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Modeling deformation processes of materials under high strain rate and impact conditions in which the strain rates vary spatiotemporally over several orders of magnitude is challenging, especially in terms of constitutive description. Visco-plastic power-law flow rule, commonly used within crystal plasticity constitutive models, usually introduces superfluous strain rate effects, entering numerically via the slip activation criterion, which result in artificially high flow stress predictions under high strain rate deformation conditions. This paper presents a novel, implicit finite element implementation of a visco-plastic self-consistent polycrystal model that is implicit in dislocation-based hardening with removed strain rate sensitivity introduced by the power-law flow rule. In the model, the strain rate dependence of the flow stress is defined solely by the rate sensitivity of Peierls stress and the evolution of dislocation density within the hardening law, enabling the prediction of high strain rate and impact deformation. As a result, the predicted flow stress is accurate in its magnitude. The physically based strain rate and temperature sensitive model also features, in addition to Schmid, the non-Schmid activation contribution for slip, which is necessary for understanding and modeling the plastic deformation of body-centered cubic metals. The model is calibrated to simulate the strain rate- and temperature-sensitive monotonic deformation of tantalum and is subsequently applied to a Taylor impact test of the same material. Since the impact simulation required remeshing and interpolation of state variables, the spherical linear interpolation algorithm in the space of quaternions is implemented to facilitate the interpolation of texture. Predictions of the simulation were found to compare favorably with experimental measurements of post-test geometrical changes and texture evolution. The implementation and insights from these predictions are presented and discussed in the paper. [ABSTRACT FROM AUTHOR]

This paper presents a high order finite element implementation of the convex multi-variable electro-elasticity for large deformations large electric fields analyses and its particularisation to the case of small strains through a staggered scheme. With an emphasis on accurate geometrical representation, a high performance curvilinear finite element framework based on an a posteriori mesh deformation technique is developed to accurately discretise the underlying displacement-potential variational formulation. The performance of the method under near incompressibility and bending actuation scenarios is analysed with extremely thin and highly stretched components and compared to the performance of mixed variational principles recently reported by Gil and Ortigosa (2016) and Ortigosa and Gil (2016). Although convex multi-variable constitutive models are elliptic hence, materially stable for the entire range of deformations and electric fields, other forms of physical instabilities are not precluded in these models. In particular, physical instabilities present in dielectric elastomers such as pull-in instability, snap-through and the formation, propagation and nucleation of wrinkles and folds are numerically studied with a detailed precision in this paper, verifying experimental findings. For the case of small strains, the essence of the approach taken lies in guaranteeing the objectivity of the resulting work conjugates, by starting from the underlying convex multi-variable internal energy, whence avoiding the need for further symmetrisation of the resulting Maxwell and Minkowski-type stresses at small strain regime. In this context, the nonlinearity with respect to electrostatic counterparts such as electric displacements is still retained, hence resulting in a formulation similar but more competitive with the existing linearised electro-elasticity approaches. Virtual prototyping of many application-oriented dielectric elastomers are carried out with an eye on pattern forming in soft robotics and other potential medical applications. [ABSTRACT FROM AUTHOR]

Abstract: The intrinsic inability to withstand compression is a challenge in numerical simulation of a membrane with the finite element method. Basically there are two possibilities to deal with this problem. On the one hand wrinkles would be detected in detail by a refined mesh of shell elements once a bending formulation is used. On the other hand this problem is computed on a coarser mesh of membrane elements augmented with a wrinkling model, which considers effects of unresolved deformation patterns. This contribution focuses on the material modification approach, which is considered as a member of the latter. Within this category, works in [P. Contri, B.A. Schrefler, A geometrically nonlinear finite element analysis of wrinkled membrane surfaces by a no-compression material model, Commun. Appl. Numer. Methods 4 (1988) 5–15; X. Liu, C.H. Jenkins, W.W. Schur, Large deflection analysis of pneumatic envelopes using a penalty parameter modified material model, Finite Elements Anal. Des. 37 (2001) 233–251; X. Liu, Fine scale analysis of wrinkled membrane, Int. J. Numer. Comput. Engrg. Sci. 1 (2) (2000) 1017–1038; R. Rossi, M. Lazzari, R. Vitaliani, E. Onate, Simulation of light-weight membrane structures by wrinkling model, Int. J. Numer, Methods Engrg. 62 (15) (2005) 2127–2153; R. Rossi, E. Onate, Convergence of the modified material model for wrinkling simulation of light-weight membrane structures, in: E. Onate, B. Kröpin (Eds.), Textile Composites and Inflatable Structures (Structural Membranes 2003), CIMNE, 2003; R. Rossi, Light Weight Structures: Structural Analysis and Coupling Issues, Ph.D. Diss., The university of Bologna, 2005] are attractive because their wrinkling models on a coarse grid are simple and intuitive. The basic idea of this model is as follows: whenever compression occurs in a membrane, the corresponding components of the constitutive tensor in the direction of the compressive stress are penalized to weaken the compressive stiffness of the membrane. However, such abrupt change causes inevitably an unphysical oscillation of stress redistribution. Therefore, an algorithm to stabilize this oscillation is required. The first objective of this paper is to give a systematic verification of a wrinkling model based on the material modification approach by means of an analogy between wrinkling and plasticity. The second one is to include this model in a static analysis for both isotropic and orthotropic membranes. This model holds following advantages: it is computationally inexpensive and virtually able to reproduce the exact stress field, caused by wrinkling, on a macroscopic scale. To the author’s knowledge applications of a material modification approach with orthotropic materials are scarcely available. This paper demonstrates the potential of the model by means of numerical examples, compared to ones in literature. [Copyright &y& Elsevier]

Within the novel approach for structural health monitoring of civil engineering structures based on smart self-sensing structural materials, Carbon NanoTube (CNT)-reinforced cement-based composites, often termed “smart concretes”, have drawn rising interest. These composites exhibit self-sensing capabilities resulting in measurable variations of their electrical properties under applied mechanical deformations. This, along with the similarity between these composites and conventional structural concrete, makes it possible to devise cost-efficient distributed monitoring systems for large-scale Reinforced Concrete (RC) structures. While several papers in the literature have focused on pilot applications of smart concrete strain sensors embedded into large RC components, the response of these sensors is not yet fully understood at the fundamental physical level and, as a consequence, their output is not properly interpreted. In order to shed some light on this issue, previous work by the authors focused on laterally unconstrained uni-axial loading conditions. This work extends the previous studies by presenting a mixed micromechanics and finite element approach for the analysis of CNT-reinforced composites subjected to arbitrary strain states. The two mechanisms that contribute to the electrical conductivity of CNT-reinforced composites, namely electron hopping and conductive networking, are contemplated within a percolation framework in the micromechanics model. On the basis of the micromechanics model, the 3D piezoresistivity matrix is determined, for the first time in the literature, by means of virtual dilation and distortion tests. Afterwards, the electro-mechanical modeling of three-dimensional composite elements is conducted by a multi-physics finite element code. The potential of the presented approach is illustrated by extensive parametric analyses, as well as a comparison against experimental data, including application of the mixed micromechanics-finite-element multi-physics formulation for electro-mechanical modeling of three-dimensional elements. [ABSTRACT FROM AUTHOR]

The main aim of the work presented in this paper was treatment of damage and deformation localisation observed in the finite element method (FEM) analysis of strain softening materials combined with local constitutive models where damage is represented using continuum damage mechanics (CDM). The CDM/FEM approach typically suffers from a number of shortcomings, including mathematical (change of the type of partial differential equations leading to ill-posed boundary value problem), numerical (pronounced mesh dependency) and physical (infinitely small softening zone with the zero dissipated energy). The approach proposed here is still based on the local constitutive model including damage, but introduces an alternative representation of damage effects in the system of linear momentum balance equations. The damage effects are included through equivalent damage force (EDF), which contributes to the right-hand side of the momentum balance equations. The main advantages of this approach are that the problem remains well posed, as the type of partial differential equations remains unchanged when the material enters softening; numerical stability is preserved without a need for regularisation measures; and significantly reduced mesh dependency. In addition, the EDF approach can be used in combination with existing local CDM damage models and does not violate symmetry of the material stiffness tensor. The EDF approach is applicable to modelling of strain softening typically observed in damaged quasi brittle materials such as fibre reinforced composites and concrete. The EDF model was implemented in the in-house developed coupled FEM-SPH code, where an explicit FEM code is coupled with a stable Total-Lagrange form of SPH. Its performance is demonstrated in the analysis of a dynamic one dimensional (1D) stress wave propagation problem, which was analytically solved by Bazant and Belytschko in 1985. For a range of loading rates that correspond to the material softening regime, the numerical results shown nonlocal character with a finite size of the damaged zone, controlled with the damage characteristic length, which can be experimentally determined and is an input parameter independent of the discretisation density. [ABSTRACT FROM AUTHOR]

Material point methods suffer from volumetric locking when modelling near incompressible materials due to the combination of a low-order computational mesh and large numbers of material points per element. However, large numbers of material points per element are required to reduce integration errors due to non-optimum placement of integration points. This restricts the ability of current material point methods in modelling realistic material behaviour. This paper presents for the first time a method to overcome finite deformation volumetric locking in standard and generalised interpolation material point methods for near-incompressible non-linear solid mechanics. The method does not place any restriction on the form of constitutive model used and is straightforward to implement into existing implicit material point method codes. The performance of the method is demonstrated on a number of two and three-dimensional examples and its correct implementation confirmed through convergence studies towards analytical solutions and by obtaining the correct order of convergence within the global Newton–Raphson equilibrium iterations. In particular, the proposed formulation has been shown to remove the over-stiff volumetric behaviour of conventional material point methods and reduce stress oscillations. It is straightforward to extend this approach to other material point methods and the presented formulation can be incorporated into all existing material point methods available in the literature. [ABSTRACT FROM AUTHOR]

Recently, Isogeometric Analysis (IGA) based on incremental methods for simulating the stamping process has been researched. To the best of our knowledge, however, few studies have combined IGA and One-step inverse approach which is based on total deformation theory of plasticity. A key step for One-step inverse IGA is to estimate a good initial solution. Traditional mesh-based initial solution algorithms for One-step inverse approach are not suitable for One-step inverse NURBS-based IGA. In this paper, we presented a method which can rapidly unfold the undevelopable NURBS surface onto a planar domain and obtain a good initial solution estimation for One-step inverse IGA. The key idea of the presented method is unfolding the control net of a NURBS surface for isogeometric analysis by energy-based initial solution estimation algorithm. In addition, we developed a “cutting-stitching” algorithm which can separate a complex control net into several parts with simple shapes. Numerical examples illustrate the initial solutions using the presented method are approaching the final results by One-step inverse finite element method. This implies that the iterative steps and computational time of One-step inverse IGA will be reduced significantly compared with that of One-step inverse finite element method. [ABSTRACT FROM AUTHOR]

Micromechanical analysis of polycrystalline microstructures of metals and alloys, using crystal plasticity finite element (CPFE) models is extensively used for predicting deformation and failure under various conditions of strain-rates, creep and fatigue loading. Many CPFE models involve a large number of degrees of freedom for accurate representation of realistic polycrystalline microstructures. This can lead to prohibitively high computational costs to conduct meaningful analyses of phenomena of interest. To overcome this limitation, the authors have recently developed a wavelet enrichment adapted finite element model in Azdoud and Ghosh (2017) for elastic materials. The method adaptively creates an optimal discretization space conforming to the solution profile by projecting the solution field onto a set of scaling and multi-resolution wavelet basis functions. This paper extends this wavelet adapted FE model to finite deformation, crystal plasticity analysis of polycrystalline microstructures. After presenting the formulations, various validation tests are conducted to examine the convergence rates and computational efficiency of this method. [ABSTRACT FROM AUTHOR]

This paper provides a finite element treatment for a generalized hyperelastic theory in which the deformation gradient is subject to the multiplicative two-factor decomposition that is commonly used in large strain plasticity. The multiplicative decomposition in the present context is motivated by a theory of balanced internal variables, one of which is the first factor in the decomposition. Previous studies have shown how the associated continuum mechanical formulation then leads not only to the stress equations of equilibrium but also to an additional tensor balance equation that serves to determine the individual factors in the decomposition. Such a theory has been shown to predict loading thresholds associated with the emergence of slip surfaces at grip locations in well-posed boundary value problems. However, previous numerical treatments focused on shooting method procedures that failed to resolve loading plateaus and other key features associated with the emergence of such singular surfaces. In the present work we describe a finite element treatment that resolves such previously obscure details. Additional finite element simulations illustrate new aspects of singular surface emergence in this type of theory, including the emergence of singular surfaces in the interior of the computational domain. [ABSTRACT FROM AUTHOR]

This paper presents the development of new three-dimensional finite elements in the finite deformation range for the numerical modeling of material failure characterized by propagating crack-type surfaces in solids, commonly referred to as strong discontinuities. The singular fields associated with such discontinuous solutions are captured by an element-based enhancement, locally within a multi-scale framework. The complex kinematics arising from the general propagating discontinuity surfaces, especially in the considered finite deformation setting, is captured by the direct identification of the element strain fields associated to the separation modes of the discontinuity, rather than attempting to interpolate directly the corresponding discontinuous deformation fields. To avoid the over-stiff response known as stress-locking, by which the discontinuities cannot open as needed by the material response, the new brick elements consider full linear interpolations of the displacement jumps. This requires a total of nine local parameters for each localized finite element, parameters that are condensed out at the element level. The resulting formulation does not change then neither the number nor the connectivity of the global degrees of freedom employed in the finite element resolution of the underlying global mechanical problem, thus leading to a computationally efficient numerical technique that captures sharply these discontinuous solutions. In particular, the localized dissipative mechanism associated to the material failure is captured objectively through the consideration of the proper cohesive law along the discontinuity surface. Special care is taken to assure the material frame indifference of the final formulation. Complete details of the implementation of the new elements, including the consistent linearization of the governing equations, are also presented. The results obtained in numerical simulations of representative benchmark problems illustrate the performance of the new finite elements with embedded strong discontinuities. [ABSTRACT FROM AUTHOR]

The basis of this work is a novel symbiosis of mechanics of solids and spherical geometry to quantify and illustrate the variation of the “non-membrane” percentage of the strain energy in the prebuckling region of linear elastic beams, arches, plates and shells, and structures assembled of such one-dimensional and two-dimensional members. The zenith angle of an arbitrary point of a specific curve on an octant of the unit sphere, called buckling sphere, is related to this energy percentage. For the limiting case of buckling from a membrane stress state this curve degenerates to a point, characterized by zero values of both spherical coordinates. For all other stress states the azimuth angle increases with the proportionally increasing load. Its magnitude at the stability limit correlates with a global quantity that depends on both the “non-membrane” deformations and the stiffness of the structure at incipient buckling. The azimuth angle is computed with the help of the so-called consistently linearized eigenproblem, which is solved by means of the Finite Element Method. This eigenvalue problem is the basis for a hypothesis for the “non-membrane” percentage of the strain energy. In the theoretical part of the paper, the concept of the buckling sphere is presented. The subsequent numerical investigation consists of four examples, referring to buckling from a membrane stress state, a pure bending stress state, and a general stress state. The practical motive for this research is the intention to investigate the influence of “non-membrane” action just before buckling on the initial postbuckling behavior of elastic structures. [ABSTRACT FROM AUTHOR]

Solving large deformation problems of hyperelastic materials by the finite element method is still a challenging problem due to the severe mesh distortion that occurs during the computation. In Leger et al. (2014), it was shown that combining an updated Lagrangian method with an efficient adaptive remeshing algorithm, an accurate transfer method for the deformation gradient tensor as well as an efficient continuation method leads to a very stable, efficient and accurate algorithm to solve two-dimensional very large deformation problems. In this paper, we show that this method can also be generalized to solve three-dimensional problems. A number of static problems will be presented and analyzed. [ABSTRACT FROM AUTHOR]

This paper presents a residual-based a posteriori error estimator for the Arnold–Winther mixed finite element that utilises a post-processing for the skew-symmetric part of the strain. Numerical experiments verify the proven reliability and efficiency for suitable approximation of the skew-symmetric deformation gradient. Numerical evidence supports that the L 2 -stress error estimator is robust in the Poisson ratio and allows stable error control even in the incompressible limit. [ABSTRACT FROM AUTHOR]

This paper presents a contact surface smoothing method combined with the node-to-segment discretization technique to solve large deformation frictional contact problems between deformable bodies. The Nagata patch interpolation is used to smooth the surface mesh, providing a master surface with quasi - G 1 continuity between patches. Moreover, the local support of the interpolation method allows to deal with surface meshes of arbitrary topology (regular and irregular finite element discretizations), as well as hybrid meshes. The non-physical oscillations in the contact force evolution, induced by the faceted contact surface representation, are reduced using the proposed smoothing method. Furthermore, the smooth representation of the master surface allows a more accurately evaluation of the resulting stresses and forces, while providing an important improvement in convergence behaviour. Four representative numerical examples are used to demonstrate the advantages of the proposed contact smoothing method. The results show a significant improvement in the accuracy, robustness and performance of the numerical simulations using the smoothing approach, when compared with the piecewise faceted contact surface description. [ABSTRACT FROM AUTHOR]

Deformation twinning is a subgrain mechanism that strongly influences the mechanical response and microstructural evolution of metals especially those with low symmetry crystal structure. In this work, we present an approach to modeling the morphological and crystallographic reorientation associated with the formation and thickening of a twin lamella within a crystal plasticity finite element (CPFE) framework. The CPFE model is modified for the first time to include the shear transformation strain associated with deformation twinning. Using this model, we study the stress–strain fields and relative activities of the active deformation modes before and after the formation of a twin and during thickening within the twin, and in the parent grain close to the twin and away from the twin boundaries. These calculations are carried out in cast uranium (U), which has an orthorhombic crystal structure and twins predominantly on the { 130 } < 3 1 ̄ 0 > systems under ambient conditions. The results show that the resolved shear stresses on a given twin system on the twin–parent grain interface and in the parent are highly inhomogeneous. We use the calculated mechanical fields to determine whether the twin evolution occurs via thickening of the existing twin lamella or formation of a second twin lamella. The analysis suggests that the driving force for thickening the existing twin lamella is low and that formation of multiple twin lamellae is energetically more favorable. The overall modeling framework and insight into why twins in U tend to be thin are described and discussed in this paper. [ABSTRACT FROM AUTHOR]

Virtual surgery simulators show a lot of advantages in the world of surgery training, where they allow to improve the quality of surgeons’ gesture. One of the current major technical difficulties for the development of surgery simulation is the possibility to perform a real-time computation of soft tissue deformation by considering the accurate modeling of their mechanical properties. However today, few models are available, they are still time consuming and limited in number of elements by algorithm complexity. We present in this paper a new method and framework that we call ‘HEML’ ( Hyper-Elastic Mass Links ), which is particularly fast. It is derived from the finite element method, can handle visco-hyperelastic and large deformation modeling. Although developed initially for medical applications, the HEML method can be used for any numerical computation of hyperelastic material deformations based on a tetrahedral mesh. A comparison with existing methods shows a much faster speed. A comparison with Mass–Spring methods, that are particularly fast but not realistic, shows that they can be considered as a degenerate case of the HEML framework. [ABSTRACT FROM AUTHOR]

The finite element implementation of a geometrically exact thin shell model is reported in the present paper. The shell kinematics is based on the Kirchhoff–Love assumption and is characterized by the deformation gradient, which yielded the generalized cross-section strain measures — stretches and curvatures, written in terms of first- and second-order derivatives of displacements. The energetically conjugate quantity, the first Piola–Kirchhoff stress tensor, is used to define the generalized stresses. The shell’s initial geometry is exactly represented using a mapping from a reference configuration. A neo-Hookean material functional, supplemented by the plane stress condition, is incorporated in the constitutive level of the model. Both configuration dependent and independent forces are considered for the domain and boundary loading. The consistent linearization of the rendered weak form is addressed. The TUBA family of plate finite elements is adopted for the discretization of the displacement vector field. These triangular elements, with necessarily straight sides in the reference configuration, provide C 1 continuous approximations, a sine qua non property for a consistent discretization of the Kirchhoff–Love model. The family contains 3 pairs of elements (primary and reduced), which provide polynomial approximants of degrees k = 5 , 6 and 7 , respectively. The reduced elements have a simpler structure and a smaller number of nodes, but still guarantee C 1 continuity. The boundary normal derivative of the approximation in these elements is constrained, such that its order is reduced by one unit, from ( k − 1 ) at the primary elements to ( k − 2 ) . Due to complexity of the elements degrees of freedom (which are the function value, its normal, first- and second-order derivatives), the imposition of the essential boundary conditions requires some preliminary analysis of the boundary behavior. Such study is presented for the most common practical cases. Six benchmark examples are presented to demonstrate the model usefulness and efficiency. [ABSTRACT FROM AUTHOR]

The paper presents a nonlinear Finite Element (FE) algorithm to simulate the inhomogeneous steady swelling of polymeric gels. The algorithm is developed based on the multiplicative decomposition of the deformation gradient into an elastic part and a swelling part. The corresponding constitutive framework and linearization of the FE equation are elaborated, which leads to an equivalent body force driving the materials to swell. A staggered iterative procedure is designed to ensure the strongly coupled chemical and mechanical fields to reach the equilibrium state simultaneously. The constitutive equations are constructed in the framework of the volumetric–isochoric splitting of the free energy function, which is helpful both for a different treatment of the incompressible part in the FE equation and for a physical interpretation of the coupling effects between the chemical and mechanical fields. In the present work, an additive free energy function, coupling the contributions of the Flory–Huggins model and the non-Gauss statistical–mechanical model, is adopted and the corresponding consistent tangent modulus in the current configuration is also derived. To implement the FE algorithm, two two-dimensional (2D) elements and one three-dimensional (3D) element are developed, which are validated by some analytical solutions, such as free swelling of gels in 2D and 3D cases, squeezing a swollen gel with uniform pressure and the constrained swelling of a single-layer and a core–shell gel ring. The algorithm is also proved to be capable of handling the surface instabilities and some engineering applications such as hydrogel-based composite beams and thin films. [ABSTRACT FROM AUTHOR]

Abstract: This paper focuses on the development of a global–local doubly-curved shell element, suitable for small-deformation implicit and/or explicit dynamic analysis of laminated composite structures. The global–local framework is based on the superposition of a global displacement field, spanning the thickness of the entire laminate, and local (layerwise) displacement fields associated with each layer of the laminate. This approach affords highly-resolved representations in regions of critical interest, and allows a smooth transition from higher to lower resolution zones. Continuity between adjacent layers is enforced by means of discontinuous Galerkin fluxes. Parasitic phenomena characteristic of bilinear shell elements, such as shear locking, are alleviated with the aid of assumed natural strain techniques. Performance characteristics of the proposed finite element are examined with the aid of several numerical examples involving static and dynamic analysis of thick as well as thin shells. [Copyright &y& Elsevier]

Abstract: In this paper, a polygonal finite element method is presented for large deformation frictionless dynamic contact-impact problems with non-conformal meshes. The geometry and interfaces of the problem are modeled independent of the background mesh based on the level set method to produce polygonal elements at the intersection of the interface with the regular FE mesh. Various polygonal shape functions are employed to investigate the capability of polygonal-FEM technique in modeling frictionless contact-impact problems. The contact constraints are imposed between polygonal elements produced along the contact surface through the node-to-surface contact algorithm. Several contact-impact problems are modeled using various polygonal interpolation functions, including the Wachspress interpolation functions, the metric coordinate shape functions, the natural neighbor based functions, and the mean value coordinate functions to demonstrate the efficiency of proposed technique in modeling contact-impact problems. [Copyright &y& Elsevier]

Abstract: Solid-shell formulations based on reduced integration with hourglass stabilization have several advantages. Among these are the smaller number of Gauss points and the direct modelling of the thickness stretch, a feature which is usually not present in standard degenerated shell elements. The latter issue is especially important for applications where contact is involved, e.g. for almost all relevant systems in production technology. Obviously this makes solid-shell formulations very attractive for their use in industrial design. A major disadvantage in the context of explicit analyses is, however, the fact that the critical time step is determined by the thickness of the solid-shell element which is usually smaller than the smallest in-plane dimension. Therefore, four-node shells (where the critical time step is determined by the in-plane dimensions) are still often preferred for explicit analysis. In the present paper we suggest several techniques to overcome this difficulty, also in the case of problems dominated by nonlinearities such as finite deformations, elastoplasticity and contact. Reference is made to an 8-node hexahedron solid-shell element recently proposed by Schwarze and Reese (2011) [32] in an implicit context. First of all, the time steps in explicit analyses are so small that it may be not necessary to update the hourglass stabilization and the implicit computation of the internal element degrees-of-freedom in every time step. Performing the update in only every hundredth step or computing an explicit rather than implicit update can reduce the computational effort up to about 50%. Another important issue is selective mass scaling which means to modify the mass matrix in such a way that the speed of sound in thickness direction is reduced. This enables the choice of a larger time step. The CPU effort can be finally noticeably decreased without changing the structural response significantly. This makes the presently used solid-shell formulation competitive to four-node shells, also for explicit analysis. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a “modified” immersed finite element method (mIFEM), a non-boundary-fitted numerical technique, to study fluid–structure interactions. Using this method, we can more precisely capture the solid dynamics by solving the solid governing equation instead of imposing it based on the fluid velocity field as in the original immersed finite element (IFEM). Using the IFEM may lead to severe solid mesh distortion because the solid deformation is been over-estimated, especially for high Reynolds number flows. In the mIFEM, the solid dynamics is solved using appropriate boundary conditions generated from the surrounding fluid, therefore produces more accurate and realistic coupled solutions. We show several 2-D and 3-D testing cases where the mIFEM has a noticeable advantage in handling complicated fluid–structure interactions when the solid behavior dominates the fluid flow. [Copyright &y& Elsevier]

Abstract: The p-version of the finite element method (p-FEM) is extended to problems in the field of biomechanics: the mechanical response of bones and arteries. These problems are extremely challenging, partly because the constitutive models governing these materials are very complex and have not been investigated by sufficiently rigorous methods. Furthermore, these biological structures have a complex geometrical description (substructures with high aspect ratios), undergo finite deformations (arteries), are anisotropic and almost incompressible (arteries). The intrinsic verification capabilities and high convergence rates demonstrated for linear problems are being exploited and enhanced here, so that validation of the results can be easily conducted by comparison to experimental observations. In the first part of the paper we present p-FE models for patient-specific femurs generated semi-automatically from quantitative computed tomography (qCT) scans with inhomogeneous linear elastic material assigned directly from the qCT scan. The FE results are being verified and thereafter validated on a cohort of 17 fresh-frozen femurs which were defrosted, qCT-scanned, and tested in an in vitro setting. The complex combined passive–active mechanical response of human arteries is considered in the second part and the enhancement of p-FEMs to these non-linear problems is detailed. We apply a new ‘p-prediction’ algorithm in the iterative scheme and demonstrate the efficiency of p-FEMs compared to traditional commercial h-FEMs as Abaqus (in respect of both degrees of freedom and CPU times). The influence of the active response is shown to be crucial if a realistic mechanical response of an artery is sought. [Copyright &y& Elsevier]

Abstract: Failure of geomaterials with pores filled with fluids is an important research area in both civil and geological engineering. Many finite element formulations for coupled problems present difficulties such as overestimating failure loads, or mesh alignment dependence resulting on spurious failure mechanisms. Moreover, the spaces where field variables are approximated have to fulfill additional requirements ensuring stability. Stress-velocity-pore pressure formulations in FE analysis provide accurate results for wave propagation and failure analysis. However, finite elements present important limitations when deformations are large. The purpose of this paper is to present a stabilized Fractional Step, SPH algorithm which combines the advantages of the SPH method for large deformation problems with those of the Taylor Galerkin algorithm used within the finite element framework. [Copyright &y& Elsevier]

Abstract: As a Lagrangian particle method, the material point method (MPM) has the potential to model extreme deformation of materials, where the traditional finite element method (FEM) often encounters mesh distortion and element entanglement which lead to numerical difficulties. However, FEM is more accurate and efficient than MPM for problems with small deformation. It is therefore desirable to model the body with extreme deformation by MPM and the body with small deformation by FEM, respectively. In this paper, a method to handle the contact interaction between the MPM body and the FEM body is proposed, which is implemented on the background grid of MPM. By this method, FEM is coupled with MPM and a hexahedral element is incorporated into our 3D explicit MPM code MPM3D®. Several numerical examples, including plate impact, sphere rolling, perforation of thick plate, and fluid–structure interaction problems, are studied and the numerical results are in good agreement with analytical solution and results available in the literature. The coupling of FEM and MPM offers advantages of both FEM and MPM. [Copyright &y& Elsevier]

Abstract: The folding of thin shells around localized lines occurs in several situations of engineering interest, such as in buckling induced deformation, as the consequence of crash, in industrial processes like metal forming, in the deployment of folded membranes, in packaging related processes. In this paper an interface element, allowing for displacement and rotation discontinuities, to be placed between two adjacent 4-node Mindlin–Reissner shell finite elements, is developed for a computationally effective simulation of this type of localized deformation. The large displacement and rotation kinematics of the interface element is discussed, arriving at a rigid-rotation free element tangent stiffness matrix. An adaptive dynamic relaxation scheme is proposed for the evolutionary analysis of the quasi-static structural response. To test the element formulation, a simple coupled membrane-bending elastoplastic behavior is assumed. The element is used for the simulation of few examples taken from the literature, where the structural response is characterized by the formation of plastic hinges, exhibiting good accuracy and computational effectiveness. [Copyright &y& Elsevier]

Abstract: We present a new multiscale, finite deformation, electromechanical formulation to capture the response of surface-dominated nanomaterials to externally applied electric fields. To do so, we develop and discretize a total energy that combines both mechanical and electrostatic terms, where the mechanical potential energy is derived from any standard interatomic atomistic potential, and where the electrostatic potential energy is derived using a Gaussian-dipole approach. By utilizing Cauchy–Born kinematics, we derive both the bulk and surface electrostatic Piola–Kirchhoff stresses that are required to evaluate the resulting electromechanical finite element equilibrium equations, where the surface Piola–Kirchhoff stress enables us to capture the non-bulk electric field-driven polarization of atoms near the surfaces of nanomaterials. Because we minimize a total energy, the present formulation has distinct advantages as compared to previous approaches, where in particular, only one governing equation is required to be solved. This is in contrast to previous approaches which require either the staggered or monolithic solution of both the mechanical and electrostatic equations, along with coupling terms that link the two domains. The present approach thus leads to a significant reduction in computational expense both in terms of fewer equations to solve and also in eliminating the need to remesh either the mechanical or electrostatic domains due to being based on a total Lagrangian formulation. Though the approach can apply to three-dimensional cases, we concentrate in this paper on the one-dimensional case. We first derive the necessary formulas, then give numerical examples to validate the proposed approach in comparison to fully atomistic electromechanical calculations. [Copyright &y& Elsevier]

Abstract: We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya–Babuska–Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress–strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields. [Copyright &y& Elsevier]

Abstract: Analytical backward Euler stress integration is presented for a volumetrically non-associated pressure-sensitive yield criterion based on a modified Reuleaux triangle. This advances previous work on associated Reuleaux plasticity using energy-mapped stress space. The analytical solution is 2–4 times faster than a standard numerical backward Euler algorithm. The merit in transforming to (and operating in) this space is that the stress return is truly the closest point on the surface to the elastic trial state. The paper includes a tension cut-off (formed by a second cone) and describes the steps necessary to allow the model’s incorporation within a finite deformation framework. Finite-element results show a 59% runtime saving for a modified Reuleaux model over a Willam–Warnke cone giving comparable accuracy in a thick-walled cylinder expansion problem. The consistent tangent provides asymptotically quadratic convergence in the Newton–Raphson scheme under both (i) small strain, infinitesimal deformation and (ii) large strain, finite deformation finite-element simulations. It is shown that the introduction of non-associated flow changes the plastic deformation field and reduces the heave predicted in a plane strain rigid strip-footing problem. The proposed model offers a significant improvement over the Drucker–Prager and Mohr–Coulomb formulations by better reproducing the material dependence on the Lode angle and intermediate principal stress, at little extra computational effort. [Copyright &y& Elsevier]

Abstract: This paper presents a non-interior point method for solving frictionless contact problems in large deformations, where we solve the problems in an incremental path-following method from warm start. We propose a novel reformulation of the nonlinear complementarity problem, which is based on the smoothed Fischer–Burmeister function but is distinguished from the conventional formulations in the following two particular aspects: (i) the smoothing parameter is considered as an independent variable; (ii) an equality constraint is added so that the smoothing parameter serves as a measure of the residual of the complementarity conditions. The reduced system of nonlinear equations is solved with a conventional Newton method for nonlinear equations from the initial point which is defined by using the solution of the preceding loading stage. Throughout numerical examples it is shown that in many cases the solution can be found within four Newton iterations. [Copyright &y& Elsevier]

Abstract: A formulation for the bending analysis of composite laminated plates with piezoelectric layers is implemented using the generalized finite element method. This formulation is derived from a mechanical description based on Higher-Order Shear Deformation Theory which allows for the use of C 0 continuous approximation functions on the domain. On the other hand, a Layerwise Theory is employed for interpolation of electric potential across the thickness of piezoelectric layers, in such a way that the kinematical hypotheses result in a mixed model. The paper presents an analysis of the approximation capability of the proposed numerical model for static analysis, using C 0 continuous Partition of Unity and polynomial enrichments to span the approximation spaces, by assessment of convergence. Analytical solutions obtained from the same kinematical hypotheses are used as references. Results for relative error in the energy norm considering p- and h-refinements for regular and distorted meshes, in addition to a pointwise evaluation of the stresses and electric field, are presented. The evaluations show that the numerical methodology is a very effective tool for improving the solution through the enrichment, even for pointwise values across the thickness, and is robust to mesh distortions. Moreover, the results furnish insight about the physical modeling for both active and sensory modes, for thick and thin plates. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a three-dimensional finite element formulation for micropolar elasticity dealing with large displacements and small strains (or equivalently small strains and finite rotations). A comprehensive outline of the theory’s characteristical features is given and we try to elucidate the set-up of a possible non-linear finite element implementation. One focus of the present study is on a sound verification process, featuring the construction of an enhanced Patch Test and the assessment of quadratic asymptotic rates of convergence. Aspects of performance and validity are discussed at a set of numerical examples. We show that, the proposed model is able to reproduce the transition between micropolar and classical continua highly accurate. Finally, we present results underlining the implementation’s applicability in the realm of finite deformation with arbitrarily large rotations. [ABSTRACT FROM AUTHOR]

Abstract: The natural element method (NEM) is one of the members of the large family of meshless methods, with clear advantages over the finite element method (FEM) in problems involving large mesh distortions or complex geometries where the design of the mesh is costly. These problems are found in many applications like, for instance, simulation of biological structures involving soft tissues, such as, articular joints. One additional advantage of NEM is that it can be easily coupled with finite elements and implemented into any FE framework, including well-known commercial packages. NEM as most other spatial approximation approaches can be applied to evolution problems in two types of time (or pseudo-time) integration schemes, namely implicit and explicit. However, the NEM explicit version has neither been implemented nor sufficiently analyzed, so a comparative study of those two types of NEM time integration schemes is still missing. The main aim of this paper is to discuss issues related to NEM accuracy and stability in its explicit version, and problems related to its implementation into an explicit FE commercial code. Finally, a comparative study addressing the main properties, advantages and disadvantages of both types of NE schemes, implicit and explicit, is presented. Several examples of application are discussed including aspects where NEM is competitive with FEM including modeling of human articular joints like the knee. Explicit NEM allows achieving accurate results for high distortions and complex contact conditions although constraints on time step still are a major drawback and comparable to those known in finite elements to keep stability and accuracy despite the less NEM sensitivity to mesh distortion. [Copyright &y& Elsevier]

Abstract: This paper presents a new formulation for thin-walled beams that includes cross-section deformation. The kinematic description of the beam emanates from the geometrically exact Reissner–Simo beam theory and is enriched with arbitrary cross-section deformation modes complying with Kirchhoff’s assumption. The inclusion of these deformation modes makes it possible to capture the cross-section in-plane distortion, wall (plate) transverse bending and out-of-plane (warping), which leads to a computationally efficient numerical implementation. Several illustrative numerical examples are presented and discussed, showing that the resulting beam finite element leads to solutions that are in very good agreement with those obtained with standard shell finite elements, albeit involving much less degrees-of-freedom. [Copyright &y& Elsevier]

Abstract: In this paper, a first-order multi-scale approach for three-dimensional structural problems is proposed. A nested solution technique enables the mutual transfer of information between global and local scales which are solved simultaneously in a finite element analysis. Therefore, apart from improving the coarse-scale solution by considering fine-scale features, a detailed simulation of the micro-level also enables the determination of local fine-scale quantities of interest. To account for transient effects, a dynamic simulation on the macro-scale is locally coupled to a quasi-static simulation of the discretized micro-level. For time integration, the generalized-α scheme is employed on both scales. The presented algorithm is applicable for large deformations, rotations and arbitrary micro-structural behavior. On the macro-level, no constitutive assumption is required since the stress–strain relationship is directly computed from the micro-scale using established computational homogenization procedures. First simulations validate the novel approach and demonstrate its suitability for lung parenchyma considering complex surfactant film dynamics as an example of fine-scale multi-physics. [Copyright &y& Elsevier]

Abstract: This paper deals with the adaptive finite element analysis of structural failure. A gradient-enhanced damage model has been chosen to simulate material degradation. Since this model is regularized in the post-peak regime, the finite element solution does not suffer from pathological mesh dependence and thus converges to an objective solution upon mesh refinement. However, the error analyses have shown that the error in the nonlocal equivalent strain field becomes dominant during the post-peak loading stages. The accuracy of the nonlocal equivalent strain field (and the corresponding damage quantity) also greatly influences the accuracy of the quantity of interest. Two error measures have been proposed. The goal-oriented error estimates have provided similar error distributions, although some small differences have been found in the softening regime. Objective error estimates, together with adaptive criteria, have been used to perform automated h-adaptivity during computation. [Copyright &y& Elsevier]

Abstract: This paper, in line with part I , is concerned with the finite element implementation of boundary potential energies and the study of their impact on the deformations of solids thereby the main thrust is the fully three-dimensional formulation and implementation incorporating anisotropic effects. Boundary effects can play a dominant role in the material behavior, the most prominent example being surface tension. However, the common modelling in continuum mechanics takes exclusively the bulk into account, nevertheless, neglecting possible contributions from the boundary. Within this contribution the boundary potentials are allowed, in general, to depend not only on the boundary deformation but also on the boundary deformation gradient and the spatial boundary normal. For the formulation of the finite element method, the concept of convected curvilinear coordinates attached to the boundary is employed and the corresponding derivations completely based on a tensorial representation are carried out. Afterwards, the discretization of the generalized weak formulation, including boundary potentials, is performed and eventually numerical examples are presented to demonstrate the boundary effects due to different proposed material models. In contrast to the previous literature on this topic, the current manuscript covers jointly the following issues related to boundary energies: (1) the formulation and implementation represents a fully three-dimensional framework at large deformations, (2) the formulation of the problem and the proposed material models are based on finite strains, however, it is shown that the linearization would lead to the small strain models proposed previously in the literature and (3) the current manuscript covers the issue of anisotropy effects on the boundary energies which to the best of our knowledge has not been exemplified earlier. [Copyright &y& Elsevier]

Abstract: This paper investigates higher-order and higher-continuity functions in isogeometric structural analysis under distortion of the control and physical meshes. First, the concepts behind isogeometric analysis are briefly reviewed. The effect of perturbing control points upon the geometrical mapping is observed in the one-dimensional setting. The concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy. A circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models. The fillet patch is tested with two problems of linear elasticity and is shown to be accurate. The investigation is further extended to the setting of large-deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed projection method. [Copyright &y& Elsevier]

Abstract: In this paper we present a hyperelastic rod model that takes into account self-contact forces. The model is based on Cosserat rod theory that incorporates shear, elongation, flexure and twist deformations. The problem of avoiding self-penetration of parts of the rod is handled by the introduction of a contact distance function and the incorporation of associated contact forces. We present a penalty method for the treatment of the multi-valued and non-differentiable contact law. We also describe an augmented Lagrangian formulation of this problem. We then give the details of the finite-element discretization of the elastic rod self-contact problem as well as some numerical examples. [Copyright &y& Elsevier]

Abstract: The paper presents a phenomenological model of rate-independent and rate-dependent thermo-plasticity in the logarithmic Lagrangean strain–entropy space at finite strains. The formulation and computational implementation are based on an additive decomposition of the strain measure into an elastic and plastic part as proposed by Green and Naghdi. A rate-independent and rate-dependent constitutive model in the logarithmic Lagrangean strain–entropy space is developed for the modelling of isotropic elastic and anisotropic plastic material behaviour. Furthermore, the notion of a plastic metric as proposed by Miehe, an isotropic hardening variable and the notion of a plastic entropy as proposed by Simo and Miehe dominate the internal variable description. The model keeps the classical general return-mapping scheme in the implicit integration algorithm of the internal variables of associative thermo-plasticity. The resulting coupled thermo-mechanical problem is solved staggeredly via an isentropic phase for the deformation and an iso-geometrical phase for the thermal field. Representative numerical simulations demonstrate the performance of the proposed model. [Copyright &y& Elsevier]

Abstract: Many researchers have proposed the use of biomechanical models for high accuracy soft organ non-rigid image registration, but one main problem in using comprehensive models is the long computation time required to obtain the solution. In this paper, we propose to use the Total Lagrangian formulation of the Finite Element method together with Dynamic Relaxation for computing intra-operative organ deformations. We study the best ways of estimating the parameters involved and we propose a termination criteria that can be used in order to obtain fast results with prescribed accuracy. The simulation results prove the accuracy and computational efficiency of the method, even in cases involving large deformations, nonlinear materials and contacts. [Copyright &y& Elsevier]

Abstract: In this paper, an exact time integration scheme is presented for von Mises elastoplasticity with linear isotropic hardening at small deformations. The method is based on the constant strain rate assumption, which is widely accepted in displacement based finite element applications. The deviatoric form of the rate equation, using the method proposed by Krieg and Krieg in [R.D. Krieg, D.B. Krieg, Accuracies of numerical solution methods for the elastic-perfectly plastic model, J. Press. Vess. Technol. Trans. ASME 99 (1977) 510–515], is rewritten to a system of nonlinear ODEs with two scalar functions: the radius of the Mises circle and an angle obtained by using the scalar product of the stress deviator and the strain rate. The time integration of these functions gives an exact solution in implicit form using the incomplete beta function. In addition, by the exact linearization of this new semi-analytical solution a stress updating algorithm and the consistent tangent operator are derived. The numerical performance and accuracy of the proposed method are illustrated on numerical examples. Moreover, a comparison of the proposed method with the well-known radial return method is included. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a novel, fully variational formulation for finite deformation analysis of inelastic membranes with wrinkling. In contrast to conventional approaches, every aspect of the physical problem derives from minimization of suitable energy functionals. A variational formulation of finite strain plasticity theory, which leads to a minimization problem for the constitutive updates, serves as the starting point for the derivations. In order to take into account the kinematics induced by wrinkles and slacks, a relaxed version of the finite strain functional is postulated. In effect, the local incremental stress–strain relations are established via differentiation of the relaxed energy functional with respect to the strains. Hence, the presented formulation is fully analogous to that of hyperelasticity with the sole exception that the aforementioned functional depends on history variables and, accordingly, it is path dependent. The advantages associated with the developed variational method are manifold. From a practical point of view, the possibility of applying standard optimization algorithms to solve the minimization problem describing inelastic membranes is remunerative. From a mathematical point of view, on the other hand, the energy of the system induces some sort of natural metric representing an essential requirement for error estimation and thus, for adaptive finite element methods. The presented derivation of the model allows to consider possible material symmetries in the elastic as well as plastic response of the material. As a prototype, a von Mises-type model is implemented. The efficiency and performance of the resulting algorithm are demonstrated by means of numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper, we study numerical locking effects in the finite element approximation of shell layers. The focus is on four-node degenerated elements that are perhaps the most widely used in practical computations. Our approach is based on simplified reformulations of the original 3D elements in the context of a classical shallow shell model, where the deformation of the shell is described in terms of the three displacements of the middle surface and two dimensionless rotations related to transverse shear deformations. Within that model we compute finite element approximations of layers generated by smooth line segments such as the boundary line. Our numerical results show that approximation failure occurs when bilinear degenerated elements are used on anisotropically refined meshes and that the accuracy depends substantially on the specific geometry of the shell. By analyzing the layers as functions of the thickness of the shell, we obtain theoretical error estimates in the energy norm framework that predict error amplification for small values of the thickness. The error estimates are then verified by conducting further numerical experiments using overrefined meshes. [Copyright &y& Elsevier]

Abstract: In this paper, the extended finite element method is presented for large elasto-plastic deformation in 3D solid mechanics problems. The X-FEM computational algorithm is presented in the framework of Lagrangian description in order to model the arbitrary interfaces in large deformations. In X-FEM, the material interface is represented independently of element boundaries and the process is accomplished by partitioning the domain with several tetrahedral sub-elements whose Gauss points are used for integration of the domain of elements. The interface between two bodies is modeled by using the X-FEM technique and applying a modified level set enrichment function. In order to simulate the nonlinear behavior of materials, the Lagrangian plasticity formulation is coupled with the X-FEM technique. Finally, several numerical examples are analyzed to demonstrate the efficiency of the X-FEM technique in large plasticity deformations. [Copyright &y& Elsevier]

In this contribution we propose an innovative approach to automatically refine shell finite element meshes in thickness direction. The procedure employs a new eight-node solid-shell element formulation based on the concept of reduced integration with hourglass stabilization. Apart from the important advantages of computational efficiency and robustness this element formulation offers the possibility to work with a variable mesh density in thickness direction. In the present paper we suggest an adaptive approach using the hanging node concept to avoid additional mesh distortion. Among different stress- and strain-based error criteria the warping of the cross-section and the residual error in the solution of the strong form of the balance of linear momentum have been proven to be suitable error measures. The former represents the deviation of the realistically non-linear dependence of the characteristic strain coefficient from its corresponding linear distribution. Applications show that the mesh refines, as expected, in loading domains, in the neighbourhood of bearings or regions with strongly accumulating inelastic deformation. On the other hand, the discretization remains rather coarse in uncritical parts of the structure which decreases the computational effort noticeably. [Copyright &y& Elsevier]

Abstract: In this paper, the contact between an elastic–viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to tension or compression, is also considered. The variational formulation leads to a coupled system of evolutionary equations for which an existence and uniqueness result is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained and, as a consequence, the convergence of the algorithm is deduced. Finally, some numerical simulations are presented to demonstrate the accuracy of the algorithm. [Copyright &y& Elsevier]

The geometric nonlinear buckling problem of a thin doubly curved shallow shell with all edges hinged is complicated and difficult to obtain an exact analytical solution. Thus, differential equations are solved incrementally by using the differential quadrature method in this paper. Detailed formulations are worked out. Convergence study is performed. Several examples with various material properties, curvatures and dimensions are investigated. Comparisons are made with existing semi-analytical data or finite element data. It is shown that the critical loads are lower than the data obtained by either so-called Adjacent Equilibrium Method or Partitioned Solution Method. The possible reasons to cause the difference are discussed. [Copyright &y& Elsevier]

Abstract: The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko–Mindlin-type (TM) shell elements has been considered in Part I of the present work. In Part II it has been developed an enhanced mixed finite element formulation that allows using load increments that are much larger than possible with existing geometrically exact displacement-based shell element formulations. In this paper the developed formulation is employed to solve frictionless contact problems for TM shells undergoing finite deformations and interacting with rigid bodies. The contact conditions are incorporated into the assumed stress–strain TM shell formulation by applying a perturbed Lagrangian procedure with the fundamental unknowns consisting of 6 displacements and 11 strains of the bottom and top surfaces of the shell, 11 conjugate stress resultants and the Lagrange multiplier, associated with a nodal contact force, through using the non-conventional technique. The efficiency and accuracy of the proposed finite element formulation are demonstrated by means of several numerical examples. [Copyright &y& Elsevier]

Abstract: The natural finite element approach introduced by Argyris in the early sixties is marked essentially by the distinction between rigid body motion and deformation, on the one hand, and by the description of the latter in compliance with the element purpose and geometry, on the other hand. For triangular and tetrahedral elements the approach suggests strain and stress measures defined along the sides, respectively the edges as homogeneous normal quantities, free of shear. In the mechanics of continua the respective infinitesimal elements represent minimum configurations to define local deformation in two- and three dimensions. The paper presents developments of the natural approach on the continuum level within a consistent theoretical framework. It is proposed to begin with a reference system of supernumerary coordinates associated with the elementary tetrahedron in the space and with the triangle in the plane. Vectorial quantities are defined, the operations of gradient and divergence are interpreted in this system. The natural deformation rate is deduced from the velocity field, the stress is introduced as work conjugate measure. The condition for local equilibrium is presented in natural quantities as well as the stress definition in association with the resultant forces. The set up of material constitutive relations is exemplified for the viscous solid. Beyond the description of the momentary kinematics as from the velocity field, the appearance of finite deformation is considered basing on displacements. Illustration of the methodology for a plane elastic case is appended. [Copyright &y& Elsevier]