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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Dynamic mass or static mass?

I am testing a cantilever beam assuming it as a single degree of freedom system, therefore it can be described by the equation $$m_d \ddot{y}(d) + c_d \dot{y}(d) + k_d y(d) = p(t) $$ Where $d$ is ...
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21 views

Understanding incompressibility (of rubber or viscoelastic material)

Literature gives a lot of explanation why rubber is incompressible. However, I still need some thinking to understand physical behavior of rubber or any such material. Often, incompressibility is ...
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19 views

Golf ball impact

A golf ball is said to be "compressed" when hit by a golf club and makes a characteristic "thwack-hiss" sound coming off of the club when impacted by professional golfers (whose impact conditions have ...
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1answer
24 views

Airfoils contradict the law of the lever?

The law of the lever says that "the less force you use, the more distance you have". It is often exemplified by referring to simple machines, but it should apply to all technical systems. But I do not ...
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23 views

Meaning of boundary conditions in solid mechanics

The Question is: A uniform horizontal beam OA, of length $a$ and weight $w$ per unit length is clamped horizontally at O and freely supported at A. The transverse displacement $y$ of the beam is ...
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43 views

Can $U_{ij}$ or $v_{ij}$ in continuum mechanics be negative?

In continuum mechanics, we have the deformation gradient $\mathbf F$ to be: $$d\mathbf x = \mathbf F d \mathbf X$$ And then, we do a polar decomposition (A good reference here would be ...
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37 views

Estimate the persistence length of a rubber band [closed]

Not much more to say here, it's all in the question. The best, most convincing estimate will be chosen as the correct answer. EDIT: Assume the rubber band is at room temperature, with thickness $t$ ...
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24 views

Relation between area elements in finite deformation theory (continuum mechanics)

There are relations for the line and volume elements in continuum mechanics. For example: \begin{align} \ \ \ \ \ \ \ \ \ \ \ \ \frac{V}{V_0}&={\rm det}(F)\tag{1}\\ \lambda^2&=(F^TFe_1\cdot ...
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32 views

Deriving general boundary conditions from first principles for elastodynamic scattering

It seems that most of the relevant books only give the linear case and the rest say something along the lines of "here are common examples of boundary conditions." What are the most general boundary ...
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1answer
21 views

Problem on bending plates in Newtonian Mechanics?

I am reading a book on interesting physics problems and demonstrations. One of the problems in the section on buildings, structures and equilibrium talks about a plate with one side attached to the ...
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2answers
31 views

On the isotropy of materials

Good morning. 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 it has. How to do it? ...
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44 views

Find the maximum allowable bending moment

A rolled steel universal I-section beam with a serial size of $406\times178$ has a mass of $60$kg/m. What is the maximum safe allowable bending moment this beam can sustain,given that the maximum ...
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616 views

what is difference between homogeneous vs isotropic material?

When we say a material is isotropic? When properties such as density, Young's modulus etc. are same in all directions. If these properties are direction dependent, then we can say that the material is ...
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13 views

Is there any viscoelastic model directly specified from strain energy function?

For most models I've seen, including the classic quasi-linear viscoelastic model, the parallel network viscoelastic model ABAQUS uses, and the model described in Holzapfel & Gasser 2000, they all ...
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2answers
51 views

Is there a way to calculate strain energy based on stress and deformation gradient?

We know that we can obtain stress from strain energy density and deformation gradient, for example: $$\mathbf P=\frac{\partial W}{\partial \mathbf F}$$ However, is there a way to calculate $W$ from ...
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27 views

Like viscoelastic polymers, why there are not storage and loss moduli for cast iron?

Viscoelastic polymers have different paths upon loading and unloading, so there is energy dissipation, so they have storage and loss moduli. Plastic behavior is also shown by cast iron: loading and ...
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1answer
43 views

Superior attachment of Möbius strip

In this DIY project, the Möbius strip is used to make a spill-proof coffee cup carrier. The author uses a Möbius strip as the handle of this carrier and says If you attach a Möbius strip to an ...
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1answer
297 views

Prove Poisson's Ratio is 0.5 [closed]

Poisson's ratio is the negative ratio of the transverse strain (_T) to the axial strain (_A). For an incompressible (density doesn't change), homogeneous (everything is the same molecule), ...
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1answer
122 views

Degree of anisotropy of crystal tensors

Does there exist a scalar that can describe how anisotropic the elasticity of a crystal is? What about other tensors such as the permittivity or susceptibility? I found a Wikipedia article that was ...
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1answer
86 views

How to define the heat current in an isotropic continuum material

I'm doing a FDTD (finite difference time domain) simulation of an isotropic continuum material. And I have several questions. How do you define the energy transferred through an isotropic continuum ...
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1answer
148 views

Divergence of Cauchy Stress Tensor

On the wikipedia page for the Cauchy Momementum Equation, it's stated that the equation can be written as $$\rho \frac{D\,\textbf{v}}{D\,t} = \nabla \cdot \sigma + \textbf{f}$$ Where $\sigma$ is ...
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1answer
103 views

Confusion in Euler-Bernoulli beam theory

Euler-Bernoulli beam equation is given by $$ EI \frac{\mathrm d^2 u}{\mathrm d x^2} = M'(x) \\ EI \frac{\mathrm d u}{\mathrm d x} = xM'(x) + C_1 $$ Where, $E$ is modulus, $I$ is second moment of ...
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106 views

Stress Force - Understanding Cauchy Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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2answers
853 views

How many atoms exist within a continuum body?

Materials, such as solids, liquids and gases, are composed of molecules separated by "empty" space. On a microscopic scale, materials have cracks and discontinuities. However, certain physical ...
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100 views

Tensorial version of Hooke's law

It is well known that $${\boldsymbol F} = k {\boldsymbol x}$$ for isotropic media. Also, according to Wikipedia $$F_k = k_{jk} x_j$$ for some elastic tensor $k_{jk}$. I'm a bit confused as to how ...
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1answer
76 views

Physics of Wrinkling: Understanding inextensibility condition

I'm reading this very cool paper on the formation of wrinkles in elastic materials. The key result of the paper is a set of scaling laws for the amplitude and wavelength of wrinkles based on the ...
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39 views

What is the functional shape assumed by a flexible rod?

Be L a flexible rod. Say that it is very difficult to significantly stretch it, so that we can uniquely identify a point on it by a parameter $l \in [0, L]$ where $L$ is its length. Be $C$ a set of ...
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27 views

Determine particle velocity from density function

I'm modeling a 1D system which consists of a large number of discrete particles distributed on a line. As a continuous approximation, I'm defining $c(x,t)$ to be the space density function of these ...
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1answer
183 views

Why plane stress condition is taken for thin plates

Why plane stress is taken for thin plates? It says in the books that the stress variation is small for thin components and is close to zero. Why is that so? Also why stress at free surface is zero? ...
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1answer
140 views

Hookes Law and Objective Stress Rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
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1answer
259 views

Relationship between the continuity equation and the wave equation

What exactly is the relationship between the continuity equation and the wave equation? Suppose $J^\mu$ is a contravariant vector that satisfies the continuity equation $\partial_\mu J^\mu=0$. Let ...
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1answer
67 views

Curve of a rod bent by force on both sides

Suppose we have a flexible rod (i.e. it can be bent without breaking apart) and we excert a force on both sides, like this: If the force $F$ is not exactly horizontal, the rod will be bent and form ...
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66 views

buckling of tube - shell thickness vs. momentum of inertia optimum

is there any simple formula (perhabs semi emperical, or aproximatively derived model) for buckling of tube under axial compression load given its crossection and wall thickness? ( and naturraly ...
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4answers
428 views

Why are stresses of continuum systems described via a tensor?

The tittle pretty much says enough. I have always been told so but no one really motivated it. So, I would like to know why do we use a tensor to describe the stresses in continuum mechanics.
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147 views

Volumetric and Deviatoric Strain Equation in 2D

Strain is defined as $$\epsilon=\frac{1}{2}\left( \nabla u + \nabla u^T\right).$$ I found a formula for the strain tensor in 3D decomposed into volumetric and deviatoric components: $$\epsilon= v + ...
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33 views

What is the criterion of stability of thick-walled spherical shell?

Is there the formula (if someone already has discovered it) or what is the algorithm (if a particular formula was not deduced), to calculate the critical pressure of thick-walled spherical shell $−$ ...
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1answer
51 views

Stress in horizontal bars

Imagine we have an horizontal bar. My teacher expresses the tensions along the longitudinal axis by this way $\sigma_{xx}=A(x)y+B(x)$ He doesn't give any motivation behind this. So, is this general? ...
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81 views

Why only vertical component of the stress tensor on vertically suspended bar?

EDIT: I am gonna rephrase the question entirely. Imagine we have a bar which we will analyze in the linear elastic regime. The shape of the cross section is irrelevant. The bar is suspended from a ...
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68 views

Is there quantitative theory of cutting with edge or blade

I wonder if there is some simple theory of what determine efficiency ( speed, energy end force required ) of cutting by edge ( blade , knife, sword ) At least something phenomenological like in ...
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1answer
146 views

Solid-body rotation of fluid in polar coordinates: How to compute the stress tensor

In a course on continuum mechanics, we are given an exercise concerning solid-body rotation of a fluid in polar coordinates. In the first parts (feel free to correct any errors here) we are tasked ...
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1answer
91 views

Why does shape of elements matter in finite elements analysis? [closed]

I have used FEA for a couple of years now, but using it and using it correctly are two different things, safety factor is not the solution to everything. I have the feeling I won't be using it right ...
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1answer
78 views

What is the difference between a linear and non-linear solution in the bending of beams?

I have been working on a simulator for bending of beams and came now to a tricky doubt: What should be the difference between a linear and non linear solution in this case (graphic at bottom)? The ...
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2answers
98 views

pure compression or pure traction?

I know that if we are given a stress tensor that is diagonal, the sign on the diagonal entries tell us whether we have traction or compression. Now, imagine that we are given a non diagonal stress ...
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2answers
130 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 ...
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35 views

Are continuous mathematical models of discrete physical phenomena messy because of a disconnect between “continuous” and “discontinuous”? [duplicate]

A copy of my question on Mathematics: Examples from statistical mechanics and continuum mechanics abound: a discrete phenomenon (e.g. kinetic energy of molecules) is "averaged" out over the ...
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1answer
281 views

Viscosity coefficients

I'm using the 2nd edition of "Transport Phenomena" by Bird and Stewart. I am having trouble with one of the equations: $$\tau_{ij} = \sum_k \sum_l \mu_{ijkl} \frac{\partial v_k}{\partial x_l} $$ ...
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2answers
189 views

What is the “discrete” analogue to “continuum” mechanics?

If I wanted to explore a discrete mathematics approach to continuum mechanics, what textbooks should I look into? I suppose a ready answer to the question might be: "computational continuum ...
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1answer
61 views

Resource(s) for developing a good understanding of surface tension?

I have read through several junior undergraduate level explanations of surface tension. Here is a typical presentation at that level: Molecules at the surface of a fluid experience approximately ...
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298 views

Continuum limit for solid mechanics

Is there a rigorous derivation of the limits for continuum properties in solid mechanics? For instance, the stress-strain relationship may be linear for large samples (the slope being the Young's ...
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184 views

Classical point particles to classical fields

I often hear that in the continuum limit we can study large numbers of particles as fields. I always imagined that by removing all bounds on the number of particles (while keeping total energy, ...