# What is Continuum mechanics: Definition and 86 Discussions

Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of 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.

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1. ### I Power done by work on a continuous body isn't the derivative of work?

The definition of work and power done over a continuous body is: $$W = \int Tn \cdot u dA + \int b \cdot u dV$$ $$P = \int Tn \cdot v dA + \int b \cdot v dV$$ ##T## is the stress tensor, ##b## is the body force, ##u## is the displacement vector, ##v## is the velocity, ##n## is the normal...
2. ### I Elastic Constants for Natural Rubber

Hi, Looking for the Elastic Constants for any rubber-like material such as Natural Rubber. It can be inorganic or organic. The constants I am looking for take the form of a fourth-rank tensor. I only need the first order elasticities, not the zeroth or higher (not Cij or Cijklmn.. just Cijkl)...
3. ### I Usage of First Order Elastic Constants in Soft Body Equations

Hi, I have some soft body equations that require first order elasticity constants. Just trying to figure out the proper indexing. From Finite Elements of Nonlinear Continua by J.T. Oden, the elastic constants I am trying to obtain are the first order, circled below: My particular constitutive...
4. ### Set up boundary conditions for a simple elasticity problem

[Mentor Note -- Thread moved to the ME forum to get better views] Let's consider an incompressible block of Neo-Hookean material. Let the initial reference geometry be described by ##B=[0,b] \times [0,b] \times [0,h]##. The professor gave me the following task: Of course there can be many...
5. ### Frame indifference and stress tensor in Newtonian fluids

During lecture today, we were given the constitutive equation for the Newtonian fluids, i.e. ##T= - \pi I + 2 \mu D## where ##D=\frac{L + L^T}{2}## is the symmetric part of the velocity gradient ##L##. Dimensionally speaking, this makes sense to me: indeed the units are the one of a pressure...
6. ### Boundary condition: null traction on the boundary of an elastic block

Hi everyone, I'm trying to understand the rationale behind the boundary condition for the problem "Finite bending of an incompressible elastic block". (See here from page 180).Here we have as Cauchy Stress tensor (see eq. (5.82)): ##T = - \pi I + \mu (\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r...
7. ### I Divergence of first Piola-Kirchoff stress tensor

Hi everyone, studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93) where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$ How is the...
8. ### Finite bending of an elastic block - Equilibrium equations

I am studying the finite bending of a rubber-like block, assuming Neo-Hookean response. In the following, ##l_0##,##h##, ##\bar{\theta}## are parameters, while the variables are ##r## and ##\theta##. The Cauchy stress tensor is ##T= - \pi I + \mu(\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r \otimes...
9. ### Extension of an elastic block of Neo-Hookean material

I'm studying elasticity from classical Gurtin's book, and my professor gave us the following example, during lecture. Unfortunately, this is not present in our references, so I'm posting it here the beginning of the solution, and I will highlight at the end my questions. First I need to state...
10. ### What is the role of Poisson's ratio in determining stress and stiffness?

Hello Can someone please tell me what is the use of poisson's ration in determinig stress cos what I know in this case we should have stress=E*strain and so now use for poison
11. ### Kinematics of deformation (Continuum mechanics)

Question is extracted from "Ellad B Tadmor, Ronald E Miller, Ryan S Elliott - Continuum mechanics and thermodynamics From fundamental concepts to governing equations". I just got stuck at part (a). I think if part(a) is solved, I may be able to do the other parts.
12. ### Boundary conditions of a bending plate

Homework Statement I'm trying to find the boundary conditions for the following problem: A plate with length 2L is placed on supports at x = L/2 and x = - L/2. The plate is deforming elastically under its own weight (maximum displacement bowing up at x = 0). Both ends of the plate are free...
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### What is the Role of Continuum Mechanics in CFD Simulations?

I am a Phd student working in a technological center. My work is related to CFD simulations by using OpenFoam coupled to discrete element method. I am very interested in mathematical and physical background of continuum mechanics both solid and fluids. I am loking forward to solve my interests...

39. ### Looking for Continuum Mechanics book using Einstein Notation

I'm taking a course in continuum mechanics this semester and the instructor is using a set of notes to teach out of it, problem is, I don't really like them. Can anyone recommend an engineering/applied physics oriented introductory continuum mechanics textbook that uses the Einstein summation...
40. ### Continuum mechanics, physical interpretation of terms in balance eq.

Hi there. I'm reading Gurtin's 'the mechanics and thermodynamics of continua', and working some exercises of his book. In the section 21: 'The first law: balance of energy', after the derivation of the balance equation, he uses an identity to rewrite the balance of energy. The balance of energy...
41. ### Action Principles in Continuum Mechanics?

Is there any book that does what Landau does in Fluid Mechanics and Theory of Elasticity, only using a Lagrangian/Action-principles the whole way through? I can really only find brief tiny descriptions like this one in books on other topics, is there nothing that does for fluids/elasticity...
42. ### Continuum mechanics and normal shear stress

Homework Statement I am self-studying this note and I am stuck in the derivation of the normal shear stress. I can't see how the relations (23) and (24) come about, i.e. I don't understand \tau'_{xx} = \frac{\tau_{xx}+\tau_{yy}}{2}+\tau_{yx} and \tau'_{yy} =...
43. ### Continuum mechanics and continuity eq

Homework Statement Hi I can't follow the derivaton in this link. It is the following equality they have in the beginning, which I don't understand: \nabla \cdot u = \frac{1}{\rho}\frac{d\rho}{dt} Following the very first equation on the page, I believe it should be \nabla \cdot u =...
44. ### Mathematical tools for continuum mechanics

Hello^^ (I'm new here) I want to know the mathematical tools i need to study continuum mechanics. It would be great if someone give me a link that contains video lectures. Thanks for help .
45. ### Continuum Mechanics Simulation; need some help with the math

Hello all, Background I've been playing with computer simulations quite a bit recently, and wrote one that crudely simulates the formation of star systems. My first version was a conventional many body simulation with about 300 small bodies; it actually tends to come up with convincing star...
46. ### Electromagnetism coupled with continuum mechanics

Are there any textbooks on something like this, a self consistent treatment of classical electromagnetism (relavistic is fine too) where the field equations are solved alongside with the matter fields.
47. ### Change in volume continuum mechanics

Let a displacement field be given by $$u_1 = \frac{1}{4}(X_3 - X_2),\quad u_2 = \frac{1}{4}(X_1 - X_3),\quad u_3 = \frac{1}{4}(X_2 - X_1).$$...
48. ### Continuum Mechanics deformation definitions

Homework Statement What do you understand by the following terms; (i) principal stretch (ii) an anisotropic material (iii) a dilatant deformation, (iv) a Lagrangian description of a deformation, and (v) a pure deformation. Homework Equations Am just trying to find descriptions for...
49. ### Book on the mathematical theory of continuum mechanics

I was wondering if anyone knows of a good book on the mathematical theory of continuum mechanics. I have looked online, and the only ones I can seem to find are like your average physics or applied mathematics book. I want something with rigorous theoretical formulation of the subject. It...
50. ### Continuum Mechanics - Deformation gradient

Hi all, I am trying to self-learn continuum mechanics, and I have a question regarding the development of the deformation gradient (which ultimately leads to green's deformation tensor). I have attached the specifics of the question in a attached photo. Ultimately, there comes a point...