An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators

The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators $$(A_1,\ldots ,A_n)$$ ( A 1 , … , A n ) . A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.

Conservation Integrals in Nonhomogeneous Materials with Flexoelectricity

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.

A Simulation Framework for Passive Acoustic Thermometry of Nonhomogeneous Materials

Purpose: Internal temperature is a significant factor for medical diagnosis. There are several thermometric methods, including IR, MRI, and active ultrasonic thermometry, which have limitations for clinical applications. The new method in this field called Passive Acoustic Thermometry (PAT), which enhanced some of this limitation. PAT is a safe method for internal temperature estimation that works based on acoustic radiation of materials with a specific temperature. Several experimental studies have been carried out so far in the field of PAT. While, to the best of our knowledge, there is no simulation-based research for nonhomogeneous materials reported yet. In this article (for the first time) we proposed a simulation framework for evaluating the PAT methodologies in nonhomogeneous materials; also we proposed a new formulation for temperature estimation in PAT algorithm. Materials and Methods: This framework supports the generation of acoustic radiation, signal processing, parameter estimation, and temperature reconstruction processes. At the moment the proposed framework estimates the temperature in the frequency domain and uses the frequency spectrum of the acquired ultrasound signals captured by a single transducer. Using the proposed framework, we tried to implement the previously practical experiments and the results of the simulation are consistent with those of the practical experiments. Also, we proposed the formulation that improves the error of temperature estimation. Results: We study 6 scenarios, including 2 environments with a target at 3 different temperatures. The average error of the proposed formulation in two different nonhomogeneous materials for three different temperatures is less than 0.25°C. Conclusion: The results show that the proposed formulation is the best estimation in the formula that has been introduced until now and compare with the previous study the accuracy is enhanced 54% (from 0.79 to 0.36 deg.). Therefore, the proposed formula enhanced PAT accuracy for temperature estimation. Also, the results show that it is possible to use this framework to evaluate the PAT in different scenarios. Therefore, this method enhances the possibility of examination of different conditions and algorithms. It also reduces the cost of practical experiment.

Random field generation of the material properties in the lattice discrete element method

The lattice discrete element method has been successfully used to simulate the evolution of damage in structural mechanics. The approach has led to new perspectives in the solution of fracture problems in nonhomogeneous materials. For such purpose, it is necessary to introduce correctly the parameters that characterize the random nature of the material. In this article, the fracture toughness of the material is considered a three-dimensional random field, characterized by a probability density and the spatial distribution of the simulated random field which is governed by the correlation length. The methodology used to separate the random field simulated from the discretization level used is depicted in detail. Examples are shown that verify the objectivity of the results obtained respecting the discretization levels. Finally, the article concludes by emphasizing the relevance of this implementation in the damage simulation process in the so-called heterogeneous materials.

Incompatible Graded Finite Elements for Analysis of Nonhomogeneous Materials

A six-node incompatible graded finite element is developed and studied. Such element is recommended for use since it is more accurate than four-node compatible element and more efficient than eight-node compatible element in two-dimensional plane elasticity. This paper presents comparison between six-node incompatible (QM6) and four-node compatible (Q4) graded elements. Numerical solution is obtained from abaqus using UMAT capability of the software and exact solution is provided as reference for comparison. A graded plate with exponential and linear gradation subjected to traction and bending load is considered. Additionally, three-node triangular (T3) and six-node triangular (T6) graded elements are compared to QM6 element. Incompatible graded element is shown to give better performance in terms of accuracy and computation time over other element formulations for functionally graded materials (FGMs).