Stress and deformation compressed cube

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Discussion Overview

The discussion revolves around the mechanical behavior of a compressed cube, specifically focusing on stress and deformation under various conditions. Participants explore theoretical aspects, mathematical modeling, and implications of material properties such as Young's modulus and Poisson's ratio.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a scenario involving a cube compressed with a specific stress and questions the relationship between strains, suggesting it may relate to volume conservation.
  • Another participant corrects the stress value in the problem statement and discusses the implications of Poisson's ratio being 0.5, indicating incompressibility and the sum of strains being zero.
  • In Part B, participants analyze whether the cube will touch the faces of an indeformable box based on calculated strains, with one participant expressing uncertainty about determining the stress and deformation tensor under these conditions.
  • Participants discuss the implications of incompressibility on stress equations and the determination of compressive stress, with one suggesting a method to find the stresses in the x and y directions based on known conditions.
  • In Part C, a participant speculates that if the box dimensions match the cube's, the deformation tensor would be zero, prompting further discussion on how to approach this without calculus.
  • Another participant raises a concern about determining εz and the implications of λ approaching infinity, questioning the validity of their approach and the necessity of certain assumptions.
  • One participant asserts confidence in their proposed method for solving the problem while expressing frustration over the lack of adherence to their advice.
  • A final post poses a question regarding the correctness of two algebraic equations related to stress and strain, inviting further scrutiny of the mathematical relationships involved.

Areas of Agreement / Disagreement

Participants express differing views on the implications of incompressibility, the validity of certain mathematical approaches, and the conditions under which the cube will touch the box. The discussion remains unresolved with multiple competing perspectives on the problem.

Contextual Notes

Participants reference specific mathematical relationships and conditions, such as the behavior of incompressible materials and the implications of Poisson's ratio, but do not reach a consensus on the correct approach or outcomes.

zDrajCa
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Hi, sorry for my bad english.
PART A
I have a cube of 10mm x 10mm x 10mm which is compressed with a stress of 0.5 Mpa on his superior face (σz=-0.6)
young modulus=20Mpa, poisson coef=0.5
I have the stress tensor:
[ 0 0 0]
[ 0 0 0]
[ 0 0 -0.6]

And then, the deformation tensor is :
[0.015 0 0]
[ 0 0.015 0]
[0 0 -0.3]

Why εx+εy+εz =0? is it a question of conservation of the volume ?

PART B
Well, now i put this cube into an indeformable box of 10.002mm (x) x 10.004 mm (y) x 10 mm (z) and this cube is again compressed with a stress of 0.5MPa;

1) do The face x and y of the cube will touch the faces of the box ?
I do ε=ΔL/L and i find εx=0.0002 and εy=0.0004. So I supposed that they will touch if we compared the deformations with the deformation in the first part.
2) Doing an hypothesis that the faces will be in contact with the face of the box, determined the stress and deformation tensor of the cube... But here i don't know how to do that... If someone can help me.. Thanks a lots

PART C
IF the box has the same dimension of the cube and if will do a stress of 0.6 MPa again, determined without calculus the deformation tensor.. I think all deformation will be equal 0 is it right ?
 
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zDrajCa said:
Hi, sorry for my bad english.
PART A
I have a cube of 10mm x 10mm x 10mm which is compressed with a stress of 0.5 Mpa on his superior face (σz=-0.6)
young modulus=20Mpa, poisson coef=0.5
I have the stress tensor:
[ 0 0 0]
[ 0 0 0]
[ 0 0 -0.6]

And then, the deformation tensor is :
[0.015 0 0]
[ 0 0.015 0]
[0 0 -0.3]

Why εx+εy+εz =0? is it a question of conservation of the volume ?
I think you meant 0.6 MPa in the problem statement, not 0.5.

Regarding your question, if Poisson's ratio is 0.5, that means that the material is incompressible, and the sum of the three strains is zero.
PART B
Well, now i put this cube into an indeformable box of 10.002mm (x) x 10.004 mm (y) x 10 mm (z) and this cube is again compressed with a stress of 0.5MPa;

1) do The face x and y of the cube will touch the faces of the box ?
I do ε=ΔL/L and i find εx=0.0002 and εy=0.0004. So I supposed that they will touch if we compared the deformations with the deformation in the first part.
2) Doing an hypothesis that the faces will be in contact with the face of the box, determined the stress and deformation tensor of the cube... But here i don't know how to do that... If someone can help me.. Thanks a lots
This is a tricky thing to analyze in the case of an incompressible material. For an incompressible material, you need to go to the incompressible limiting form of Hooke's law which is:

$$σ_x=-p+2Gε_x$$
$$σ_y=-p+2Gε_y$$
$$σ_z=-p+2Gε_z$$
where p is an arbitrary compressive stress that can be determined from the problem conditions. Since you know that the material is compressible and you know the strains in the x and y directions, what is the strain in the z direction? Once you know the strain in the z direction (and you know the stress in the z direction from the problem statement), you can use the z equation to determine the compressive stress p. What value do you get? Once you know p, you can get the stresses in the x and y directions.
PART C
IF the box has the same dimension of the cube and if will do a stress of 0.6 MPa again, determined without calculus the deformation tensor.. I think all deformation will be equal 0 is it right ?
You do this the same way you did part B, but here, all three strains is zero. So all you need to do is determine the compressive stress p.
 
yes it was what I have written for the Part A and the part C. In the part B I would like to determine εz with what i have written but λ→∞...

It is that: http://image.noelshack.com/fichiers/2015/45/1446679886-20151105-002818.jpgSO i can't use this method.

I have to do that ?
http://image.noelshack.com/fichiers/2015/45/1446680189-20151105-003354.jpg
But as we can see σx and σy are positive then i have to take them equal to 0 if i want them to touche the box right?
 
zDrajCa said:
yes it was what I have written for the Part A and the part C. In the part B I would like to determine εz with what i have written but λ→∞...

It is that: http://image.noelshack.com/fichiers/2015/45/1446679886-20151105-002818.jpg

So λ→∞, but the sum of the strains ---> 0, and their product is finite. That finite product is what I call p.

Now, you asked me for my help, and I've given you a method for solving this problem which I know is 100% correct. If you don't want to follow my advice, that's fine. Have a nice day.:smile:
 
Irrespective of the value of λ, are the following 2 algebraic equations (a) correct of (b) incorrect:
$$σ_x-σ_z=2μ(ε_x-ε_z)$$
$$σ_y-σ_z=2μ(ε_y-ε_z)$$
 

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