Penalty Stiffness Research Articles

Layerwise third-order shear deformation theory with transverse shear stress continuity for piezolaminated plates

Regarding laminated structures, an electromechanically coupled Finite Element (FE) model based on Layerwise Third-Order Shear Deformation (LW-TOSD) theory is proposed for static and dynamic analysis. LW-TOSD ensures the continuity of in-plane displacements and transverse shear stresses. The current LW-TOSD can be applied to arbitrary multi-layer laminated structures with only seven Degrees of Freedom (DOFs) for each element node and eliminates the use of the shear correction factors. Moreover, a shear penalty stiffness matrix is constructed to satisfy artificial constraints to optimize the structural shear strain. A dynamic finite element model is obtained based on LW-TOSD using the Hamilton’s principle. First, the accuracy of the current model is validated by comparing with literature and ABAQUS results. Then, this study carries out numerical investigations of piezolaminated structures for different width-to-thickness ratios, length-to-width ratios, penalty stiffness matrix, boundary conditions, electric fields and dynamics.

Numerical implementation of a modified cohesive zone model for HCF behavior of adhesively bonded composite laminates under mixed mode loading

In this paper, the cohesive zone model for high-cycle fatigue (HCF) behavior of adhesively bonded composite laminates subjected to mixed mode loading is implemented in Abaqus/Standard 6.14. The cyclic cohesive zone model takes into account mode-dependent penalty stiffness and fracture process zone size in adhesively bonded orthotropic composite laminates. Unlike earlier cyclic cohesive-based models, the developed UEL/UMAT is made available to the readers and end users for their specific needs. The model is validated extensively against the available experimental results in both load as well as displacement control. The model exhibited superior accuracy, positioning it as a valuable research tool. Furthermore, a novel parametric study was conducted to investigate the impact of fracture toughness and strength on HCF behavior. Finally, numerical simulation on a single fiber composite system is performed to explore micro-scale debonding using the developed model.

A unified buckling formulation for linear and nonlinear analysis of laminated plates using penalty based [formula omitted] FEM-HSDT model

In this paper, the effect of pre-buckling boundary conditions and the type of nonlinearity for stress stiffening used in different linear and nonlinear buckling approaches is studied for laminated composite plates. The study is conducted using a C0 finite element (FE) plate model, employing a unified C1 higher-order shear deformation theory (HSDT). The set of governing equations is derived using the principle of virtual displacement and solved using the tangent-based arc-length method in conjunction with a simple branch switching technique. The performance of the present C0 FE model is assessed through a validation exercise and comparison with results obtained via the use of ANSYS and, for linear analysis, Navier solution, as well as solutions available in the literature. The influence of the different in-plane loads, boundary conditions, side-to-thickness ratio, fiber orientation, types of imperfection and penalty stiffness matrix are also examined. The results show that the same boundary conditions must be utilized in both pre-buckling and linear eigenvalue analyses for accurate and realistic predictions of critical buckling loads, as confirmed from the nonlinear buckling analyses. Furthermore, the critical buckling loads obtained using Green–Lagrange nonlinearity are observed to be more conservative than those obtained using von Kármán nonlinearity. The nonlinear buckling approach is a generalized approach while the nonlinear eigenvalue approach has a limited range of application.

Simulation and experimental studies of mechanical delamination characteristics of REBCO tapes and lap joints under transverse tension

In this study, we investigate the delamination properties of high-temperature superconducting REBa2Cu3O7−x (REBCO) coated conductor (CC) tapes and lap joints under transverse tension by a finite element model (FEM) based on the cohesive zone model. The interfacial cohesive strength and the initial penalty stiffness used in the FEM calculations are adopted based on the experimental data obtained from the anvil tensile tests. Compared to previous theoretical studies, our calculation results of the delamination behaviors for the REBCO CC tapes and the lap joints are more approaching to the experimental data, and the abrupt failure behavior of the transverse load–displacement curves, which appears in the anvil tests, is reproduced by the simulations. Moreover, the effects of welding strength between the upper and lower tapes and internal tape damage during the welding process on the delamination characteristics, including the delamination strength, initial separation, delamination locations, and damage distribution of the CC lap joints, are presented.

A numerical study on rotational stiffness characteristics of the disk lock joint

Rotational stiffness characteristics of the disk lock joint have a significant effect on the load bearing capacity and stability of disk lock scaffolds. However, experimental researches on scaffold joints are usually expensive and time consuming, especially when different joint connection configurations and loading modes are involved. This paper explores the feasibility and effectiveness of the 3D nonlinear finite element analysis method for studying the rotational stiffness characteristics of the disk lock joint. Detailed geometric models of the disk lock joints are established and the geometric models are meshed with finite elements. The elastoplastic constitutive model based on the isotropic hardening law and the associated flow rule is used to describe the complete deformation process of the material, and the penalty stiffness method is applied to accurately simulate the interaction between different components. Thus the complete finite element models of the disk lock joints are established. Firstly, the loading processes of the one-way disk lock joint are simulated and verified by experiments. To further demonstrate the robustness and effectiveness of the finite element method, the loading processes of disk lock joints under different connection configurations and loading modes are simulated. The influences of connection configurations and loading modes on joint rotational stiffness are also investigated to reveal the load bearing mechanism of the disk lock joint, and thereby contribute to better predictions of the load bearing capacity of the disk lock scaffolds.

Superposed element method for the numerical simulation of structure crack

Crack is a key factor for the normal operation of structures. In this paper, a new method called Superposed Element Method (SEM) is proposed for the numerical simulation of crack analysis. Firstly, the structure containing crack is divided into two independent meshes. One is the global mesh without crack, the other is the local mesh of the crack and its vicinity. Then, the two independent meshes are coupled together, finite element method (FEM) is used for the local mesh and the complete elements outside of the coverage of the local mesh, and numerical manifold technique is applied to the incomplete elements. Finally, in order to couple the global mesh with the local mesh, the penalty stiffness is imposed on the boundaries of local mesh. The results of the numerical examples show that the pre-processing of SEM is quite simple, and the accuracy of the solution is relatively high. The method proposed is expected to be widely used for dynamic simulation of crack propagation.

Transient dynamic response of brick masonry walls under low velocity repeated impact load

An experimental and numerical investigation have been carried out on the clay brick masonry walls against repeated impact load in order to estimate the transient dynamic response. The experiments were carried out on 110 mm thick masonry walls against 60 kg mass impactor with hemisphere nose shape with the velocity of 1.98 m/s. In the first phase, the experiment on the wall of 1.2 × 1.2 m (height (h) x breadth (b)) was conducted considering the bed (W1) and head (W2) joint of walls parallel to the boundary in order to verify the conservative design of the joint interface. It was observed that the W2 found resistance up to 3 hits whereas W1 offered resistance up to 11 hits and it was concluded that the W2 found to be sensitive. In the second phase, the aspect ratio of sensitive wall W2 was varied by reducing the breadth from 1.2 to 0.25 m, corresponding aspect ratio (h/b), 1.33 – 4.8 (W3-W6) whereas, the height of the wall remains constant, i.e. 1.2 m. It was observed that the aspect ratio was found to have marginal effect on the peak force however it changes the failure pattern significantly. In the third phase, the influence of mortar strength has been studied and it was observed that the mortar strength does not impart any stiffness into the wall. As a fourth phase, the response of W2, W3 and W4 walls was studied under repeated loading and the resistance offered up to 2, 3 and 2 hits by W2, W3 and W4 walls, respectively. Further, numerical simulations on W1, W2, W3 and W4 walls were carried out using ABAQUS and the Drucker Prager model and traction separation law were used to simulate the brick and interface joints, respectively. The results thus predicted were compared with the experimental results and found in good agreement however, a maximum deviation of 14% on the prediction of peak force. Further, the sensitive parameters were identified which are sensitive to the qualitative numerical results. It was concluded that the parameters such as cohesion, fracture energy, friction and penalty stiffness found to affect the numerical results significantly. Further, wall W2 with externally bonded engineered cementitious composite (ECC) layers has been studied in order to estimate the resistance of wall with backing plate. It was concluded that the ECC backing plate was found to enhance the response of masonry wall in terms of contact force, damage and failure pattern, significantly.

Bending Behavior of a Frictional Single-Layered Spiral Strand Subjected to Multi-Axial Loads: Numerical and Experimental Investigation

Bending deformation gives rise to interwire slippage for spiral strands subjected to multi-axial loads, and further induces wear or fatigue phenomena in practice. The interwire friction would resist bending deformation and lead to uneven tension distribution of individual constituent wires but little research has quantified these effects. To figure out this issue, a beam finite element (FE) is established, into which a penalty stiffness algorithm and a Coulomb friction model are incorporated. A series of free bending simulations are developed for parametric study on deflection near the terminations and tension distribution of individual wire for strands with different levels of length and friction coefficient as well as external loads. Based on the simulation results, it is found that strand length has little influence on bending deformation and tension distribution if the strand length exceeds six times the pitch length. A deflection formula extended from the classical Euler beam model well predicts the sag deflections and the relative error with respect to experimental measurements is less than 10%. Furthermore, additional axial tension induced by the friction is clearly characterized and an approximate expression is proposed to estimate tension distribution for outer layer wires. Its predictions are encouraging for longer strands.

Dynamically synergistic regulation mechanism for rotation energy harvesting

To realize the sustainable energy supply of wireless sensor networks, a novel dynamically synergistic regulation mechanism is proposed for rotation energy harvesting under the guidance of the dynamic regulation evolution. The connotation of synergyincludes: (1) Nonlinear magnetic force and variable stiffness (centrifugal stiffening effect and penalty stiffness of adaptive-anastomotic barricades) synergistically regulate the dynamic behavior of the system; (2) The adaptive-anastomotic barricadesalso convertmechanical energy into electrical energy when regulating the dynamic behavior of the system, i.e., the triboelectric nanogeneratorsynergistically generateselectricity with piezoelectric energy harvesterin different states (collision and vibration) of the system, respectively.Based on the energy method of Hamiltonian principle, the electromechanical coupling dynamic model of the system is developed. The parameters analysis is performed for the discussion of dynamic characteristics and electrical performance of the system in a wide rotation speed range. The prototype is fabricated and the experiments under different conditions are carried out. The experimental data are in good agreement with the simulation results, which verifies the effectiveness of the theoretical model. The experimental results show that the proposed harvester can harvest energy effectively in a wide speed range of 0–1000 r/min. The peak-peak voltage and the average power for PEH and TENG can reach 122.1 V, 1600 μW; 774.3 V, 205.1 μW, respectively. The proposed dynamically synergistic regulation mechanism provides a new perspective to improve the comprehensive performance of the rotation energy harvester, which exhibits a promising application prospect in the self-powered IoT applications.

Bridging the gap between local and nonlocal numerical methods—A unified variational framework for non-ordinary state-based peridynamics

The paper aims to develop a unified variational framework to bridge the gap between the non-ordinary state-based peridynamics (NOSB-PD) and the classical continuum mechanics (CCM). First, a new force state vector is proposed by introducing the first Piola–Kirchhoff stress. This new force state vector enables the stress divergence of each material point to be expressed by averaging all the force state vectors in its support domain. The new force state vector also ensures the mathematical consistency between the strong form of PD and CCM when the horizon of a material point approaches to zero. Second, the displacement and traction boundaries in CCM are transformed into the non-local fictious boundary layers in PD, and a non-local Gauss’s formulation is presented by transforming the displacement and traction boundaries in CCM into the non-local fictious boundary layers in PD, and this formulation unifies the variational framework and boundary conditions of PD and CCM. Third, a fully implicit algorithm is developed to obtain the general nonlinear problems such as fracture and large deformation of solid materials. Further, a penalty method is employed to eliminate the zero-energy mode oscillation inherently observed in NOSB-PD, and the penalty force and penalty stiffness matrix are derived for the proposed implicit algorithm and numerical implementation. Numerical results demonstrate that the proposed method is accurate and can well capture the fracture and large deformation of solid materials. Results also indicate that the method can effectively prevent the zero-mode oscillations inherently observed in the original NOSB-PD, and thus ensures the computational stability.

Nonlinear 3D Model of Double Shear Lap Tests for the Bond of Near-surface Mounted FRP Rods in Concrete Considering Different Embedment Depth

The utilization of near-surface mounted Fiber Reinforced Polymer (FRP) reinforcement as a method of strengthening in reinforced concrete structures has increased considerably in recent years. Moreover, the application of double-shear lap tests for this rein-forcement method leads to the achievement of a local bond-slip behavior in a bonded joint. This research, therefore, focused on 3-D modeling of this type of test to suitably characterize the bond mechanics between FRP rods and concrete at various embedment depth. The use of different alternatives to represent the interface between the FRP rod and concrete were analyzed after which a comparison was drawn between the numerical finite element (FE) simulations and experimental measurements. The results showed the prediction of the load–slip corresponded with the data obtained from the experiment. Finally, the proposed model has the ability to express the relationship between the penalty stiffness parameters in shear direction Kss = (Ktt) and the embedment depth of FRP rods.

On the robustness of the stabilized finite element method for delamination analysis of composites using cohesive elements

We demonstrate the ability of a stabilized finite element method, inspired by the weighted Nitsche approach, to alleviate spurious traction oscillations at interlaminar interfaces during composite delamination. The method allows for the use of any value for the cohesive stiffness and obviates the need for ad hoc approaches to estimate the minimum penalty stiffness for accurate analysis. Simulation studies involving constant strain patch tests and benchmark delamination tests under mode-I, mode-II, and mixed-mode loadings show evidence of traction oscillations with the standard method. In contrast, the stabilized method is able to alleviate traction oscillations, which illustrates its efficacy and robustness.

A numerical sensitivity analysis on the cohesive parameters of a carbon-steel single lap joint

One of the key issues concerning the use of composite materials in the construction of steel ships, or vice versa, the use of steel in the construction of composite ships, is related to the ways of joining composite parts to the adjacent steel structure. Carbon-steel joints are of increasing interest for the maritime industry, but knowledge is scarce. Applications of adhesive joints as an alternative method to welding or riveting has numerous advantages. Still, gathering more information about the behaviour of such joints through numerical sensitivity analysis and simulation are necessary to speed up applications on ships and marine structures.The present investigation aims at assessing the sensitivity of the parameters that define the traction separation law on the numerical simulation of the de-bonding of a steel to composite single lap joint, following a previous experimental campaign. The test is intended to induce a mode II separation, although the slight bending moment caused by the misalignment of the two ends leads to mixed mode effects, as well.Two finite element software were compared, ADINA and LS-DYNA.The influence of peak traction, penalty stiffness and energy release rate were investigated, along with the effect of a mixed mode opening behaviour.

A detailed finite element analysis of composite bolted joint dynamics with multiscale modeling of contacts between rough surfaces

This paper presents the multi-scale modeling and finite element analysis of the dynamics of a composite structure with bolted joints. Microscale contact surface effects are considered and a single-degree-of-freedom (SDOF) stick-slip friction model is proposed. Interface mechanics parameters were solved using fractal contact theory and damping properties of the composite material were determined according to the elastic-viscoelastic correspondence principle. Then, by inputting calculated values for the penalty stiffness, damping coefficient, and material parameters, a finite element analysis of the overall and local dynamic response of the joint was carried out in ABAQUS. Simulated values of the resonance frequency and damping ratio of the structure were consistent with experimental results, thus validating the proposed model. Presence of the joint interface reduced the structural stiffness by 2% and increased energy dissipation by 30–40% compared to a rigid joint. Furthermore, the local interface energy dissipation was found to be influenced by both the excitation amplitude and preload. Finally, analysis of the interface roughness parameters showed that rougher surfaces facilitate preload relaxation.

Effects of surface contact on the dynamic responses of delaminated composite plates

This paper investigates the modal parameters such as the damping ratio, frequency and shape of delaminated composite plates with contact properties. The contact damping is analyzed using a viscoelastic Coulomb friction model with sliding phase, which combines energy dissipation of sliding friction and sticky friction. Finite element method (FEM) based models simulate the dynamic responses of delaminated composite plates, in which the constant normal contact stiffness and contact damping are introduced via penalty stiffness method and equivalent viscous damping, respectively. Relevant vibration experiments are set up to obtain the modal parameters of delaminated composite plates. The results show that the modal damping ratio of the composite laminated plates increases significantly with increase in the delamination percentage, and the contact damping plays a major role in the increase of the modal damping ratio. Furthermore, the contact stiffness also has a minor influence on the modal frequency and shape of delaminated composite plates. The experimental data is consistent with simulations, thereby establishing the accuracy of the FEM model.

A simple procedure for the determination of the cohesive law in 4-ENF test with consideration of the friction and R-curve effect

The measurement of initiation and propagation fracture toughness is important for characterizing the delamination behavior of composite laminates, especially in the case with distinct R-curve behavior which can be caused by fiber bridging, shear deformation of matrix, etc. Four-Point End Notched Flexure (4-ENF) is one of the widely adopted tests to measure the mode II toughness values, while the friction between the crack surfaces may have a significant impact on the results. In this paper, a simple procedure to determine the cohesive law in 4-ENF test with consideration of the friction and R-curve effect is proposed. Firstly, determining the equivalent crack advance according to the compliance variation of the specimen is performed, without the need for monitoring the crack length during the test. Next, the Energy Release Rate in mode II (GII) and crack sliding displacement (δt) are derived based on the point-friction assumption and beam theory. Finally, the cohesive law is obtained via the J-integral approach. To obtain suitable values of the penalty stiffness and interface strength, the influence of these parameters on the prediction of 4-ENF test is studied. Comparisons of the numerical results and the experimental data show a good correlation. The effect of variation of friction coefficients on derived cohesive stress and initiate energy release rate is discussed as well.

On cohesive element parameters and delamination modelling

The cohesive element (CE) has been widely used to model delamination in laminated composites. The penalty stiffness, interfacial strength and fracture toughness are three CE parameters that often determine the accuracy of the numerical analysis by finite elements (FEs). The fracture toughness may be experimentally determined through standard fracture tests. The interfacial strength is difficult to obtain experimentally but also plays a crucial role in predictive numerical modelling. The penalty stiffness is a numerical parameter whose range of values may influence computational results. Recently, the CE has been deployed to model the initiation and propagation of delamination in composite structures where no pre-existing cracks or delamination has been assumed. The parameters are often calibrated, but comprehensive studies on the influence of CE parameters and their selection criteria remain limited in the literature. In the present paper, the effects of CE parameters on delamination modelling are systematically examined. The sensitivity of the overall progressive damage process and failure load to the assumed CE strengths and stiffness is studied based on selected examples of fracture and coupon-level tests. With varying CE strengths employed in the FE analyses, two distinct kinds of failure behaviour, namely the strength-sensitive and strength-insensitive delamination processes, have been observed, depending on the geometry and ply layups of composite materials. On the other hand, the penalty stiffness of CE mainly affects the computational efficiency and accuracy of delamination modelling. The failure mechanisms and physics behind are investigated to get better understanding on the numerical effects of CE parameters, and thus provide the selection guidelines in practical analyses.

Rotational Constraint between Beams in 3-D Space

Abstract. In this paper, we develop two alternative formulations for the rotational constraint between the tangents to connected beams with large deformations in 3-D space. Such a formulation is useful for modeling bonded/welded connections between beams. The first formulation is derived by consistently linearizing the variation of the strain energy and by assuming linear shape functions for the beam elements. This formulation can be used with both the Lagrange multiplier and the penalty stiffness method. The second non-consistent formulation assumes that the contact normal is independent of the nodal displacements at each iteration, and is updated consistently between iterations. In other words, we ignore the contribution due to the change of the contact normal in the linearization of the contact gap function. This assumption yields simpler equations and requires no specific assumption regarding the shape functions for the underlying beam elements. However, it is limited to the penalty method. We demonstrate the performance of the presented formulations in solving problems using implicit time integration. We also present a case showing the implications of ignoring this rotational constraint in modeling a network of beams.

Efficient isogeometric shell element with through-thickness stretch: application to incremental sheet forming

An isogeometric shell element with through-thickness stretch is applied to a two-point incremental forming problem. The shell element supports full three-dimensional constitutive laws and therefore does not make the plane stress assumption. An anisotropic material model is implemented to account for the sheet rolling direction. Automatically adjusting penalty stiffness is proposed for modeling the contact between the stylus tool and the sheet, whereas the die contact algorithm uses traditional constant penalty stiffness. A comparison is made between experimental results as well as results from a conventional shell formulation.

A length scale insensitive phase-field damage model for brittle fracture

Being able to model complex nucleation, propagation, branching and merging of cracks in solids within a unified framework, the classical phase-field models for brittle fracture fail in predicting length scale independent global responses for a solid lacking elastic singularities (e.g., corners, notches, etc.). Motivated from Barenblatt’s approximation of Griffith’s brittle fracture with a vanishing Irwin’s internal length, this paper extends our recent work in quasi-brittle failure (Wu, 2017, 2018a) and presents for the first time a length scale insensitive phase-field damage model for brittle fracture. More specifically, with a set of optimal characteristic functions, a phase-field regularized cohesive zone model (CZM) with linear softening law is addressed and applied to brittle fracture. Both the failure strength and the traction – separation law are independent of the incorporated length scale parameter. Compared to other phase-field models and CZM based discontinuous approaches for brittle fracture, the proposed phase-field regularized CZM is of several merits. On the one hand, being theoretically equivalent to Barenblatt’s CZM (at least in the 1-D case), it needs neither the explicit crack representation/tracking nor the elastic penalty stiffness which both are necessary but cumbersome for discontinuous approaches. On the other hand, it gives length scale independent global responses for problems with or without elastic singularities while preserving the expected Γ-convergence property of phase-field models. Representative numerical examples of several well-known benchmark tests support the above conclusions, validating its capability of modeling both mode-I and mixed-mode brittle fracture.