Consistent Tangent Stiffness Research Articles

Nonlinear Analysis of Plane Frames Using a Co-Rotating Timoshenko Beam Element

The present work describes a co-rotating shear flexible beam element without shear locking and integrating Euler-Bernoulli's and Timoshenko's beam theories. The co-rotational kinematics is based on the separation of the motion in deformational and rigid body components. The deformation of the beam element is composed by three natural modes of deformation: the extension mode, the symmetric bending mode, and the anti-symmetric bending mode. The respective generalized stresses from these natural modes are self-balanced, allowing the achievement of a consistent tangent stiffness matrix. In this paper, it is detailed and deduced all the algebraic steps for the deduction of the elastic stiffness matrix, the geometric stiffness matrix, and the co-rotation stiffness matrix. Some examples are presented and the numerical results demonstrate that the beam element here presented is able to handle large rotations.

A Thermodynamically Consistent Anisotropic Continuum Damage Mechanics Model With Damage-Dependent Equivalent Strain

Abstract This paper proposes a new thermodynamically consistent anisotropic continuum damage mechanics model that the degradation of material is described by a set of damage variables. We adopt a set of novel equivalent strains which are functions of both the strain and the damage state variables as well. A viscous stabilization method is considered to improve the convergence during the material softening. The evolution of damage state variables is updated through a Newton–Raphson iterative process. The expression of the consistent tangent stiffness matrix is also derived. A smoothed exponential damage evolution shape function with rescaling regularization method is proposed to describe the material softening process and the mesh dependency phenomenon is relieved. Several sets of numerical examples including three-point-bending of a notched beam, uniaxial tension of unnotched and notched plate of composite material with multiple layups are presented to verify the anisotropic damage models.

Modeling confined concrete behavior in finite element with a non-orthogonal elastoplastic model

The accuracy of a constitutive model for confined concrete largely relies on its capability to capture concrete's dilatancy behavior. In this paper, a non-orthogonal flow rule (NFR) is used to reasonably characterize the concrete's volume change law in relation to the stress state without the necessity for a plastic potential function. Then, the non-orthogonal plastic model is implemented in the finite element (FE) software ABAQUS using the implicit stress update algorithm, which employs the line search method and the numerical consistent tangent stiffness matrix to ensure the robustness and computational efficiency of FE analysis. Finally, FE simulations are performed to evaluate the constitutive model's capabilities in actively and passively confined concrete. In the latter case, steel tubes restrict the concrete's lateral deformation. An analysis and discussion are conducted regarding the impact of dilatancy behavior on concrete-filled steel tube (CFST) columns. The consistency between experimental data and simulation results demonstrates that the FE modeling with a non-orthogonal constitutive model provides an effective tool to describe the behavior of confined concrete.

A Constitutive Model for Stud Connection in Composite Structures

The complexity of finite element analysis for composite structures can be significantly reduced by representing the connector and adjacent concrete as a macroscopic element. Nevertheless, the prevailing macroscopic models for shear connections predominantly employ nonlinear elastic theory. This approach introduces inaccuracies in estimating structural stiffness and load-bearing capabilities, primarily due to its inability to precisely capture the cumulative effects of plastic damage. In response, this study introduces a novel macroscopic elastoplastic model grounded in plasticity theory, aimed at accurately characterizing the nonlinear behavior of stud connections subjected to concurrent shear and tensile forces. This paper meticulously delineates the implementation of the elastoplastic constitutive model using the backward Euler method for numerical integration. It further articulates the derivation of the consistent tangent stiffness, which aligns with the convergence efficiency of the Newton–Raphson iterative approach. The computation of the element stiffness matrix for a two-node element is executed via the governing equation inherent to the finite element method. An exemplar macroelement test conducted in ABAQUS affirms the implicit backward Euler scheme’s stability and consistency across varying tolerances. Validation of the elastoplastic model against empirical test outcomes corroborates its efficacy, demonstrating the model’s precision in predicting the load–displacement behavior of stud connections under the influence of shear and tensile forces.

Computational multiscale modelling of material interfaces in electrical conductors

Material interfaces occur at various length scales and may exhibit significantly different properties than the surrounding bulk. Motivated by their importance for electrical engineering applications such as wire bonds and electrically conductive adhesives, the focus of the present work is on material interfaces in electrical conductors. In order to approximate the physical interphase (of finite thickness) as a (zero-thickness) cohesive zone-type interface in macroscale simulations, scale-bridging relations are established that relate the apparent electro-mechanical interface properties to the underlying microstructure. A finite element-based implementation is discussed with particular focus lying on the efficient calculation of the flux-type macroscale quantities and the associated generalised algorithmic consistent tangent stiffness tensors. Analytical solutions are derived for validation purposes and representative boundary value problems are studied.

An Anisotropic Damage-Plasticity Constitutive Model of Continuous Fiber-Reinforced Polymers.

Accurate structural analyses of continuous fiber-reinforced polymers (FRPs) are imperative for diverse engineering applications, demanding efficient material constitutive models. Nonetheless, the constitutive modeling of FRPs is complicated by the nonlinear behavior resulting from internal damages and the inherent plasticity. Consequently, this study presents an innovative anisotropic constitutive model for FRPs, designed to adeptly capture both the damage evolution and plasticity. All requisite parameters can be easily obtained through fundamental mechanical tests, rendering the model practical and user-friendly. The model utilizes the three-dimensional Puck criteria to determine damages, initiating the evolution process through a combination of continuum damage mechanics and linear stiffness attenuation methods. This evolution is coupled with a one-parameter plastic model. Subsequently, the numerical implementation method, integrated into ANSYS, is detailed. This emphasizes the Cauchy stress and consistent tangent stiffness solution strategy. Finally, the effectiveness of the developed model is demonstrated through comprehensive verification, encompassing existing biaxial tension and open-hole-tension tests conducted on carbon and glass FRP laminates. The simulation results exhibit a remarkable correspondence with the experimental data, validating the reliability and accuracy of the proposed model.

An exact consistent tangent stiffness matrix for a second-gradient model for porous plastic solids: Derivation and assessment

It is well known that the use of a consistent tangent stiffness matrix is critical to obtain quadratic convergence of the global Newton iterations in the finite-element simulations of problems involving elasto-plastic deformation of metals, especially for large-scale metallic structure problems. In this article, we derive an exact consistent stiffness matrix for a porous material model, the GLPD model developed by Gologanu, Leblond, Perrin, and Devaux for ductile fracture of porous solids based on generalized continuum mechanics assumptions. Full expressions for the derivatives of the Cauchy stress tensor and the generalized moments stress tensor the model involved are provided and implemented into a finite-element code. The effectiveness and robustness of the proposed tangent stiffness moduli are assessed by applying the formulation in the finite-element simulations of typical ductile fracture problems. Comparisons between the performance of our stiffness matrix and the standard ones are also provided.

A novel machine learning based tangent stiffness calculation method for 3D wheel-rail interaction element

A machine learning (ML) based method is presented in this paper for obtaining tangent stiffness of a complicated three-dimensional wheel-rail interaction element that is able to practically and effectively simulate the complicated dynamic responses of vehicle-track problems. The element tangent stiffness, defined as differentiation of element insisting force to nodal displacement, is important in improving efficiency when Newton’s method is used to solve implicit nonlinear finite element equations. However, deriving and software implementing the tangent stiffness require significant efforts, and calculating the tangent stiffness in each iteration of the Newton method is usually time consuming. On the other hand, ML can directly obtain the implicit mapping between inputs and outputs of complex calculation process with limited programming effort and high computational efficiency, and is potentially a good alternative way to calculate the tangent stiffness of complicated element. In this paper, a feedforward artificial neural network is trained for obtaining the tangent stiffness, while inputs are the displacement and velocity of the element and outputs are the entries of the tangent stiffness matrix. The ML based tangent stiffness are implemented in an open source finite element software framework, OpenSees, and verified by application examples of a wheelset and a light rail vehicle running on straight rigid rail. The accuracy and efficiency are compared between the use of ML based tangent stiffness (MLTS) and the consistent tangent stiffness obtained at different speeds. The results demonstrate the MLTS can ensure the calculation accuracy and significantly improve the calculation efficiency.

An Effective Tangent Stiffness of Train–Track–Bridge Systems Based on Artificial Neural Network

Dynamic response analysis of a train–track–bridge (TTB) system is a challenging task for researchers and engineers, partially due to the complicated nature of the wheel–rail interaction (WRI). When Newton’s method is used to solve implicit nonlinear finite element equations of a TTB system, consistent tangent stiffness (CTS) is essential to guarantee the quadratic convergence rate. However, the derivation and software implementation of CTS for the WRI element require significant efforts. Artificial neural network (ANN) can directly obtain a potentially good tangent stiffness by a trained relationship between input nodal displacement/velocity and output tangent stiffness. In this paper, the backpropagation neural-network-based tangent stiffness (BPTS) of the WRI element is derived and implemented into a general finite element software, OpenSees, and verified by dynamic response analysis of a high-speed train running on a seven span simply supported beam bridge. The accuracy and efficiency are compared between the use of BPTS and CTS. The results demonstrate that BPTS can not only save the significant efforts of deriving and software implementing CTS but also improve computational efficiency while ensuring good 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.

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.

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.

Cable structures: An exact geometric analysis using catenary curve and considering the material nonlinearity and temperature effect

Structures formed by cables constitute structural systems of great practical application in engineering, such as suspension bridges, cable-stayed bridges, cable cars, transmission lines in which case they are used as supporting elements that conduct electricity, overcoming the spans between the towers or in stays for the so-called cable-stayed towers, among other applications. Cable structures have a highly non-linear behavior both from a geometric point of view, arising from their initial equilibrium configuration, and from a material point of view, since the constitutive laws of the cables present an elastoplastic behavior. The objective of this work is to present a geometrically exact formulation for static analysis of cable structures considering the catenary configuration. For this, both geometric and material nonlinearities are included in the development of the the catenary cable element formulation. The effect of temperature variation is also considered. A rigorous updated Lagrangian description combined with the corotational approach is employed to obtain the consistent tangent stiffness matrix of the exact cable element. The proposed methodology was implemented in the software program called Advanced Structural Analysis System (ASTRAS), which is an academic software based on the Finite Element Method (FEM). Finally, numerical examples are analyzed using the present procedure and the obtained results are compared with those found in the literature and given by commercial softwares in order to demonstrate the consistency and precision of the formulation.

Consistent Tangent Stiffness of a Three-Dimensional Wheel–Rail Interaction Element

Consistent tangent stiffness plays a crucial role in delivering a quadratic rate of convergence when using Newton’s method in solving nonlinear equations of motion. In this paper, consistent tangent stiffness is derived for a three-dimensional (3D) wheel–rail interaction element (WRI element for short) originally developed by the authors and co-workers. The algorithm has been implemented in finite element (FE) software framework (OpenSees in this paper) and proven to be effective. Application examples of wheelset and light rail vehicle are provided to validate the consistent tangent stiffness. The quadratic convergence rate is verified. The speeds of calculation are compared between the use of consistent tangent stiffness and the tangent by perturbation method. The results demonstrate the improved computational efficiency of WRI element when consistent tangent stiffness is used.

The direct force correction based framework for general co-rotational analysis

The use of nonlinear projector matrix in co-rotational (CR) analysis was pioneered by Rankin and Nour-Omid in 1990s (Rankin and Nour-Omid, 1988; Nour-Omid and Rankin, 1991), and has almost became a standard manner for CR formulations deduction over the past thirty years. This matrix however relies heavily on a hysterical and sophisticated derivation of the variation of the local displacements to the global ones, leading to complicated expressions for the internal force vector and the tangent stiffness matrix, which may devalue the simplicity and convenience for the original intention of using CR approach. This paper begins by making a discussion on existing element independent CR formulation and the objective is to develop a new and simpler framework for general CR analysis that avoids using conventional nonlinear projector matrix. The methodology consists of two steps in the element internal force calculation. The first one is to obtain a preliminary result of the internal force. This is done by following the conventional element-independent CR formulation but dropping the terms involving projector matrix and therefore yields simple formulations of the internal force and the tangent stiffness matrix. The second one is a correction step to obtain a new internal force vector that satisfies the element self-equilibrium condition. This step inherits the spirit of using projector matrix but is conducted directly in the global frame, thus avoiding complicated entanglement of local–global rotation and is independent of the choice of the local CR frame used in the CR analysis. This further leads to a simple and unified formulation for different kinds of elements that can be cooperated in CR framework. Closed formulation of the correction force as well as the related consistent tangent stiffness matrix is derived for different correction approaches. It is also shown that the existing linear projector matrix used for infinitesimal rotation analysis is a special case of the current correction approaches. Multiple numerical examples involving various kinds of elements and different choices of element local CR frame are presented to demonstrate the performance of the proposed framework. The outcomes show that for all the examples the accuracy of the results is comparable with those obtained in conjunction with conventional nonlinear projector matrix.

Geometrically exact beam element with rational shear stress distribution for nonlinear analysis of FG curved beams

Based on the geometrically exact beam theory, a first-order shear deformable curved beam element is developed for geometrically nonlinear analysis of functionally graded (FG) curved beams. In order to accurately predict the distribution of transverse shear stress, the stress equilibrium condition is introduced into the element formulation by using the mixed finite element method with displacements and internal forces as independent fields. The element nodal force vector and consistent tangent stiffness matrix for nonlinear iterative solution are obtained by going through a consistent linearization procedure. Numerical examples are presented to validate the present formulation. As indicated by the numerical results, the proposed element demonstrates a high level of solution accuracy and good applicability. Furthermore, using the proposed element, an investigation is conducted into the nonlinear stability of FG structures with different material distribution parameters, and the accuracy loss caused by unreasonable distribution of shear stress is discussed.

Numerical solutions for strain-softening surrounding rock under three-dimensional principal stress condition

Abstract In this study, elastoplastic model for the surrounding rock of axisymmetric circular tunnel is investigated under three-dimensional (3D) principal stress states. Novel numerical solutions for strain-softening surrounding rock were first proposed based on the modified 3D Hoek–Brown criterion and the associated flow rule. Under a 3D axisymmetric coordinate system, the distributions for stresses and displacement can be effectively determined on the basis of the redeveloped stress increment approach. The modified 3D Hoek–Brown strength criterion is also embedded into finite element software to characterize the yielding state of surrounding rock based on the modified yield surface and stress renewal algorithm. The Euler implicit constitutive integral algorithm and the consistent tangent stiffness matrix are reconstructed in terms of the 3D Hoek–Brown strength criterion. Therefore, the numerical solutions and finite element method (FEM) models for the deep buried tunnel under 3D principal stress condition are presented, so that the stability analysis of surrounding rock can be conducted in a direct and convenient way. The reliability of the proposed solutions was verified by comparison of the principal stresses obtained by the developed numerical approach and FEM model. From a practical point of view, the proposed approach can also be applied for the determination of ground response curve of the tunnel, which shows a satisfying accuracy compared with the measuring data.

On the co-rotational method for geometrically nonlinear topology optimization

This paper investigates the application of the co-rotational method to solve geometrically nonlinear topology optimization problems. The main benefit of this approach is that the tangent stiffness matrix is naturally positive definite, which avoids some numerical issues encountered when using other approaches. Three different methods for constructing the tangent stiffness matrix are investigated: a simplified method, where the linear elastic stiffness matrix is simply rotated; the consistent method, where the tangent stiffness is derived by differentiating residual forces by displacements; and a symmetrized method, where the consistent tangent stiffness is approximated by a symmetric matrix. The co-rotational method is implemented for 2D plane quadrilateral elements and 3-node shell elements. Matlab code is given in the appendix to modify an existing, freely available, density-based topology optimization code so it can solve 2D problems with geometric nonlinear analysis using the co-rotational method. The approach is used to solve four benchmark problems from the literature, including optimizing for stiffness, compliant mechanism design, and a plate problem. The solutions are comparable with those obtained with other methods, demonstrating the potential of the co-rotational method as an alternative approach for geometrically nonlinear topology optimization. However, there are differences between the methods in terms of implementation effort, computational cost, final design, and objective value. In summary, schemes involving the symmetrized tangent stiffness did not outperform the other schemes. For problems where the optimal design has relatively small displacements, then the simplified method is suitable. Otherwise, it is recommended to use the consistent method, as it is the most accurate.

Semi-Implicit Numerical Strategy for Implementing a Nonlocal Finite-Deformation Gurson Model for Glassy Polymer Using Finite Element Analysis

Post-peak strain softening for some thermoplastic resins often leads to localized deformation with plastic flow instability, spurious energy dissipation, and numerical solutions with mesh dependency using finite element analysis (FEA) due to the lack of a length scale. This paper introduces the nonlocal integral theory with a length scale into the Gurson model to capture objective void evolution and shear bands for glassy polymer under quasi-static loading. Nonlocal numerical technique using the finite-deformation Gurson model is developed, where the nonlocal implicit stress return algorithm and the nonlocal consistent tangent stiffness are presented in the corotational configuration by performing nonlocal averaging on the void volume fraction in a semi-implicit way. The numerical strategy is implemented by combining the finite element software ABAQUS-UMAT, USDFLD, and UEXTERNALDB subroutines. For plane tension problem, the developed nonlocal numerical technique circumvents mesh dependency well by studying the void growth, the evolution of shear bands, and the load responses.

An anisotropic viscoelastic-viscoplastic model for short-fiber composites

In this paper, an anisotropic viscoelastic-viscoplastic macro-mechanical model is presented for short-fiber reinforced polymers. In addition to the rate-dependent response of the polymer matrix, fiber orientation leads to elastic and plastic anisotropy in short-fiber composites. The dependence of the yield stress on the strain rate and on the orientation is modeled by use of the Hill equivalent stress and the Eyring flow rule. Uniaxial tests at various strain rates were performed on injection molded samples cut at different orientations with respect to the mold flow direction. The test results show that the effects of strain rate and material orientation on yield stress are factorizable. The model aims to capture this behavior, which simplifies the characterization process. First, the model with a single relaxation time is presented and then the model is extended to multiple relaxation times to improve the predictions in the pre-yield regime. An efficient method for finding the model parameters for different modes is presented. An implicit scheme is used for the integration of the constitutive equations and the derivation of the consistent tangent stiffness tensor is presented. The model is implemented as an ABAQUS user material (UMAT) subroutine and is validated through comparison of the simulation results with the experiments.