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Hot answers tagged solid-mechanics

10 votes
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4th order tensor rotation - sources to refer

There are two main ways to write stress/strain tensors as 6 components vectors: Voigt notation, that is the most common; and Mandel-Kelvin notation, that has the advantage of writing stress and ...
• 8,510
9 votes
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Why is the FVM traditionally used in CFD, and FEM in computational structures?

The finite element method is actually quite widely used in fluid flow problems, for example for the Stokes and Navier-Stokes equations. The delineation between the methods is more along the following ...
• 55.5k
9 votes

Is it really necessary to solve a system of linear equations in the Finite Element Method?

I think your question is actually pretty fundamental and deserves a thoughtful answer. Paraphrasing a bit, your question is perhaps motivated by the observation that engineering design is often ...
• 4,862
7 votes
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Derivative of the inverse of the Right Cauchy-Green Deformation Tensor wrt itself

Nick Alger gives a nice explanation. Here is another one, possibly slightly simpler because it avoids the "should stay roughly the same" part. Let's say you want to compute the derivative of any ...
• 55.5k
7 votes
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• 1,522
5 votes
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Why the solid FEM problem can not be solved after constraining 3 degrees of freedom?

The particular set of constraints you have chosen does not prevent a rigid body rotation about node 1. Thus the stiffness matrix is singular, as you have noted. One way to prevent this rigid body ...
• 6,064
5 votes
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Uniaxial stretching solution not uniform in FEM code

The reason that this particular mesh does not give the correct, uniform displacement solution to this problem is that it is "non-conforming." Specifically, at the intersection of the two ...
• 6,064
5 votes
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Transition from 2D to 3D finite element code, what are the inevitable modifications to be implemented?

So so many places you have to rewrite. The whole mesh handling (accessing faces and edges from cells, neighbors from cells, ...). Shape functions. Dealing with the question of how the normal vector of ...
• 55.5k
5 votes
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Why FEM for incompressible materials is ill-posed?

For incompressible materials, as the extent of incompressibility increases, the bulk modulus approaches infinity (for the isotropic case). This is what causes ill-conditioning. Consider the case of ...
• 934
5 votes

If FEM is exact at the nodes, why do first and second-order elements give very different results?

They should both converge to the same limit solution, but are expected to do so at different rates of convergence. The best way to verify the convergence rates is to use a test-setting with known ...
• 2,935
4 votes

Developing a C++ solid mechanics program

Specific answers to this question are probably time-limited. However, the following general approach (from the great Eric S. Raymond) works very well: Rule of Modularity: Write simple parts ...
4 votes

Simulation of a complicated projectile motion

You need to know which equations you need to solve inc initial conditions. IMO, a disadvantage of click&result software is that it's not transparent to what you are actually solving. Why don't ...
• 277
4 votes
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Structural mechanics - traction free = Zero displacement gradient?

The traction is defined as $$\mathbf{t} = \mathbf{n} \cdot \boldsymbol{\sigma}$$ In terms of components, the zero-traction condition is $$t_j = \sum_i n_i \sigma_{ij} = 0$$ From the above ...
4 votes

Time step relationship with number of elements or material properties

The most commonly used explicit ODE solver in structural analysis is the central difference method. Because it is explicit, the solution becomes unstable if the time step is larger than a so-called ...
• 6,064
4 votes

How does one calculate reaction force in FEA?

Once the solution of the problem is known, i.e. you know displacement vector, to calculate reaction/internal forces an integral is evaluated \mathbf{f}^\textrm{int} = \sum_{e=1}^{...
• 906
4 votes
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1D analytical solution vs FEM solution for a bar under compression

It is a matter of boundary conditions on the longitudinal faces. As you noted, the axial stress and strain for a linear isotropic material will satisfy the following relation: \...
• 472
4 votes
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I wrote a 2D Finite Element program for Axial Loaded Plates, but the results are unexpected

The primary problem is that the CST approximation has a different displacement response depending on the orientation of mesh elements relative to the applied element loading (you're only allowed to ...
• 2,749
4 votes

How to determine global stiffness matrix is constrained or not

You already know that at least theoretically, unconstrained matrices have a null space and consequently eigenvalues that are equal to zero. But, in practice, this is a meaningless condition because it ...
• 55.5k
4 votes

Global stiffness matrix from element stiffness matrices for a thin rectangular plate (Kirchhoff plate)

What you are looking for is a Discrete Kirchhoff Quadrilateral plate or DKQ plate. Seems you are looking for a very straight ...
• 141
4 votes

How are the classical set of equilibrium equations for linear elasticity derived?

You take an arbitrary volume $V$ and use the translational and rotational equilibrium equations over it. They read \begin{align*} \int\limits_A \mathbf{t}\mathrm{d}A + \int\limits_V \mathbf{f} \mathrm{...
• 8,510
4 votes
Accepted

How to solve a linear problem A x = b in PETSC when matrix A has zero diagonal enteries?

Use of Lagrange multipliers produces a saddle-point problem,  \begin{pmatrix} A & B^T \\ B & 0 \end{pmatrix} \begin{pmatrix} u \\ \lambda \end{pmatrix} = \begin{pmatrix} b \\ 0 \end{pmatrix} ...
• 25.7k
4 votes

Modelling question: example of a physical phenomenon with this jump condition at an interface?

In addition to Wolfgang Bangerth's explanation of temperature and concentration, let me give an other application where such interface conditions arise: (linear) elasticity, which has a similar ...
• 822

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