Sufficient conditions of objectivity are obtained for convective rates of objective tensors. Time rates of tensors are determined using convective rates. In this case, the lie derivatives of lagrangian densities by vector elds which call into play the stressenergymomentum tensors fail to be equal to zero in general. An eulerianlagrangian approach to the navierstokes. Representations of strain rate and spin tensors in lagrangian and eulerian triads. Applications of coupled eulerianlagrangian method to. Cauchy stress tensor and almansi strain tensor are generally used as a conjugate pair in this approach for constitutive equations. May 31, 2009 weve only just started learning a little bit about tensors, and i am finding questions like this quite difficult. It is also possible to follow a socalled arbitrarylagrangeeuler approach. A coordinate independent derivation of the eulerian and lagrangian strain tensors of finite deformation theory is given based on the parallel propaga. This work presents an eulerianlagrangian approach to the navierstokes equation.
Green strain tensor in displacements in cartesian coordinates. The eulerian strain tensor e is defined as 1 1 2 eiff. Continuum mechanics and nonlinear elasticity stefano giordano. There are two main ways of approaching problems that involve the motion of deformable materials the lagrangian way and the eulerian way. To this end, consider the following strain tensors.
Armyresearch laboratory 2800powdermillroad adelphi, maryland, usa207831197 february 20, 2001 abstract a coordinate independent derivation of the eulerian and lagrangian strain tensors of. The derivation explicitly shows that the eulerian and lagrangian strain tensors are twopoint tensors, each a function of both the spatial and material. Transformation properties of the lagrangian and eulerian strain. Topic 3 lagrangian continuum mechanics variables for analysis. The deformation and rigid rotation will be separated shortly see 2. Lagrangian and eulerian coordinates for the reference configuration. They do not contain information about the rigid body rotation, i. With the proper constitutive model, any of these strains can be related to a choice of stress tensor chapter 4. Displacement field, displacement gradient tensor and transformation tensor. The greenlagrange strain tensor is directly defined in function of the right strain tensor by e c.
Transformationproperties of the lagrangian and eulerian strain tensors thomas b. Following ogden, a class of objective lagrangian and eulerian tensors is identified among the secondrank tensors characterizing continuum deformation, but a more general definition of objectivity than that used by ogden is introduced. There, the authors proposed a smooth fourth order optical flow model which directly computes the. For nonuniform stretch all these are average measures of strain for the entire bar that. A coordinate independent derivation of the eulerian and lagrangian strain tensors of finite deformation theory is given based on the parallel propagator, the world function, and the displacement vector field as a threepoint. Rotation consider two particles initially separated by dai note. Objective tensor rates and applications in formulation of. Pdf natural lagrangian strain measures of the nonlinear. Eulerian points of view in fluid mechanics we describe the motion of liquids and gases such as water and air using the approach of continuum mechanics, wherein the fluid is characterized by properties that are aggregates over a large number of individual molecules. 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. The standard derivation of these strain tensors is done with the help of shifter tensors 8,9. Lagrangian strain an overview sciencedirect topics. A coordinate independent derivation of the eulerian and lagrangian strain tensors of finite deformation theory is given based on the.
We base our analysis of di erential conservation laws on the canonical decomposition of the lie derivative of a lagrangian density lby a projectable vector eld on a bundle and. Lagrange strain tensors to explain some important properties of the 2nd piolakirchhoffstress tensor and the greenlagrange strain tensor, we consider the deformation gradient tensor this tensor captures the straining and the rigid body rotations of the material fibers. In this chapter we have discussed three strain tensors lagrangian strain, eulerian strain and logarithmic strain. Strain and strain rate tensors introduction to finite. A deformation gradient tensor and strain tensors for. I provide a coordinate independent derivation of the lagrangian and eulerian strain tensors, where i keep track of the positions of the basis vectors. A coordinate independent derivation of the eulerian and lagrangian strain tensors of finite deformation theory is given based on the parallel. Objective symmetrically physical strain tensors, conjugate. A coordinate independent derivation of the eulerian and lagrangian strain tensors of finite deformation theory is given based on the parallel propagator, the world function, and the displacement vector field as a threepoint tensor. The lagrangian strain is the difference between the square of the deformed length and the square of the original length divided by twice the square of the original length.
Such an equivalency verifies that the proposed eulerian and lagrangian models are unified. Stretch of a material in 1d general definition of strains in 1d. It is a very fundamental quantity used in continuum mechanics. Both lagrange and eulerian strain tensors contain information about the strain, i. For the further analysis, five scale functions are chosen which generate strain tensors belonging to the families of strain tensors previously introduced by other authors i. Most of those studies have focused primarily on how to model its lagrangian evolution. The lagrangian and eulerian specifications of the kinematics and dynamics of the flow field are related by the material derivative also called the lagrangian derivative, convective derivative, substantial derivative, or particle derivative. The derivation explicitly shows that the eulerian and lagrangian strain tensors are twopoint tensors, each a function of both the spatial and material coordinates. Stressenergymomentum tensors in lagrangian field theory. My appreciation for mechanics was nucleated by professors douglas amarasekara and munidasa ranaweera of the then university of ceylon, and was subsequently shaped and grew substantially under the in uence of professors james k. Pdf lagrangian strain tensor computation with higher order. In the eulerian method, the fundamental property is the velocity. The displacement and velocity field, examples of deformations and motions, eulerian and lagrangian descriptions of motion.
Green lagrangian strain almansi eulerian strain logarithmic strain conventional notions of strain in 1d consider a uniform bar of some material before and after motiondeformation. These approaches are distinguished by three important aspects. These strain and strain rate measures, which are mixed tensor fields, were mapped from the body to space1 in both the eulerian and lagrangian configurations and were then transformed from. The eulerian strain is a twopoint tensor that transforms as a second rank tensor under transformation of spatial coordinates and transforms as a scalar under transformation of the. Lagrangian and eulerian specification of the flow field. The concept of strain is used to evaluate how much a given displacement differs locally from a rigid body displacement. Strain tensors lagrange strain and eulerian strain, cauchy green strain, infinitesimal strains, compatibility. In the lectures we have not covered cases when the field is also a tensor. Euler and lagrange descriptions euler approach the. For infinitesimal deformations of a continuum body, in which the displacement gradient 2nd order tensor is small compared to unity, i. When the deformation process is time or rateindependent, the.
In order to solve a problem based on the energy viewpoint, the mechanical, thermal and thermomechanical tensors conjugate to the sethhill strains, and a general class of lagrangian, eulerian. Pdf lagrangian strain tensor computation with higher. The full lagrangian dynamics of the eulerian velocity gradient tensor have been modelled and studied extensively meneveau2011. Through changing the way to approach the derivation of a constitutive relation, we have veri. A lagrangian model for hardening behaviour of materials at. Consider an eulerian quantity, temperature, in a room at points a and b where the temperature is di. I know about raising and lowering indices using the metric tensor, and combining. Transformation properties of the lagrangian and eulerian strain tensors. Note, however, that the cauchygreen strain tensors that encode the lagrangian stretching are obtained.
In this research it is shown that both the proposed lagrangian and unified eulerian rate models are equivalent and results obtained from both models are identical for the same finite strain loading path. I derive the strain tensors using the concept of absolute tensors, where a tensor is associated with a point in the spacerather than the coordinates in a given moving coordinate system. Decomposition of deformation gradient tensor into isochoric and volumetric parts. Eulerian coordinates xed euclidean coordinates are natural for both analysis and laboratory experiment. Continuum mechanicsstrains and deformations wikiversity. Notes on strain and deformation tensors marcel frehner september 2006. I2, where i is the identity tensor, and its components are noted e ij with i, j 1, 3. We have examined these three because they are most pertinent to oil.
Suppose we have a flow field u, and we are also given a generic field with eulerian specification fx,t. The computation of the lagrangian strain tensor by a variational method was addressed in 17. Green lagrange strain tensor an overview sciencedirect. Deformation, stress, and conservation laws in this chapter, we will develop a mathematical description of deformation. What is the difference between lagrangian and eulerian. Transformation properties of the lagrangian and eulerian. As the strain tensor components, values depend on the basis in which they are written, some use the strain invariants to express the constitutive law.
Green lagrange strain tensor an overview sciencedirect topics. These strain and strainrate measures, which are mixed tensor fields, were mapped from the body to space1 in both the eulerian and lagrangian configurations and were then transformed from. These positions can be described as functions of time in two ways. The lagrangian and eulerian specifications of the kinematics and dynamics of the flow field are related by the material derivative also called the lagrangian derivative, convective derivative, substantial derivative, or particle derivative suppose we have a flow field u, and we are also given a generic field with eulerian specification fx,t. In order to relate two second rank tensors, a fourth rank tensor is necessary. Our focus is on relating deformation to quantities that can be measured in the. For an eulerian rate formulation of elastoplasticity, the strain rate tensor can be additively decomposed into its elastic and inelastic parts 4,7,8,10,18. Specifically, the left cauchygreen strain and right cauchygreen strain tensors give a measure of how the lengths of line elements and. The undeformed state is indicated as the state at time t0 and the deformed state as the state at the current time t. One of such strains for large deformations is the lagrangian finite strain tensor, also called the green lagrangian strain tensor or green stvenant strain tensor, defined as. We have examined these three because they are most pertinent to oil well tubular analyses appearing in the literature.
Pattillo, in elements of oil and gas well tubular design, 2018. Sep 25, 2018 we introduce a new family of strain tensorsa family of symmetrically physical sp strain tensorswhich is also a subfamily of the wellknown hill family of strain tensors. Greenlagrangian strain almansieulerian strain logarithmic strain conventional notions of strain in 1d consider a uniform bar of some material before and after motiondeformation. An eulerianlagrangian description of the euler equations has been used in 4, 5 for local existence results and constraints on blowup.
761 1438 543 571 1177 905 445 697 1210 523 1091 716 210 1410 964 663 823 417 1572 1283 118 1358 1297 708 634 102 1052 616 1426 1304 710 145 1124 165 1338 293 126 571 474