On a representation of the linear strain tensor

1999 ◽  
Vol 31 (3) ◽  
pp. 286-291
Author(s):  
N. M. Borodachev
Keyword(s):  
Strain Tensor ◽  
Linear Strain ◽  
Linear Strain Tensor

Description of Large Deformation Problem Using Means of Visual Strain

2013 ◽  
Vol 275-277 ◽  
pp. 16-22
Author(s):  
You Liang Xu
Keyword(s):  
Constitutive Equation ◽  
Large Deformation ◽  
Strain Tensor ◽  
Visual Image ◽  
Linear Strain ◽  
Deformation Problem ◽  
Practical Engineering ◽  
Large Deformation Problem ◽  
Strain Tensors ◽  
Linear Strain Tensor

The constitutive equation of large deformation problem is closely related to geometric description. Nowadays, linear strain tensor is no longer unsuitable to describe large deformation. However, the existing non-linear strain tensors have complicated forms as well as no apparent geometric or physical meaning. While, the increment method is used to solve, however, convergence and efficiency are low sometimes. Thus the idea of visual strain tensor is proposed, with distinct meaning and visual image. Beside, it is likely to be used in engineering measurement, and it can be connected with measured constitutive equation directly. Thus this research provides a new idea and method for solving large-deformation problems in practical engineering.

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.

scholarly journals Using silicon-vacancy centers in diamond to probe the full strain tensor

2021 ◽  
Vol 130 (2) ◽  
pp. 024301
Author(s):  
Kelsey M. Bates ◽  
Matthew W. Day ◽  
Christopher L. Smallwood ◽  
Rachel C. Owen ◽  
Tim Schröder ◽  
...  
Keyword(s):  
Strain Tensor ◽  
Silicon Vacancy

scholarly journals Origin and Application of the Twinned Calcite Strain Gauge

Geosciences ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 296
Author(s):  
Richard H. Groshong
Keyword(s):  
Stress Analysis ◽  
Strain Gauge ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Constant Strain Rate ◽  
Axis Orientation ◽  
Order Of Magnitude ◽  
Rate Experiment ◽  
Strain Axis ◽  
Expected Values

This paper is a personal account of the origin and development of the twinned-calcite strain gauge, its experimental verification, and its relationship to stress analysis. The method allows the calculation of the three-dimensional deviatoric strain tensor based on five or more twin sets. A minimum of about 25 twin sets should provide a reasonably accurate result for the magnitude and orientation of the strain tensor. The opposite-signed strain axis orientation is the most accurately located. Where one strain axis is appreciably different from the other two, that axis is generally within about 10° of the correct value. Experiments confirm a magnitude accuracy of 1% strain over the range of 1–12% axial shortening and that samples with more than 40% negative expected values imply multiple or rotational deformations. If two deformations are at a high angle to one another, the strain calculated from the positive and negative expected values separately provides a good estimate of both deformations. Most stress analysis techniques do not provide useful magnitudes, although most provide a good estimate of the principal strain axis directions. Stress analysis based on the number of twin sets per grain provides a better than order-of-magnitude approximation to the differential stress magnitude in a constant strain rate experiment.

scholarly journals Strain Tensor Imaging: Cardiac-induced brain tissue deformation in humans quantified with high-field MRI

NeuroImage ◽  
2021 ◽  
pp. 118078
Author(s):  
Jacob-Jan Sloots ◽  
Geert Jan Biessels ◽  
Alberto de Luca ◽  
Jaco J.M. Zwanenburg
Keyword(s):  
Brain Tissue ◽  
High Field ◽  
Strain Tensor ◽  
Tissue Deformation ◽  
High Field Mri

Opto-mechanical modeling and experimental implementation of an FBG-based piezoelectric bimorph actuator for high voltage sensing applications

2021 ◽  
pp. 1045389X2110282
Author(s):  
Raúl E Jiménez ◽  
José P Montoya ◽  
Rodrigo Acuna Herrera
Keyword(s):  
High Voltage ◽  
Good Accuracy ◽  
Correction Term ◽  
Experimental Results ◽  
Mechanical Modeling ◽  
Piezoelectric Bimorph ◽  
Linear Strain ◽  
Voltage Sensing ◽  
Sensing Applications ◽  
Experimental Implementation

This paper proposes a highly simplified optical voltage sensor by using a piezoelectric bimorph and a Fiber Bragg Grating (FBG) that can be used for high voltage applications with a relatively good accuracy and stability. In this work the theoretical framework for the whole opto-mechanical operation of the optical sensor is detailed and compared to experimental results. In the analysis, a correction term to the electric field is derived to account for the linear strain distribution across the piezoelectric layer improving the designing equations and giving more criteria for future developments. Finally, some experimental results from a laboratory scale optical-based high voltage sensing setup are discussed, and shown to be in excellent agreement with theoretical expected behavior for different voltage magnitudes.

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