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Cauchy's first law of motion

Hi All

I'm a bioengineering PhD student, I just started reading on the finite elasticity theory and have a question regarding to the governing equation.

As I understand, Cauchy's first law of motion is the governing equation for finite elasticity. For steady-state (no acceleration), the equation is:

dσ/dx=F

where the dσ/dx is the tress gradient and is the F body force. This equation suggests that the stress shall be spatially constant with the absense of body force. However, I'm not sure where does the boundary forces (surface tractions) comes in? I know the stress may not be constant in the presence of the external force, but I just can not see the relationship between the surface tractionsand the governing equation.

Another question is, I know stress and strain is a relative measure, and I have often seen people to assume the stress is measured from an unstrained reference configuration such that zero stress occurs at zero strain. Why is this a common assumption? Is it just for convenience? or is it more to do with the fact that reference configuration shall be considered as a state in which the material is free of all forces (both bodyand surface forces), hence zero stress shall be present?

 

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You have differential equation. If F = 0 then σ = const. But what is this constant? This constant is determined by boundary conditions.

Math Example
Eq: y'(x) = 0
BC: y(0) = y0
Solution: y(x) = const = y0. 

Hi alex

 Thank you very much for the reply,  I understand that if the body force F=0, then  σ=constant from Cauchy's first law of motion.

 However, my question is that, some boundary conditions can cause non-constant σ without another body forces. Specifically, if I apply a boundary moment to a body (but no body force) so that it bends, obviously, the internal σ will not be constant in this case (due to inhomogeneous strain). However, this will not show up in the Cauchy's first law of motion, since according to the equation,  σ = const whenever there is no body force.

 Am I overlooking some critical information?

 

Of course I was not right! I am stupid! :-/

The Cauchy's Law may be written in the form

div σ = 0 (F = 0).

Here div is not usual derivation.

The solution of this equation can be more complex.

Second question.

I think that you are talking about initial stresses.

Initial stresses are stresses that appear in the part or the whole structure due to the conditions of its manufacture, for example: in casting, forging, stamping, assembly, etc.

In this case the external forces are not applied, but there are internal stresses.

Hi alex

Thank you again for answering my second question :-)

 So is the initial stresses usually constant inside the body? or can there be a stress concentration locally even where there are no external forces or body forces.

 Thank you again for answering my questions!

 

 

 

Hi again.

Of course initial stresses are not constant usually.

Consider thin rod. We bend it in a ring and glue the ends of rod. Now ring is the initial configuration. External forces are absent but there are initial stresses. These stresses are not constant in the ring. In this case the initial stresses are not local. But in other cases they may be local.

Usually initial stresses are associated with defects. These are dislocations and disclinations in elasticity theory.

The nonlinear theory of dislocations and disclinations we may see in "Nonlinear theory of dislocations and disclinations in elastic bodies" by Leonid M. Zubov.  

Just like to remind the following before you go too far.

d_sigma/d_x+f=0 is generally a partial differential equation, where sigma is a tensor and f is a vector. f=0 doesn't necessarily lead to sigma = constant.

Bests.

Thank you...

In my first comment I was very wrong!.. It was clouding my mind! :-)

thank you alex and Bo for your comments

  I see it now. Thank you both very much :-)

Mohammad Aramfard's picture

Sorry for my stupid question but it makes me busy: 

Is its dimension agreed?

Dsigma/Dx = Force/Length^3 

Dsigma - Force/Length^2

Dx - Length 

F - Force/Leght^3

Hi, ttme!

I'd like to share my opinoion about the 2nd question with you.

you know, σ、ε are relative variables, that's right. In mechanical analysis, people alway select a initial  configuration as reference configuration, the basis of analysis, so that we can see how the material will change after loading, compared with the zero-loading state. In order to understand the response of materials under the external/internal load(bosy force, surface force, thermal, and so on), it is understandable to compare loading state with zero-loading state. Not for  convenience, but for the purpose of mechanical analysis.

reference configuration is an important concepet in Continuum mechanics。

In mechanics we may introduce different configurations: reference (initial), undeformed, current (deformed).
Usually reference configuration is undeformed. This is easier and more common.
But we do not always know the undeformed configuration. We have ring and we are not know as it was made. This is undeformed ring or deformed rod. 
Also not always the undeformed state is not stress-free state. There are initial stresses or pre-stresses which occur due to defects in microscale. There is no the stress-free configuration in this case. Usually pre-stresses are unknown.  

Hi, alex_kam :)

Thank you for your discussion and pointing out the link among different configurations. Yeah, I know what you mean about the configurations. There are no necessary conection between the reference and undeformed configurations. We can choose any state as the reference configuration, but always regard it as undeformed one acquiescently.

How to characterize and ''measure'' the intrinsic stress is also a key problem caused by thermal mismatch or/and defects, which is a hot research point in microscale.

Mohammad Aramfard's picture

Thank you Alex, it was force per volumeCool

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