Continuum mechanics: General principles, constitutive relations

2017 ◽  
pp. 77-86
Keyword(s):  
Continuum Mechanics ◽  
Constitutive Relations

Atomistic Aspects of Fracture Modelling in the Framework of Continuum Mechanics

1998 ◽  
Vol 538 ◽  
Author(s):  
F. Cleri
Keyword(s):  
Continuum Mechanics ◽  
Constitutive Relations ◽  
Point Of View ◽  
Atomic Scale ◽  
Continuum Models ◽  
Cohesive Law ◽  
Macroscopic Models ◽  
Scale Models ◽  
Level Information ◽  
Set Up

AbstractThe validity and predictive capability of continuum models of fracture rests on basic informations whose origin lies at the atomic scale. Examples of such crucial informations are, e.g., the explicit form of the cohesive law in the Barenblatt model and the shear-displacement relation in the Rice-Peierls-Nabarro model. Modem approaches to incorporate atomic-level information into fracture modelling require to increase the size of atomic-scale models up to millions of atoms and more; or to connect directly atomistic and macroscopic, e.g. finite-elements, models; or to pass information from atomistic to continuum models in the form of constitutive relations. A main drawback of the atomistic methods is the complexity of the simulation results, which can be rather difficult to rationalize in the framework of classical, continuum fracture mechanics. We critically discuss the main issues in the atomistic simulation of fracture problems (and dislocations, to some extent); our objective is to indicate how to set up atomistic simulations which represent well-posed problems also from the point of view of continuum mechanics, so as to ease the connection between atomistic information and macroscopic models of fracture.

scholarly journals State Variables in Saturated-Unsaturated Soil Mechanics

2016 ◽  
pp. 3-17
Author(s):  
Delwyn G. Fredlund
Keyword(s):  
Stress State ◽  
Continuum Mechanics ◽  
Unsaturated Soil ◽  
Material Properties ◽  
Soil Mechanics ◽  
Constitutive Relations ◽  
Saturated Soil ◽  
Clear Distinction ◽  
State Variables ◽  
Unsaturated Soil Mechanics

The description of the stress state in soils is the foundational point around which an applied science should be built for engineering practice. The stress state description has proven to be pivotal for saturated soil mechanics and the same should be true for unsaturated soil mechanics. Continuum mechanics sets forth a series of principles upon which a common science base can be developed for a wide range of materials. The principles require that there be a clear distinction between state variables and constitutive relations. Constitutive relations relate state variables and incorporate material properties. State variables, on the other hand, are independent of the material properties. It has been possible to maintain a clear distinction between variables of state and constitutive relations in the development of saturated soil mechanics and the same should be true for unsaturated soil mechanics. This paper presents a description of the source and character of stress state variables for saturated and unsaturated soils. The descriptions are consistent with the principles of multiphase continuum mechanics and provide an understanding of the source and importance of stress state variables.

scholarly journals Stress theory for classical fields

2017 ◽  
Vol 25 (7) ◽  
pp. 1472-1503 ◽  
Author(s):  
Raz Kupferman ◽  
Elihu Olami ◽  
Reuven Segev
Keyword(s):  
Continuum Mechanics ◽  
Configuration Space ◽  
Fiber Bundle ◽  
Constitutive Relations ◽  
General Setting ◽  
The Body ◽  
Dimensional Manifold ◽  
Stress Theory ◽  
Infinite Dimensional ◽  
Geometric Setting

Classical field theories, together with the Lagrangian and Eulerian approaches to continuum mechanics, are embraced under a geometric setting of a fiber bundle. The base manifold can be either the body manifold of continuum mechanics, the space manifold, or space–time. Differentiable sections of the fiber bundle represent configurations of the system and the configuration space containing them is given the structure of an infinite-dimensional manifold. Elements of the cotangent bundle of the configuration space are interpreted as generalized forces and a representation theorem implies that there exists a stress object representing forces, non-uniquely. The properties of stresses are studied, as well as the role of constitutive relations in this general setting.

scholarly journals Tensors in Newtonian Physics and the Foundations of Classical Continuum Mechanics

2019 ◽  
Vol 24 (3) ◽  
pp. 79
Author(s):  
George L. Brovko
Keyword(s):  
Mechanical Properties ◽  
Continuum Mechanics ◽  
Constitutive Relations ◽  
Physical And Mechanical Properties ◽  
The Body ◽  
Newtonian Physics ◽  
Axiomatic Construction ◽  
Strain Tensors ◽  
Objective Tensors

In the Newtonian approach to mechanics, the concepts of objective tensors of various ranks and types are introduced. The tough classification of objective tensors is given, including tensors of material and spatial types. The diagrams are constructed for non-degenerate (“analogous”) relations between tensors of one and the same (any) rank, and of various types of objectivity. Mappings expressing dependence between objective tensor processes of various ranks and types are considered. The fundamental concept of frame-independence of such mappings is introduced as being inherent to constitutive relations of various physical and mechanical properties in the Newtonian approach. The criteria are established for such frame-independence. The mathematical restrictions imposed on the frame-independent mappings by the objectivity types of connected tensors are simultaneously revealed. The absence of such restrictions is established exclusively for mappings and equations linking tensors of material types. Using this, a generalizing concept of objective differentiation of tensor processes in time, and a new concept of objective integration, are introduced. The axiomatic construction of the generalized theory of stress and strain tensors in continuum mechanics is given, which leads to the emergence of continuum classes and families of new tensor measures. The axioms are proposed and a variant of the general theory of constitutive relations of mechanical properties of continuous media is constructed, generalizing the known approaches by Ilyushin and Noll, taking into account the possible presence of internal kinematic constraints and internal body-forces in the body. The concepts of the process image and the properties of the five-dimensional Ilyushin’s isotropy are generalized on the range of finite strains.

Viscoelastic Model of Finitely Deforming Rubber and Its Finite Element Analysis

1997 ◽  
Vol 64 (4) ◽  
pp. 835-841 ◽  
Author(s):  
Seung Jo Kim ◽  
Kyeong Su Kim ◽  
Jin Yeon Cho
Keyword(s):  
Finite Element ◽  
Continuum Mechanics ◽  
Constitutive Relations ◽  
Viscoelastic Model ◽  
Finite Element Approximation ◽  
Nonlinear Finite Element ◽  
Element Approximation ◽  
Extended Model ◽  
Element Analysis ◽  
Highly Nonlinear

A viscoelastic model of finitely deforming rubber is proposed and its nonlinear finite element approximation and numerical simulation are carried out. This viscoelastic model based on continuum mechanics is an extended model of Johnson and Quigley’s one-dimensional model. In the extended model, the kinematic configurations and measures based on continuum mechanics are rigorously defined and by using these kinematic measures, constitutive relations are introduced. The obtained highly nonlinear equations are approximated by the nonlinear finite element method, where a mixture of the total and updated Lagrangian descriptions is used. To verify the theory and the computer code, uniaxial stretch tests are simulated for various stretch rates and compared with actual experiments. As a practical example, an axisymmetric rubber plate under various time-dependent pressure loading conditions is analyzed.

scholarly journals Extension of Boley’s continuum mechanics-based successive approximation method to two-layer rectangular beams

2021 ◽  
Author(s):  
Johannes Gahleitner ◽  
Hans Irschik
Keyword(s):  
Boundary Conditions ◽  
Successive Approximation ◽  
Continuum Mechanics ◽  
Approximation Method ◽  
Constitutive Relations ◽  
Single Layer ◽  
Successive Approximation Method ◽  
Field Equations ◽  
Compact Formulation ◽  
Stress Boundary Conditions

AbstractAn extension of Boley’s continuum mechanics-based successive approximation method is presented for rectangular beams composed of two isotropic linear elastic layers. The solution is cast into the form of tables, in complete analogy to the tables originally presented by Boley and Tolins for single-layer strips. The first column in these tables corresponds to the classical Bernoulli–Euler theory of beams. The further columns represent comparatively fast converging correction terms of an increasing refinement. Our two-layer formulation automatically satisfies the stress continuity conditions at the interface of the two layers. Enforcing displacement continuity at the interface between the layers, we derive results that do satisfy the equilibrium field equations, the stress continuity conditions at the interface and the stress boundary conditions at the upper and lower edges. When converged, the field constitutive relations and the displacement continuity at the interface between the two layers are also satisfied. We present a compact formulation, which allows writing down the results for more than the three successive steps considered by Boley and Tolins. The elasticity solutions presented subsequently can be used as novel analytic benchmarks for comparison with refined structural mechanics beam theories. Interior solutions for beams with a finite axial extent can be obtained by assigning approximate boundary conditions at the lateral ends. Comparisons to finite element computations for a clamped–clamped beam give strong evidence for the correctness of our analytic results.

scholarly journals About applicability of the winkler model for contact interaction of cylindrical elastoplastic shells with an elastic filler at external pressure

2020 ◽  
pp. 3-8
Author(s):  
Валентин Георгиевич Баженов ◽  
Елена Владимировна Нагорных ◽  
Дарья Анатольевна Самсонова
Keyword(s):  
Contact Interaction ◽  
Continuum Mechanics ◽  
Constitutive Relations ◽  
Equations Of Motion ◽  
External Pressure ◽  
Isotropic Hardening ◽  
Shells Of Revolution ◽  
Plastic Properties ◽  
Wide Range ◽  
Elastic Filler

Представлено сравнение результатов расчетов контактного взаимодействия и потери устойчивости упругопластических цилиндрических оболочек с упругим толстостенным заполнителем, выполненных на основе двух подходов: с позиций механики сплошных сред и теории оболочек типа Тимошенко с основанием Винклера. Оба подхода позволяют решать задачи деформирования и устойчивости непологих оболочек с учетом геометрических нелинейностей. Постановка с позиций механики сплошных сред позволяет аппроксимировать оболочку по толщине рядом слоев конечных элементов. Определяющие соотношения формулируются в переменных Лагранжа с использованием в качестве отсчетной неподвижной декартовой или цилиндрической системы координат. Кинематические соотношения записываются в метрике текущего состояния. Упругопластические свойства оболочек описываются теорией пластического течения с изотропным упрочнением. Уравнения движения следуют из баланса виртуальных мощностей работ. В первом подходе контактное взаимодействие оболочки и упругого тела моделируется условиями непроникания по нормали и свободного проскальзывания вдоль касательной. Во втором подходе контактное взаимодействие упругого заполнителя с оболочкой моделируется основанием Винклера. Оба подхода позволяют описать нелинейное докритическое деформирование оболочек вращения с упругим заполнителем, определить предельные (критические) нагрузки в широком диапазоне скоростей нагружения с учетом геометрических несовершенств формы. Оценивается область применимости гипотезы Винклера при контактном взаимодействии оболочки с упругой средой в зависимости от жесткости и толщины основания. Comparison of the results of calculations of contact interaction and loss of stability of elastoplastic cylindrical shells with an elastic thick-walled filler, performed on the basis of two approaches: from the standpoint of continuum mechanics and the theory of Timoshenko-type shells with a Winkler base is presented. Both approaches allow solving the problems of deformation and stability of non-sloping shells, taking into account geometric nonlinearities. The statement from the perspective of continuum mechanics makes it possible to approximate the shell in thickness by a number of layers of finite elements. The constitutive relations are formulated in Lagrange variables using a fixed Cartesian or cylindrical coordinate system as a reference. Kinematic relations are recorded in the metric of the current state. The elastic-plastic properties of shells are described by the theory of plastic flow with isotropic hardening. The equations of motion follow from the balance of the virtual powers of the jobs. In the first approach, the contact interaction of a shell and an elastic body is modeled by the conditions of nonpenetration along the normal and free slip along the tangent. In the second approach, the contact interaction of the elastic filler with the shell is modeled by the Winkler base. Both approaches allow one to describe the nonlinear subcritical deformation of shells of revolution with an elastic filler, to determine the limiting (critical) loads in a wide range of loading rates, taking into account the geometric imperfections of the shape. The area of applicability of the Winkler hypothesis is estimated for the contact interaction of a shell with an elastic medium, depending on the stiffness and thickness of the base.

scholarly journals Analysis of the fracture toughness parameters at the free edge in layered composites

2020 ◽  
pp. 49-59
Author(s):  
D. A Bondarchuk ◽  
B. N Fedulov ◽  
A. N Fedorenko ◽  
E. V Lomakin
Keyword(s):  
Contact Interaction ◽  
Continuum Mechanics ◽  
Constitutive Relations ◽  
External Pressure ◽  
Isotropic Hardening ◽  
Winkler Foundation ◽  
Shells Of Revolution ◽  
Plastic Properties ◽  
Elastic Core ◽  
Wide Range

The problem of deformation and elastoplastic buckling of shells of revolution with a thick-walled elastic core under combined static and dynamic loading is formulated in a two-dimensional planar formulation based on two approaches: full-scale modeling within the framework of continuum mechanics and a simplified formulation based on the hypotheses of the theory of shells of the Timoshenko type and the Winkler foundation. Both approaches allow solving the problems of deformation and stability of non-shallow shells on the basis of Timoshenko's hypotheses, taking into account geometric nonlinearities. The statement from the perspective of continuum mechanics makes it possible to approximate the shell in thickness by a number of layers of finite elements. The constitutive relations are formulated in Lagrange variables using a fixed Cartesian coordinate system as a reference one. Kinematic relations are recorded in the metric of the current state. The elastic-plastic properties of shells are described by the theory of plastic flow with isotropic hardening. The equations of motion follow from the balance of the virtual powers of the work. In the first approach, the contact interaction of a shell and an elastic body is modeled by the conditions of nonpenetration along the normal and free slip along the tangent. The nonpenetration conditions are satisfied only in the active phase of the contact interaction; if the contact is broken, they are replaced by conditions on the free surface. In the second approach, the contact interaction of the elastic core with the shell is modeled by the Winkler foundation. Both approaches allow one to describe the nonlinear subcritical deformation of shells of revolution with an elastic core, to determine the limiting (critical) loads in a wide range of loading rates, taking into account the geometric imperfections of the shape. Using both approaches, a numerical simulation of contact interaction problem of an elastoplastic cylindrical shell with a thick-walled elastic core at a quasi-static uniform external pressure is carried out. The study of the influence of the thickness and initial deflection of the shell, as well as the stiffness and thickness of the core, on the value of the critical pressure and the form of buckling has been carried out. Based on these calculations, a conclusion was made about a wide range of applicability of the Winkler foundation model.

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