Questions tagged [continuum-mechanics]
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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How could we estimate the number of molecules below which a fluid cannot be adequately modeled by the Navier-Stokes equations?
As is known, the Navier-Stokes equations are an approximation in the continuum mechanics to model an aggregate of independent molecules, which, although they can move freely, interact strongly with ...
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Transformation of density with finite deformations
Initially, there is a infinite continuous medium with a given mass density $\rho(\vec r)$, which is a smooth function that vanishes outside of a sphere of radius R. Then, a deformation of medium is ...
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How to get local density field of a compressible solid Neo-Hookean material undergoing large deformation?
My understanding of a compressible material in the context of non-linear elasticity in continuum mechanics is that the volume of the body can change due to the applied forces. Due to the law of ...
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What is the hydrodynamic limit exactly and why is it called that?
Hydrodynamics is one of those words which are used everywhere in the literature, but I can not seem to find a clear definition! My idea (which could be wrong) is it is a continuum limit of a theory ...
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Do fluids flow from high pressure areas to low pressure areas? What theory explains that?
Do fluids flow from high pressure areas to low ones? Why ?
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Recommend a classic physics book to prep for uni course [closed]
Biology/premed major here, always been pretty poop at physics despite being good at math, but I have a physics course next year concerning classical mechanics and was wondering if you if someone could ...
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What curve does a rod form when bent to intersect 3 or more points?
Suppose that we have a sufficiently thin, flexible cylindrical rod of length $L$ made from a homogeneous, isotropic material, and that initially [at rest?] the central axis of the rod is a straight ...
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Is there a Lagrangian $L$ (equivalently an action functional $S$) which yields the Navier-Stokes equation?
The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws.
However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\...
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Question to multi-component fluids
I have read these lecture notes:
http://geo.mff.cuni.cz/~soucek/vyuka/materials/theory-of-mixtures/theory_of_mixtures-lecture-notes.pdf
Now look at page 41: There are 4 different types of mixture ...
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How to obtain nodal forces from stress tensor?
I am working on a FEM code in which I need to obtain the force vector on each node of a triangular linear element from the stress tensor in 2D. I have the shape functions, and the stress is constant ...
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$F=m*a$ accounting for pressure wave propagation
Imagine a long deformable rod which has just been hammered on the top end (the bottom end is clamped to Earth). Consider a time interval $dt$ = $t_{2}$ - $t_{1}$ in which the pressure wave is ...
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Requesting good reference to understand the Gurtin-Murdoch model of surface elasticity
I am searching for a good and understandable reference about the Gurtin-Mordoch model for surface elasticity. I was wondering if anyone can help me with the preliminaries and good references to fully ...
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Equilibrium and constitutive partial and algebraic equations describing stresses and deformation of an axisymmetric elastic thin shell over a hole
I want to design a vacuum table to clamp down a very thin plate and I want to know the stresses and deformations due to the atmospheric pressure. Consider the simplified model below:
...
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Confusion in Derivation of Excess Pressure in a Cylindrical Drop
I have recently learnt about surface tension and have developed a list of key points to solve problems:-
Surface tension acts on the surface where a surface is defined as the interface(flat or curved)...
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Inconsistent units in modal analysis of vibrating beam
(read the last sentence for the actual question).
Consider an elastically suspended solid body from a massless cantilever beam
The equations of motion of such a system are
$$\begin{bmatrix}m\\
& ...
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In Reynolds transport theorem, is the integral region simply-connected?
Reynolds transport theorem for a material element (parcels of fluids or solids which no material enters or leaves) reads $^1$
$$
(1):\frac{d}{dt}\left(\int_{\Omega(t)} \mathbf{f}\,dV\right) = \int_{\...
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Why do we use cubic fluid elements when deriving fluid equations?
Recently I've been delving into fluid dynamics but is frequently troubled by a conceptual issue that I could never get my head around with.
To derive fluid equations, we have to separate fluid into ...
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How does the shape of an object affect the coefficient of restitution?
I can understand how elasticity of an object plays a role in determining the coefficient of restitution, but various sources say that the shape of the object is important as well (and in particular I ...
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Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model?
We know that the Standard Model is a theory about almost everything (except gravity). So it should be the basis of fluid mechanics, which is a macroscopic theory from experiences. So is it possible ...
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Why $\det(\mathbf{F}J^{-1/3}) = \det(\mathbf{F})J^{-1}$?
$J$ is the jacobian, and $F$ is the deformation gradient.
As illustrated in the picture above, I don't understand why is the $\det(J^{-1/3}) = \det(J^{-1})$ ?
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Rate of deformation tensor in cylindrical coordinates
A question regarding the rate of deformation in cylindrical coordinates.
The rate of deformation tensor, $D_{ij}$, is the symmetric component of the velocity gradient tensor, i.e.
$D_{ij} = (\frac{\...
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Oscillation of heavy spring
I am trying to derive the solution for oscillation of a mass of a heavy spring. I have classical spring-pendulum (harmonic pendulum) in mind with one side of spring attached to ceiling and suspended ...
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What physical state of a wound string corresponds to a 'tuned' string?
Many instruments can be strung with strings comprised of a solid core and wound with another metal wire, e.g. the viola. I was wondering about the physical state of wound strings when they are 'in ...
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Confusion with equation in the introduction to fluid-solid interaction
First, the pictures are taken from "Cardiovascular Mathematics_Modeling and simulation of the circulatory system" book.
So, below are equations for the fluid and solid before applying the ...
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What is Tension in a string? How is it produced at a molecular level?
I can't understand the direction of tension. Why is the direction of tension at the ends of a string away from the object or block of mass? Can someone tell me what happens internally in a string? ps: ...
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Homogeneous strain-Atomic position in deformed state
I encountered today an equation in a quite old paper, which I am not able to derive.
For the notation they define $\vec{\mathring{x}}$ as the reference position for any material point and $\vec{x}(\...
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Euler-Bernoulli equation for a periodically supported static beam
The Euler-Bernoulli equation for a homogeneous beam is
$$ EI w^{(4)}(x) = q(x),$$
where $w$ is beam height and $q$ is load density.
Inspired by the deflection in a multi-support cantilever bridge ...
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Concept of a point particle in physics
How is mechanics, which deals with theoretical point particles, applied to real objects? For example, a force acting on a point particle is reasonable, but for an extended object, how is it natural to ...
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Why does a uniform flexible rod sag?
Imagine a long flexible rod with uniform mass. The rod is supported by a pivot at its center and it is at equilibrium. How do we explain why the rod sags at the far ends?
If we attempt to draw a free ...
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What is the derivation of the generic form of the strain energy density in the Saint Venant-Kirchhoff model?
Following my question here, Why is the generic form of the strain energy density in the Saint Venant-Kirchhoff model equal to $$W = \frac{1}{2}C_{ijkl}E_{ij}E_{kl}$$
I understand the answer but there ...
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Stress tensor and equality of normal stresses on opposite faces
Consider a body arbitrarily loaded as shown,
At a particular point in the body, I take an element and show all the stresses acting on its faces.
To specify a plane I will be using the the axis which ...
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Does material time derivative physically make sense?
where $u$ is the velocity field.
So one term of the material time derivative tells us how much the field is changing with its motion along particle, and another when it's fixed in position and ...
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How to transform from a discrete to integral equation?
In this picture, the red curve is an elastic rod that has resistance to bending and extension. I am trying to model the adhesions (contact) between the rod and the substrate (glass): the green dashed ...
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Why is the stress tensor contravariant?
The stress tensor relates the traction $\vec{t}$ (force per area) on a surface with surface normal $\vec{n}$ usually written as (when disregarding co- and contravariance)
$$ t_j = \sigma_{ij} n_i.$$
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Why is the divergence of Cauchy stress equal to zero in the reference domain?
Below are three configurations of a body: The current one, the reference, and the natural one. The natural reference is when the configuration is pre-stressed. Why is the divergence of Cauchy stress ...
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Saint Venant–Kirchhoff model
Why is the second Piola–Kirchhoff stress tensor $S$ equal to $\lambda$ $tr$($E$) $I$ $+$ $2$ $\mu$ $E$ $?$ Is there a derivation of it? By other means, from where does the assumption of strain ...
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Why does the stress of fluid depend on rate of deformation unlike stress of solid that depends on deformation itself?
So as stated in the picture above, stress behavior in fluids and solids isn't the same. Why is it physically that way?
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Why is bulk modulus large for a nearly Incompresible material?
Source of the picture
I don't understand why do we consider a large value of bulk modulus when we want to model incompressibe material? I mean bulk modulus is the bulk change in volume and the ...
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Why do we decompose the boundary surface in Initial boundary-value problems?
Just started to learn about initial boundary-value problem, and I have came across the concept of decomposing the boundary surface into to disjoint parts, one for stress tensor field and another for ...
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How could we have a non-isothermal deformation without any heat source or heat flux?
I don't understand what does it mean that "no time remains for isothermal removal of heat"? The term "isothermal" means temperature is constant, so how does isothermal removal of ...
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Why do I get different Euler-Almansi strains depending on how I calculate them?
I'm trying to calculate the Euler-Almansi strain for a simple bar with one end fixed (see drawing below). I know that the Almansi strain can be calculated by either directly using the displacement ...
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Why does entropy decreases with the increase of helmholtz free energy?
I came across the following constitutive equation relating entropy with the change of Helmholtz free energy with respect to change in temperature while holding deformation tensor fixed.
Source of ...
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Stress-strain relationship for linear viscoelastic solid
I am a bit confused about the definition of a linear viscoelastic (isotropic) solid.
Following Landau and Lifshitz (Theory of Elasticity, section "viscosity of solids"), I would say that in ...
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What does it mean for a quantity to be immaterial and why can we replace it by its tensor product?
So if we have a composite material that has only one type of fiber that has a specified direction expressed with the vector a, then why can we replace the diretion vector a by its tensor product when ...
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Relation between Youngs modulus $Y$ and Bulks modulus $B$ for a cube
lets suppose a cube of side $l$, Young's modulus $Y$, bulk's modulus $K$ under a force F across all sides.
so $$Y=\frac{F*l}{\Delta l*l^2}$$
now
$$\Delta v=l^3-(l-\Delta l)^3$$
now ignoring powers of $...
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Will the equation of continuity be applicable when there are more than one path where the liquid may flow like in a junction?
Consider a situation of a container having cross-section A and two holes of area "a" are at some point on the sides of the container diametrically opposite to each other . If fluid is filled ...
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What is the difference between Kirchhoff stress tensor and Cauchy stress tensor?
The Kirchhoff stress tensor is the Cauchy stress tensor multiplied by the Jacobian.
Normally Jacobian which is the determinant of deformation gradient tensor is used when transforming between material ...
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Difference between objectivity and isotropy
For a material property to be objective (frame-indifferent) it should not change if another observer is measuring it from his own reference frame.
For a material property to be isotropic means that it ...
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Does equal deformation gradient tensor imply same material configuration?
If we consider a dynamical process (deformation taking place) that takes place between two instants of time, and if we had the deformation gradient at the initial and final time to be equal, does that ...
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Why do we consider left Cauchy-Green tensor in isotropic elasticity?
Isotropic elasticity starts with the assumption that the constitutive equation of stress depends on a response function that is expressed in terms of left Cauchy green tensor.
Why do we do this ...
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