# Tagged Questions

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles.

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### What equation predicts at what point a stretched object comes apart?

I am creating a simulation and am interested in pulling stretchy things and when they break, like taffy. I imagine this is a bit tougher then a simple equation like gravity, but I have no idea. Is ...
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### Derivation of continuum expression of the first law of thermodynamics

Continuum expression of first law of thermodynamics: $$\frac{D E_t}{D t}=\nabla\cdot({\bf \sigma\cdot v}) - \nabla\cdot{\bf q}$$ (I've seen it in my physics book) How this equation is derived? ...
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### Objective time derivative that is no Lie derivative

Summary Led by an interest into the concept of "Material Objectivity", I am asking myself: Are there objective time rates that are not Lie derivatives? The long read I am trying to understand the ...
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### Equivalent beam rigidity for a 1D lattice structure

To model the behavior of continua, often discrete lattice models with nodes joined by 2-point spring elements (which resist tensile forces) and 3-point beam elements (which resist bending moments) are ...
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### Momentum equation in a Lagrangian configuration

When writing the momentum equation in a lagrangian configuration is the the stress tensor used the first Piola-Kirchhoff stress tensor or the nominal stress tensor (which is the transpose of the 1st P-...
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Let us consider a homogeneous rope hanging from the ceiling. I will call the vertical direction $x$ and the horizontal displacement $y$. When we apply the second Newton's Law to a portion of mass $\... 1answer 35 views ### Confused about shear elasticity and complementary shear stress I am a self learner of continuum mechanic. I am confused about simple shear stress in situation similar to figure 1, in case$F_\textrm{ext}$is caused by external perturbation by i.e., human, what ... 2answers 848 views ### In wave motion of a string both kinetic energy and potential energy are minimum at$y=y_\text{max}$then why does the string comes down again? In wave motion of a string both kinetic energy and potential energy are minimum at$y=y_\text{max}$then why does the string comes down again? As everything in tries to attain lowest energy possible ... 2answers 61 views ### Why an impact exerts so much force? [closed] If an object of velocity$v$and mass$m$moves towards a resting object of mass$M$, then if the object which is hit might break. Why? What is the reason that a collision has more power than a ... 0answers 9 views ### Warping function for torsion of non-circular prism I have a few questions regarding the case of torsion of a prism, as encountered in continuum mechanics. Specifically, a prism (which can be a cylinder, a rectangular prism, elliptical prism, etc.) has ... 2answers 312 views ### Jaumann deviatoric stress rate Background about terms in this question: Hookes law and objective stress rates From my understading, the Jaumann rate of deviatoric stress is written as: $$dS/dt = \overset{\bigtriangleup}{{S}} = {\... 0answers 20 views ### Gradient effects in continuum mechanics What I have learned is that inhomogenous materials (materials with different material properties over space and time) can be treated by the homogenization technique (https://en.wikipedia.org/wiki/... 1answer 26 views ### Liquid Flow In A Vertical Tube A cylindrical vertical tube has uniform cross section A_1, and length l. It is open at both ends. Water enters from the top with a constant velocity v_1, and allowed to flow out from the bottom. ... 0answers 17 views ### Linear viscoelastic differential operators I am starting with differential operators: P = \sum_{i=0}^{N}p_i \cfrac{d^i}{dt^i} Q = \sum_{i=0}^{N}q_i \cfrac{d^i}{dt^i} p_i and q_i are functions of time only. K is a constant that ... 3answers 409 views ### Is it possible that Cauchy stress be asymmetric? According to conservation of linear momentum and angular momentum, one can derive that Cauchy stress tensor is symmetric and hence has only 6 independent components. Is it possible that, when breaking ... 0answers 230 views ### Derivative of deformation gradient with respect to Green-Lagrangian strain? For hyperelastic material, the elastic energy \Psi is related to the deformation gradient F and other internal variables (e.g. temperature \theta) In many literatures (including Malvern's and ... 1answer 131 views ### Current density in phase space \newcommand{\dd}{{\rm d}} I have a question which arises from looking at the impact free Boltzmann equation. Let (\vec{x},\vec{v}) be a vector in our phase space \Gamma^N = \mathbb{R}^{6N}. The ... 3answers 158 views ### Why is it said that standing waves do not transfer energy? The author of my physics textbook writes that standing waves, unlike travelling waves, do not transfer energy. He says that this is because a standing wave is composed of two travelling waves carrying ... 2answers 53 views ### On the isotropy of materials I am working on honeycomb structures and first of all I would like to understand whether it is isotropic or not, and, if the latter holds, which kind of anisotropy does it have? How to do it? I don't ... 0answers 41 views ### How to calculate Lagrangian density function in classical field theory In Lagrangian mechanics observing the possible degrees of freedom we first write down our Lagrangian. Then we use E-L equation to determine equation of motion and using sufficient boundary condition ... 0answers 35 views ### Identity in continuum mechanics [closed] For a problem in the textbook I am reading, I need to prove that$$\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS,$$where S is the boundary of the volume V, v_i is the velocity vector field of a ... 1answer 67 views ### Infinite elastic half-space with point load (Mindlin's problem) What is the equilibrium deformation of an infinite half-space (that is, an isotropic and homogeneous linearly-elastic three-dimensional medium, with a single planar surface) produced by a force which ... 0answers 29 views ### Pressure vessel analysis of transversely isotropic multilayer material Suppose I have a transversely isotropic, hyperelastic material with known strain energy that is a fibrous composite. I am interested in an explicit formula for the displacements (so I can get the ... 1answer 37 views ### Uniqueness of a stress (only) boundary value problem A static problem in linear elasticity is typically written as the following boundary value problem: find \boldsymbol u and \boldsymbol \sigma such that: \text{div} \boldsymbol \sigma + \... 0answers 51 views ### Question regarding the use of the Green-Naghdi Objective Stress Rate The equation for the Green-Naghdi Stress Rate reads: \boldsymbol{\sigma}^{GN} = \dot{\boldsymbol{\sigma}} + \boldsymbol{\sigma}\cdot\boldsymbol\Omega - \boldsymbol\Omega\cdot\boldsymbol{\sigma} ... 1answer 26 views ### How is a unidirectional lamina transversely isotropic? What I don't understand specifically is that if there happen to be more fibers in the x_2 direction than the x_3 direction, wouldn't that make the material properties in those directions different?... 1answer 43 views ### How to prove this material derivative formula rigoriously with experiment or prove it without the chain rule? [closed] How to prove this rigoriously by experiment or prove it with other mechanics law but without the chain rule? This is coming from reddy's introduction to continuum mechanics. Also please explain ... 1answer 2k views ### Good books on elasticity Can someone suggest good books/textbooks/treatises/etc on elasticity? 1answer 40 views ### What is the intuition behind this acceleration formula? What is the intuition behind this acceleration formula? In another word, how to demonstrate this by using common sense without using chain rule? 2answers 195 views ### Does the definition of compressibility depend on the frame of reference? According to many authors, a fluid is defined to be incompressible if the material derivative of the density \frac{D\rho}{Dt} is zero, that is to say, that in an frame of reference following the ... 0answers 84 views ### Why is maximum shear stress at 45 degrees instead of near 0? I think I might havr an immensely stupid question but it's really bothering me so please be patient. I am not physicist smart. Look at this guy's question: https://www.physicsforums.com/threads/mohrs-... 0answers 8 views ### Equation Governing Small Lateral Deflections z of a Uniform Membrane The equation governing small lateral deflections z of a uniform membrane subjected to a lateral (dimensionless) pressure p is given by$$ \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\... 1answer 11 views ### Is deviatoric strain associated with thermal effects? Does temperature have any effects on deviatoric strain for a linearly elastic isotropic material? 0answers 32 views ### From in-plane strain to Poisson ratio I have recently been trying to simulate a graphene cantilever and thus I need to know the Poisson ratio, Young's modulus, and density. From literature it is easy to find the Young's modulus and ... 1answer 29 views ### Traveling wave solutions for an irregular “waveguide” I'm looking at solutions for the wave equation $$\frac{\partial^2 z}{\partial t^2}=c^2 \nabla^2z,$$ in a finite 2D domain. Say that I have periodic boundary conditions on the left and right edges ... 2answers 370 views ### In continuum mechanics, what is work potential in the context of total potential energy? I'm reading a book on the finite element method. Specifically I'm looking at the background material where they are discussing potential energy, equilibrium, and the Rayleigh–Ritz method. The book ... 0answers 38 views ### How to derive an expression for entropy generation in a diffusive, reacting continuum I'm trying to understand a derivation from "The Thermodynamics of Linear Fluids and Fluid Mixtures," by Miloslav Pekař and Ivan Samohýl (2014). The derivation produces an expression for the entropy ... 0answers 32 views ### The force of a spring I am new in continuum mechanics and I want to prove the formula which gives the force given by a spring : $$F_{max}= \frac{Ed^4(L-nd)}{16(1+\nu)(D-d)^3 n}$$ where :$E$– Young's modulus$d$– ... 1answer 58 views ### physical meaning of major symmetry of the stiffness tensor What happens if a stiffness tensor does not have the "major symmetry"$C_{ijkl}=C_{klij}$? Background: In linear elasticity (generalising Hooke's law from a spring to a continuous medium), the ... 1answer 814 views ### How to determine plastic strain rate Equivalent plastic strain rate is defined as $$\dot{\bar{\epsilon}}=\sqrt{\frac{2}{3}\dot{\epsilon_{ij}}^{p}\dot{\epsilon_{ij}}^{p} }$$ Where,$ \dot{\bar{\epsilon}}\$ is equivalent plastic strain ...
When deriving the sound wave equation: $${1 \over c^2} {\partial^2 p' \over \partial t^2 }= \Delta^2 p'$$ by linearizing the Euler equation: $$\rho {d v \over dt }= - \nabla p$$ and the continuity ...