Dealing with Stress! Anurag Gupta, IIT Kanpur Augustus Edward Hough Love (1863‐1940) Cliffor d AAbmbrose TdllTruesdell III (1919‐2000)

(Portrait by Joseph Sheppard) What is Stress?

“The 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

“…the physical reality of such modes of action is, in this view, admitted as part of the conceptual scheme”

(Quoted from Love) Cauchy’s stress principle

Upon the separang surface ∂Ω, there exists an integrable field equivalent in effect to the acon exerted by the maer outside ∂Ω to tha t whic h is iidinside ∂Ω and contiguous to it.

This field is given by the traction vector t.

30th September 1822

t n ∂Ω

Ω Augustin Louis Cauchy (1789‐1857) Elementary examples

(i) Bar under tension

A: area of the c.s.

s1= F/A

s2= F/A√2

(ii) Hydrostatic pressure

Styrofoam cups after experiencing deep‐sea hydrostatic pressure http://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

Total force acting on Ω F(Ω) = Fc(Ω,B\ Ω) + Fd(Ω) Ω

Mc(Ω,B\ Ω, c) = ∫∂Ω(X –c) x t dA moment due to contact action

B 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)

Linear momentum of Ω L(Ω) = ∫Ωρv dV

Moment of momentum of Ω G(Ω, c) = ∫Ω (X –c) x ρv dV Euler’s laws

F = dL /dt

M = dG /dt Euler, 1752, 1776

Or eqqyuivalently

∫∂Ωt dA + ∫Ωρb dV = d/dt(∫Ωρv dV)

∫∂Ω(X –c) x t dA + ∫Ω(X –c) x ρb dV = d/dt(∫Ω(X –c) x ρv dV)

Leonhard Euler (1707‐1783) Kirchhoff, 1876 portrait by Emanuel Handmann Upon using mass balance these can be rewritten as

∫∂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 p 2 2 A(da) = πR + o(R ), 2 A(fa) = o(R ), 2 V (Pa) = o(R ).

Subtract 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

2 ∫d1 t dA ‐∫d2 t dA = o(R )

2 2 2 Divide by πR to write (1/ A(d1)) ∫d1 t dA = (1/ A(d2)) ∫d2 t dA + (1/πR ) o(R )

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)

n To prove it consider a pillbox of thickness ε.

ε The linear momentum balance for ε→0 reduces to

Limε→0 ∫∂Pε t dA = 0 P ε ∫ (t(X,n)+ t(X,‐n)) dA = 0 S ‐n S 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 that t(X,n) = σ(X)n

Note that: ∫∂T t dA = ∫S t(X,m)dA‐∑a=1 to 3 (∫Sa t(X, ea)dA)

NtNext recalling the bblalance 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 get Cauchy’s tetrahedron T

V(T) = cδ3 As δ→0, t(X0;m) = ∑a=1 to 3 (m∙ ea) s(X0; ea) A(()s) = dδ2 A(sa) = (m ∙ ea) A(s) Hence t is linear in m. Cauchy’s equations of motion

Using Cauchy’s theorem we can obtain the fllfollow ing from the blbalances of momentums:

Div σ + ρb = ρa (σij,j + ρbi = ρai)

T σ = σ (σij = σji)

From Cauchy’s Exercices de Mathématiques, 1829. Euler’s hydrodynamics

Euler’s equations are recovered when

t=‐pn, which implies σ = ‐pI

‐Grad p + ρb = ρa

These are among the first partial differential 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 Mémoires de l’ Académie des Sciences, Berlin , 1755. Cauchy, 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 given poi”int”

(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 angular momentum balance. Euler’s theory of deformable lines

1770‐1774 Euler’s equations of motion:

Appearance of shear force

Can 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 NthiNothing ihis har der to surmoun ttht than a corpus of true but too special knowledge; to reforge the tradition of his forbears is the greatest originality a man can have. ‐ C. Truesdell

A page from Galileo’s discussion on the breaking strength of a cantilever, 1638.