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. Klaus3

    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. D

    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. D

    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. B

    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. B

    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. B

    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. B

    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. B

    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. B

    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. M

    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. J

    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. K

    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...
  13. A

    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...
  14. C

    Continuum mechanical analogous of Maxwell stress tensor

    Maxwell stress tensor ##\bar{\bar{\mathbf{T}}}## in the static case can be used to determine the total force ##\mathbf{f}## acting on a system of charges contanined in the volume bounded by ##S## $$ \int_{S} \bar{\bar{\mathbf{T}}} \cdot \mathbf n \,\,d S=\mathbf{f}= \frac{d}{dt} \mathbf...
  15. V

    Continuum mechanics in physics education

    I came across this article about the near absence of continuum mechanics in university-level physics education: http://www.troian.caltech.edu/papers/Gollub_PhysToday_Dec03.pdf I have wondered this issue myself: why is continuum mechanics mainly studied by engineers rather than physicists, even...
  16. Mike J

    Internal stresses of an accelerating body

    Homework Statement Refer to image attached. Lets say I have a deformable solid that is being accelerated by a force that is equally distributed along the back face of the Main Body that is drawn in the picture. Attached to this Main Body is a Wing. At high accelerations, there will be inertial...
  17. S

    Definition(s) of entropy in continuum mechanics

    Those treatments of Entropy in continuum mechanics that I've viewed on the web introduce Entropy abruptly, as if it is a fundamental property of matter. For example the current Wikepedia article on continuum mechanics ( https://en.wikipedia.org/wiki/Continuum_mechanics ) says: Are other...
  18. V

    I Infinitesimal cube and the stress tensor

    The Cauchy stress tensor at a material point is usually visualized using an infinitesimal cube. The stress vectors (traction vectors) on opposite sides of the cube are equal in magnitude and opposite in direction. As a result, the infinitesimal cube is in equilibrium. However, when we derive...
  19. V

    Physical interpretation of the "total potential energy"

    The principle of minimum total potential energy is frequently used in solid mechanics as an elegant way of deriving the equilibrium equations for an elastic body under conservative forces. This method states that out of all the possible displacement fields that fulfill the boundary conditions...
  20. H

    Classical Modern Tensor Calculus/Continuum Mech Textbook

    Hi, I'm looking for a modern, colourful, illustrative introductory textbook to work through on tensor calculus/continuum mechanics. I'd like one with lots of physical examples, exercises, summaries, etc. I'd like an emphasis on engineering. Something in the mould of Frank White's Fluid...
  21. S

    Engineering Continuum Mechanics or Design/Manufacturing?

    Hi, people of PF I'm trying to decide between concentrating on continuum mechanics or design/manufacturing for my master's degree. My goal is to ultimately work in the industry, so design/manufacturing seems to make a lot of sense. However at the same time, continuum mechanics (and physics in...
  22. A

    I What is physicial interpretation of deviatoric strain rate

    How these properties are related to velocity fluid. The https://postimg.org/image/674a6sw4t/ https://postimg.org/image/674a6sw4t/ figure shows an area of Earth's mantle where upwelling of hot semi-liquid mantle is occurring in middle and then two downwelling currents on two sides (forming...
  23. Remixex

    Is It Possible to Solve This Diffusion Equation via Separation of Variables?

    Homework Statement $$\frac{\partial U}{\partial t}=\nu \frac{\partial^{2} U}{\partial y^{2}}$$ $$U(0,t)=U_0 \quad for \quad t>0$$ $$U(y,0)=0 \quad for \quad y>0$$ $$U(y,t) \rightarrow {0} \quad \forall t \quad and \quad y \rightarrow \infty$$ Homework Equations This is a diffusion problem on...
  24. Remixex

    Understanding the Equation for Velocity Field in Cylindrical Coordinates

    Homework Statement $$\bar{v}=\nabla \times \psi \hat{k}$$ The problem is much bigger, i know how a rotor or curl is calculated in cylindrical coordinates, but I'm just asking to see what would be the "determinant" rule for this specific curl. Homework Equations $$\psi$$ is in cylindrical...
  25. popbatman

    Solve Plane Elasticity w/ Airy Stress Function

    Hello to everyone, I have a problem with the solution of plane elasticity problems with the method of Airy stress functions. For instance I can solve a problem of uniaxial or biaxial uniform tension with a 2nd order polynomial, but if I add shear on only two opposite sides the problem seems to...
  26. F

    I Difference between 'Field' (algebra) and 'Field' (geometry)

    I am trying to build up a kind of mind map of the following: Module (eg. vector space) Ring (eg Field) Linear algebra (concerning vectors and vector spaces, from what I understood) Multilinear Algebra (analogously concerning tensors and multi-linear maps) Linear maps & Multilinear maps The...
  27. W

    The elasticity/stiffness tensor for an isotropic materials

    Hi PF, As you may know, is the the elasticity/stiffness tensor for isotropic and homogeneous materials characterized by two independant material parameters (λ and μ) and is given by the bellow representation. C_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu(\delta_{ik}\delta_{jl} +...
  28. L

    Continuum Mechanics: Rigid Heat Conductor

    Homework Statement Determine the thermodynamic restrictions for a rigid heat conductor defined by the constitutive equations: \DeclareMathOperator{\grad}{grad}\psi = \hat{\psi}\left(\theta,\grad \theta, \grad \grad \theta\right) \\ \eta = \hat{\eta}\left(\theta,\grad \theta, \grad \grad...
  29. H

    Thermodynamics of hyperelastic solid

    Can someone please help with hyperelastic theory, I need to know how changes in reversible work are related to energy density function and then to strain and stress tensors. A reference that explains the theories simply would also be appreciated. So far I have failed to find anything relatively...
  30. H

    Relation between energy conservation and numerical stability

    Hi, Consider the conservation laws for an isothermal linear incompressible flow governed by the mass and momentum equation. The kinetic energy equation is then solved to see if energy conserved. Can anyone tell me if once it is shown energy is conserved, it implies that convergence is obtained...
  31. S

    Deviation after completion of the doctorate

    At the moment I'm in the final stages of my doctorate in mathematics. (My background is a BS in physics and an MS in mathematics.) My focus and interest have been in applied functional analysis in general and various kinds of abstract and concrete delay equations in particular. These are...
  32. M

    Exponential behavior in elasticity?

    Hi! I know some constitutive models for elastic materials like Neo-Hooke or Mooney-Rivlin, which give a relation between elongation ##\lambda=y/y_o## (where ##y## and ##y_o## are the length of the elastic material in a uniaxial compression test in the direction of the compression at stress ##P##...
  33. G

    Exercises in continuum mechanics

    Hello Please forgive me if i am not posting in the correct forum. Also you may find my English a bit rusty since i am basically French Ok so i want to solve some exercises in continuum mechanics . The first exercise states : we have a stress tensor in a Cartesian coordinate system with the...
  34. H

    Understanding 2D Tensors: Covariant, Contravariant, and Physical Components

    Explain, plase to me: 2D covariant, conravariant and physical components of vector. I can not find this thematic in my official study book.
  35. H

    Exploring the Applications of Tensors in Mechanical Engineering Research

    My name is Hrvoje Zorić. I live in Republic Croatia. I am living in capital town Zagreb. Professors do not help me. In this time: I am Mechanical Enginering PhD research student
  36. N

    General Relativity and continuum mechanics

    Hello! I started learning relativity recently and I'm an engineering student. I can't stop drawing similarities between the nature of gravity and behaviour of continuous media in the field of continuum mechanics. Is there some direct connection and if so, has something like this been explored...
  37. Sivasakthi

    Continuum mechanics in solids

    Hi all.. I am currently doing some works on the continuum mechanics. And trying to study the macroscopic behavior of solids ( for simplicity, taken homogeneous materials) upon the action of external force ( which is the stress; pressure). How is it possible to account for the changes that can be...
  38. Matt atkinson

    Streamlines - Continuum mechanics

    Homework Statement In Cartesian coordinates ##x##, ##y##, where ##x## is the horizontal and ##y## the vertical coordinate, the velocity in a small-amplitude standing surface wave on water of depth ##h## is given by; $$v_x = v_0 sin(\omega t) cos(kx) cosh[k(y + h)]$$ $$v_y = v_0 sin(\omega t)...
  39. M

    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. Telemachus

    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. B

    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. N

    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. N

    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. P

    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. F

    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. H

    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. D

    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. E

    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. E

    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. T

    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...