And so, let us spend a little bit of time on this tensor, to get also a bit of a physical feel for what it means, what it stands for, and what can we do with it. Conservation laws in einsteins gravitation theory 15. Electromagnetic potentials making use of the homogeneous maxwell equations we want to introduce the electromagnetic potentials. Technical question ive been reading about how to generate the stress energymomentum tensor t\mu \nu. Stressenergy tensor and conservation of energy and momentum christopher m. If you look at them in tensor forms, the engineering strain corresponding to f i in tensor will not exclude rigid body motions, while the other two will. Topic 3 lagrangian continuum mechanics variables for analysis. I hope anybody will derive the most profit from comparing that to his own notes. Conjugate stresses of the sethhill strain tensors request pdf. The elastic energy momentum tensor in special relativity. The 2nd piolakirchhoff stress tensor and the green lagrange strain tensor are energetically conjugate. Stress energy tensor and conservation of energy and momentum christopher m. The quadratic terms are what gives the green strain tensor its rotation independence. Similarly principal values of the lagrangian strain tensor.
The standard derivation of these strain tensors is done with the help of shifter tensors 8,9. In general, bodiesstructures movedeform when forces are acting on them, but we are not concerned for now about the causes of this motiondeformation. A deformation gradient tensor and strain tensors for. The derivation explicitly shows that the eulerian and lagrangian strain tensors are twopoint tensors, each a function of both the spatial and material coordinates. Module 2 kinematics of deformation and strain learning objectives develop a mathematical description of the local state of deformation at a material point understand the tensorial character of the resulting strain tensor. Most of the descriptions are taken from the internet site.
That is multiplying this stress tensor with the green lagrange tensor will not be equal to the product of the cauchy stress tensor multiplied by the deformation strain tensor. What is the physical meaning of green lagrangian strain and eulerianalmansi strain measures. In this video, the fundamental terms leading up to the green lagrange strain are provided. Overview in this lecture, we will consider the spatial distribution of energy and momentum and their transport and conservation laws. Derivation of maxwell stress tensor from em lagrangian.
An isoparametric finite element model for largestrain. Pdf on mindlins isotropic strain gradient elasticity. Section vi contains a detailed calculation of the stress energymomentum. The eulerian strain is a twopoint tensor that transforms as a. T1 rst piolakirchho stress tensor t2 second piolakirchho stress tensor v volume at v volume at 0 v velocity at v lagrangian description of v x eulerian cartesian coordinates x lagrangian cartesian coordinates e total internal energy h total entropy k total kinetic energy w total mechanical power internal energy density per unit mass. The emphasis is mostly on the lagrangian formulation of classical electrodynamics and the solution of maxwells equations by greens function methods.
To satisfy the objectivity requirement and to obtain exact cauchy stress tensor, we first have to calculate the incremental green lagrange strain, from which, the incremental second piolakirchhoff stress can be. Again, they all describe the same stress state of the material, but add to no extra physical meanings you can argue that nominal stress is the force devided by the area in the reference state while true stress is that devided. Cauchygreen tensor an overview sciencedirect topics. Specifically, the left cauchy green strain and right cauchy green strain tensors give a measure of how the lengths of line elements and angles between line elements through the vector dot product change between configurations. Continuum mechanicsstrains and deformations wikiversity. The green lagrange strain tensor is inherently obtained in material. One of such strains for large deformations is the lagrangian finite strain tensor, also called the greenlagrangian strain tensor or green stvenant strain tensor, defined as or the. 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. Nov 21, 20 in structural mechanics you will come across a plethora of stress and strain definitions.
Cartesian basis vectors in reference configuration. Since the pk2 stress tensor and its workconjugate green lagrange strain is defined in the material coordinate system, it does not change at all under rigid body rotations bathe says invariant. Therefore, it is imperative that the stress and strain tensors be conjugate. Both the lagrangian and eulerian finite strain tensors can be conveniently expressed in terms of the displacement gradient tensor. A continuum is an ideal material body, where the neighborhood of a material point. They do not contain a lot of words but mainly mathematical equations. If the material rotates without a change in stress state rigid rotation, the components of the 2nd piolakirchhoff stress tensor remain constant, irrespective of material orientation.
The tensor c can be equivalently substituted by the tensor c. Technical question ive been reading about how to generate the stressenergymomentum tensor t\mu u. Buttimore replaced by ma3431 classical field theory and ma3432 classical electrodynamics, the former corresponding to at least the rst four sections of these notes. The greenlagrange strain of a line element with a known direction e0 in the. Green lagrange strain tensor an overview sciencedirect topics. As long as you keep consistency when applying the constitutive model in your study, you can use any form of the stress strain. The lagrangian strain is the difference between the square of the deformed length and the. One of such strains for large deformations is the lagrangian finite strain tensor, also called the green lagrangian strain tensor or green stvenant. Finite deformations from statics and kinematics to. The emphasis is mostly on the lagrangian formulation of classical electrodynamics and the solution of maxwells equations by green s function methods.
It may be a second piolakirchhoff stress or a logarithmic strain. A configuration is a set containing the positions of all particles of the body. Oct 16, 2014 lectures for transport phenomena course at olin college this lecture describes what the stress tensor is. A number of forces are acting on this body in different directions but the net force the vector sum of the forces on the body is 0. Stressenergymomentum tensors in lagrangian field theory. The deformation gradient tensor is actually something very, very basic in continuum mechanics. The action principle determines the bulk equations of motion as well as the natural boundary conditions for the body. In this paper, a general class of eshelbylike or weighted stress tensors based on the right stretch tensor is defined as the product of a general type of lagrangian stress tensor and a class of. Other stress tensors effort or surface can be considered in other congifurations ii stress 1 stress tensors pierre badel emse ds 0 ds n n df df df. The finite element method for the analysis of nonlinear. F ru or a sequence of rotation and stretching operations. Particularities of the strain tensors the green lagrange and the euleralmansi strain tensors are different tensors. The notion of conjugation in this context was introduced in the last century. Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration.
The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it. Section vi contains a detailed calculation of the stressenergymomentum. Stress energymomentum tensors of gauge potentials 11. Lagrangian formulation the stress tensor does not have. What is the physical meaning of greenlagrangian strain and. Lagrangian stresses piola kirchoff stress tensors elasticity. The action principle for a relativistic hyperelastic body is presented in sec. Deformation in lagrangian coordinates our notation is similar to, but not identical with, that of green and zerna 2. For the lagrangian strain tensor, first we differentiate the displacement vector, with respect to the material coordinates to obtain the material displacement gradient tensor. Derivative of deformation gradient with respect to green. Derivative of deformation gradient with respect to green lagrangian strain. Finite deformations in geomechanics are common modelling failure mechanisms. What is the physical meaning of greenlagrangian strain.
Mse203 3d stress tensors finding eigenvalues and eigenvectors duration. The tensor captures the straining of the body and the rotation of the body. Cooper the attached pdf provides a derivation of greenlagrange straindisplacement relations in twodimensional cartesian coordinates. This gadget is the thing that appears on the right side of einsteins equation for general relativity. The canonical stress energy tensor from noethers theorem is not necessarily symmetric, and often needs to be improved with appropriate improvements terms. Transformation properties of the lagrangian and eulerian. N ds df s n ds i ij j s piola kirchhoff 2 stress tensor pk2. They are affected by different vectors dx and dx when measuring distances. 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. General relativity tutorial the stressenergy tensor john baez in local coordinates, the stressenergy tensor may be regarded as a 4x4 matrix t ab at each point of spacetime. All the rules for transformations, principal values, hydrostatic and deviatoric components, etc, are the same in rectangular coordinates as in cylindrical coordinates. To satisfy the objectivity requirement and to obtain exact cauchy stress tensor, we first have to calculate the incremental greenlagrange strain, from which, the incremental second piolakirchhoff stress can be obtained. Ma432 classical field theory trinity college, dublin.
To satisfy the objectivity requirement and to obtain exact cauchy stress tensor, we first have to calculate the incremental green lagrange strain, from which, the incremental second piolakirchhoff stress can be obtained. May 05, 2018 in the spatial reference frame, we ought to see a change of the stress tensor for any element that undergoes a rotation. However, we might be examining a problem of large displacements with small strains. Chapter 1 kinematics the motion and deformation of a threedimensional continuum is studied in continuum mechanics. Green tensor of the threedimensional navier opera tor, namely the. Our focus is on relating deformation to quantities that can be measured in the. The 2nd piolakirchhoff stress tensor is energy conjugate to the green lagrange finite strain tensor. J a stress tensor work conjugate to the green strain tensor g must be referred to the initial undeformed configuration as is the green strain tensor.
It may be shown that the 2nd piolakirchhoff pk stress tensor s that gives the transformed current force df per. They are not the material and spatial descriptions of a same strain tensor. Introduction to continuum mechanics computational solid. Green lagrange strain tensor an overview sciencedirect. This is followed by the analysis of strainsdisplacement relations in beams 1d and plates 2d. Right cauchy green deformation tensor lagrangian tensor c. The green deformation tensor c is cf tf a6 the stretch of the line oriented in the direction n of the undeformed configuration can then be computed as o 2n n x cn a7 equation a7 holds for any curve with dzds n. I in the initial tensor basis made up of the vectors gi defined by. Tl formulation second piolakirchho stress tensor, green lagrange strain tensor. Derivative of j with respect to the right cauchy green deformation tensor the derivative of j with respect to the right cauchy green deformation tensor c \displaystyle \boldsymbol c is also often encountered in continuum mechanics. The right cauchygreen deformation tensor is fully defined in the undeformed.
Distance variation, cauchy green tensor, green lagrange tensor 3 strain. Deformation, stress, and conservation laws in this chapter, we will develop a mathematical description of deformation. Continuum mechanics and nonlinear elasticity stefano giordano. General relativity tutorial the stressenergy tensor. Kinematics of deformation in this chapter, we will study how bodiesstructures movedeform and how can this motiondeformation be described mathematically. A very useful interpretation of the deformation gradient is that it causes simultaneous stretching and rotation of tangent vectors. Lagrangian and euler coordinates kinematics is the description of motion without regard to forces. Imagine an arbitrary solid body oriented in a cartesian coordinate system. Stressenergymomentum tensors of gauge potentials 11. Lagrangian continuum mechanics variables for general nonlinear. Stress used in lagrangian equation for solid mechanics. Geometry of logarithmic strain measures in solid mechanics patrizio ne 1, bernhard eidel 2 and robert j. We begin by considering the motion of a uid particle in lagrangian coordinates, the coordinates familiar from classical mechanics.
Geometry of logarithmic strain measures in solid mechanics. As an example of conjugate pairs, the mechanical work produced by combining second piolakirchhoff stress with green lagrange strain must match that obtained by combining cauchy stress with. A stress tensor work conjugate to the green strain tensor. In this blog post we will investigate these quantities, discuss why there is a need for so many variations of stresses and strains, and illuminate the consequences for you as a finite element analyst.
1143 381 3 758 509 1277 233 1017 114 1505 559 894 647 120 897 411 140 148 855 683 984 165 795 1268 912 811 1261 1482 405