scholarly journals Conservation Integrals in Nonhomogeneous Materials with Flexoelectricity

2021 ◽  
Vol 11 (2) ◽  
pp. 681
Author(s):  
Pengfei Yu ◽  
Weifeng Leng ◽  
Yaohong Suo
Keyword(s):  
Electromechanical Coupling ◽  
Energy Momentum Tensor ◽  
Great Influence ◽  
Operator Method ◽  
Electric Polarization ◽  
Momentum Tensor ◽  
Noether Theorem ◽  
Strain Gradients ◽  
Flexoelectric Effect ◽  
Nonhomogeneous Materials

The flexoelectricity, which is a new electromechanical coupling phenomenon between strain gradients and electric polarization, has a great influence on the fracture analysis of flexoelectric solids due to the large gradients near the cracks. On the other hand, although the flexoelectricity has been extensively investigated in recent decades, the study on flexoelectricity in nonhomogeneous materials is still rare, especially the fracture problems. Therefore, in this manuscript, the conservation integrals for nonhomogeneous flexoelectric materials are obtained to solve the fracture problem. Application of operators such as grad, div, and curl to electric Gibbs free energy and internal energy, the energy-momentum tensor, angular momentum tensor, and dilatation flux can also be derived. We examine the correctness of the conservation integrals by comparing with the previous work and discuss the operator method here and Noether theorem in the previous work. Finally, considering the flexoelectric effect, a nonhomogeneous beam problem with crack is solved to show the application of the conservation integrals.

scholarly journals On the Significance of the Fields’ Energy-Momentum Tensor

2019 ◽  
pp. 1-9
Author(s):  
E. Comay
Keyword(s):  
Quantum Electrodynamics ◽  
Lagrangian Density ◽  
Order Differential Equation ◽  
Energy Momentum Tensor ◽  
Massive Particle ◽  
Momentum Tensor ◽  
Noether Theorem ◽  
Consistency Test ◽  
Independent Analysis ◽  
Physical Fields

This work discusses the significance of the energy-momentum tensor of physical fields of an elementary particle. The Noether theorem shows how this tensor can be derived from the Lagrangian density of a given field. This work proves that the energy-momentum tensor can also be used for a consistency test of a field theory. The results show that the Dirac Lagrangian density of a spin-1/2 massive particle yields consistent results. On the other hand, problems exist with the present structure of quantum electrodynamics, and with quantum fields of massive particles that are described by a second order differential equation. All problematic results are confirmed by an independent analysis.

scholarly journals Construction of energy–momentum tensor of gravitation

2016 ◽  
Vol 13 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Kazuharu Bamba ◽  
Katsutaro Shimizu
Keyword(s):  
General Relativity ◽  
Energy Momentum Tensor ◽  
Momentum Conservation ◽  
Momentum Tensor ◽  
Gravitational Energy ◽  
Lorentz Frame ◽  
Noether Theorem ◽  
General Coordinate Transformation ◽  
Two Indices ◽  
Energy Momentum Conservation

We construct the gravitational energy–momentum tensor in general relativity through the Noether theorem. In particular, we explicitly demonstrate that the constructed quantity can vary as a tensor under the general coordinate transformation. Furthermore, we verify that the energy–momentum conservation is satisfied because one of the two indices of the energy–momentum tensor should be in the local Lorentz frame. It is also shown that the gravitational energy and the matter one cancel out in certain space-times.

The electroelastic energy–momentum tensor

1991 ◽  
Vol 433 (1888) ◽  
pp. 299-312 ◽  
Keyword(s):  
Electric Fields ◽  
Energy Momentum Tensor ◽  
Electric Polarization ◽  
Momentum Tensor ◽  
Cauchy Stress ◽  
Material Forces ◽  
Differential Identity ◽  
Material Force ◽  
Electric Effects ◽  
Independent Integral

Eshelby’s energy–momentum tensor useful for studying material forces acting on various kinds of inhomogeneities is constructed in the exact nonlinear theory of deformable dielectrics. This is achieved by examining the possible changes of reference configurations relative to fixed, locally defined, ‘reference crystals'. The electroelastic energy–momentum tensor thus obtained does not involve the Maxwell stress of free electric fields. Electric effects manifest themselves through the ultimate decomposition of the Cauchy stress in a symmetric ‘elastic’ part and an interaction part involving electric polarization. When the electroelastic body is made of the same material at all points, the electroelastic energy–momentum is shown to satisfy a remarkable differential identity involving the torsion of the material connection. In the quasi-linear approximation, the material force thus defined leads to the notion of path-independent integral which should be useful in studying cracks in electrodeformable ceramics. Various extensions and generalizations are briefly discussed, and the Peach–Koehler force acting on a dislocation element is found by an independent method in an appendix.

Size-Dependent Flexoelectric Response of a Truncated Cone and the Consequent Ramifications for the Experimental Measurement of Flexoelectric Properties

2017 ◽  
Vol 84 (10) ◽  
Author(s):  
Qian Deng
Keyword(s):  
Scale Effects ◽  
Piezoelectric Materials ◽  
Electrical Response ◽  
Electric Polarization ◽  
Strain Gradients ◽  
Size Dependent ◽  
Flexoelectric Effect ◽  
Truncated Cone ◽  
Shape Effects ◽  
Typical Model

The flexoelectric effect is an electromechanical phenomenon that is universally present in all dielectrics and exhibits a strong size-dependency. Through a judicious exploitation of scale effects and symmetry, flexoelectricity has been used to design novel types of structures and materials including piezoelectric materials without using piezoelectric. Flexoelectricity links electric polarization with strain gradients and is rather difficult to estimate experimentally. One well-acknowledged approach is to fabricate truncated pyramids and/or cones and examine their electrical response. A theoretical model is then used to relate the measured experimental response to estimate the flexoelectric properties. In this work, we revisit the typical model that is used in the literature and solve the problem of a truncated cone under compression or tension. We obtained closed-form analytical solutions to this problem and examine the size and shape effects of flexoelectric response of the aforementioned structure. In particular, we emphasize the regime in which the existing models are likely to incur significant error.

scholarly journals Engineering of atomic-scale flexoelectricity at grain boundaries

2022 ◽  
Vol 13 (1) ◽  
Author(s):  
Mei Wu ◽  
Xiaowei Zhang ◽  
Xiaomei Li ◽  
Ke Qu ◽  
Yuanwei Sun ◽  
...  
Keyword(s):  
Grain Boundary ◽  
Grain Boundaries ◽  
Strain Gradient ◽  
Electromechanical Coupling ◽  
Mechanical Strain ◽  
Atomic Scale ◽  
Strain Gradients ◽  
Flexoelectric Effect ◽  
Unit Cells ◽  
Large Elastic Deformation

AbstractFlexoelectricity is a type of ubiquitous and prominent electromechanical coupling, pertaining to the electrical polarization response to mechanical strain gradients that is not restricted by the symmetry of materials. However, large elastic deformation is usually difficult to achieve in most solids, and the strain gradient at minuscule is challenging to control. Here, we exploit the exotic structural inhomogeneity of grain boundary to achieve a huge strain gradient (~1.2 nm−1) within 3–4-unit cells, and thus obtain atomic-scale flexoelectric polarization of up to ~38 μC cm−2 at a 24° LaAlO3 grain boundary. Accompanied by the generation of the nanoscale flexoelectricity, the electronic structures of grain boundaries also become different. Hence, the flexoelectric effect at grain boundaries is essential to understand the electrical activities of oxide ceramics. We further demonstrate that for different materials, altering the misorientation angles of grain boundaries enables tunable strain gradients at the atomic scale. The engineering of grain boundaries thus provides a general and feasible pathway to achieve tunable flexoelectricity.

Spinorielle Feldtheorien und Noether-Theorem in der gekrümmten Raum-Zeit

1964 ◽  
Vol 19 (9) ◽  
pp. 1027-1031 ◽  
Author(s):  
Ernst Schmutzer
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Space Time ◽  
Point Of View ◽  
Coordinate Transformations ◽  
Noether Theorem ◽  
Metric Tensor ◽  
Curved Space ◽  
Spinor Fields ◽  
Tensor Expression

On the basis of a curved space-time with RIEMANNEAN geometry the conception of spinors is analyzed. It is shown that a consequent treatment of spinors as invariants with respect to coordinate transformations (SOMMERFELD’S first point of view) gives the well known energy-momentum-tensor and the correct spin integral. For this purpose it is necessary to develop NOETHER’S theorem in such a way that not the metric tensor gmn but the metric spintensor is the fundamental metrical quantity. This fact is the cause that the BELINFANTE tensor expression cannot be applied. A new tensor expression for spinor fields is derived. In this connection DIRAC’S theory and HEISENBERG’S theory are investigated.

CASIMIR EFFECT FOR THE ROBIN SURFACES IN STATIC ROBERTSON–WALKER SPACE–TIME

2011 ◽  
Vol 20 (02) ◽  
pp. 161-168 ◽  
Author(s):  
MOHAMMAD R. SETARE ◽  
M. DEHGHANI
Keyword(s):  
Scalar Field ◽  
Energy Momentum Tensor ◽  
Casimir Effect ◽  
Vacuum Expectation ◽  
Robin Boundary Condition ◽  
Momentum Tensor ◽  
Space Time ◽  
Conformally Invariant ◽  
Time Space ◽  
Robin Boundary

We investigate the energy–momentum tensor for a massless conformally coupled scalar field in the region between two curved surfaces in k = -1 static Robertson–Walker space–time. We assume that the scalar field satisfies the Robin boundary condition on the surfaces. Robertson–Walker space–time space is conformally related to Rindler space; as a result we can obtain vacuum expectation values of the energy–momentum tensor for a conformally invariant field in Robertson–Walker space–time space from the corresponding Rindler counterpart by the conformal transformation.

scholarly journals Cutoff AdS3 versus $$ T\overline{T} $$ CFT2 in the large central charge sector: correlators of energy-momentum tensor

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yi Li ◽  
Yang Zhou
Keyword(s):  
Field Theory ◽  
Euclidean Plane ◽  
Energy Momentum Tensor ◽  
Central Charge ◽  
Momentum Tensor ◽  
Two Dimensional ◽  
Conformal Field ◽  
Operator Identity ◽  
Pure Gravity ◽  
Charge Sector

Abstract In this article we probe the proposed holographic duality between $$ T\overline{T} $$ T T ¯ deformed two dimensional conformal field theory and the gravity theory of AdS3 with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the $$ T\overline{T} $$ T T ¯ CFT in a Euclidean plane and a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in Euclidean AdS3 with the corresponding cutoff surface, given a holographic dictionary which identifies gravity parameters with $$ T\overline{T} $$ T T ¯ CFT parameters.

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