Lagrangian approach
The Lagrangian point of view describes a special perspective when observing a movement of a body , i.e. it represents a certain observer's point of view. In the Lagrangian point of view or in the Lagrange picture , the movement of the body is analyzed from one of its material points (particles), which is why this point of view is also called material . For example, a buoy floating freely with the water in a river would perceive the current in the Lagrangian way. An example from solid mechanics is shown in the figure on the right. The question here is what conditions, e.g. B. what pressure or what temperature are present in a particular particle. The Lagrangian approach is used by solid mechanics for small to not too large deformations.
The Lagrangian approach was introduced by Leonhard Euler in 1762.
description
In the Lagrange image, the observer of a movement stands at a fixed material point or particle. By marking all points in space that a particle passes over time, a trajectory is created , which is associated with the Lagrangian approach. In the Lagrangian way of looking at things, all physical quantities related to the initial configuration , which represent the body in a fixed time for calculations, are shown. Whenever the same initial configuration is used, it is referred to as the total Lagrange approach, otherwise it is referred to as the updated Lagrange approach (total lagrange or updated lagrange). The equations derived with the Lagrangian approach are then available in the Lagrangian version or Lagrangian representation . In continuum mechanics , the quantities related to the initial configuration are usually written in capital letters or given the index (.) 0 in the equations .
advantages
Because the physical quantities are present on the particle with the Lagrangian approach, their time derivative has no convective component and is easy to calculate. A mass transport over the boundary of the body cannot take place, so that the mass balance as well as the other balance equations are easy to formulate. Many of the initial configuration related stress and strain tensors are objective, as are their time derivatives . The specification of secondary conditions on free areas does not cause any difficulties. With small deformations and linear material behavior, which is the case in many cases, especially in the technical area, the equations are simplified in such a way that analytical solutions are available or can be derived for many important problems.
disadvantage
Because of the relationship to the initial configuration, large deformations such as B. in forming processes, considerable numerical difficulties. The change in the properties of a body associated with large deformations, e.g. B. in a constriction, cause geometrically non-linear effects that increase the computational effort by a multiple compared to a geometrically linear one. The incompressibility of a material creates additional difficulties.
Summary
The properties of the Lagrangian approach are summarized again in the table.
| property | Occupancy |
|---|---|
| Namesake | Joseph-Louis Lagrange |
| Originator | Leonhard Euler (1762) |
| Observer location | Material point |
| application | Solid mechanics with not too large deformations |
| Visualization | Railway line |
| Cause of the kinematic non-linearity | Changes in shape of the body ( geometric non-linearity ) |
| Time derivative | Partial derivative , contains no convective part |
| Effort for balances of field sizes | low |
| Associated configuration | Initial and / or reference configuration |
| Designation of the variables in continuum mechanics | Uppercase letters or index zero |
| Kinematic unknowns | Shifts |
| Capability for incompressibility | limited |
| Suitability for boundary conditions on open areas | high |
See also
Footnotes
- ^ A b C. Truesdell: A First Course in Rational Continuum Mechanics . Academic Press, 1977, ISBN 0-12-701300-8 .
literature
- H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .