Ch. 1. Movement Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 3 Overview (cont’d) Stress Tensor in Different Coordinate Systems

CH.4. STRESS Continuum Mechanics Course (MMC)

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Overview

 Forces Acting on a Continuum Body

 Cauchy’s Postulates

 Stress Tensor

 Stress Tensor Components  Scientific Notation  Engineering Notation  Sign Criterion

 Properties of the Cauchy Stress Tensor  Cauchy’s Equation of Motion  Principal Stresses and Principal Stress Directions  Mean Stress and Mean Pressure  Spherical and Deviatoric Parts of a Stress Tensor  Stress Invariants

Lecture 1 Lecture 1

Lecture 2

Lecture 3 Lecture 4

Lecture 5

Lecture 6

Lecture 7

https://youtu.be/cmwjCla3wbY?t=00m00s https://youtu.be/Pkg5fAtpQ5s?t=00m00s https://youtu.be/9c9jhomP13A?t=00m00s https://youtu.be/S-2-rYkEfho?t=00m00s https://youtu.be/AaO37g-0zYo?t=00m00s https://youtu.be/WHEmfz8ks4Y?t=00m00s https://youtu.be/H0AZ81WPLZw?t=00m00s https://youtu.be/sq0625Wk5PM?t=00m00s

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Overview (cont’d)

 Stress Tensor in Different Coordinate Systems  Cylindrical Coordinate System  Spherical Coordinate System

 Mohr’s Circle

 Mohr’s Circle for a 3D State of Stress  Determination of the Mohr’s Circle

 Mohr’s Circle for a 2D State of Stress  2D State of Stress  Stresses in Oblique Plane  Direct Problem  Inverse Problem  Mohr´s Circle for a 2D State of Stress

Lecture 8

Lecture 9

Lecture 10

Lecture 11

Lecture 12

Lecture 13

https://youtu.be/4yBrjDzm82M?t=00m00s https://youtu.be/79rbGpRn_jE?t=00m00s https://youtu.be/JNM93noleY4?t=00m00s https://youtu.be/Tx_eR_SBHrs?t=00m00s https://youtu.be/cOTJfCvAGqI?t=00m00s https://youtu.be/3xIGagdCfZo?t=00m00s

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Overview (cont’d)

 Mohr’s Circle a 2D State of Stress (cont’d)  Construction of Mohr’s Circle  Mohr´s Circle Properties  The Pole or the Origin of Planes  Sign Convention in Soil Mechanics

 Particular Cases of Mohr’s Circle

Lecture 13

Lecture 14

Lecture 15

Lecture 16

Lecture 17

https://youtu.be/3xIGagdCfZo?t=00m00s https://youtu.be/k5Co-Arw-IQ?t=00m00s https://youtu.be/NbyBSEslOz8?t=00m00s https://youtu.be/4D1XCZ97bPE?t=00m00s https://youtu.be/4oHsHfLbgKg?t=00m00s

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Ch.4. Stress

4.1. Forces on a Continuum Body

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Forces acting on a continuum body:  Body forces.  Act on the elements of volume or mass inside the body.  “Action-at-a-distance” force.  E.g.: gravity, electrostatic forces, magnetic forces

 Surface forces.  Contact forces acting on the body at its boundary surface.  E.g.: contact forces between bodies, applied point or distributed

loads on the surface of a body

Forces Acting on a Continuum Body

( ),V V t dVρ= ∫f b x

( ),S V t dS∂= ∫f xt

body force per unit mass

(specific body forces)

surface force per unit surface (traction vector)

https://youtu.be/Pkg5fAtpQ5s?t=00m00s

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Ch.4. Stress

4.2. Cauchy’s Postulates

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1. Cauchy’s 1st postulate. The traction vector remains unchanged for all surfaces passing through the point and having the same normal vector at .

2. Cauchy’s fundamental lemma (Cauchy reciprocal theorem)

The traction vectors acting at point on opposite sides of the same surface are equal in magnitude and opposite in direction.

Cauchy’s Postulates

t

n P P

( ),P=t t n

REMARK The traction vector (generalized to internal points) is not influenced by the curvature of the internal surfaces.

( ) ( ), ,P P= − −t n t n

P

REMARK Cauchy’s fundamental lemma is equivalent to Newton's 3rd law (action and reaction).

https://youtu.be/9c9jhomP13A?t=00m00s

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Ch.4. Stress

4.3. Stress Tensor

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 The areas of the faces of the tetrahedron are:

 The “mean” stress vectors acting on these faces are

 The surface normal vectors of the planes perpendicular to the axes are

 Following Cauchy’s fundamental lemma:

Stress Tensor

1 1

2 2

3 3

S n S S n S S n S

= = =

( ) ( ) ( ) 1 2 3

1 * 2 * 3 ** * * * * 1 2 3

* *

ˆ ˆ ˆ( , ), ( , ), ( , ), ( , )

1, 2,3 ;

t t x n t t x e t t x e t t x e

x x i

S S S S

S i SS i S

 = − = − − = − − = − 

∈ = ∈ → mean value theorem

REMARK

The asterisk indicates an mean value over the area.

{ }T1 2 3n , n , n≡nwith

1 1 2 2 3 3ˆ ˆ ˆ; ;= − = − = −n e n e n e

( ) ( ) ( ) ( ) { } not

i i iˆ ˆ, , i 1, 2,3t x e t x e t x− = − =− ∈

https://youtu.be/S-2-rYkEfho?t=00m00s

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 Let be a continuous function on the closed interval , and differentiable on the open interval , where .

Then, there exists some in such that:  I.e.: gets its “mean value” at the interior of

Mean Value Theorem

[ ]: a, bf → R

[ ]a,b ( )a,b ( )a,b

a b< *x

( ) ( )* 1 df x f x Ω

= Ω Ω ∫

[ ]: a, bf → R

( )*f x [ ]a,b

https://youtu.be/S-2-rYkEfho?t=02m10s

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 From equilibrium of forces, i.e. Newton’s 2nd law of motion:

 Considering the mean value theorem,

 Introducing and ,

Stress Tensor

( ) ( ) ( )

1 2 3

1 2 3 V S S S S V

dV dS dS dS dS dVρ ρ+ + − + − + − =∫ ∫ ∫ ∫ ∫ ∫b t t t t a

( ) ( ) ( )1 2 3* * * * * * 1 2 3( ) V S S S S ( ) Vρ ρ+ − − − =b t t t t a

resultant body forces

resultant surface forces

{ }1,2,3i iS n S i= ∈ 1 3

V Sh=

( ) ( ) ( )1 2 3* * * * * * 1 2 3

1 1( ) S S S S ( ) 3 3

h S n n n hSρ ρ+ − − − =b t t t t a



i i i i i V V V V

m dV dS dV dV dm

ρ ρ ρ ∂

= = + = =∑ ∑ ∫ ∫ ∫ ∫R f a b t a a

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 If the tetrahedron shrinks to point O,

 The limit of the expression for the equilibrium of forces becomes,

Stress Tensor

( ) ( ), i iO n− =t n t 0( ) ( ) ( ) 1 2 3* * * * * *

1 2 3 1 1( ) ( ) 3 3

h n n n hρ ρ+ − − − =b t t t t a

* *

h 0 h 0

1 1lim ( ) lim ( ) 3 3

h hρ ρ → →

   = =       

b a 0

( ),= t nO

( )1= t ( )2= t ( )3= t

( ) ( ) ( ) ( ) { } ( ) ( )

i i

i i* * * S S i i

* * * S S

h 0

h 0

ˆ ˆlim , i 1, 2,3

lim , ,

x x t x e t e

x x t x n t n

O

O

O,

O →

→

 → = ∈   → = 

https://youtu.be/S-2-rYkEfho?t=14m14s

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P

 Considering the traction vector’s Cartesian components :

 In the matrix form:

( ) ( )

( ) 

( ) ( )

( )

,

,

,

i i

i j j i i ij

ij

P n

P n n

P P

σ σ

= ⇒

 = =    = ⋅

t n t

t n t

t n n σ

( ) ( ) ( ) ( ) ( )

{ } ( ) ˆ ˆ( )

, 1, 2,3 i i

j j ij j

i ij j

P t P i j

P t P

σ

σ

 = = ∈ =

t e e

ˆ ˆσ= ⊗e eij i jσ Cauchy’s Stress Tensor

[ ] [ ] [ ] {1,2,3}

T j i ij ji i

T

t n n j

σ σ = =  ∈ 

= t nσ

( )1t ( )2t ( )3t

( )1 1t

( )2 1t

( )3 1t

Stress Tensor

https://youtu.be/AaO37g-0zYo?t=00m00s

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REMARK 2 The Cauchy stress tensor is constructed from the traction vectors on three coordinate planes passing through point P.

Yet, this tensor contains information on the traction vectors acting on any plane (identified by its normal n) which passes through point P.

Stress Tensor

REMARK 1 The expression is consistent with Cauchy’s postulates:

( ) ( ),t n n σP P= ⋅ ( ),t n n σP = ⋅ ( ),t n n σP − = − ⋅ ( ) ( ), ,P P= − −t n t n

11 12 13

21 22 23

31 32 33

σ σ σ σ σ σ σ σ σ σ

   ≡     

https://youtu.be/AaO37g-0zYo?t=06m30s

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Ch.4. Stress

4.4.Stress Tensor Components

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 Cauchy’s stress tensor in scientific notation

 Each component is characterized by its sub-indices:  Index i designates the coordinate plane on which the component acts.  Index j identifies the coordinate direction in which the component acts.

Scie