Strain tensor field in proteins

1996 ◽  
Vol 14 (2) ◽  
pp. 105-107 ◽  
Author(s):  
Takahisa Yamato
Keyword(s):  
Tensor Field ◽  
Strain Tensor

Intrinsic formulation of the displacement-traction problem in linearized elasticity

2014 ◽  
Vol 24 (06) ◽  
pp. 1197-1216 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Elastic Body ◽  
Displacement Field ◽  
Tensor Field ◽  
Strain Tensor ◽  
Linearized Strain ◽  
Linearized Elasticity

The displacement-traction problem of linearized elasticity is a system of partial differential equations and boundary conditions whose unknown is the displacement field inside a linearly elastic body. We explicitly identify here the corresponding boundary conditions satisfied by the linearized strain tensor field associated with such a displacement field. Using this identification, we are then able to provide an intrinsic formulation of the displacement-traction problem of linearized elasticity, by showing how it can be recast into a boundary value problem whose unknown is the linearized strain tensor field.

scholarly journals The intrinsic theory of linearly elastic plates

2018 ◽  
Vol 24 (4) ◽  
pp. 1182-1203 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Cristinel Mardare
Keyword(s):  
Boundary Condition ◽  
Elastic Plate ◽  
Tensor Field ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Middle Surface ◽  
Curvilinear Coordinates ◽  
Constant Thickness ◽  
Classical Approach ◽  
Tensor Fields

In an intrinsic approach to a problem in elasticity, the only unknown is a tensor field representing an appropriate ‘measure of strain’, instead of the displacement vector field in the classical approach. The objective of this paper is to study the displacement traction problem in the special case where the elastic body is a linearly elastic plate of constant thickness, clamped over a portion of its lateral face. In this respect, we first explicitly compute the intrinsic three-dimensional boundary condition of place in terms of the Cartesian components of the linearized strain tensor field, thus avoiding the recourse to covariant components in curvilinear coordinates and providing an interesting example of actual computation of an intrinsic boundary condition of place in three-dimensional elasticity. Second, we perform a rigorous asymptotic analysis of the three-dimensional equations as the thickness of the plate, considered as a parameter, approaches zero. As a result, we identify the intrinsic two-dimensional equations of a linearly elastic plate modelled by the Kirchhoff–Love theory, with the linearized change of metric and change of curvature tensor fields of the middle surface of the plate as the new unknowns, instead of the displacement field of the middle surface in the classical approach.

scholarly journals A NEW DUALITY APPROACH TO ELASTICITY

2012 ◽  
Vol 22 (01) ◽  
pp. 1150003 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
GIUSEPPE GEYMONAT ◽  
FRANÇOISE KRASUCKI
Keyword(s):  
Duality Theory ◽  
Tensor Field ◽  
Displacement Vector ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Minimization Problems ◽  
Displacement Vector Field ◽  
Linearized Elasticity ◽  
Duality Approach ◽  
New Perspective

The displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field, or the stress tensor field, or the strain tensor field, is the unknown. The objective of this paper is to put these three different formulations of the same problem in a new perspective, by means of Legendre–Fenchel duality theory. More specifically, we show that both the displacement and strain formulations can be viewed as Legendre–Fenchel dual problems to the stress formulation. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new duality approach to elasticity.

FE Simulation of Cross Roll Straightening: a Strain Tensor Field Approach

2010 ◽  
Author(s):  
A. Mutrux ◽  
B. Berisha ◽  
P. Hora ◽  
F. Barlat ◽  
Y. H. Moon ◽  
...  
Keyword(s):  
Tensor Field ◽  
Fe Simulation ◽  
Strain Tensor ◽  
Field Approach

LAGRANGE MULTIPLIERS IN INTRINSIC ELASTICITY

2011 ◽  
Vol 21 (04) ◽  
pp. 651-666 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
PATRICK CIARLET ◽  
OANA IOSIFESCU ◽  
STEFAN SAUTER ◽  
JUN ZOU
Keyword(s):  
Saddle Point ◽  
Minimization Problem ◽  
Tensor Field ◽  
Ad Hoc ◽  
Displacement Vector ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Quadratic Minimization ◽  
Linearized Elasticity ◽  
Potential Applications

In an intrinsic approach to three-dimensional linearized elasticity, the unknown is the linearized strain tensor field (or equivalently the stress tensor field by means of the constitutive equation), instead of the displacement vector field in the classical approach. We consider here the pure traction problem and the pure displacement problem and we show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations (the form of which depends on the type of boundary conditions considered). Using the Babuška-Brezzi inf-sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints. Such results have potential applications to the numerical analysis and simulation of the intrinsic approach to three-dimensional linearized elasticity.

Rubber Composition–Properties Relationships during Tire Numerical Simulation and Design Optimization

2017 ◽  
Vol 45 (1) ◽  
pp. 71-84 ◽  
Author(s):  
Alexey Mazin ◽  
Alexander Kapustin ◽  
Mikhail Soloviev ◽  
Alexander Karanets
Keyword(s):  
Numerical Simulation ◽  
Design Optimization ◽  
Strain Tensor ◽  
General Purpose ◽  
Orthogonal Functions ◽  
Element Analysis ◽  
Time Investment ◽  
Footprint Analysis ◽  
Composition Properties ◽  
Nonlinear Elastic

ABSTRACT Numerical simulation based on finite element analysis is now widely used during the design optimization of tires, thereby drastically reducing the time investment in the design process and improving tire performance because it is obtained from the optimized solution. Rubber material models that are used in numerical calculations of stress–strain distributions are nonlinear and may include several parameters. The relations of these parameters with rubber formulations are usually unknown, so the designer has no information on whether the optimal set of parameters is reachable by the rubber technological possibilities. The aim of this work was to develop such relations. The most common approach to derive the equation of the state of rubber is based on the expansion of the strain energy in a series of invariants of the strain tensor. Here, we show that this approach has several drawbacks, one of which is problems that arise when trying to build on its basis the quantitative relations between the rubber composition and its properties. An alternative is to use a series expansion in orthogonal functions, thereby ensuring the linear independence of the coefficients of elasticity in evaluation of the experimental data and the possibility of constructing continuous maps of “the composition to the property.” In the case of orthogonal Legendre polynomials, the technique for constructing such maps is considered, and a set of empirical functions is proposed to adequately describe the dependence of the parameters of nonlinear elastic properties of general-purpose rubbers on the content of the main ingredients. The calculated sets of parameters were used in numerical tire simulations including static loading, footprint analysis, braking/acceleration, and cornering and also in design optimization procedures.

A new approach to curvature measures in linear shell theories

2020 ◽  
pp. 108128652097275
Author(s):  
Miroslav Šilhavý
Keyword(s):  
Strain Tensor ◽  
Middle Surface ◽  
Surface Strain ◽  
Affine Function ◽  
Shear Deformations ◽  
Strain Measure ◽  
Bending Strain ◽  
Curvature Measures ◽  
Tensor Formula ◽  
Strain Measures

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor [Formula: see text], which is different from the widely used Naghdi’s bending strain tensor [Formula: see text]. In the particular case of Kirchhoff–Love deformations, the tensor [Formula: see text] reduces to a tensor [Formula: see text] introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, [Formula: see text] is different from Koiter’s bending strain tensor [Formula: see text] (frequently used in this context). AMS 2010 classification: 74B99

scholarly journals Investigation of Deformation Inhomogeneity and Low-Cycle Fatigue of a Polycrystalline Material

2021 ◽  
Vol 11 (6) ◽  
pp. 2673
Author(s):  
Mu-Hang Zhang ◽  
Xiao-Hong Shen ◽  
Lei He ◽  
Ke-Shi Zhang
Keyword(s):  
Cyclic Loading ◽  
Low Cycle Fatigue ◽  
Strain Tensor ◽  
Equivalent Strain ◽  
Differential Entropy ◽  
Strain Components ◽  
Deformation Inhomogeneity ◽  
Standard Deviations ◽  
Number Of Cycles ◽  
The Relationship

Considering the relationship between inhomogeneous plastic deformation and fatigue damage, deformation inhomogeneity evolution and fatigue failure of superalloy GH4169 under temperature 500 °C and macro tension compression cyclic loading are studied, by using crystal plasticity calculation associated with polycrystalline representative Voronoi volume element (RVE). Different statistical standard deviation and differential entropy of meso strain are used to measure the inhomogeneity of deformation, and the relationship between the inhomogeneity and strain cycle is explored by cyclic numerical simulation. It is found from the research that the standard deviations of each component of the strain tensor at the cyclic peak increase monotonically with the cyclic loading, and they are similar to each other. The differential entropy of each component of the strain tensor also increases with the number of cycles, and the law is similar. On this basis, the critical values determined by statistical standard deviations of the strain components and the equivalent strain, and that by differential entropy of strain components, are, respectively, used as fatigue criteria, then predict the fatigue–life curves of the material. The predictions are verified with reference to the measured results, and their deviations are proved to be in a reasonable range.

Sign in / Sign up

Export Citation Format

Share Document