How are the classical set of equilibrium equations for linear elasticity derived? - Computational Science Stack Exchange most recent 30 from scicomp.stackexchange.com 2021-07-29T10:59:21Z https://scicomp.stackexchange.com/feeds/question/36248 https://creativecommons.org/licenses/by-sa/4.0/rdf https://scicomp.stackexchange.com/q/36248 2 How are the classical set of equilibrium equations for linear elasticity derived? student010101 https://scicomp.stackexchange.com/users/37428 2020-11-07T17:09:57Z 2020-11-09T03:07:02Z <p>In linear elasticity, the governing PDE is the equilibrium equations (absent of vibration considerations):</p> <p><span class="math-container">$$-\nabla \cdot \sigma = F$$</span></p> <p>Is this equation simply derived from the sum of forces and moments?</p> <p>In most linear elasticity papers, I see these governing equations. Is there an original source for where these equations came from? I'm looking for a more fundamental citation, but it seems so ubiquitously used and known that it's difficult for me to find the original source.</p> https://scicomp.stackexchange.com/questions/36248/-/36249#36249 2 Answer by nicoguaro for How are the classical set of equilibrium equations for linear elasticity derived? nicoguaro https://scicomp.stackexchange.com/users/9667 2020-11-07T19:08:26Z 2020-11-08T19:07:22Z <p>You take an arbitrary volume <span class="math-container">$V$</span> and use the translational and rotational equilibrium equations over it. Then, due to the arbitrariness of the volume the integrals should equal 0 and you get the equation you present and the symmetry for the stress tensor (in classic elasticity).</p> <p>According to @BiswajitBanerjee's comment, the first publications to discuss the topic were:</p> <ul> <li><p>Navier, C. L. M. H. (1821). Sur les lois des mouvement des fluides, en ayant egard a l’adhesion des molecules. Ann. Chimie, 19, 244-260.</p> </li> <li><p>Cauchy, A. L. B. (1822). Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques.</p> </li> </ul> <p>You can find a recent discussion on</p> <ul> <li><p>Mase, George Thomas, Ronald M. Smelser, y George E. Mase. 2010. Continuum mechanics for engineers. 3rd ed. Boca Raton: CRC Press. (Chapter 5).</p> </li> <li><p>Reddy, J. N. (2013). An introduction to continuum mechanics. Cambridge university press. (Chapter 5).</p> </li> </ul> https://scicomp.stackexchange.com/questions/36248/-/36251#36251 0 Answer by Nachiket for How are the classical set of equilibrium equations for linear elasticity derived? Nachiket https://scicomp.stackexchange.com/users/36539 2020-11-08T13:57:27Z 2020-11-09T03:07:02Z <p>You can find a discussion in AF Bower's book.</p> <p>Applied Mechanics of Solids 1st Edition ISBN-13: 978-1439802472, ISBN-10: 1439802475</p> <p>The book is available online at AF Bower's website</p> <p><a href="http://solidmechanics.org/Text/Chapter2_3/Chapter2_3.php#Section2_3_1" rel="nofollow noreferrer">http://solidmechanics.org/Text/Chapter2_3/Chapter2_3.php#Section2_3_1</a></p> https://scicomp.stackexchange.com/questions/36248/-/36254#36254 1 Answer by Wolfgang Bangerth for How are the classical set of equilibrium equations for linear elasticity derived? Wolfgang Bangerth https://scicomp.stackexchange.com/users/393 2020-11-09T01:58:29Z 2020-11-09T01:58:29Z <p>A different perspective on the question is this: Newton's law says that mass times acceleration equals the sum of all forces. You are interested in the steady state case, so the acceleration is zero and as a consequence, the sum of all forces is zero. This has to hold at each point of the solid if you want the body to not move.</p> <p>The sum of all forces equals the external forces <span class="math-container">$F$</span> (actually, a force density, because we're looking at individual points) acting at each point of the body plus the internal forces <span class="math-container">$\nabla \cdot \sigma$</span> due to the stresses.</p> <p>In other words, the equation you quote is simply a <em>force balance</em>.</p>