisen, Leontiev and Belomestnykh--Tesleva relations is considered. The possibility of a correlation between the harmonic and anharmonic characteristics of solids is discussed. Keywords: elastic properties, Gruneisen parameter, formulas of Belomestnykh--Tesleva, Leont'ev, Gruneisen equation, tangential and normal stiffness of interatomic bond, crystals, glasses.

Effects of Microstructure on the Buckling Fatigue and Functional Degradation Characteristics of the Tape-shaped Cu-Al-Mn Shape Memory Alloy Element

Shape memory alloys (SMAs) are functional materials with high shape memory and super elasticity effects that are used in a wide range of fields, including medical and engineering fields. Our research group has focused on the negative stiffness characteristics of plate-shaped SMA during buckling deformation and has attempted to apply them to vibration-isolation devices. The buckling properties of Cu-Al-Mn SMA, which exhibits almost the same super elastic properties as Ti-Ni SMA, were evaluated in comparison with those of Ti-Ni SMA. The results showed that Cu-Al-Mn SMA is more suitable for vibration isolators because of its smaller change in post buckling properties due to temperature change and smaller hysteresis compared with Ti-Ni SMA. In addition, the buckling fatigue and functional degradation properties of Cu-Al-Mn SMA were studied. As a result, it was clarified that fatigue failure is basically caused by the propagation of cracks originating from grain boundaries, and that the increase of these cracks is the cause of the decrease in buckling load. In this study, the buckling fatigue properties of Cu-Al-Mn SMA material in which bainite phase is formed at the grain boundary by changing the heat treatment conditions were investigated, and the effects of the differences in internal microstructures on the buckling fatigue and functional degradation properties of plate-like Cu-Al-Mn SMA with different internal microstructures were compared with those of conventional Cu-Al-Mn material. The effects of different internal microstructures on buckling fatigue and functional degradation properties of plate Cu-Al-Mn SMA with different internal structures were investigated. The results show that the specimens with bainite phase have superior fatigue properties, but in terms of functional degradation properties, they increase the rate of decrease in Pcr and increase the rate of increase in tangential stiffness after buckling.

Упругие свойства и ангармонизм твердых тел

The squares of the velocities of the longitudinal and transverse acoustic waves separately are practically not associated with anharmonicity, and their ratio (vL2 / vS2) turns out to be a linear function of the Grüneisen parameter γ - the measure of anharmonicity. The obtained dependence of (vL2 / vS2) on γ is in satisfactory agreement with the experimental data. The relationship between the quantity (vL2 / vS2) and anharmonicity is explained through its dependence on the ratio of the tangential and normal stiffness of the interatomic bond λ, which is a single-valued function of the Grüneisen parameter λ (γ). The relationship between Poisson's ratio μ and Grüneisen parameter γ, established by Belomestnykh and Tesleva, can be substantiated within the framework of Pineda's theory. Attention is drawn to the nature of the Leont'ev formula, derived directly from the definition of the Grüneisen parameter by averaging the frequency of normal lattice vibration modes. The connection between Grüneisen, Leontiev and Belomestnykh-Tesleva relations is considered. The possibility of a correlation between the harmonic and anharmonic characteristics of solids is discussed.

Research on solid shell element based on penalty method

Since the angle degrees of freedom do not have the vector property, it is critical to handle this for nonlinear geometrical behavior. To simulate the nonlinear geometrical behavior of shell structure efficiently, we developed a solid‐shell element type without drilling degrees of freedom. In this paper, the element satisfies the assumption of the straight normal of the shell and we established penalty functions in the thickness direction of the shell. The formulations of equivalent node force and tangent stiffness obtained by 3D isoparametric element theory can be expressed by matrix forms clearly. The constitutive relationship in the global system is derived from the local constitutive matrix according to the tensor coordinate transform. Finally, taking a series of classic flat plate problems and curved shell problems as examples, the solid shell element using the penalty method is compared with the solid shell element using the artificial stiffness method and other shell elements. The results show that the solid element proposed in this paper can effectively eliminate shear self-locking after using the reduced integral scheme and obtain satisfactory accuracy for thin shells and medium-thickness•A type of solid shell element without rotational degrees of freedom is introduced using the penalty method.•A suitable penalty factor of this element is determined through numerical experiments.•This element is more efficient than the traditional shell elements and can be programmed with less time.