Kirchhoff Material Research Articles

A Hidden Convexity of Nonlinear Elasticity

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive ‘Hadamard instability;’ this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response.

Flow-induced reconfiguration and cross-flow vibrations of an elastic plate and implications to energy harvesting

We numerically study flow-induced vibrations (FIV) of an elastic cantilevered plate in a cross-flow configuration. The plate is kept upright in an open domain with a fixed bottom end. Such a configuration eliminates energy losses in the wall boundary. Due to allowed vortex shedding from its bottom end, the resulting novel dynamics of the fluid-structure interaction (FSI) system is explored. An in-house sharp-interface immersed boundary method-based flow solver, coupled with a Saint-Venant Kirchhoff material model-based open-source finite element solver, is employed. Two sets of simulations were carried out in two-dimensional Cartesian coordinates at Reynolds number of 100 for a wide range of reduced velocity (UR), by performing a parametric sweep of dimensionless Young’s modulus (E) and mass ratio (M). In general, FIV characteristics of the plate are similar to the transverse vortex-induced vibration of an elastically-mounted cylinder. The plate deforms to a curved mean position depending on E and M and exhibits angular oscillations. In particular, the following FIV regimes are obtained with increasing UR: initial excitation, lock-in with the first mode, desynchronization after the first lock-in, lock-in with the second mode, and desynchronization after the second lock-in. During lock-in, the plate oscillation and vortex shedding frequency are closer to the first or second mode natural frequency of plate. A larger angular amplitude and increased energy exchange between the structure and fluid characterize the lock-in. We quantify the modal contributions of Euler-Bernoulli modes in the plate’s vibration response with a curved mean position. The spatiotemporal variation of energy exchange between the fluid and plate is analyzed to understand the observed plate dynamics. The flow physics of the wake is discussed using wake patterns, vortex formation length, and vortex strength. Drag over the plate reduces due to flow-induced reconfiguration and strongly correlates with these parameters. Lastly, based on the spatial distribution of strain energy density over the plate, we present a potential design for an ambient fluid energy harvester to harness the plate’s vibration energy with optimized piezoelectric patching. The optimum location and height of a piezoelectric patch on the plate for different cases are reported.

Inferring displacement fields from sparse measurements using the statistical finite element method

Nowadays, strain and displacement can be measured using techniques such as electronic speckle pattern interferometry and digital image correlation. However, usually, only some part of the domain of interest is accessible for measurement devices. In order to assess also inaccessible areas of the structure, a fundamental task for structural health monitoring is the inference of full-field displacements from sparse measurements. Another important task is the inference of mechanical stress from displacement or strain data. In computational mechanics, the link between such data and stress is established via constitutive models.A well-established approach for inferring full displacement and stress fields from possibly sparse data is to calibrate the parameter of a given constitutive model using a Bayesian update. After calibration, a (stochastic) forward simulation is conducted with the identified model parameters to resolve physical fields in regions that were not accessible to the measurement device. A shortcoming of model calibration is that the model is deemed to best represent reality, which is only sometimes the case, especially in the context of the aging of structures and materials. While this issue is often addressed with repeated model calibration, a different approach is followed in the recently proposed statistical Finite Element Method (statFEM). Instead of using Bayes’ theorem to update model parameters, the displacement is chosen as the stochastic prior and updated to fit the measurement data more closely. For this purpose, the statFEM framework introduces a so-called model-reality mismatch, parametrized by only three hyperparameters. This makes the inference of full-field data computationally efficient in an online stage: If the stochastic prior can be computed offline, solving the underlying partial differential equation (PDE) online is unnecessary. Compared to solving a PDE, identifying only three hyperparameters and conditioning the state on the sensor data requires much fewer computational resources. Computational efficiency is an essential requirement for online applications. It is noted that conditioning a model-based prior field on sensor data represents a variant of physics-based regression.This paper presents two contributions to the existing statFEM approach: First, we use a non-intrusive polynomial chaos method to compute the prior, enabling the use of complex mechanical models in deterministic formulations. Second, we examine the influence of prior material models (linear elastic and St.Venant Kirchhoff material with uncertain Young’s modulus) on the updated solution. We present statFEM results for 1D and 2D examples, while an extension to 3D is straightforward.

Limitations of the St. Venant–Kirchhoff material model in large strain regimes

The St. Venant–Kirchhoff law is a widely used constitutive relation in engineering applications. However, its limitation to small strains is frequently mentioned in the literature but rarely elaborated on. Therefore, we present a comprehensive study discussing the St. Venant–Kirchhoff law and its shortcomings in large stain regimes, focusing on compressive loading based on one-dimensional stress and strain states. The linear stress–strain relation in combination with the associated Green–Lagrange strain yields a nonphysical structural behavior for large compressive strains, which is reflected in instabilities and nonphysical softening. Common misconceptions concerning the St. Venant–Kirchhoff law and alternative constitutive models are presented and evaluated based on a two-bar snap-through problem.

Accuracy of Föppl–von Karman membrane theory for determining elastic constants of monolayer graphene

For interpreting results of an indentation test on a single-layer graphene sheet (SLGS), it is frequently modeled as an elastic, isotropic, and homogeneous membrane. With a small thickness (∼10−10 m) as compared to a characteristic length (∼10−6 m) of specimens used in experiments, the non-dimensional von-Karman factor (vKf ∼ 108) for a SLGS predicts negligible flexural rigidity (D ∼ 10−19 Nm). Since the membrane undergoes large deformations under a small force, a direct relationship between the resulting force-displacement curves and elasticities of the graphene sheet is still elusive. Here we compare predicted load-displacement curves for nanoindentation of a SLGS by using a full non-linear theory with St. Venant–Kirchhoff material, and the Föppl–von Karman nonlinear equations with a Hookean material. The nonlinear governing equations in each formulation are numerically integrated by the shooting method. Even though the radial variations of the membrane deflections computed with the two theories differ by at most 1.5%, the stresses, the declination angles, the curvatures and the strain energy densities may reach differences as high as 6.4%, 7.4%, 70% and 25%, respectively. These differences are significant enough to reexamine using the FvK derived solutions to find elastic constants from the test data.

OpenFPCI: A parallel fluid–structure interaction framework

This paper presents OpenFPCI, a framework for coupling the C++ toolbox OpenFOAM-Extend, a computational fluid dynamics package, with the general purpose finite element package ParaFEM, written in Fortran and used to solve structural mechanics problems. The coupling of these two open source and scalable toolboxes, facilitates the use of high performance computing resources for the solution of fluid–structure interaction problems. The framework uses a master–slave approach, with OpenFOAM-Extend acting as the master and calling OpenFPCI plugins. The plugins are composed of a series of subroutines used to initialise and solve a specific engineering problem and make use of ParaFEM’s highly parallel implementation. The plugins are wrapped by C constructs such that OpenFOAM-Extend can call these Fortran subroutines consistently and when the solution from ParaFEM is required. Each plugin solves a different solid mechanics problem, with the current features including the deformation of a linear-elastic structure undergoing small strain and the deformation of a St. Venant–Kirchhoff material. Throughout this paper the focus will lie on the large strain plugin, considering the implementation and its validation for a benchmark problem, along with assessment of parallel capabilities, which are shown to scale to three thousand cores. This paper will be of interest to OpenFOAM and ParaFEM practitioners looking to utilise multiphysics simulations for their research, along with researchers looking to integrate fluid–structure interaction into their studies. Program summaryProgram Title:OpenFPCIProgram Files doi:http://dx.doi.org/10.17632/ntprzxk477.1Licensing provisions: BSD 2-ClauseProgramming language: Fortran, C and C++External libraries:OpenFOAM and ParaFEMSupplementary material: Example test cases are available within the OpenFPCI repository.Nature of problem: OpenFPCI was developed to solve computationally expensive fluid structure interaction problems by running on high performance computing systems. The framework was designed to enable the coupling of advanced ParaFEM capabilities to OpenFOAM-Extend.Solution method: OpenFOAM-Extend uses the classic Arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations to deal with moving boundaries. The moving boundary is defined by using an OpenFPCI plugin, using ParaFEM’s libraries, to solve the deformation of the adjoining structure.

Dynamic simulation of machining composites using the explicit element-free Galerkin method

Machining operations are performed on composite parts to obtain the final geometry. However, machining composites is challenging due to their low machinability and high cost. Numerical modelling of machining presents a valuable tool for cost reduction and a better understanding of the cutting process. Meshfree methods are an attractive choice to model machining problems due to their capability in modelling large deformations. This work presents an explicit meshfree model for orthogonal cutting of unidirectional composites based on the element-free Galerkin (EFG) Method. Advantages of the proposed model include: simple and automated preprocessing, advanced material modelling and ability to model high-speed machining. Workpiece material is modelled as orthotropic Kirchhoff material with a choice of three failure criteria: maximum stress, Hashin and LaRC02. Frictional contact calculations are performed based on central differencing, therefore avoiding the use of penalty parameters. Validation of the EFG model is conducted by comparing cutting forces against orthogonal cutting experiments on GFRP samples using a vertical milling machine. It is found that while the numerical cutting forces are in good agreement with experimental ones, the numerical thrust forces are significantly under-estimated. Analysis of failure showed that chip is formed along the fibre direction in the studied range.

Higher-order beam elements based on the absolute nodal coordinate formulation for three-dimensional elasticity

This study thoroughly examines various higher-order three and four-node beam elements for use in the absolute nodal coordinate formulation (ANCF). The paper carefully investigates which potential benefits and drawbacks the utilization of higher-order ANCF beam elements without in-slope vectors has in the case of the usage of full three-dimensional elasticity. When the elastic forces for shear-deformable ANCF beam elements are calculated using full three-dimensional elasticity—especially in the form of the St. Venant–Kirchhoff material law—Poisson locking severely deteriorates the accuracy of the numeric results. As shown in this paper, an existing approach to preventing this locking phenomenon for three-node beam elements can still produce unsatisfying results in load cases involving bidirectional bending. The results of this study show that enriching the polynomial basis used to approximate the beam kinematics provides a natural solution to this issue. As will be seen, these findings for three-node elements can also be extended to four-node elements. When using a sufficient approximation order in transverse directions, satisfying accuracy can be achieved both in conventional one-dimensional bending and in the above-mentioned bidirectional load case.

Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green–Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant–Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied.

A parallel monolithic algorithm for the numerical simulation of large‐scale fluid structure interaction problems

SummaryA novel parallel monolithic algorithm has been developed for the numerical simulation of large‐scale fluid structure interaction problems. The governing incompressible Navier–Stokes equations for the fluid domain are discretized using the arbitrary Lagrangian–Eulerian formulation‐based side‐centered unstructured finite volume method. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant–Kirchhoff material, and the classical Galerkin finite element method is used to discretize the governing equations in a Lagrangian frame. A special attention is given to construct an algorithm with exact total fluid volume conservation while obeying both the global and the local discrete geometric conservation law. The resulting large‐scale algebraic nonlinear equations are multiplied with an upper triangular right preconditioner that results in a scaled discrete Laplacian instead of a zero block in the original system. Then, a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned sub‐domains is utilized for the modified system. The accuracy and performance of the proposed algorithm are verified for the several benchmark problems including a pressure pulse in a flexible circular tube, a flag interacting with an incompressible viscous flow, and so on. John Wiley & Sons, Ltd.

RETRACTED: Analytic solutions to three-dimensional finite deformation problems governed by St Venant–Kirchhoff material

The following article has been included in a multiple retraction: Analytic Solutions to three-dimensional Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract Global Optimization Solutions to a Class of Non-convex Quadratic Minimization Problems with Quadratic Constraints; Yubo Yuan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591086.abstract On Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao. http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstract The Editor-in-Chief and SAGE strive to uphold the very highest standards of publication ethics and are committed to supporting the high standards of integrity of Mathematics & Mechanics of Solids. Authors, reviewers, editors and interested readers are encouraged to consult SAGE’s ethics statements and the Committee on Publication Ethics (COPE) website for guidelines on publication ethics.

Joint Segmentation/Registration Model by Shape Alignment via Weighted Total Variation Minimization and Nonlinear Elasticity

This paper falls within the scope of joint segmentation-registration using nonlinear elasticity principles. Because Saint Venant--Kirchhoff materials are the simplest hyperelastic materials (hyperelasticity being a suitable framework when dealing with large and nonlinear deformations), we propose viewing the shapes to be matched as such materials. Then we introduce a variational model combining a measure of dissimilarity based on weighted total variation and a regularizer based on the stored energy function of a Saint Venant--Kirchhoff material. Adding a weighted total variation--based criterion enables us to align the edges of the objects even when the modalities are different. We derive a relaxed problem associated to the initial one for which we are able to provide a result of existence of minimizers. A description and analysis of a numerical method of resolution based on a decoupling principle is then provided including a theoretical result of $\Gamma$-convergence. Applications are illustrated in academic and biological images.

An orthotropic material model for steel fibre reinforced concrete based on the orientation distribution of fibres

In the present paper the authors focus on constitutive mappings for steel fibre reinforced concrete, SFRC. The anisotropic properties of this composite are caused by the orientation distribution of fibres. The constitutive relation is developed for one meso-volume element of SFRC as a combination of isotropic and orthotropic St. Venant–Kirchhoff material models, which are applied to concrete matrix and to steel fibres, respectively. The alignment tensors and orientation distribution function adopted from the mesoscopic continuum theory are utilised to identify the material meso-symmetry axes and to asses the contribution of fibres in the symmetry axes defined. While assessing the orthotropic meso-elasticity for fibres, the elasticity of an individual fibre in its local coordinates is transformed into the material meso-symmetry axes and weighted with the orientation distribution function of fibres. The advantage of the material model developed for SFRC is that it uses complete orientation information of fibres (two angles in spherical coordinates) and utilises tensor quantities complying with material objectivity.

A Relaxed Problem of Registration Based on the Saint Venant--Kirchhoff Material Stored Energy for the Mapping of Mouse Brain Gene Expression Data to a Neuroanatomical Mouse Atlas

In this paper, we address the issue of designing a theoretically well-motivated registration model capable of handling large deformations. Motivated by the fine properties of the Saint Venant--Kirchhoff material, we propose a variational model combining a measure of dissimilarity and a regularizer based on the stored energy of such a material. We prove the existence of generalized solutions of this problem. We then describe and analyze a numerical method of resolution based on the introduction of an associated decoupled problem in which an auxiliary variable simulates the Jacobian matrix of the deformation field. A theoretical result is established, and we investigate the efficiency of the proposed matching model for the registration of mouse brain gene expression data to a neuroanatomical mouse atlas.

Finite deformations of curved laminated St. Venant–Kirchhoff beam using layer-wise third order shear and normal deformable beam theory (TSNDT)

A layer-wise third order shear and normal deformable plate/shell theory (TSNDT) incorporating all geometric nonlinearities is used to study finite transient deformations of a curved laminated beam composed of a St. Venant–Kirchhoff material. In the TSNDT all displacement components of a point are expressed as 3rd order polynomials in the thickness coordinate in each layer while maintaining the displacement continuity across adjoining layers. No shear correction factor is used. Transverse shear and transverse normal stresses are found from the computed displacement fields and the constitutive relation (i.e., no stress recovery technique is employed). For the St. Venant–Kirchhoff material the strain energy density is a quadratic function of the Green-St. Venant strain tensor appropriate for finite deformations. The software based on the finite element method (FEM) capable of solving static and transient nonlinear problems has been verified by using the method of manufactured solutions. Furthermore, results computed with the TSNDT have been found to agree well with those obtained using the commercial software ABAQUS, and C3D20 elements. Significant contributions of the work include developing a TSNDT considering all geometric nonlinearities and a materially objective constitutive relation, using the method of manufactured solutions to verify the numerical solution of transient nonlinear problems, and showing that results from the plate theory agree well with those from the analysis of plane strain nonlinear problems using the finite elasticity theory. Plate problems using the TSNDT can be analyzed with piecewise linear basis functions in the FEM.

Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Static and Linearized Dynamic Examples

In the present paper, a three-dimensional shear deformable beam finite element is presented, which is based on the absolute nodal coordinate formulation (ANCF). The orientation of the beam’s cross section is parameterized by means of slope vectors. Both a structural mechanics based formulation of the elastic forces based on Reissner’s nonlinear rod theory, as well as a continuum mechanics based formulation for a St. Venant Kirchhoff material are presented in this paper. The performance of the proposed finite beam element is investigated by the analysis of several static and linearized dynamic problems. A comparison to results provided in the literature, to analytical solutions, and to the solution found by commercial finite element software shows high accuracy and high order of convergence, and therefore the present element has high potential for geometrically nonlinear problems.

Large deformation and stability of an extensible elastica with an unknown length

A beam resting on spatially fixed supports may slide relatively to these as soon as external forces are applied. Consequently, the length of the portion of the reference configuration, which is currently located between the supports, depends on the loading and therefore is not known in advance. In the present paper, the problem of a slender beam under a uniformly distributed force is investigated, which is clamped at one side but may slide through another clamping device in axial direction at the opposite side. In combination with a suitable coordinate transformation by which the numerical treatment is simplified, a finite element approach is utilized to determine the equilibrium shape for the maximum critical load that can be imposed on the beam. In the course of this, the influence of the extensibility of the beam axis is studied. A theory based on Reissner’s geometrically exact relations for the plane deformation of beams is adapted such that it allows constitutive relations on stress–strain level to be integrated consistently. In addition to the classical equations of the extensible elastica, a constitutive model derived from the St. Venant–Kirchhoff material of non-linear continuum mechanics is studied. The results obtained in this survey are finally compared to those from the linear beam theory, which turns out to be incapable of describing the problem under consideration in a satisfactory manner.

Steady Flow of a Navier–Stokes Liquid Past an Elastic Body

We perform a mathematical analysis of the steady flow of a viscous liquid, $${\mathcal{L}}$$ , past a three-dimensional elastic body, $${\mathcal{B}}$$ . We assume that $${\mathcal{L}}$$ fills the whole space exterior to $${\mathcal{B}}$$ , and that its motion is governed by the Navier–Stokes equations corresponding to non-zero velocity at infinity, v ∞. As for $${\mathcal{B}}$$ , we suppose that it is a St. Venant–Kirchhoff material, held in equilibrium either by keeping an interior portion of it attached to a rigid body or by means of appropriate control body force and surface traction. We treat the problem as a coupled steady state fluid-structure problem with the surface of $${\mathcal{B}}$$ as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently small |v ∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier–Stokes equation in the reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and to the elasticity equations.

Algorithms by Design: Part III—A Novel Normalized Time Weighted Residual Methodology and Design of Optimal Symplectic-Momentum Based Controllable Numerical Dissipative Algorithms for Nonlinear Structural Dynamics

While in Part I (see [1]) and Part II (see [2]) of this three-part exposition we focused attention on non-dissipative symplectic-momentum conserving designs of time operators for applicability to nonlinear dynamics, here in Part III of this exposition we demonstrate how to further advance the theoretical developments and introduce controllable numerical dissipation via a novel time weighted residual approach we have previously described. The unique aspects of the algorithmic designs are such that when the numerical dissipative features are turned off, the resulting time operators readily recover the original designs of algorithms that are inherently symplectic-momentum conserving (we defer to those that are energy-momentum conserving elsewhere [3, 4]). The focus of the current work is on the theoretical developments of the new formulation termed the displacement based normalized time weighted residual approach for nonlinear dynamics applications. In particular, we provide extensions of the well—known Generalized Single Solve Single Step (GSSSS) framework, which comprises two distinct classifications, namely constrained U and V algorithmic architectures, and was originally developed for solving linear dynamic problems. Starting with the generalization of the classical time weighted residual approach, the GSSSS framework that was previously developed encompasses the general class of LMS methods represented by the single field form of the second order ordinary differential equations in time involving a single solve, and also covers most of the developments to date in the literature, including providing new avenues towards optimal designs of algorithms. However, the classical time weighted residual approach fails to adequately provide proper extensions to nonlinear dynamics applications. Consequently, the basic premise and argument that is herein advanced is that controllable numerical dissipative time operators designed for linear dynamic problems are very valuable, and are indeed the basis and can be readily employed as the basic parent algorithms, such that when implemented appropriately via the present new time weighted residual representation, they are now readily suitable for extensions to nonlinear dynamics applications. To demonstrate the basic concepts, we consider applications to the Saint Venant Kirchhoff material model simply for illustration, although the method can be readily extended to general material models. The numerical examples presented show that in contrast to the classical time weighted residual approach, which fails to recover the original designs of symplectic-momentum conservation when numerical dissipation is turned off, this new approach readily accomplishes this feature naturally and without enforcing any added constraints, and is the more appropriate way to design a particular class of controllable numerical dissipative schemes. It also leads to algorithm designs that yield fewer numerical oscillations in the energy and angular momentum in contrast to the classical approach, thus additionally confirming the improved effectiveness of the proposed approach. Further, we also show that amongst all the controllable numerical dissipative schemes considered in the sense of and under the framework of LMS methods in the single field form and involving a single solve and second-order time accuracy, the U0− V0 optimal is the preferred choice of this particular class of symplectic-momentum conserving based controllable numerical dissipative schemes since: (i) it yields least amount of energy dissipation; and (ii) it is ideal for any given set of initial conditions in the sense that it possesses the highly desirable attributes involving zero order displacement and velocity overshooting behavior. Simple numerical examples are provided that illustrate the fundamental ideas for applications to nonlinear dynamics problems.

DECOMPOSITION OF DEFORMATIONS OF THIN RODS: APPLICATION TO NONLINEAR ELASTICITY

This paper deals with the introduction of a decomposition of the deformations of curved thin beams, with section of order δ, which takes into account the specific geometry of such beams. A deformation v is split into an elementary deformation and a warping. The elementary deformation is the analog of a Bernoulli–Navier's displacement for linearized deformations replacing the infinitesimal rotation by a rotation in SO(3) in each cross section of the rod. Each part of the decomposition is estimated with respect to the L2norm of the distance from gradient v to SO(3). This result relies on revisiting the rigidity theorem of Friesecke–James–Müller in which we estimate the constant for a bounded open set star-shaped with respect to a ball. Then we use the decomposition of the deformations to derive a few types of asymptotic geometrical behavior: large deformations of extensional type, inextensional deformations and linearized deformations. To illustrate the use of our decomposition in nonlinear elasticity, we consider a St Venant–Kirchhoff material and upon various scalings on the applied forces we obtain the Γ-limit of the rescaled elastic energy. We first analyze the case of bending forces of order δ2which leads to a nonlinear extensible model. Smaller pure bending forces give the classical linearized model. A coupled extentional-bending model is obtained for a class of forces of order δ2in traction and of order δ3in bending.