Compact Basis Free Relations for Stress Tensors Conjugate to Hill’s Strain Measures

2006 ◽  
Author(s):  
Reza Naghdabadi ◽  
Mohsen Asghari ◽  
Kamyar Ghavam
Keyword(s):  
Strain Tensor ◽  
Three Dimensional ◽  
Inner Product ◽  
Stress And Strain ◽  
Conjugate Pair ◽  
Stress Tensors ◽  
Plastic Materials ◽  
The Right ◽  
Elastic Plastic Materials ◽  
Strain Measures

If the double contraction of a stress tensor such as T and rate of a Lagrangean strain tensor such as E, i.e. T : E˙, produces the stress power then these stress and strain tensors are called a conjugate pair. The applications of the conjugate stress and strain measures are in the development of the basic relations in nonlinear continuum mechanics analysis such as modeling of constitutive equations of elastic-plastic materials. In this paper relations for stress tensors conjugate to an arbitrary Lagrangean strain measure of Hill’s class are obtained. The results of this paper are more compact and simpler in compare with those available in the literature. The results are valid for the three dimensional Euclidean inner product space and the case of distinct eigenvalues of the right stretch tensor U.

Unified Basis-Free Relation Between Two Stress Tensors Conjugate to Arbitrary Hill’s Strain Measures

2006 ◽  
Author(s):  
Mohsen Asghari ◽  
Reza Naghdabadi
Keyword(s):  
Arbitrary Dimension ◽  
Rate Of Change ◽  
The Body ◽  
Inner Product ◽  
Adiabatic Process ◽  
Stress And Strain ◽  
Stress Tensors ◽  
Two Measures ◽  
The Right ◽  
Strain Measures

The concept of energy conjugacy for stress and strain measures states that a stress tensor T is conjugate to a strain measure E if T: E˙ provides the rate of change of the internal energy per unit reference volume of the body in an adiabatic process. The applications of the conjugate stress and strain measures are in the development of the basic relations in nonlinear analysis of solids. In this paper using eigenprojection method, unified explicit basis-free relation between two arbitrary stress tensors T(f) and T(g), respectively conjugate to two measures of Hill’s strains is determined. The result is valid for arbitrary dimension of the Euclidean inner product space and for all cases of distinct and repeated eigenvalues of the right stretch tensor U.

The Propagation of Plasticity in Uniaxial Compression

1948 ◽  
Vol 15 (3) ◽  
pp. 256-260 ◽  
Author(s):  
M. P. White ◽  
LeVan Griffis
Keyword(s):  
Uniaxial Compression ◽  
Thin Sheet ◽  
High Stress ◽  
Stress And Strain ◽  
Velocity Range ◽  
Plastic Materials ◽  
The Face ◽  
Elastic Plastic Materials ◽  
The Impact ◽  
Type Of Behavior

Abstract A theoretical investigation of the mechanism of uniaxial compression impact on elastic-plastic materials is described in this paper. The method of analysis is similar in some respects to that previously given for tension impact on such materials. It is concluded that four different kinds of behavior can occur, depending upon the impact velocity. In the lowest velocity range the behavior in compression is similar to that found in tension. In this case stress and strain are propagated from the point of impact as a zone or wave front of ever-increasing length. This type of behavior ends at a velocity corresponding to the “critical” velocity found in tension impact. Within the next higher velocity range, stress and strain are propagated as a shock-type wave, or wave of very small length in which the transition from low to high stress and strain is very abrupt. At still higher impact velocities, there occurs “flowing deformation” in which the material is too weak to maintain coherency. Here there is a steady flow of the material toward and against the hammer, after which it flows in a thin sheet radially outward over the face of the hammer. The final possible state occurs at impact velocities greater than the speed of an elastic wave, so that no disturbance can escape from the hammer into the medium. Here the behavior is essentially that of a fluid, impact force being independent of strength of material.

Constrained Elastic-Plastic Materials

1994 ◽  
Vol 61 (3) ◽  
pp. 511-518 ◽  
Author(s):  
H. C. Lin ◽  
P. M. Naghdi
Keyword(s):  
Detailed Examination ◽  
Stress And Strain ◽  
Response Functions ◽  
Elastic Plastic ◽  
Plastic Materials ◽  
Yield Functions ◽  
Space Formulation ◽  
Elastic Plastic Materials ◽  
Strain Space

The main purpose of this paper is to present a general (purely mechanical) constrained theory of finitely deforming elastic-plastic materials. Our development is based on a strain-space formulation of plasticity and requires a detailed examination of the effect of constraint on various constitutive ingredients in the unconstrained theory, including the yield functions (in both the stress and strain spaces), the loading criteria, and various response functions. Also examined is the effect of constraint on the restrictions arising from the work inequality of Naghdi and Trapp (1975b).

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 Right-sided Bochdalek hernia in an adult with hepatic malformation and intestinal malrotation

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Naoki Enomoto ◽  
Kazuhiko Yamada ◽  
Daiki Kato ◽  
Shusuke Yagi ◽  
Hitomi Wake ◽  
...  
Keyword(s):  
Elective Surgery ◽  
Abdominal Cavity ◽  
Three Dimensional ◽  
Simulation Software ◽  
3D Simulation ◽  
Diaphragmatic Defect ◽  
Intestinal Malrotation ◽  
Intestinal Loop ◽  
Bochdalek Hernia ◽  
The Right

Abstract Background Bochdalek hernia is a common congenital diaphragmatic defect that usually manifests with cardiopulmonary insufficiency in neonates. It is very rare in adults, and symptomatic cases are mostly left-sided. Diaphragmatic defects generally warrant immediate surgical intervention to reduce the risk of incarceration or strangulation of the displaced viscera. Case presentation A 47-year-old woman presented with dyspnea on exertion. Computed tomography revealed that a large part of the intestinal loop with superior mesenteric vessels and the right kidney were displaced into the right thoracic cavity. Preoperative three-dimensional (3D) simulation software visualized detailed anatomy of displaced viscera and the precise location and size of the diaphragmatic defect. She underwent elective surgery after concomitant pulmonary hypertension was stabilized preoperatively. The laparotomic approach was adopted. Malformation of the liver and the presence of intestinal malrotation were confirmed during the operation. The distal part of the duodenum, jejunum, ileum, colon, and right kidney were reduced into the abdominal cavity consecutively. A large-sized oval defect was closed with monofilament polypropylene mesh. No complications occurred postoperatively. Conclusion Symptomatic right-sided Bochdalek hernia in adults is exceedingly rare and is frequently accompanied by various visceral anomalies. Accurate diagnosis and appropriate surgical repair are crucial to prevent possible incarceration or strangulation. The preoperative 3D simulation provided comprehensive information on anatomy and concomitant anomalies and helped surgeons plan the operation meticulously and perform procedures safely.

scholarly journals Alteration within the Hippocampal Volume in Patients with LHON Disease—7 Tesla MRI Study

2020 ◽  
Vol 10 (1) ◽  
pp. 14
Author(s):  
Cezary Grochowski ◽  
Kamil Jonak ◽  
Marcin Maciejewski ◽  
Andrzej Stępniewski ◽  
Mansur Rahnama-Hezavah
Keyword(s):  
Magnetic Resonance Imaging ◽  
Magnetic Resonance ◽  
Molecular Layer ◽  
Three Dimensional ◽  
Hippocampal Volume ◽  
7 Tesla ◽  
Diagnostic Process ◽  
Resonance Imaging ◽  
The Right

Purpose: The aim of this study was to assess the volumetry of the hippocampus in the Leber’s hereditary optic neuropathy (LHON) of blind patients. Methods: A total of 25 patients with LHON were randomly included into the study from the national health database. A total of 15 patients were selected according to the inclusion criteria. The submillimeter segmentation of the hippocampus was based on three-dimensional spoiled gradient recalled acquisition in steady state (3D-SPGR) BRAVO 7T magnetic resonance imaging (MRI) protocol. Results: Statistical analysis revealed that compared to healthy controls (HC), LHON subjects had multiple significant differences only in the right hippocampus, including a significantly higher volume of hippocampal tail (p = 0.009), subiculum body (p = 0.018), CA1 body (p = 0.002), hippocampal fissure (p = 0.046), molecular layer hippocampus (HP) body (p = 0.014), CA3 body (p = 0.006), Granule Cell (GC) and Molecular Layer (ML) of the Dentate Gyrus (DG)–GC ML DG body (p = 0.003), CA4 body (p = 0.001), whole hippocampal body (p = 0.018), and the whole hippocampus volume (p = 0.023). Discussion: The ultra-high-field magnetic resonance imaging allowed hippocampus quality visualization and analysis, serving as a powerful in vivo diagnostic tool in the diagnostic process and LHON disease course assessment. The study confirmed previous reports regarding volumetry of hippocampus in blind individuals.

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.

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