Assumed Natural Strain Research Articles

Reduced integration‐based solid and solid‐shell finite elements for gradient‐extended damage

AbstractThe present contribution is concerned with the incorporation of gradient‐extended damage into reduced integration‐based continuum finite elements. To this end, the purely mechanical low‐order solid and solid‐shell elements based on the isoparametric concept are combined with a gradient extended two‐surface damage plasticity model. Due to a tailored combination of reduced integration with hourglass stabilization, the enhanced assumed strain (EAS) method and in case of the solid‐shell the assumed natural strain (ANS) method, the most dominant locking phenomena are eliminated. A polynomial approximation of the strain‐like as well as the stress‐like quantities within the weak forms enables the definition of a suitable hourglass stabilization. In this way, the element stiffness contributions coming from the hourglass stabilization can be determined analytically, since they represent polynomials with respect to Cartesian coordinates. Two representative numerical examples of an elasto‐plastic asymmetrically notched specimen as well as an elastic thin annular plate reveal the accuracy and efficiency of the proposed methodology. Besides the ability to deliver mesh independent results, the framework is especially suitable for constrained situations in which conventional low‐order finite elements suffer from well‐known locking phenomena.

Gradient-extended damage analysis with reduced integration-based solid-shells at large deformations

The present contribution is concerned with the extension of a recently presented non-local damage analysis of shells to the geometrically non-linear regime. For this purpose, a gradient-extended two-surface damage-plasticity model for finite strains is incorporated into a low-order solid-shell finite element formulation for large deformations which is based on reduced integration with hourglass stabilization. Due to a specific combination of the assumed natural strain (ANS) and the enhanced assumed strain (EAS) method, the most relevant locking pathologies which stem from the linear interpolation of the displacement field are eliminated. Furthermore, an additional micromorphic nodal degree of freedom which comes from the gradient extension is considered. A polynomial approximation of the kinematic as well as the constitutively dependent quantities within the weak forms provides a suitable hourglass stabilization which is computationally highly efficient, since the corresponding element residual vectors and stiffness matrices can be determined analytically. Three numerical examples of different elastic as well as elasto-plastic plate and shell configurations, reveal the ability of the present framework to efficiently and accurately predict the damage processes within both geometrically non-linear thin and thick-walled structures.

Three-dimensional thermal buckling analysis of functionally graded material structures using a modified FSDT-based solid-shell element

- In the current paper, an attempt is made to analyze the thermal buckling behavior of functionally graded (FG) shell structures by using a modified first-order enhanced solid-shell element formulation. The material properties of functionally graded shell are presumed to vary continuously in the thickness direction according to a simple power law distribution. Temperature-independent (TID) and temperature-dependent (TD) material properties are both considered. The modified theory takes into account the shear strains through the FGM shell thickness with a parabolic shape function imposed in the compatible strain part, and it verifies the zero shear stresses condition at the top and bottom surfaces of shell. To subdue locking problems, the assumed natural strain (ANS) method and the enhanced assumed strain (EAS) method with a minimal number of internal parameters are employed. Numerical results of the present research are compared and validated with the existing studies on the thermal buckling of FGM shells. Both uniform and nonuniform temperature distributions are considered. The effect of different parameters on the thermal buckling temperature of FGM structures is highlighted by solving numerous examples.

Towards forming simulations by means of reduced integration-based solid-shell elements considering gradient-extended damage

The present contribution is concerned with the non-local damage analysis of geometrically non-linear shells. To this end, a low-order displacement-based solid-shell finite element formulation is combined with a gradient-extended damage-plasticity model. Due to a tailored combination of reduced integration with hourglass stabilization, the enhanced assumed strain (EAS) method and the assumed natural strain (ANS) method, the most dominant locking phenomena are eliminated. A polynomial approximation of the strain-like as well as the stress-like quantities within the weak forms enables the definition of a suitable and efficient hourglass stabilization. In this way, the internal element force vectors as well as the element stiffness contributions coming from the hourglass stabilization can be determined analytically. A numerical example of a circumferentially notched cylinder considering plasticity coupled with damage reveals the potential of the proposed methodology. Besides the ability to deliver mesh independent results within the softening regime, the framework is especially suitable for thin-walled structures, in which conventional low-order continuum elements suffer from well-known locking phenomena.

An evaluation of MITC and ANS elements in the nonlinear analysis of shell structures

The recently emerged idea of incorporating higher-order strain or displacement discontinuities into standard finite element interpolations has triggered the development of robust techniques, which allow efficient modeling of large deflections and rotations. A number of studies on efficient elements with embedded high-order discontinuities, including mixed interpolation tensorial components (MITC) and assumed natural strain (ANS) elements, were published during the past decade. It was verified that local enrichments of the displacement and/or strain interpolation can enhance the responses of displacements and stresses by efficient models. The multitude of methods proposed in the literature calls for a comparative study that would present the diverse approaches in a unified framework, point out their common features and differences, and find their limits of applicability. In this regard, by proposing two robust triangular elements based on the higher-order strain field, mixed and strain-based approaches are compared. The strengths and weaknesses of individual formulations are elucidated by analyzing the convergence behavior. After determining the optimal condition, the nonlinear theoretical analysis of functionally graded (FG) shells has been performed by employing the weighted arc-length method. The equilibrium path for load-bearing factor versus post-buckling capacity, the error rate path and the residual force norm are also presented in different material and boundary conditions.

A reduced integration-based solid-shell finite element formulation for gradient-extended damage

The present contribution is concerned with the incorporation of gradient-extended damage into a reduced integration-based solid-shell finite element formulation. To this end, a purely mechanical low-order solid-shell element based on the isoparametric concept is combined with a gradient-extended two-surface damage plasticity model. Due to a tailored combination of the assumed natural strain (ANS) as well as the enhanced assumed strain (EAS) method, the most important locking phenomena are eliminated. A polynomial approximation of the kinematic as well as the constitutively dependent quantities within the weak forms enables the definition of a suitable hourglass stabilization. In this way, the element stiffness contributions coming from the hourglass stabilization can be determined analytically, since they represent polynomials with respect to Cartesian coordinates. Several numerical examples on elastic as well as elasto-plastic plates and shells under various loading scenarios show the ability of the present methodology to predict various degradation processes such as damage initiation, propagation, merging as well as branching.

Three-dimensional stress analysis of structures in instability conditions using nonlinear displacement-based and hybrid-mixed quadrilaterals based on SaS formulation

In this paper, the three-dimensional (3D) stress analysis of plate-type structures in instability conditions is presented. The displacement-based and hybrid-mixed four-node quadrilateral elements are developed taking the advantages of the sampling surfaces (SaS) method. The SaS formulation is based on considering inside the plate N not equally spaced SaS parallel to the middle surface to specify the displacements of these surfaces as primary plate unknowns. The displacements, strains and stresses are assumed to be distributed through the thickness using Lagrange polynomials of degree N–1 that lead to a well-set higher-order plate theory. The locations of SaS are based on the use of Chebyshev polynomial nodes that allow us to minimize uniformly the error due to Lagrange interpolation. To circumvent shear locking and have no spurious zero energy modes, the assumed transverse shear strains are employed. The nonlinear equilibrium equations are solved by the Newton–Raphson iterative method combined with the Crisfield arc-length algorithm. The accuracy and efficiency of both elements in different conditions such as coarse and distorted meshes are investigated. The developed assumed-natural strain (ANS) elements can be useful for the 3D stress analysis of thin and thick plates in whole states of equilibrium path involving bifurcation, snap-through, and/or snap-back phenomena.

Phase field model for fracture analysis of functionally graded power-based shell structures

The phase field (PF) approach to fracture has emerged as a promising modeling tool that regularizes the variational fracture theory by Griffith via the introduction of a nonlocal damage-like field variable in the corresponding formulation. In this work, we outline a PF formulation for triggering brittle fracture phenomena in shell structures made of Functionally Graded Materials (FGMs). This model relies on the 6-parameter shell formulation complying with a solid shell kinematic description and incorporates the use of the Enhanced Assumed Strain (EAS) and Assumed Natural Strain (ANS) methods in order to alleviate different locking pathologies. The corresponding multi-field variational formalisms is consistently derived and discretized within the context of the Finite Element Method (FEM). Details regarding the implementation in the general purpose FE packages are outlined. The applicability of this model is demonstrated by means of several numerical applications.

An efficient ABAQUS solid shell element implementation for low velocity impact analysis of FGM plates

The main objective of this paper is to develop a numerical model susceptible to solve the numerical locking problems that may appear when applying the conventional solid and shell finite elements of ABAQUS. This model is based on a hexahedral solid shell element. The formulation of this element relays on the combination of the enhanced assumed strain (EAS) and assumed natural strain (ANS) methods with modified First Shear Deformation Theory (FSDT). The developed element is implemented into the ABAQUS user element (UEL) interface. The performance of this element is demonstrated by different benchmark tests from the literature. Our contribution consists on applying a single solid shell element through the thickness direction to predict the low velocity impact behavior on functionally graded material (FGM) circular plates.

Reduced-Integrated 8-Node Hexahedral Solid-Shell Element for the Macroscopic Forming Simulation of Continuous Fibre-Reinforced Polymers

Finite element (FE) forming simulation offers the possibility of a detailed analysis of the deformation behaviour of continuously fibre-reinforced polymers (CFRPs) during forming, in order to predict possible manufacturing effects such as wrinkling or local changes in fibre volume content. The majority of macroscopic simulations are based on conventional two-dimensional shell elements with large aspect ratios to model the membrane and bending behaviour of thin fibrous reinforcements efficiently. However, without a three-dimensional element approach, stresses and strains in thickness direction cannot be modelled accurately. Commercially available linear 3D solid elements for this purpose are rarely suitable for forming simulations since they are subjected to several locking phenomena under bending deformation, especially with large aspect ratios. To alleviate this problem, so-called solid-shell elements based on the assumed natural strain (ANS) and enhanced assumed strain (EAS) method can be used. Therefore, a locking-free explicit reduced-integrated 8-node-hexahedron solid-shell element, based on the initial work of Schwarze and Reese (2011), is implemented in the commercially available FE solver Abaqus. Its suitability for macroscopic modelling of the forming behaviour of fibrous reinforcements is outlined in this work. The presented element combines the advantages of a locking-free out-of-plane deformation behaviour of conventional thin shell elements with the advantage of maintaining a fully three-dimensional material model and geometry description.

Hybrid-mixed shell quadrilateral that allows for large solution steps and is low-sensitive to mesh distortion

We compare three nearly optimal quadrilateral finite elements for geometrically exact inextensible-director shell model. Two of them are revisited and one is novel. The assumed natural strain (ANS) element of Ko et al. (Comput Struct 185:1–14, 2017) shows low sensitivity to mesh distortion and excellent convergence behavior for most types of shell problems. The Hu–Washizu element with ANS shear strains of Wagner and Gruttmann (Int J Numer Methods Eng, 64:635–666, 2005) allows for large solution steps and is computationally fast. However, both formulations have undesirable weak spots, which we clearly identify by a comprehensive set of numerical examples. We show that a straightforward combination of both formulations results in a novel element that synergizes the positive features and eliminates the weak spots of its predecessors.

A mixed edge-based smoothed solid-shell finite element method (MES-FEM) for laminated shell structures

We in this study present a mixed solid-shell finite element formulation for laminated shell analysis. With the commonly adopted truncations applied to the strain components and the Jacobian determinant, a generalized stiffness matrix is established. Discrete shear gap (DSG) technique together with assumed natural strain (ANS) method is employed in order to alleviate shear locking and trapezoidal locking. Stiffness matrix formulation is analytically calculated based on the edge-based cell backgrounds, which result in simple and efficient computation. The proposed formulation does not require using cross-diagonal meshes, which can be an obstacle for several existing three-node degenerated shell elements. This aspect makes the present method much more appealing than others through examples tested. Numerical validations show that the present method performs well for both structured and unstructured triangular meshes.

A GPU Based Explicit Solid-Shell Finite Element Solver

In this work we present a co-rotational/updated Lagrangian, strain-rate based explicit finite element code which uses hexahedral solid-shell tri-linear elements, intended for simulation of the incremental sheet forming (ISF) process. This element is based heavily on the elements described in [1, 2, 3]: it is under-integrated with a single stack of stress integration points in the thickness direction passing through the elements center; it uses Assumed Natural Strain (ANS) interpolates for the thickness and transverse shear strains; it uses a single parameter Enhanced Assumed Strain (EAS) for the thickness strain; and it selectively scales the mass in the through thickness direction to increase the stable time-step. We combine these methods with a hypo-elastic constitutive model to simulate the ISF process. Initial results obtained with a GPU implementation of the element are presented.

Exact geometry SaS solid-shell element for 3D stress analysis of FGM piezoelectric structures

Abstract A hybrid-mixed functionally graded material (FGM) piezoelectric four-node solid-shell element through the sampling surfaces (SaS) method is proposed. The SaS formulation is based on choosing inside the shell N SaS parallel to the middle surface in order to introduce the displacements and electric potentials of these surfaces as fundamental shell unknowns. Such choice of unknowns with the use of Lagrange polynomials of degree N-1 in through-thickness interpolations of the displacements, strains, electric potential, electric field and material properties leads to a robust FGM piezoelectric shell formulation. The inner SaS are located at Chebyshev polynomial nodes that make it possible to minimize uniformly the error due to Lagrange interpolation. To implement the effective analytical integration throughout the element, the extended assumed natural strain (ANS) method is employed. As a result, the piezoelectric four-node solid-shell element exhibits a superior performance in the case of coarse meshes. To circumvent shear and membrane locking, the hybrid stress-strain solid-shell formulation via the Hu-Washizu variational principle is employed. The developed solid-shell element could be useful for the 3D stress analysis of FGMstructures because the SaS method allows obtaining the solutions with a prescribed accuracy, which asymptotically approach the exact solutions of electroelasticity as the number of SaS tends to infinity.

Surface-based and solid shell formulations of the 7-parameter shell model for layered CFRP and functionally graded power-based composite structures

ABSTRACTIn this study, we present the extension of the so-called 7-parameter shell formulation to layered CFRP and functionally graded power-based composite structures using two different parametrizations: (i) the three-dimensional shell formulation, and (ii) the solid shell approach. Both numerical strategies incorporate the use of the Enhanced Assumed Strain (EAS) and the Assumed Natural Strain (ANS) methods to alleviate locking pathologies and are implemented into the FE code ABAQUS. The applicability of the current developments is demonstrated by means of several benchmark examples, whose results are compared with reference solutions using shell elements of ABAQUS, exhibiting an excellent level of accuracy

Numerical investigations of CMC nozzle structures: constitutive modelling and finite element technology

AbstractThis work deals with the development of numerical tools predicting the lifetime of nozzle structures made of ceramic matrix composites (CMC). These materials are favorable for thermostructural components due to their excellent specific mechanical properties, high thermal conductivity and thermoshock resistance. The anisotropic constitutive behaviour of the CMC is reflected by a continuum model which is micro‐mechanically motivated. The model is based on structural tensors representing different fibre orientations. Furthermore, the thin structure of the nozzle requires an appropriate finite element technology in order to overcome locking phenomena. For this reason, a solid‐shell formulation with reduced integration is utilized, for which the implementation of the fibre orientation is crucial. The enhanced assumed strain (EAS) concept is used to avoid volumetric locking as well as Poisson thickness locking. The transverse shear and curvature thickness locking are cured by the assumed natural strain (ANS) concept. Using reduced integration together with hourglass stabilization leads to high computational efficiency. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Linear shell elements for active piezoelectric laminates

Piezoelectric composite laminates are a powerful material system that offers vast options to improve structural behavior. Successful design of piezoelectric adaptive structures and testing of control laws call for highly accurate, reliable and numerically efficient numerical tools. This paper puts focus onto linear and geometrically nonlinear static and dynamic analysis of smart structures made of such a material system. For this purpose, highly efficient linear 3-node and 4-node finite shell elements are proposed. Both elements employ the Mindlin-Reissner kinematics. The shear locking effect is treated by the discrete shear gap (DSG) technique with the 3-node element and by the assumed natural strain (ANS) approach with the 4-node element. Geometrically nonlinear effects are considered using the co-rotational approach. Static and dynamic examples involving actuator and sensor function of piezoelectric layers are considered.

Reduced-order modeling analysis of shell structures buckling using a co-rotational solid-shell element

The solid-shell elements can simplify the modeling process of the three-dimensional thin-walled structures that are commonly used in aerospace engineering. In this work, an efficient locking-free 8-node solid-shell element, termed SolSh8, is developed for geometrically nonlinear buckling analysis of shell structures. The solid-shell element is developed based on the conventional brick solid element with only translational degrees of freedom. The Assumed Natural Strain (ANS) method and the Enhanced Assumed strain (EAS) method are used to improve the linear performance of the element formulation. The proposed element is implemented into the co-rotational formulation of the Koiter–Newton method. The geometrically nonlinear equilibrium path of the structure can be traced efficiently benefiting from the reduced order model used in the Koiter–Newton method. Numerical examples demonstrate the excellent performance of the proposed method.

Concurrently coupled solid shell-based adaptive multiscale method for fracture

A solid shell-based adaptive atomistic–continuum numerical method is herein proposed to simulate complex crack growth patterns in thin-walled structures. A hybrid solid shell formulation relying on the combined use of the enhanced assumed strain (EAS) and the assumed natural strain (ANS) methods has been considered to efficiently model the material in thin structures at the continuum level. The phantom node method (PNM) is employed to model the discontinuities in the bulk. The discontinuous solid shell element is then concurrently coupled with a molecular statics model placed around the crack tip. The coupling between the coarse scale and the fine scale is realized through the use of ghost atoms, whose positions are interpolated from the coarse scale solution and enforced as boundary conditions to the fine scale model. In the proposed numerical scheme, the fine scale region is adaptively enlarged as the crack propagates and the region behind the crack tip is adaptively coarsened in order to reduce the computation costs. An energy criterion is used to detect the crack tip location. All the atomistic simulations are carried out using the LAMMPS software. A computational framework has been developed in MATLAB to trigger LAMMPS through system command. This allows a two way interaction between the coarse and fine scales in MATLAB platform, where the boundary conditions to the fine region are extracted from the coarse scale, and the crack tip location from the atomistic model is transferred back to the continuum scale. The developed framework has been applied to study crack growth in the energy minimization problems. Inspired by the influence of fracture on current–voltage characteristics of thin Silicon photovoltaic cells, the cubic diamond lattice structure of Silicon is used to model the material in the fine scale region, whilst the Tersoff potential function is employed to model the atom–atom interactions. The versatility and robustness of the proposed methodology is demonstrated by means of several fracture applications.

Isogeometric phase-field modeling of brittle and ductile fracture in shell structures

Phase-field modeling of brittle and ductile fracture is a modern promising approach that enables a unified description of complicated failure processes (including crack initiation, propagation, branching, merging), as well as its efficient numerical treatment [1-4]. In the present work, we apply this approach to model fracture in shell structures, considering both thin and thick shells. For thin shells, we use an isogeometric Kirchhoff-Love shell formulation [5-6], which exploits the high continuity of the isogeometric shape functions in order to avoid rotational degrees of freedom, i.e., the shell geometry is modeled as a surface and its deformation is fully described by the displacements of this surface. For thick shells, we use an isogeometric assumed natural strain (ANS) solid shell formulation [7], i.e., a 3D solid formulation enhanced with the ANS method in order to alleviate geometrical locking effects. According to the discretization of the structural formulations, an isogeometric basis is also used for the phase-field. While the phase-field fracture formulation for solid shells is basically the same as for standard solids, some reformulation is necessary for thin shells, accounting for the interaction of stresses devoted to membrane and bending deformation. We test both formulations on several numerical examples and perform comparisons of the results obtained by the two methods to each other as well as to reference solutions, which confirm the validity and applicability of the presented methods.