dealing with stress - iit kanpurhome.iitk.ac.in/~ag/papers/stress_anurag.pdf · 2013. 10. 7. ·...
TRANSCRIPT
Dealing with Stress!Dealing with Stress!Anurag Gupta, IIT Kanpur
Augustus Edward Hough Love (1863‐1940)Augustus Edward Hough Love (1863‐1940)
Cliff d A b T d ll III (1919 2000)Clifford Ambrose Truesdell III (1919‐2000)
(Portrait by Joseph Sheppard)
What is Stress?What is Stress?“The notion (of stress) is simply that of mutual action between twoThe notion (of stress) is simply that of mutual action between two bodies in contact, or between two parts of the same body separated by an imagined surface…”
∂Ω ∂Ω∂Ω
Ω Ω
∂Ω ∂Ω
“…the physical reality of such modes of action is, in this view, admitted as part of the conceptual scheme”part of the conceptual scheme
(Quoted from Love)
Cauchy’s stress principleCauchy s stress principle
h f ∂Ω h iUpon the separa ng surface ∂Ω, there exists an integrable field equivalent in effect to the ac on exerted by the ma er outside ∂Ω to th t hi h i i id ∂Ω d ti t itthat which is inside ∂Ω and contiguous to it.
This field is given by the traction vector t.
30th September 1822
∂Ωnt
Augustin Louis Cauchy (1789‐1857) Ω
Elementary examples(i) Bar under tension
A: area of the c.s.
s1= F/A1 /
s2= F/A√2
(ii) Hydrostatic pressure(ii) Hydrostatic pressure
Styrofoam cups after experiencing deep‐sea hydrostatic pressurehttp://www.expeditions.udel.edu/
Contact action
Fc(Ω,B\ Ω) = ∫∂Ωt dA net force through contact action
Fd(Ω) = ∫Ωρb dV net force through distant action
ΩTotal force acting on Ω F(Ω) = Fc(Ω,B\ Ω) + Fd(Ω)
B
Mc(Ω,B\ Ω, c) = ∫∂Ω(X – c) x t dA moment due to contact action
Md(Ω, c) = ∫Ω(X – c) x ρb dV moment due to distant actionB Md(Ω, c) ∫Ω(X c) x ρb dV moment due to distant action
Total moment acting on Ω M(Ω) = Mc(Ω,B\ Ω) + Md(Ω)g ( ) c( , \ ) d( )
Linear momentum of Ω L(Ω) = ∫Ωρv dV
Moment of momentum of Ω G(Ω, c) = ∫Ω (X – c) x ρv dV
Euler’s lawsF = dL /dt
M = dG /dtM = dG /dtEuler, 1752, 1776
Or equivalentlyq y
∫∂Ωt dA + ∫Ωρb dV = d/dt(∫Ωρv dV)
Leonhard Euler (1707‐1783)
∫∂Ω(X – c) x t dA + ∫Ω(X – c) x ρb dV = d/dt(∫Ω(X – c) x ρv dV)
Kirchhoff, 1876Leonhard Euler (1707 1783)portrait by Emanuel Handmann
Upon using mass balance these can be rewritten as
∫ t dA ∫ b dV ∫ dV∫∂Ωt dA + ∫Ωρb dV = ∫Ωρa dV
∫∂Ω(X – c) x t dA + ∫Ω(X – c) x ρb dV = ∫Ω(X – c) x ρa dV
Emergence of stress(A) Cauchy’s hypothesis
At a fixed point, traction depends on the surface of interaction only through the normal, i.e.
t(X,∂Ω) = t(X,n)
Cauchy, 1823,1827
Proof by Walter Noll (in 1957 at a symposium on axiomatic methods in Berkeley)
Walter Noll (1925‐)http://www.math.cmu.edu/~wn0g/noll/
Proof of Cauchy’s hypothesis
∂P1 = d1 ∪ f1 ∪ e∂P2 = d2 ∪ f2 ∪ e
A(da) = πR2 + o(R2),A(fa) = o(R2), ( ) ( 2)
p
V (Pa) = o(R2).
Subtract the momentum balance equations for P1 and P2 to getSubtract the momentum balance equations for P1 and P2 to get
∫d1 t dA ‐ ∫d2 t dA = ∫P1 ρ(a‐b) dV ‐ ∫P2 ρ(a‐b) dV + ∫f1 t dA ‐ ∫f2 t dA
∫d1 t dA ‐ ∫d2 t dA = o(R2)
Divide by πR2 to write (1/ A(d1)) ∫d1 t dA = (1/ A(d2)) ∫d2 t dA + (1/πR2) o(R2)y ( 1 ) ∫d1 ( 2 ) ∫d2 ( ) ( )
Finally, use Mean value theorem to get (for continuous t)
t(p, d1) = t(p, d2)
Emergence of stress(B) Cauchy’s lemma
t(X n) = t(X n)t(X,n) = ‐ t(X,‐n)
nTo prove it consider a pillbox of thickness ε
ε
To prove it consider a pillbox of thickness ε.
The linear momentum balance for ε→0 reduces to
Pε‐n
Limε→0 ∫∂Pε t dA = 0
∫S (t(X,n) + t(X,‐n)) dA = 0S n
which, using the continuity of t, localizes to the required result.
Emergence of stress(C) Cauchy’s theorem
There exists a linear transformation (tensor) σ(X) such thatt(X,n) = σ(X)n
Note that: ∫∂T t dA = ∫S t(X,m) dA ‐ ∑a=1 to 3 (∫Sa t(X, ea) dA)
N t lli th b l f li t hNext recalling the balance of linear momentum we have
| ∫∂T t dA |= |∫T ρ(a‐b) dV| ≤ ∫T |ρ(a‐b)|dV ≤ kV (T)
Finally use continuity of t and the Mean value theorem to getCauchy’s tetrahedron T
V(T) = cδ3A(s) = dδ2
Finally use continuity of t and the Mean value theorem to get
As δ→0, t(X0;m) = ∑a=1 to 3 (m∙ ea) s(X0; ea)( )
A(sa) = (m ∙ ea) A(s) Hence t is linear in m.
Cauchy’s equationsof motionof motion
Using Cauchy’s theorem we can obtainth f ll i f th b l fthe following from the balances of momentums:
Div σ + ρb = ρa (σ + ρb = ρa )Div σ + ρb = ρa (σij,j + ρbi = ρai)
σ = σT (σij = σji)
From Cauchy’s Exercices de Mathématiques, 1829.
Euler’s hydrodynamics
Euler’s equations are recovered when
t= pn which implies σ = pIt=‐pn, which implies σ = ‐pI
‐Grad p + ρb = ρa
These are among the first partial differential equations to be writtendifferential equations to be written
Pressure is introduced as a field
Dynamical equations for perfect fluids are given for the first time
Euler isolated parts of the body and studied action of the exterior on the interior
Euler’s fundamental equations of hydrodynamics from Mémoiresde l’ Académie des Sciences, Berlin , 1755.
Cauchy in introducing traction wrote that it is “of the sameCauchy, in introducing traction, wrote that it is of the same nature as the hydrostatic pressure exerted by a fluid at rest against the exterior surface of the body. Only, the new pressure does not always remain perpendicular to the planes subjected to it, nor the same in all directions at a i i ”given point”
(Translated by Truesdell)
In hydrostatics, however, there are no shear stresses (the idea of stress is constrained by geometry) and there is no place of angularis constrained by geometry) and there is no place of angular momentum balance.
Euler’s theory of deformable linesdeformable lines
1770‐1774
Euler’s equations of motion:Euler’s equations of motion:
Appearance of shear force
Can be seen as one dimensional formCan be seen as one dimensional form of Cauchy’s equations
From Novi Commentarii Academiae Scientiarum Petropolitanae, 1770
p.s.The term “stress” was used for the first time by William Rankine (1820‐1872)in a paper which appeared in Phil. Trans. Roy. Soc. London, 1856
Painted by R. C.Crawford
N thi i h d t t thNothing is harder to surmount than a corpus of true but too special knowledge;to reforge the tradition of his forbears is theto reforge the tradition of his forbears is thegreatest originality a man can have.
‐ C. Truesdell
A page from Galileo’s discussion on the breaking strengthof a cantilever, 1638.