Nonlocal theories in continuum mechanics article pdf available in acta polytechnica 4456. Deformable bodies occupy regions in threedimensional euclidean space e, and a given body will occupy di. However, all these objectives begin with the same first step quantifying the objects. The concept of strain strain is a fundamental concept in continuum and structural mechanics. Continuum mechanics ch 2 lecture 2 deformation gradient. The material deformation gradient tensor characterizes the local deformation at a material point with position vector, i. Even in small deformation linear elasticity, the eshelby stress tensor t is quadratic in f. So, the orthogonal rotation tensor do has intrinsic mechanical significance.
In small deformation linear elasticity where p, s and s are all equivalent, the stress tensor s is linear in the deformation gradient f. Chung, general continuum mechanics, cambridge university press, 2007. Continuum mechanics ch 2 lecture 2 deformation gradient tensor multimedia course. Cartesian basis vectors in reference configuration. Continuum mechanicsvolume change and area change wikiversity. The deformation gradient f is the fundamental measure of deformation in continuum mechanics. Continuum mechanics concepts joseph teran 15 min continuum description of material motion kinematics, deformation gradient, strain stress and hyperelasticity governing equations, conservation of massmomentum 4. The deformation gradient is used to separate rigid body translations and rotations from deformations, which are. Vectors, tensors, tensor properties, vector and tensor fields, configurations, motion, the lagrangian description of motion, stretch and strain tensors, the polar decomposition, velocity gradients, and rates of deformation, balance of mass, reynolds transport theorem, linear and angular momentum, models of viscous fluids. The mathematics of deformation continuum mechanics deals with the mathem atics, mechanics, kinematics and physics of deformable bodies. Be able to describe motion, deformation and forces in a continuum. The deformation gradient is used to separate rigid body translations and rotations from deformations, which are the source of stresses.
The operator f is known as the deformation gradient. It also carries information about the rotation of the vector dzds. A very useful interpretation of the deformation gradient is that it causes simultaneous stretching and rotation of tangent vectors. Deformation and strain online course on continuum mechanics fundamentals gl strain duration. Continuum deformation gradient as motivation for a discrete deformation gradient, we. The approach taken in this text is to use continuum mechanics as a universal tool in which to formulate the polymer mechanics theories that will be. Our focus is on relating deformation to quantities that can be measured in the. In tissue mechanics, hard tissues fit under the small deformation model, but most soft tissues typically undergo large deformation strains 5%. In this section, we will discuss and derive deformation and strain measures for both small and large deformation. Evolution of continuum from elastic deformation to flow arxiv. A deformation gradient tensor and strain tensors for. Continuum mechanics variables for general nonlinear analysis the principle ofvirtualworkinterms ofthe 2nd piolakirchhoffstress and greenlagrangestraintensors deformation gradient tensor physicalinterpretationofthe deformation gradient change ofmass density polardecomposition ofdeformation gradient green. A material body b fxgis a compact measurable set of an in nite number of material elements x, called the material particles or material points, that can be placed in a.
And this page and the next, which cover the deformation gradient, are the. A survey of the application of the multiplicative decomposition in these three areas of. Displacement elds and strains can be directly measured using gauge clips or the digital image correlation dic method. The deformation gradient f is a tensor with the coordinate representation.
And this page and the next, which cover the deformation gradient, are the center of that heart. Continuum mechanics an overview sciencedirect topics. Continuum mechanics is a mathematical framework for studying the transmission of force through and deformation of materials of all types. Dec 27, 2012 hi all, i am trying to selflearn continuum mechanics, and i have a question regarding the development of the deformation gradient which ultimately leads to greens deformation tensor. I have attached the specifics of the question in a attached photo. Deformation gradient visualizer file exchange matlab. Continuum mechanics of any material operates with some fundamental concepts characterizing dynamic stresses, geometrical deformations and kinematic deformation rates situation at a point site. A continuum disregards the molecular structure of matter.
In this chapter we concentrate upon the conceptual and mathematical details necessary to generalize the problem structure introduced previously for kinematically linear problems, such that our formulation for contact interaction becomes suitable for large deformation, large motion behavior. Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. Anand, the mechanics and thermodynamics of continua, cambridge university press, 2010. The material point method for simulating continuum materials. Rotation and stretch polar decomposition f r u v r however one can also express the effect of fin terms of a sequence of stretching and rotation operations. Continuum mechanics models the physical universe as a collection of deformable bodies, a concept that is easily accepted from our everyday experiences with observable phenomena. For continuum, the displacement gradient can be divided into symmetry part and. Continuum mechanics is a subject that unifies solid mechanics, fluid mechanics, thermodynamics, and heat transfer, all of which are core subjects of mechanical engineering. For many materials it is convenient to decompose the deformation gradient in a volumetric part and a distortional part. A configuration is a set containing the positions of all particles of the body. A deformation may be caused by external loads, body forces such as gravity or electromagnetic forces, or changes in temperature, moisture. Gurtin, an introduction to continuum mechanics, academic press, 1981.
Topic 3 lagrangian continuum mechanics variables for analysis. Theory manuals 92298 nonlinear continuum mechanics measures of deformation measures of deformation 5 deformation tensor as any other nonsymmetric secondorder tensor, the velocity gradient can be decomposed into a symmetric l and skew part. The goal is to construct a framework that is free of special assumptions about the type of material, the size of deformations, the geometry of the problem and so forth. Continuum mechanicsstrains and deformations wikiversity.
The assumption that the deformation gradient is a driving factor in describing the constitutive behavior of elastic materials, embodied in eq. The deformation gradient and its inverse play key roles in describing the local deformation in the neighborhood of a continuum particle. The ultimate goal is often the determination of the stress, strength, fatigue, and fracture properties of an object or material. It also carries information about the rotation of the vector. It is the second order tensor which maps line elements in the reference configuration into line elements consisting of the same material particles in the current configuration. The deformation gradient carries the information about the stretching in the infinitesimal neighborhood of the point z. Continuum mechanics is a combination of mathematics and physical laws that approximate the largescale behavior of matter that is subjected to mechanical loading. Vectors, tensors, tensor properties, vector and tensor fields, configurations, motion, the lagrangian description of motion, stretch and strain tensors, the polar decomposition, velocity gradients, and rates of deformation, balance of mass, reynolds transport theorem, linear and angular momentum, models of viscous. The deformation gradient and displacement gradient tensors therefore do provide information on the presence. This is a set of notes written as part of teaching me185, an elective senioryear under. The book introduces the invariant form as well as the component form of the basic equations and their applications to problems in elasticity. This is particularly useful when there is no volume change in the material when it deforms for example in muscles, rubber tires, metal plasticity, etc. The deformation gradient and displacement gradient tensors therefore provide no information on the presence or absence of a rigid body translation. The deformation of the body maps the material vector in the reference state to the material vector in the current state.
Hi all, i am trying to selflearn continuum mechanics, and i have a question regarding the development of the deformation gradient which ultimately leads to greens deformation tensor. Theory manuals 92298 nonlinear continuum mechanics measures of deformation measures of deformation 5 stress, and conservation laws in this chapter, we will develop a mathematical description of deformation. It is a generalization of newtonian particle dynamics, and starts with the same physical assumptions inherent to newtonian mechanics. To calculate the derivative of j det f \displaystyle j\det \boldsymbol f with respect to c \displaystyle \boldsymbol c, we recall that for any second order tensor t \displaystyle \boldsymbol t. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations the directional derivative provides a systematic way of finding these derivatives. The deformation gradient f describes stretches and rigid body motion of the material fibers from 0 to n in contrast to a linear analysis, where we may apply a linear strain measure e. Deformation gradient tensor an overview sciencedirect topics. This chapter on displacements and deformations is the heart of continuum mechanics.
An introduction to continuum mechanics, second edition. Mpm algorithm explicit integration chenfanfu jiang 15 min particlegrid transfers. Deformation gradient tensor an overview sciencedirect. F ru or a sequence of rotation and stretching operations. The derivatives of scalars, vectors, and secondorder tensors with respect to secondorder tensors are of considerable use in continuum mechanics. The color may be changed so that multiple plots can be overlaid. D for cases of no deformation and rigid body rotation the respective tensors are different.
A continuum disregards the molecular structure of matter and pictures it as being without gaps or empty spaces. Mass spring vs continuum mechanics mass spring systems. Continuum mechanics deformation gradient physics forums. Be familiar with linear vector spaces relevant to continuum mechanics and able to perform vector and tensor manipulations in cartesian and curvilinear coordinate systems. It starts with a revision of tensor analysis that discusses the definition of tensor and coordinate transformations. Continuum mechanics ch 2 lecture 5 strain tensors youtube. A deformation may be caused by external loads, body forces such as gravity or electromagnetic forces, or changes in temperature, moisture content, or chemical reactions, etc. In this approach, a point is always understood in a mathematical sense as infinitesimal small objects. By considering the various weakforms of the balance law for t itself, we derive a variety of con. The derivative of j with respect to the right cauchygreen deformation tensor is also often encountered in continuum mechanics. Continuum mechanics of large deformation contact springerlink.
Distortional component of the deformation gradient. The deformation of the body is homogeneous if f is a linear operator, and is the same for any choices of the two material particles. This function simply plots the deformation that would occur to a unit cube if a prescribed deformation gradient is applied to it. Summary this section gets to the heart of what continuum mechanics is all about dealing with large displacements and deformations of objects. The word kinematics refers to the mathematics of motion. The deformation gradient tensor quantifies the changes in these edge vectors by assembling the three deformed edge vectors into columns of a.
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