Continuum Mechanics Of Solids Research Articles

Extending reliability of FEM simulations, based on optically assisted tensile tests and digital twins

As it is well known, Finite Element simulations are widely used in research and industry to predict the performance of components. Most challenging task of these simulations is predicting the Structural Integrity limits of components and systems. Despite the fact that nonlinear Continuum Mechanics of Solids is a promising approach for ductile materials, the potential predictive capabilities of these methods are difficult to utilize. Generally, in applications, a common problem is that plastic flow curves obtained from the measurement results are often not reliable enough, especially for large deformations. In the present work, standard tensile tests were carried out on specimens with square cross-section. A combination of contact strain gauge and an optical system were used in these experiments. Square grids were printed on the surface of the specimens. Evaluation of the image sequences provided the possibility to follow the specimen contours as well as the deformation of its surfaces. Based on the contours retrieved from the images, it became possible to determine nominal stresses at the minimum cross-section of the specimen. Corrections were applied to the flow curve, using analytical formulae. The corrected flow curve was used in Finite Element simulations on the Digital Twins of the specimen. The geometry changes of the surface and the selected points of the Finite Element model were compared to the measured changes of the surface and the corresponding selected points of the test specimen.

Dispersion relations of in-plane elastic waves in nano-scale one dimensional piezoelectric semiconductor/piezoelectric dielectric phononic crystal with the consideration of interface effect

In this paper, the propagation properties of in-plane elastic waves in a nano-scale N-type piezoelectric semiconductor material/piezoelectric dielectric material layered periodic composite are theoretically investigated. Under the background of classical continuum mechanics of solids, the transfer matrix method and the Bloch theorem, the transfer matrices of the N-type piezoelectric semiconductor material, the piezoelectric dielectric material and the interface and the dispersion equation of in-plane Bloch waves are derived mathematically. Basing on the numerical calculation results, we investigate the physical influences of the semiconductor effect, biasing electric fields and the interface effect on the dispersion curves of in-plane elastic waves. It is found that the electron carrier mobility and diffusion will reduce the equivalent rigidity of the N-type piezoelectric semiconductor material, which will be limited by the existence of biasing electric fields. On the basis of the propagation of in-plane elastic waves, both the mathematical model and the physical mechanisms are fundamental to the analysis and optimization of nano-scale energy harvesters.

New understanding on the principle of earthquake

Earthquake is a natural disaster that causes enormous losses to human society and its prediction is a major scientific challenge widely concerned by the society. However, the mechanism of earthquake is far from clear, and the mainstream view in the international seismology community is that earthquakes are unpredictable. Based on some new concepts and new knowledge developed in physics, this study scrutinizes the incubation and occurrence of earthquake from a novel perspective, and introduces a new understanding of earthquake principle. It is found that the view of earthquake unpredictability originates from the incorrect understanding of both earthquake principle and the self-organized criticality (SOC). That is to say, earthquake is consistent with the laws of SOC, which means it would be impossible to make a medium- or long-term prediction, yet the short-term prediction should still be possible. The preconditions for successful prediction include understanding correctly of earthquake principle, obtaining sufficient characteristic precursory information, and gathering relevant geological data. Traditional seismology is based on the solid continuum mechanics which holds the view that earthquakes are caused by brittle fracture of crustal rocks through the so-called “elastic rebound” mechanism. This point of view is seriously inconsistent with many field observations, cannot account for many seismic phenomena. It therefore cannot obtain and understand the earthquake precursory information correctly, and naturally reach the false conclusion that earthquake is unpredictable. Based on the simple fact that the crust is composed of rock blocks with fault gouges filling in between them, we treat the crust as a discrete system and understand the earthquake incubation process by means of granular physics. The new understanding gained is that the tectonic force propagates through force chains formed by the rock blocks, and the rock blocks move in the manner of stick-slip. Furthermore, by carefully analyzing how the strength of crustal rocks and the distributions of tectonic force vary with depth, we propose that the physical mechanism of earthquake is plastic sliding of rock and a jamming — unjamming transition of rock motion. Our novel theory on the earthquake principle and the earthquake processes can explain many seismological phenomena that could not be understood in terms of traditional seismology, such as the heat-flow paradox and the cause of deep-focus earthquake, etc. Based on this new understanding of earthquake principle, we put forward suggestions on how to obtain the earthquake precursory information correctly, so as to realize the goal of short-term prediction of earthquake.

A phase-field modeling for brittle fracture and crack propagation based on the cell-based smoothed finite element method

This work develops a phase field model in the framework of cell-based smoothed finite element method (CS-FEM) aiming to solve problems of brittle fracture in solids. The phase-field method is employed to model complex discontinuous behaviors including crack initiation, crack coalescence, crack branching without using any ad hoc models. The CS-FEM which has been proven softer than the standard FEM is used to solve the equations that govern the continuum mechanics of solids. The current CS-FEM phase field method offers advantages of both S-FEM and phase field method in accurately predicting the evolution process of complex fracture patterns in solids. This CS-FEM phase-field technique is implemented in the commercial software ABAQUS using its user-element (UEL) and user-material (UMAT) modules. The coupled non-linear system of equations is solved by the built-in solver by decoupling the phase and displacement field parameters. Several examples have been presented to demonstrate the feasibility and effectiveness of the present method.

Energy-Conserving Time Discretization of Abstract Dynamic Problems with Applications in Continuum Mechanics of Solids

ABSTRACTAn abstract second-order evolution equation or inclusion is discretized in time in such a way that the energy is conserved at least in qualified cases, typically in the cases when the governing energy is component-wise quadratic or “slightly perturbed” quadratic. Specific applications in continuum mechanics of solids possibly with various internal variables cover vibrations or waves in linear viscoelastic materials at small strains, coupled with some inelastic processes as plasticity, damage, or phase transformations, and also some surface variants related to contact mechanics. The applicability is illustrated by numerical simulations of vibrations interacting with a frictional contact or waves emitted by an adhesive contact of a two-dimensional viscoelastic body.

Sparse direct solver for large finite element problems based on the minimum degree algorithm

A sparse direct solver for large problems from solid continuum mechanics based on the minimum degree algorithm is proposed and tested. The solver is designed to take advantage of the properties of the finite element method, particularly the structure of the finite element mesh. For the minimization of the fill-in in the matrix factors a modification of the approximate minimum degree ordering algorithm of Amestoy, Davis and Duff is utilized. The employed sparse matrix storage format and the algorithms for each of the solver phases are also described. The results of numerical tests of the solver on large real-world finite element problems are presented and its performance is compared to a frontal solver and the PARDISO sparse direct solver.

Weak Lower Semicontinuity of Integral Functionals and Applications

Minimization is a reoccurring theme in many mathematical disciplines ranging from pure to applied ones. Of particular importance is the minimization of integral functionals that is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the involved functional called weak lower semicontinuity. While early studies of lower semicontinuity go back to the beginning of the 20th century the milestones of the modern theory were set by C.B. Morrey Jr. in 1952 and N.G. Meyers in 1965. We recapitulate the development on this topic from then on. Special attention is paid to signed integrands and to applications in continuum mechanics of solids. In particular, we review the concept of polyconvexity and special properties of (sub)determinants with respect to weak lower semicontinuity. Besides, we emphasize some recent progress in lower semicontinuity of functionals along sequences satisfying differential and algebraic constraints which have applications in elasticity to ensure injectivity and orientation-preservation of deformations. Finally, we outline generalization of these results to more general first-order partial differential operators and make some suggestions for further reading.

On Finsler Geometry and Applications in Mechanics: Review and New Perspectives

In Finsler geometry, each point of a base manifold can be endowed with coordinates describing its position as well as a set of one or more vectors describing directions, for example. The associated metric tensor may generally depend on direction as well as position, and a number of connections emerge associated with various covariant derivatives involving affine and nonlinear coefficients. Finsler geometry encompasses Riemannian, Euclidean, and Minkowskian geometries as special cases, and thus it affords great generality for describing a number of phenomena in physics. Here, descriptions of finite deformation of continuous media are of primary focus. After a review of necessary mathematical definitions and derivations, prior work involving application of Finsler geometry in continuum mechanics of solids is reviewed. A new theoretical description of continua with microstructure is then outlined, merging concepts from Finsler geometry and phase field theories of materials science.

AutoMT, a library for tensor operations and its performance evaluation for solid continuum mechanics applications

AutoMT, a library for tensor operations and its performance evaluation for solid continuum mechanics applications

The quest for truth, particularly in mechanics

Received 2 September 2013, in revised form 23 September 2013 Abstract. The author ponders about things that necessarily come into engineering mind when the results obtained by theoretical, numerical and experimental approaches in solid continuum mechanics are correlated and compared with a pious wish to ascertain which of them are 'truer' or closer to 'reality'. This invokes many questions. How ancient and contemporary philosophers viewed the truth? How is truth related to consistency and validity of theoretical, numerical and experimental models we are inventing and employing? What is the role of threshold in physics, engineering, computation and in experiment? How are the basic quantities like time, force, stress, etc. defined? Do we properly understand them? What is the role of singularity in mathematics, physics and in engineering? The doubts stemming from uneasy answers to above pertinent questions are complemented by discussing examples from theoretical, numerical and experimental results obtained by solving dynamical problems in solid continuum mechanics. It should be stressed out that the role of doubts in our understanding the World plays a significantly positive role.

A novel analytical model and energy analysis of thermal stresses in two-phase composites

This paper deals with analytical modelling of thermal stresses in a multi-particle-matrix system with isotropic spherical particles. These particles are periodically distributed in an isotropic infinite matrix. This model system which is characterized by microstructural parameters (particle volume fraction, particle radius) is applicable to two-phase composites of a precipitate-matrix type with isotropic phases. The thermal stresses originate during a cooling process due to a difference in thermal expansion coefficients. The analytical modelling which is based on fundamental equations of solid continuum mechanics represents a combination of different mathematical procedures applied to equilibrium and compatibility equations. This novel analytical model is compared with that which is based on mathematical procedures applied to the equilibrium equations only. The energy analysis of both analytical models which is applied to the SiC-Al2O3 composite is presented.

Mechanics of variable-mass systems—Part 1: Balance of mass and linear momentum

The equations of balance of momentum and energy usually are formulated under the assumption of conservation of mass. However, mass is not conserved when sources of mass are present or when the equations of balance are applied to a non-material volume. Mass then is said to be variable for the system under consideration. It is the scope of the present contribution to review the mechanical equations of balance for variable-mass systems. Our review remains within the framework of the classical, non-relativistic continuum mechanics of solids and fluids. We present general formulations and refer to various fields of applications, such as astronomy, machine dynamics, biomechanics, rocketry, or fluid dynamics. Also discussed are the equations for a single constituent of a multiphase mixture. The present review thus might be of interest to workers in the field of heterogeneous media as well. We first summarize the general balance law and review the Reynolds transport theorem for a non-material volume. Then the latter general formulations are used to derive and to review the equations of balance of mass and linear momentum in the presence of sources of mass in the interior of a material volume. We also discuss the appropriate modeling of such sources of mass. Subsequently, we treat the equations of balance of mass and linear momentum when mass is flowing through the surface of a non-material volume in the absence of sources of mass in the interior, and we point out some analogies to the previously presented relations. A strong emphasis is given in this article to the historical evolution of the balance equations and the physical situations to which they have been applied. The equations of balance of angular momentum and energy for variable-mass systems will be treated separately, in a second part of the review, to be published later. This review article cites 96 references.

Book Review: Nonlinear Continuum Mechanics of Solids. By Y. Basar and D. Weichert.

Book Review: Nonlinear Continuum Mechanics of Solids. By Y. Basar and D. Weichert.

Erratum: Nonlinear Continuum Mechanics of Solids: Fundamental Mathematical and Physical Concepts

11R2. Nonlinear Continuum Mechanics of Solids: Fundamental Mathematical and Physical Concepts. - Y Basar (Inst fur Statik und Dynamik, Ruhr-Univ Bochum, Universitatsstr 150, Bochum, 44780, Germany) and D Weichert (Inst fur Allgemeine Mech, RWTH Aachen, Templergraben 62, Aachen, 52056, Germany). Springer-Verlag, Berlin. 2000. 193 pp. ISBN 3-540-66601-X. $59.95.Reviewed by J Petrolito (Sch of Sci and Eng, La Trobe Univ, PO Box 199, Bendigo, Vic 3350, Australia).This is a graduate-level textbook on the fundamental concepts of nonlinear solid mechanics. The authors’ stated goal is to provide students with sufficient background to carry out research in this area. As such, the book emphasizes the theoretical aspects of the subject rather than dealing with specific applications. However, given that this is a textbook, it is surprising that there are not many exercises to help students reinforce their learning.The book, divided into six chapters and an appendix, makes extensive use of tensor analysis, and the first chapter discusses the basic aspects of this theory. This chapter is complemented by the appendix on index notation and general coordinate systems. The combination serves as a concise introduction to tensor analysis in its own right. The second chapter is concerned with the concepts of deformation and strain. There is considerable discussion on the distinction between material and spatial coordinates and the conversions between the two systems. Various nonlinear strain measures are defined, and these are shown to be particular cases of a unified definition of strain. In total, the first two chapters take up around half the book. Chapter 3 discusses the concept of stress and shows how different stress measures are associated with appropriate energy-conjugate strain measures. This is followed by a very short chapter on time derivatives that could have been incorporated into Chapter 2. The next chapter discusses the various balance laws in mechanics and includes a brief discussion on the principle of virtual work. The final chapter details the requirements of constitutive relationships, with particular emphasis on elastic materials.It is interesting to compare this book with another recent book that broadly covers the same area 1. Both books emphasize theory over practical applications, with 1 taking a more abstract approach to the subject. The current book, being around twice the length, devotes considerably more attention to tensor analysis and its use in solid mechanics. But ultimately, both books stop too soon. While they clearly develop the basic concepts, neither book provides any significant insight on how to apply the theory in practice. Even a short chapter on some typical applications would have enhanced both books.In summary, the current book, Nonlinear Continuum Mechanics of Solids: Fundamental Mathematical and Physical Concepts, provides a useful introduction to the theoretical aspects of nonlinear solid mechanics at the graduate level. It will appeal to students who want to quickly gain an understanding of the basic principles. Students who need guidance on particular problems in the area will need to look to other books for this.

Synergetic principles of physical mesomechanics

A synergetic methodology of physical mesomechanics is developed. Fundamental principles underlying the approach are discussed and formulated. The methodology is shown to account for and substantiate well-known plastic-deformation and fracture mechanisms in solids under various loading conditions. An analysis of the available experimental evidence suggests that the key meso- and macroscale processes developed in solids under load occur at maximum tangential stresses. Microscale deformation caused by dislocations is self-adaptable. Physical mesomechanics establishes natural relations between the physics of dislocation-induced deformation, continuum mechanics of solids, and fracture mechanics.

Pulmonary interstitial resistance.

The mechanical properties of the perivascular interstitium surrounding large pulmonary blood vessels are defined in terms of interstitial fluid pressure, interstitial compliance, and interstitial hydraulic resistance. Interstitial pressure is one of the main forces which determine liquid filtration across the microvascular barrier. Interstitial compliance is a measure of the ability of the interstitium to swell with hydration which increases interstitial pressure and reduces the filtration rate. Interstitial pressure and compliance are functions of the elastic properties of the surrounding lung parenchyma and the vessel wall. Solid continuum mechanics are used to describe the behavior of the lung parenchyma. The transport properties of the interstitium are described in terms of a porous material whose fluid resistance is determined by a permeability constant. The dynamics of interstitial fluid are governed by the coupling of the flow with the elastic environment. An electrical analog model is developed to predict the growth of interstitial fluid cuffs during edema formation.

On a variational formulation of problems in the classical continuum mechanics of solids

The basic equations of linear elasticity, of mildly non-linear elasticity and of plastic flow are reduced to a generalization of Hamilton's canonical formalism. Complementary variational principles are deduced. Numerical applications of these principles are demonstrated in some simple cases.