Material Frame Research Articles

Finite Particle Method for Progressive Failure Simulation of Truss Structures

A structural analysis framework called the finite particle method (FPM) for structure failure simulation is presented in this paper. The traditional finite-element method is generated from continuum mechanics and the variational principle; vector mechanics form the basis of FPM. It discretizes the domain with finite particles whose motions are described by Newton’s second law. Instead of imposing a global equilibrium of the entire continuous system, FPM enforces equilibrium on each particle. Thus, particles are free to separate from one another, which is advantageous in the simulation of structural failure. One of the features of this approach is that no iterations to follow nonlinear laws are necessary, and no global matrices are formed or solved in this method. A convected material frame is used to evaluate the structure deformation and internal force. The explicit time integration is adopted to solve the equation of motion. To simulate the truss structure failure, a failure criterion on the basis of th...

A physical interpretation of softening of pressure-sensitive and anisotropic materials

Several new dynamic models are proposed to explain the mechanical behaviour of softening of pressure-sensitive and anisotropic materials at a macroscopic level. If a pressure-sensitive material is loaded by a force and a variable pressure or an anisotropic material is subjected to a load with a changeable loading direction relative to the material frame, their stress–strain relationships become more complicated. Mechanical behaviours of these stress–strain relationships have to cover the feature concerning the change of pressure or loading direction, i.e. mechanical properties of pressure-sensitive material corresponding to different pressure state or anisotropic material relating to different loading direction will play an important role in deciding their stress–strain relationships. Such shift of material properties due to the variable pressure or loading history may significantly expand the traditional concept of the stability of material deformation, and the second order of plastic work being negative may be a response of stable plastic deformation, which is commonly called softening.

Entropy, entropy flux and entropy rate of granular materials

The aim of this work is to analyze the entropy, entropy flux and entropy rate of granular materials within the frameworks of the Boltzmann equation and continuum thermodynamics. It is shown that the entropy inequality for a granular gas that follows from the Boltzmann equation differs from the one of a simple fluid due to the presence of a term which can be identified as the entropy density rate. From the knowledge of a non-equilibrium distribution function–valid for processes closed to equilibrium–it is obtained that the entropy density rate is proportional to the internal energy density rate divided by the temperature, while the entropy flux is equal to the heat flux vector divided by the temperature. A thermodynamic theory of a granular material is also developed whose objective is the determination of the basic fields of mass density, momentum density and internal energy density. The constitutive laws are restricted by the principle of material frame indifference and by the entropy principle. Through the exploitation of the entropy principle with Lagrange multipliers, it is shown that the results obtained from the kinetic theory for granular gases concerning the entropy density rate and entropy flux are valid in general for processes close to equilibrium of granular materials, where linearized constitutive equations hold.

Prediction of flange wrinkling in deep drawing process using bifurcation criterion

Surface distortions in the form of wrinkles are often observed in sheet metals during stamping and other forming operations. Because of the trend in recent years towards thinner, higher-strength sheet metals, wrinkling is increasingly becoming a more common and troublesome mode of failure in sheet metal forming. The prediction and prevention of wrinkling during a sheet forming process are important issues for the design of part geometry and processing parameters. This paper treats the phenomenon of flange wrinkling as a bifurcated solution of the equations governing the deep drawing problem when the flat position of the flange becomes unstable. Hill’s bifurcation criterion is used to predict the onset of flange wrinkling in circular and square cup drawing. In particular, the maximum cup height that can be drawn without the onset of flange wrinkling is predicted for the given set of process parameters. A parametric study of the maximum cup height is also carried out with respect to various geometric, material and process parameters. Finite element formulation, based on the updated Lagrangian approach, is employed for the analysis. The incremental logarithmic strain measure, which allows the use of a large incremental deformation, is used. The stresses are updated in a material frame. The material is assumed to be elastic–plastic, strain hardening, yielding according to an anisotropic yield criterion of Barlat et al. (2005) [23] (named as Yld2004-18p). Isotropic power law hardening is assumed. Inertia forces are neglected due to small accelerations. Modified Newton–Raphson iterative technique is used to solve the nonlinear incremental equations.

El prólogo de Amadís (1508) y las Estorias de Troya. Transferencias

This article analyses Montalvo’s Amadís “compositio” on the frame of Trojan material, its literary tradition and its generic context in the Castilian and European Middle Ages. The article puts the question, always through the texts, about how Montalvo assumes this tradition and how he revitalized it for his readers. The matière antique and its formulation in the Middle Ages provide to Montalvo a poetic plot where to fixe his Amadís as a crossroads between fiction (fabula) and history-story (historia-estoria). In this way Montalvo scapes to the moral limitations and censures against chivalric fiction, parallels to the fiction development.

A material frame approach for evaluating continuum variables in atomistic simulations

We present a material frame formulation analogous to the spatial frame formulation developed by Hardy, whereby expressions for continuum mechanical variables such as stress and heat flux are derived from atomic-scale quantities intrinsic to molecular simulation. This formulation is ideally suited for developing an atomistic-to-continuum correspondence for solid mechanics problems. We derive expressions for the first Piola–Kirchhoff (P–K) stress tensor and the material frame heat flux vector directly from the momentum and energy balances using localization functions in a reference configuration. The resulting P–K stress tensor, unlike the Cauchy expression, has no explicit kinetic contribution. The referential heat flux vector likewise lacks the kinetic contribution appearing in its spatial frame counterpart. Using a proof for a special case and molecular dynamics simulations, we show that our P–K stress expression nonetheless represents a full measure of stress that is consistent with both the system virial and the Cauchy stress expression developed by Hardy. We also present an expanded formulation to define continuum variables from micromorphic continuum theory, which is suitable for the analysis of materials represented by directional bonding at the atomic scale.

Nonequilibrium molecular dynamics for bulk materials and nanostructures

We describe a method of constructing exact solutions of the equations of molecular dynamics in non-equilibrium settings. These solutions correspond to some viscometric flows, and to certain analogs of viscometric flows for fibers and membranes that have one or more dimensions of atomic scale. This work generalizes the method of objective molecular dynamics (OMD) ( Dumitrică and James, 2007). It allows us to calculate viscometric properties from a molecular-level simulation in the absence of a constitutive equation, and to relate viscometric properties directly to molecular properties. The form of the solutions is partly independent of the form of the force laws between atoms, and therefore these solutions have implications for coarse-grained theories. We show that there is an exact reduction of the Boltzmann equation corresponding to one family of OMD solutions. This reduction includes most known exact solutions of the equations of the moments for special kinds of molecules and gives the form of the molecular density function corresponding to such flows. This and other consequences leads us to propose an addition to the principle of material frame indifference, a cornerstone of nonlinear continuum mechanics. The method is applied to the failure of carbon nanotubes at an imposed strain rate, using the Tersoff potential for carbon. A large set of simulations with various strain rates, initial conditions and two choices of fundamental domain (unit cell) give the following unexpected results: Stone–Wales defects play no role in the failure (though Stone–Wales partials are sometimes seen just prior to failure), a variety of failure mechanisms is observed, and most simulations give a strain at failure of 15–20%, except those done with initial temperature above about 1200 K and at the lower strain rates. The latter have a strain at failure of 1–2%.

Entgrenzte Zeitregime und wirtschaftsräumliche Konzentrationen

Disenclosed time regimes and spatial concentrations of economies. The creative sector in Hamburg’s Schanzenviertel from a time-geographical perspective. In recent years the spatial and temporal organization of everyday life has been transformed by the flexibilisation of economic relations and a growing variety of the ways how individuals coordinate their professional and private spheres and activities. Central elements of this transformation are (1) disenclosures of fixed socio-structural, temporal and spatial boundaries, (2) technical, social and cultural accelerations and (3) a growing subject-orientation within work and life conditions. Cities offer places where these transformations are well observable. These places provide both diverse infrastructures and images as material and symbolic frames for local socioeconomic processes: multiple encounters, mutual learning processes and work-life-combinations. The analysis of Hamburg’s Schanzenviertel shows the explanatory power of a broadened time geographical concept in elucidating those new socioeconomic spaces. We add the concept of coupling opportunities to the traditional constraints-approach in order to show why “the urban” under the label of the creative city should be understood as a specific form of organizing the everyday as well as the work life. Interrelations of subjective and structural conditions clarify the subject-orientation within spatial economies as both system-related constraints and chances for individual self-determination. This analysis opens up a critical perspective on the selectivity and spatial inequalities within the creative city debate.

Constitutive theory of anisotropic rigid heat conductors

For rigid heat conductors, the constitutive equation of the heat flux is usually assumed to be independent of the deformation of the body, or equivalently by setting the deformation gradient as the identity tensor in the constitutive equations. However, the notion of material symmetry would have involved change in reference configurations, hence also change in deformation gradients. Consequently it would be impossible to consider anisotropic rigid heat conductors from such a formulation. In this paper, rigid heat conductors are treated with rigidity as internal constraint and a constitutive theory consistent with the principle of material frame indifference, material symmetry, and entropy principle is presented. Constitutive restrictions for a transversely isotropic rigid heat conductor are carefully analyzed.

Prediction of momentum and scalar transport in turbulent swirling flows with an objective Reynolds-stress transport closure

The accurate prediction of turbulent swirling flows requires the use of a differential Reynolds-stress transport model to close the time-averaged Navier–Stokes equations. The performance of such model is largely determined by the way in which the fluctuating pressure–strain correlations are approximated. A number of alternative approximations are available, all of which depend explicitly on the mean vorticity tensor. Such dependence renders a constitutive relation inconsistent with the principle of Material Frame Indifference (MFI). In this paper, an objective model (i.e. one which is consistent with MFI) for the pressure–strain correlations is presented. This model, which was developed using Tensor Representation Theory, has fewer terms than the conventional alternatives and is therefore easier to implement in computational codes. Moreover, the model was calibrated to correctly reproduce the relative stress levels in both free and wall-bounded flows without the need to employ wall-damping corrections. The performance of this model is assessed using experimental data from both weakly- and strongly-swirled jets. Comparisons are also made with results obtained using three widely-used alternative models for the pressure–strain correlations. It is found that the objective model, although simpler in formulation than the others, yields results that are generally in closer correspondence with the data. The paper also reports on the prediction of mass transfer in a swirling jet. The case considered was that of a co-axial, strongly-swirled flow with an outer annular air stream and an inner helium jet. Swirl was imparted to the outer stream only. The concentration of helium was predicted using a differential scalar-flux transport closure. Close agreement was obtained with the measured concentrations. Analysis of the predicted mass fluxes revealed that the turbulent diffusivity is strongly anisotropic in this flow.

Finite particle method for kinematically indeterminate bar assemblies

This study presents a structural analysis algorithm called the finite particle method (FPM) for kinematically indeterminate bar assemblies. Different from the traditional analysis method, FPM is based on the combination of the vector mechanics and numerical calculations. It models the analyzed domain composed of finite particles. Newton’s second law is adopted to describe the motions of all particles. A convected material frame and explicit time integration for the solution procedure is also adopted in this method. By using the FPM, there is no need to solve any nonlinear equations, to calculate the stiffness matrix or equilibrium matrix, which is very helpful in the analysis of kinematically indeterminate structures. The basic formulations for the space bar are derived, following its solution procedures for bar assemblies. Three numerical examples are analyzed using the FPM. Results obtained from both the straight pretension cable and the suspension cable assembly show that the FPM can produce a more accurate analysis result. The motion simulation of the four-bar space assembly demonstrates the capability of this method in the analysis of kinematically indeterminate structures.

A frame-invariant formulation of Fung elasticity

Fung elasticity refers to the hyperelasticity constitutive relation proposed by Fung and co-workers for describing the pseudo-elastic behavior of biological soft tissues undergoing finite deformation. A frame-invariant formulation of Fung elasticity is provided for material symmetries ranging from orthotropy to isotropy, which uses Lamé-like material constants. In the orthotropic case, three orthonormal vectors are used to define mutually orthogonal planes of symmetry and associated texture tensors. The strain energy density is then formulated as an isotropic function of the Lagrangian strain and texture tensors. The cases of isotropy and transverse isotropy are derived from the orthotropic case. Formulations are provided for both material and spatial frames. These formulations are suitable for implementation into finite element codes. It is also shown that the strain energy function can be naturally uncoupled into a dilatational and a distortional part, to facilitate the computational implementation of incompressibility.

Finite element simulation of earing defect in deep drawing

Deep drawing is an extensively used press working process since it eliminates expensive machining and welding operations and enables the production of components at a very high rate. The workpiece material used in a deep drawing process is anisotropic in nature, due to a prior thermomechanical treatment. Earing is one of the major defects observed in a deep drawing process due to the anisotropic nature of the sheet material. Knowledge about the ear formation in deep drawing allows a prior modification of the process, which can result in a defect-free final product with financial savings. In this paper, a recently proposed anisotropic yield criteria by Barlat et al. for rolled sheets is used to model the anisotropy for simulating the earing defect in square and circular cup drawing processes. The effect of the tooling geometry and process parameters on the ear formation is studied. It is shown that, in the square cup, the uneven metal flow rate, rather than the material anisotropy, is mainly responsible for the flange earing. Finite element formulation, based on the updated Lagrangian approach, is employed for the analysis. The stresses are updated in a material frame and the logarithmic strain measure is used, which allows the use of a large increment size. Isotropic hardening is assumed, and it follows a power law. Inertia forces are neglected due to small accelerations. The modified Newton–Raphson iterative technique is used to solve the nonlinear incremental equations.

From Non-Linear Elasticity to Linearized Theory: Examples Defying Intuition

Material frame indifference implies that the solution in non-linear elasticity theory for a connected body rigidly rotated at its border is a rigid, stress-free, deformation. If the same problem is considered within linear elasticity theory, considered as an approximation to the true elastic situation, one should expect that if the angle of rotation is small, the body still undergoes a rigid deformation while the corresponding stress, though not zero, remains consistently small. Here, we show that this is true, in general, only for homogeneous bodies. Counterexamples of inhomogeneous bodies are presented for which, whatever small the angle of rotation is, the linear elastic solution is by no means a rigid rotation (in a particular case it is an “explosion”) while the stress may even become infinite. If the same examples are re-interpreted as problems in an elasticity theory based upon genuinely linear constitutive relations which retain their validity also for finite deformations, it is shown that they would deliver constraint reaction forces that are not in equilibrium in the actual, deformed, state. This furnishes another characterization of the impossibility of an exact linear constitutive theory for elastic solids with zero residual stress.

Behavior of a rhombus frame of nonlinear elastic material under large deflection

The large-deflection spring behavior of an inextensible thin rhombus frame is studied in this work. The frame has a rectangular cross-section and supports two equal and opposite diagonal forces that act on its two opposite corners with a given apex angle γ. Both geometrical and material nonlinearities have been considered in the study. The beam was assumed to be made of a material having a nonlinear stress–strain relationship of the Ludwick type. A semi-analytical solution has been formulated for the relation between the large displacement at the corners of the frame and the applied forces. The deformed shapes and the nonlinear secant stiffnesses of the rhombus frame were plotted against applied loads. The frame behavior was fully investigated for different apex angles and nonlinear elastic material constants, n. In order to check the accuracy and efficiency of the proposed analytical solution, a large-displacement finite-element analysis utilizing the multi-purpose computer program ADINA was performed. Each side of the rhombus frame was divided into 10 moment–curvature beam elements. Results obtained using the analytical solution were identical to the results obtained using the software ADINA.

A POSTERIORI TESTS OF ONE-EQUATION LES MODELING OF ROTATING TURBULENCE

Eight subgrid-scale (SGS) models were evaluated using two flow configurations: homogeneous decaying turbulence, and rotating turbulence forced at large or intermediate scales. Testing was performed for the first configuration through a systematic comparison between direct numerical simulation results and large eddy simulation results of many characteristics, including resolved kinetic energy, SGS energy production, molecular dissipation, and kinetic energy spectrum. The new models, which are based on dynamic structure model and satisfy the consistency of material frame indifference with the SGS stress, showed more accurate results than traditional models. In the forced testing, the new models were better able to capture essential features of rotating turbulence, including cyclonic/anti-cyclonic asymmetry, quasi-2D at large scales, and reverse kinetic energy transfer from small to large scales.

Consumer attitudes towards timber as a construction material and towards timber frame houses – selected findings of a representative survey among the German population

This paper presents selected findings from a representative consumer survey of the German population exploring the image of timber as a construction material in general, and of timber frame houses in particular. The study found that prejudice regarding the deficiency of timber as a construction material and of timber frame houses, in terms of fire resistance, durability and stability, persists in the minds of consumers. On the other hand, timber as a construction material and timber frame houses have a positive association with well-being, aesthetics and eco-friendliness. Although these “soft” criteria are important factors in the choice of a certain building construction mode, these attributes are not sufficient, on their own, to trigger the choice of timber as a construction material.

Discrete elastic rods

We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description---we represent the material frame by its angular deviation from the natural Bishop frame---as well as in the dynamical treatment---we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons.

Motion analysis of structures (MAS) for flexible multibody systems: planar motion of solids

In this paper, an approach named Motion Analysis of Structures (MAS) is introduced to analyze planar motion of solids. The theoretical basis of MAS is constructed on vector mechanics instead of analytical mechanics. Four procedures are involved to embody this approach. Namely, (a) discretization of structure, (b) discretization of particle path, (c) evaluation of deformations and internal forces, and (d) time integration. The first three procedures are not involved in solving the governing equations of continuum, instead directly formulating the equations of motion of a particle set via Newton’s law. Therefore, MAS is a particle approach. In procedure (c), the vector form intrinsic finite element (VFIFE) is employed, where a description of kinematics to dissect rigid body motion and deformation, a set of deformation coordinates for each time increment to describe element deformation, and the convected material frame solution procedure are included. The algorithms of coupling with rigid bodies and modeling of constraints are presented as well. Numerical examples for large rotation and benchmark verifications are performed to demonstrate the capability and accuracy of the approach in analysis of multibody system dynamics.

Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics

Complete dynamical PDE systems of one-dimensional nonlinear elasticity satisfying the principle of material frame indifference are derived in Eulerian and Lagrangian formulations. These systems are considered within the framework of equivalent nonlocally related PDE systems. Consequently, a direct relation between the Euler and Lagrange systems is obtained. Moreover, other equivalent PDE systems nonlocally related to both of these familiar systems are obtained. Point symmetries of three of these nonlocally related PDE systems of nonlinear elasticity are classified with respect to constitutive and loading functions. Consequently, new symmetries are computed that are: nonlocal for the Euler system and local for the Lagrange system; local for the Euler system and nonlocal for the Lagrange system; nonlocal for both the Euler and Lagrange systems. For realistic constitutive functions and boundary conditions, new dynamical solutions are constructed for the Euler system that only arise as symmetry reductions from invariance under nonlocal symmetries.