Concept# Continuum mechanics

Summary

Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.
A continuum model assumes that the substance of the object completely fills the space it occupies. This ignores the fact that matter is made of atoms, however provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships.
Continuum mechanics treats the physical properties of solids and fluids independ

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications (89)

Loading

Loading

Loading

Related people (9)

, , , , , , , ,

Related concepts (75)

Force

In physics, a force is an influence that can cause an object to change its velocity, i.e., to accelerate, unless counterbalanced by other forces. The concept of force makes the everyday notion of pus

Mechanics

Mechanics (from Ancient Greek: μηχανική, mēkhanikḗ, "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Fo

Stress (mechanics)

In continuum mechanics, stress is a physical quantity that describes forces present during deformation. An object being pulled apart, such as a stretched elastic band, is subject to tensile stress an

Related courses (66)

ME-201: Continuum mechanics

Continuum conservation laws (e.g. mass, momentum and energy) will be introduced. Mathematical tools, including basic algebra and calculus of vectors and Cartesian tensors will be taught. Stress and deformation tensors will be applied to examples drawn from linear elastic solid mechanics.

ME-104: Introduction to structural mechanics

The student will acquire the basis for the analysis of static structures and deformation of simple structural elements. The focus is given to problem-solving skills in the context of engineering design.

PHYS-423: Plasma I

Following an introduction of the main plasma properties, the fundamental concepts of the fluid and kinetic theory of plasmas are introduced. Applications concerning laboratory, space, and astrophysical plasmas are discussed throughout the course.

Related units (7)

Related lectures (101)

Minimally Invasive Surgery (MIS) represents one of the major progresses in surgery in the last decade. In general, it is based on the application of small body-cuts through which instruments are inserted into the patient's body allowing the surgeon to carry out their job. The advantages are numerous, mainly related to the patient's health, and economic advantages. The rather small size of the necessary cuts means a decreased risk of trauma and a faster recovery. It follows that hospitalization time and costs are reduced significantly. The conditions in which such surgery takes place are considerably different from traditional surgery. First of all, the extent to which the surgical instruments may be moved freely is reduced, as they have to pass through a fixed point. Then, visualisation of the surgery site takes place on a computer screen. In laparoscopy, the screen is connected to a micro-camera whereas in interventional radiology, an image created by x-rays is displayed on the screen. As more new surgical techniques are continuously invented, it is expected that these techniques can be practiced by the surgeons and, more importantly, preferred by patients. Amongst these surgical techniques, the suturing during laparoscopic surgery requires a certain skill in instrument handling to realize the required knots. In interventional radiology, during an angiography, the manipulation of the catheter requires a highly trained hand as the catheter has to pass through a complex, 3-dimensional network of blood vessels. The catheter is only visible when a radio-opaque marker is injected while the patient is under the emission of x-rays. Furthermore, such pictures are 2D. To help the surgeons to improve their abilities, we propose a haptic simulator to learn how to realize the knots for the suturing and a virtual environment for the angiography. For both training modules, we developed a model of filament based on the Cosserat theory applied to one-dimensional structures, which can describe the behaviour of a string as well as the behaviour of a rigid and flexible rod (catheter). This model starts from the energetic formulation of the filament considering the Hook laws of continuum mechanics. The Lagrange equations provide us with the equations of motion which describe the model deformation. This model takes collisions and auto-collisions into account and it is revealed to be very efficient for interactive applications. The training module dedicated to angiography permits to carry out the most usual gesture: beside the catheter navigation, a marker can be injected to visualize the vessels, to drive the balloon angioplasty towards the pathology and to set stents. Moreover, the visual rendering is very realistic and the heart beatings as well as breathing are also simulated.

The explosive growth of machine learning in the age of data has led to a new probabilistic and data-driven approach to solving very different types of problems. In this paper we study the feasibility of using such data-driven algorithms to solve classic physical and mathematical problems. In particular, we try to model the solution of an inverse continuum mechanics problem in the context of linear elasticity using deep neural networks. To better address the inverse function, we start first by studying the simplest related task,consisting of a building block of the actual composite problem. By empirically proving the learnability of simpler functions, we aim to draw conclusions with respect to the initial problem.The basic inverse problem that motivates this paper is that of a 2D plate with inclusion under specific loading and boundary conditions. From measurements at static equilibrium,we wish to recover the position of the hole. Although some analytical solutions have been formulated for 3D-infinite solids - most notably Eshelby’s inclusion problems - finite problems with particular geometries, material inhomogeneities, loading and boundary conditions require the use of numerical methods which are most often efficient solutions to the forward problem, the mapping from the parameter space to the measurement/signal space, i.e. in our case computing displacements and stresses knowing the size and position of the inclusion. Using numerical data generated from the well-defined forward problem,we train a neural network to approximate the inverse function relating displacements and stresses to the position of the inclusion. The preliminary results on the 2D-finite problem are promising, but the black-box nature of neural networks is a huge issue when it comes to understanding the solution.For this reason, we study a 3D-infinite continuous isotropic medium with unique concentrated load, for which the Green’s function gives an analytical mathematical expression relating relative position of the point force and the displacements in the solid. After de-riving the expression of the inverse, namely recovering the relative position of the force from the Green’s matrix computed at a given point in the medium, we are able to study the sensitivity of the inverse function. From both the expression of the Green’s function and its inverse, we highlight what issues might arise when training neural networks to solve the inverse problem. As the Green’s function is not bijective, bijection must been forced when training for regression. Moreover, due to displacements growing to infinity as we approach the singularity at zero, the training domain must be constrained to be some distance away from the singularity. As we train a neural network to fit the inverse of the Green’s function, we show that the input parameters should include the least possible redundant information to ensure the most efficient training.We then extend our analysis to two point forces. As more loads are added, bijection is harder to enforce as permutations of forces must be taken into account and more collisions may arise, i.e. multiple specific combinations of forces might yield the same measurements.One obvious solution is to increase the number of nodes where displacements are measured to limit the possibility of collision. Through new experiments, we show again that the best training is achieved for the least possible amount of nodes, as long as the training data generated is indeed bijective. As the medium is elastic, we propose a neural network architecture that matches the composite nature of the inverse problem. We also present another formulation of the problem which is invariant to permutations of the forces,namely multilabel classification, and yields good performance in the two-load case.Finally, we study the composite inverse function for 2, 3, 4 and 5 forces. By comparing training and accuracy for different neural network architectures, we expose the model easiest to train. Moreover, the evolution of the final accuracy as more loads are added indicates that deep-neural networks (DNNs) are not well suited to fit a non-linear mapping from and to a high-dimensional space. The results are more convincing for multilabel classification.

2020