scholarly journals Lagrangian Strain Tensor Computation with Higher Order Variational Models

2013 ◽  
Author(s):  
Alexander Hewer ◽  
Joachim Weickert ◽  
Henning Seibert ◽  
Tobias Scheffer ◽  
Stefan Diebels
Keyword(s):  
Strain Tensor ◽  
Higher Order ◽  
Variational Models ◽  
Lagrangian Strain

Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
Wooseok Ji ◽  
Anthony M. Waas ◽  
Zdeněk P. Bažant
Keyword(s):  
Finite Element ◽  
Fiber Composite ◽  
Strain Tensor ◽  
Stress And Strain ◽  
Hencky Strain ◽  
Tangential Stiffness ◽  
Jaumann Rate ◽  
Lagrangian Strain ◽  
Strain Formulation ◽  
Accurate Expression

Many finite element programs including standard commercial software such as ABAQUS use an incremental finite strain formulation that is not fully work-conjugate, i.e., the work of stress increments on the strain increments does not give a second-order accurate expression for work. In particular, the stress increments based on the Jaumann rate of Kirchhoff stress are work-conjugate with the increments of the Hencky (logarithmic) strain tensor but are paired in many finite element programs with the increments of Green’s Lagrangian strain tensor. Although this problem was pointed out as early 1971, a demonstration of its significance in realistic situations has been lacking. Here it is shown that, in buckling of compressed highly orthotropic columns or sandwich columns that are very “soft” in shear, the use of such nonconjugate stress and strain increments can cause large errors, as high as 100% of the critical load, even if the strains are small. A similar situation may arise when severe damage such as distributed cracking leads to a highly anisotropic tangential stiffness matrix, or when axial cracks between fibers severely weaken a uniaxial fiber composite or wood. A revision of these finite element programs is advisable, and will in fact be easy—it will suffice to replace the Jaumann rate with the Truesdell rate. Alternatively, the Green’s Lagrangian strain could be replaced with the Hencky strain.

Thermomechanical Buckling of Laminated Composite Plates Using Mixed, Higher-Order Analytical Formulation

2002 ◽  
Vol 69 (6) ◽  
pp. 790-799 ◽  
Author(s):  
J. B. Dafedar ◽  
Y. M. Desai
Keyword(s):  
Strain Tensor ◽  
Single Layer ◽  
Laminated Composite ◽  
Composite Plates ◽  
Higher Order ◽  
Laminated Composite Plates ◽  
Individual Layer ◽  
Energy Principle ◽  
Transverse Stresses ◽  
Benchmark Solutions

A novel, analytical mixed theory based on the potential energy principle has been presented in this paper to investigate buckling response of laminated composite plates subjected to mechanical and hygrothermal loads. Two sets of higher-order mixed models have been proposed on the basis of an individual layer as well as equivalent single layer theories by selectively incorporating nonlinear components of Green’s strain tensor. Displacements, as well as transverse stress continuities, have been enforced in the formulation of models by incorporating displacements and transverse stresses as the degrees-of-freedom. The modal transverse stresses have been obtained as eigenvectors and thus their separate calculations have been advantageously avoided. Solutions from the models have been shown to be in excellent agreement with the available three-dimensional elasticity solutions. Few benchmark solutions have also been presented for the bi-axial compression-tension loading.

A New Higher-Order Euler-Bernoulli Beam Element of Absolute Nodal Coordinate Formulation

2020 ◽  
Author(s):  
Peng Zhang ◽  
Jianmin Ma ◽  
Menglan Duan
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Strain Tensor ◽  
Beam Element ◽  
Higher Order ◽  
Position Vector ◽  
Traditional Theory ◽  
Bernoulli Beam ◽  
Geometrically Nonlinear Analysis ◽  
Absolute Nodal Coordinate ◽  
Euler Bernoulli Beam

Abstract In this study, a new higher-order Euler-Bernoulli beam element of Absolute Nodal Coordinate Formulation (ANCF) is developed for geometrically nonlinear analysis of planar structures. The strain energy of the beam element is derived by applying the definition of the Green–Lagrange strain tensor in continuum mechanics. The first contribution of this research is to realize the accurate calculation of curvature on the beam element node by additionally considering the second derivative of the position vector obtained by quintic Hermite interpolation function. Furthermore, in traditional theory, the independent variable of finite formulation is arc-length coordinate s, while in this work, a correction is come up with and proven that it is actually an equivalent parameter. Some benchmark problems of straight beams are solved by the proposed element and accurate results are obtained by just fewer elements when compared with the other works including the traditional ANCF element and B23 element of ABAQUS. What leads to this accuracy result is that the precise calculation of nodal curvature is obtained from higher order interpolation scheme. The correctness and accuracy of the proposed element are validated in this work and it can be further developed for tackling large deformation and large rotation problems of spatial curved beams.

Time Micro Scales of Lagrangian Strain Tensor in Turbulence

1993 ◽  
Vol 62 (2) ◽  
pp. 506-513 ◽  
Author(s):  
Takashi Ishihara ◽  
Yukio Kaneda
Keyword(s):  
Strain Tensor ◽  
Lagrangian Strain

scholarly journals A Unified Approach to the Timoshenko Geometric Stiffness Matrix Considering Higher-Order Terms in the Strain Tensor

2019 ◽  
Vol 16 (4) ◽  
Author(s):  
Marcos Antonio Campos Rodrigues ◽  
Rodrigo Bird Burgos ◽  
Luiz Fernando Martha
Keyword(s):  
Stiffness Matrix ◽  
Strain Tensor ◽  
Higher Order ◽  
Unified Approach ◽  
Geometric Stiffness ◽  
Higher Order Terms

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Quantitative EPMA of nitrogen : A tricky element in the electron-probe microanalyzer

1992 ◽  
Vol 50 (2) ◽  
pp. 1622-1623
Author(s):  
G.F. Bastin ◽  
H.J.M. Heijligers
Keyword(s):  
Background Subtraction ◽  
Electron Probe ◽  
Detection System ◽  
Higher Order ◽  
Light Elements ◽  
Strong Suppression ◽  
Electron Probe Microanalyzer ◽  
Quantitative Epma ◽  
Peak Count ◽  
Lead Stearate

Among the ultra-light elements B, C, N, and O nitrogen is the most difficult element to deal with in the electron probe microanalyzer. This is mainly caused by the severe absorption that N-Kα radiation suffers in carbon which is abundantly present in the detection system (lead-stearate crystal, carbonaceous counter window). As a result the peak-to-background ratios for N-Kα measured with a conventional lead-stearate crystal can attain values well below unity in many binary nitrides . An additional complication can be caused by the presence of interfering higher-order reflections from the metal partner in the nitride specimen; notorious examples are elements such as Zr and Nb. In nitrides containing these elements is is virtually impossible to carry out an accurate background subtraction which becomes increasingly important with lower and lower peak-to-background ratios. The use of a synthetic multilayer crystal such as W/Si (2d-spacing 59.8 Å) can bring significant improvements in terms of both higher peak count rates as well as a strong suppression of higher-order reflections.

Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

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