CH.4. STRESS - UPC Universitat Politècnica de...

CH.4. STRESSContinuum Mechanics Course (MMC) - ETSECCPB - UPC

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

03/11/2014MMC - ETSECCPB - UPC

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

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

Ch.4. Stress

4.1. Forces on a Continuum Body

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 Vt dV f b x

,S Vt dS

body force per unit mass

(specific body forces)

surface force per unit surface (traction vector)

Ch.4. Stress

4.2. Cauchy’s Postulates

1. Cauchy’s 1st postulate.The traction vector remains unchanged for all surfaces passing through the pointand 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

,Pt 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

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

Ch.4. Stress

4.3. Stress Tensor

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

S n SS n SS n S

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

ˆ ˆ ˆ( ), ( , ), ( , ), ( , )

1,2,3 ; mean value theoremi

S S S S

S i SS i S

t t x t t x e t t x e t t x e

REMARK

The asterisk indicates an mean value over the area.

T1 2 3n , n ,nnwith

1 1 2 2 3 3ˆ ˆ ˆ; ; n e n e n e

ii iˆ ˆ, , i 1, 2,3t x e t x e t x

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 interiorof

Mean Value Theorem

: a, bf R

a, b a,b a,b

* 1 df x f x

: a, bf R

*f x a,b

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

Considering the mean value theorem,

Introducing and ,

Stress Tensor

1 2 3 V S S S S V

dV dS dS dS dS dV b t t t t a

1 2 3* * * * * *1 2 3( ) V S S S S ( ) V b t t t t a

resultant body forces

resultant surface forces

1, 2,3i iS n S i 13

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

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

h S n n n hS b t t t t a

i i ii i V V V V

m dV dS dV dVdm

R f a b t a a

If the tetrahedron shrinks to point O,

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

Stress Tensor

, iiO n t n t 0 1 2 3* * * * * *

1 2 31 1( ) ( )3 3

h n n n h b t t t t a

h 0 h 0

1 1lim ( ) lim ( )3 3

, t nO

1 t 2 t 3 t

i i* * *S S i

* * *S S

ˆlim i 1,2,3

lim , ,

x x t x t e

x x t x n t n

Considering the traction vector’s Cartesian components :

In the matrix form:

ij j i i ij

( ) ˆ ˆ( )

, 1,2,3i i

j j ij j

P t Pi j

ˆ ˆ e eij i jCauchy’s Stress Tensor

{1,2,3}

Tj i ij ji i

11 21 31

12 22 32

13 23 33

1t 2t 3t

Stress Tensor

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: ,P P t n n

,P t n n ,P t n n , ,P P t n t n

11 12 13

21 22 23

31 32 33

Ch.4. Stress

4.4.Stress Tensor Components

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.

Scientific Notation

11 12 13

21 22 23

31 32 33

Cauchy’s stress tensor in engineering notation

Where: is the normal stress acting on plane a. is the tangential (shear) stress acting on the plane perpendicular to

the a-axis in the direction of the b-axis.

Engineering Notation

x xy xz

yx y yz

zx zy z

The stress vector acting on point P of an arbitrary plane may be resolved into:

a vector normal to the plane

an in-plane (shear) component which acts on the plane.

The sense of with respect to defines the normal stress character:

The sign criterion for the stress components is:

Tension and compression

( )n n

( ; )n n

n n<0 compressive stress (compression)>0 tensile stress (tension)

ij a ortensile stresscompressive stress

positive (+)negative (−)

positive direction of the b-axisnegative direction of the b-axis

Ch.4. Stress

4.5.Properties of the Cauchy Stress Tensor

Consider an arbitrary material volume, Cauchy’s equation of motion is:

In engineering notation:

Cauchy’s Equation of Motion

1,2,3ijj j

b a jx

yxx zxx x

xy y zyy y

yzxz zz z

b ax y z

REMARK Cauchy’s equation of motion is derived from the principle of balance of linear momentum.

For a body in equilibrium , Cauchy’s equation of motion becomes

The traction vector is now known at the boundary

The stress tensor symmetry is derived from the principle of balance of angular momentum:

Equilibrium Equations

0 1,2,3ijj

, , ,1,2,3i ij j

t t t Vn t j

n x x t x x equilibrium equation at the boundary

internal equilibrium equation

, 1,2,3T

ij ji i j

Taking into account the symmetry of theCauchy Stress Tensor, Cauchy’s equation of motion

Boundary conditions

Cauchy’s Equation of Motion

1,2,3ij jij j j

b b a jx x

b b a x

( , ), , 1,2,3i ij ji i j

t Vn n t t V i j

n n t x xx x

Regardless of the state of stress, it is always possible to choose a special set of axes (principal axes of stress or principal stress directions) so that the shear stress components vanish when the stress components are referred to this system.

The three planes perpendicular to the principal axes are the principal planes.

The normal stress components in the principal planes are the principal stresses.

Principal Stresses and Principal Stress Directions

0 00 00 0

333231

The Cauchy stress tensor is a symmetric 2nd order tensor so it will diagonalize in an orthonormal basis and its eigenvalues are real numbers. For the eigenvalue and its corresponding eigenvector :

Principal Stresses and Principal Stress Directions

v v v v 0 1

det 0 not

3 21 2 3( ) ( ) ( ) 0I I I σ σ σ characteristic

equation

INVARIANTS

REMARK The invariants associated with a tensor are values which do not change with the coordinate system being used.

Given the Cauchy stress tensor and its principal stresses, the following is defined: Mean stress

Mean pressure

A spherical or hydrostatic state of stress:

Mean Stress and Mean Pressure

1 2 31 1 13 3 3m iiTr

1 2 313mp

1 2 3 0 0

0 00 0

REMARK In a hydrostatic state of stress, the stress tensor is isotropic and, thus, its components are the same in any Cartesian coordinate system.As a consequence, any direction is a principal direction and the stress state (traction vector) is the same in any plane.

11 12 13

21 22 23

31 32 33

The Cauchy stress tensor can be split into:

The spherical stress tensor: Also named mean hydrostatic stress tensor or volumetric stress tensor or

mean normal stress tensor. Is an isotropic tensor and defines a hydrostatic state of stress. Tends to change the volume of the stressed body

The stress deviator tensor: Is an indicator of how far from a hydrostatic state of stress the state is. Tends to distort the volume of the stressed body

Spherical and Deviatoric Parts of a Stress Tensor

1 1:3 3sph m iiTr 1 1 1

dev m 1

REMARK The principal directions of a stress tensor and its deviator stress component coincide.

Principal stresses are invariants of the stress state: invariant w.r.t. rotation of the coordinate axes to which the stresses are

referred.

The principal stresses are combined to form the stress invariants I :

These invariants are combined, in turn, to obtain the invariants J :

Stress Invariants

1 1 2 3iiI Tr

22 1 1 2 1 3 2 3

3 detI

1 1 iiJ I

22 1 2

1 1 12 :2 2 2ij jiJ I I

33 1 1 2 3

1 1 13 33 3 3 ij jk kiJ I I I I Tr

REMARK The J invariants can be expressed in the unified form:

1 1, 2,3iiJ Tr i

REMARK The I invariants are obtained from the characteristic equation of the eigenvalue problem.

The stress invariants of the stress deviator tensor:

These correspond exactly with the invariants J of the same stress deviator tensor:

Stress Invariants of the Stress Deviator Tensor

1 1 0J I

J I 2 212 :2

J I 1 23I I 3 31 133 3 ij jk kiI I Tr

1 0I Tr 2

2 11 :2

I I 12 12 13 13 23 23

2 2 23 11 22 33 12 23 13 12 33 23 11 13 22

1det 23 ij jk kiI

Ch.4. Stress

4.6. Stress Tensor in Different Coordinate Systems

The cylindrical coordinate system is defined by:

The components of the stress tensor are then:

Stress Tensor in a Cylindrical Coordinate System

´ ´ ´ ´ ´

´ ´ ´ ´

´ ´ ´ ´ ´

x x y x z r r rz

x y y y z r z

x z y z z rz z z

cos ( , , ) sin

x rr z y r

dV r d dr dz

The cylindrical coordinate system is defined by:

The components of the stress tensor are then:

Stress Tensor in a Spherical Coordinate System

2 sen dV r dr d d

sen cos

, , sen sen cos

x rr y r

´ ´ ´ ´ ´

´ ´ ´ ´

´ ´ ´ ´ ´

x x y x z r r r

x y y y z r

x z y z z r

Ch.4. Stress

4.7. Mohr´s Circle

Introduced by Otto Mohr in 1882. Mohr´s Circle is a two-dimensional graphical representation of

the state of stress at a point that: will differ in form for a state of stress in 2D or 3D. illustrates principal stresses and maximum shear stresses as well as stress

transformations. is a useful tool to rapidly grasp

the relation between stresses for a given state of stress.

Mohr’s Circle

Ch.4. Stress

4.8. Mohr´s Circle for a 3D State of Stress

Consider the system of Cartesian axes linked to the principal directions of the stress tensor at an arbitrary point P of a continuous medium: The components of the stress tensor are

The components of the traction vector are

where is the unit normal to the base associated to the principal directions

Determination of Mohr’s Circle

0 00 00 0

1 1 1 1

2 2 2 2

3 3 3 3

0 0 0 0 0 0

n nn nn n

The normal component of stress is

The modulus of the traction vector is

The unit vector must satisfy

Locus of all possible points?

12 2 2

1 1 2 2 3 3 2 1 1 2 2 3 3

nn n n n n n n

1n 2 2 21 2 3 1n n n

2 2 2 2 2 2 21 1 2 2 3 3 2 2 2 2 2 2 2 2

1 1 2 2 3 32 2 2 : : n

n n nn n n

Mohr's 3D problem half space

The previous system of equations can be written as a matrix equation which can be solved for any couple

A feasible solution for requires that for theexpression to hold true.

Every couple of numbers which leads to a solution , will be considered a feasible point of the half-space. The feasible point is representative of the traction vector on a

plane of normal which passes through point P.

The locus of all feasible points is called the feasible region.

2 2 2 2 2 21 2 3 1

21 2 3 2

231 1 1

1 2 3, ,T

n n n n

212223

0 10 10 1

2 2 21 2 3 1n n n

2 2 21 2 3, ,

Tn n n x

The system

can be re-written as

1 2 2 3 1 3A with

2 2 21 3 1 3 1

2 2 22 3 2 3 2

2 2 21 2 1 2 3

AIII n

2 2 2 2 2 21 2 3 1

21 2 3 2

231 1 1

Consider now equation :

It can be written as:

which is the equation of a semicircle of center and radius :

2 2 2a R

2 23 1 2 2 3 1 3 3

1 , 02

1 2 1 2 31 2

1 2 2 3 1 3A with( )III

2 21 2 2 3 1 3 3

REMARK A set of concentric semi-circles is obtained with the different values of

with center and radius :3C 3 3R n3n

3 1 2 3

Following a similar procedure with and , a total of three semi-annuli with the following centers and radii are obtained:

( )II( )I

1C [ ,0]2

Superposing the three annuli,

The final feasible region must be the intersection of these three semi-annuli.

The final feasible region must be the intersection of these semi-annuli

Every point of the feasible region in the Mohr’s space, corresponds to the stress (traction vector) state on a certain plane at the considered point

Ch.4. Stress

4.9. Mohr´s Circle for a 2D State of Stress

2D State of Stress

3D general state of stress 2D state of stress

x xy xz

yx y yz

zx zy z

REMARK In 2D state of stress problems, the principal stress in the disregarded direction is known (or assumed) a priori.

3D problem

2D (plane) problem

Given a plane whose unit normal forms an angle with the axis, Traction vector

Normal stress

Shear stress

Tangential stress is now endowed with sign Pay attention to the “positive” senses given in the figure

Stresses in a oblique plane

sincos

cos sincoscos sinsin

x xy x xy

xy y xy y

cos 2 sin 22 2

x y x yxy

sin 2 cos 22

cossin

( 0 0)or

Direct Problem: Find the principal stresses and principal stress directions given in a certain set of axes.

Inverse Problem: Find the stress state on any plane, given the principal stresses and principal stress directions.

Direct and Inverse Problems

equivalent stresses

In the and axes, then,

Using known trigonometric relations,

Direct Problem

sin 2 cos 2 02

1sin 211

tg 2 2

1 2cos 21 tg 2

x́ ´y 0

This equation has two solutions:

These define the principal stress directions.(The third direction is perpendicular to the plane of analysis.)

1 ( " ")sign

2 1 ( " ")2

2cos 2 xy sin 2

2sin 2 xy cos 2

The angles and are then introduced into the equation

to obtain the principal stresses (orthogonal to the plane of analysis):

Direct Problem

x y x yxy

cos 2 sin 22 2

x y x yxy

2cos 2 xy sin 2

2sin 2 xy cos 2

Given the directions and principal stresses and , to find the stresses in a plane characterized by the angle : Take the equations

Replace , , and to obtain:

Inverse Problem

x 1 y 2 xy 0

1 2 1 2

cos 22 2

sin 22

2cos 2 xy sin 2

2sin 2 xy cos 2

Considering a reference system and characterizing the inclination of a plane by , From the inverse problem equations:

Squaring both equations and adding them:

Mohr’s Circle for a 2D State of Stress

2 221 2 1 2

´ ´x y

1 2 1 2

cos 22 2

sin 22

REMARK This expression is valid for any value of .

1 2 , 02

Eq. of a circle with center and radius .

Mohr’s CircleRC

The locus of the points representative of the state of stress on any of the planes passing through a given point P is a circle.

(Mohr’s Circle) The inverse is also true: Given a point in Mohr’s Circle, there is a plane passing through

P whose normal and tangential stresses are and , respectively.

Mohr’s Circle for a 2D State of Stress

1 2 , 02

2cos 2

Mohr's 2D problemspace

Interactive applets and animations: by M. Bergdorf:

from MIT OpenCourseware:

from Virginia Tech:

From Pennsilvania State University:

Construction of Mohr’s Circle

http://www.zfm.ethz.ch/meca/applets/mohr/Mohrcircle.htm

http://ocw.mit.edu/ans7870/3/3.11/tools/mohrscircleapplet.html

http://web.njit.edu/~ala/keith/JAVA/Mohr.html

http://www.esm.psu.edu/courses/emch13d/design/animation/animation.htm

A. To obtain the point in Mohr’s Circle representative of the state of stress on a plane which forms an angle with the principal stress direction :

1. Begin at the point on the circle (representative of the plane where acts).

2. Rotate twice the angle in the sense . 3. This point represents the shear and normal stresses at the desired plane

(representative of the stress state at the plane where acts).

Mohr’s Circle’s Properties

( 1,0)

B. The representative points of the state of stress on two orthogonal planes are aligned with the centre of Mohr’s Circle:

This is a consequence of property A as .

C. If the state of stress on two orthogonal planes is known, Mohr’s Circle can be easily drawn:

1. Following property B, the two points representative of these planes will be aligned with the centre of Mohr’s Circle.

2. Joining the points, the intersection with the axis will give the centre of Mohr’s Circle.

3. Mohr’s Circle can be drawn.

D. Given the components of the stress tensor in a particular orthonormal base, Mohr’s Circle can be easily drawn:

This is a particular case of property C in which the points representative of the state of stress on the Cartesian planes is known.

1. Following property B, the two points representative of these planes will be aligned with the centre of Mohr’s Circle.

2. Joining the points, the intersection with the axis will give the centre of Mohr’s Circle.

3. Mohr’s Circle can be drawn.

The radius and the diametric points of the circle can be obtained:

2x y 2

1 a R x y

2 a R x y

Note that the application of property A for the point representative of the vertical plane implies rotating in the sense contrary to angle.

The point called pole or origin of planes in Mohr’s circle has the following characteristics: Any straight line drawn from the pole will intersect the Mohr circle at a

point that represents the state of stress on a plane parallel in space to that line.

The Pole or the Origin of Planes

The point called pole or origin of planes in Mohr’s circle has the following characteristics: If a straight line, parallel to a given plane, is drawn from the pole, the

intersection point represents the state of stress on this particular plane.

The Pole or the Origin of Planes

03/11/2014

MMC - ETSECCPB - UPC

The sign criterion used in soil mechanics, is the inverse of the one used in continuum mechanics: In soil mechanics,

But the sign criterion for angles is the same: positive angles are measured counterclockwise

Sign Convention in Soil Mechanics

continuum mechanics

soil mechanics

tensile stresscompressive stresspositive (+)

negative (−)

negative (-)positive (+) counterclockwise rotation

clockwise rotation

For the same stress state, the principal stresses will be inverted.

The expressions for the normal and shear stresses are

The Mohr’s circle construction and properties are the same in both cases

Sign Convention in Soil Mechanics

continuum mechanics soil mechanics

* * * ** *2 1 2 1

* * * ** *1 2 1 2 1 2 1 2

* * * *1 2 * * *2 1 1 2

cos 22 2

cos 2 cos 22 2 2 2

sin 2 sin 2 sin 22 2 2

like incontinuum mechanics

Ch.4. Stress

4.10. Particular Cases of Mohr’s Circle

Hydrostatic state of stress

Mohr’s circles of a stress tensor and its deviator

Pure shear state of stress

Particular Cases of Mohr’s Circles

sph 1 m 1

sph m 1

Ch.4. Stress

Summary

Forces acting on a continuum body:

Body forces:

Surface forces:

Cauchy’s Postulates:1.2.

The traction vector:

Summary

,V Vt dV f b x

,S Vt dS

,Pt t n

, ,P P t n t n

ˆ( ) , 1,2,3iij jP i j t e

Cauchy stress tensor:

The stress vector acting on point P of an arbitrary plane may be resolved into:

a component normal to the plane .

a shear component which acts on the plane .

Summary

11 12 13

21 22 23

31 32 33

x xy xz

yx y yz

zx zy z

Scientific notationEngineering notation

333231

Cauchy’s equation of motion:

Boundary conditions

The principal stress directions correspond to the set of axes that make the shear stress components vanish when the stress components are referred to this system. The normal stress components in the principal planes are the principal stresses.

Summary

V b b a x

*( , )t V n n t x x

0 00 00 0

Mean stress:

Mean pressure:

A spherical or hydrostatic state of stress:

The Cauchy stress tensor can be split into:

Stress invariants:

Summary

1 2 31 1 13 3 3m iiTr

1 2 3 1

sph :sph m 1dev m 1

spherical stress tensor:stress deviator tensor:

1 1 2 3iiI Tr

22 1 1 2 1 3 2 3

3 detI

1 1 iiJ I

22 1 2

1 1 12 :2 2 2ij jiJ I I

33 1 1 2 3

1 1 13 33 3 3 ij jk kiJ I I I I Tr

The components of the stress tensor in the cylindrical coordinate system:

The components of the stress tensor in the spherical coordinate system:

Summary

´ ´ ´ ´ ´

´ ´ ´ ´

´ ´ ´ ´ ´

x x y x z r r rz

x y y y z r z

x z y z z rz z z

dV r d dr dz

´ ´ ´ ´ ´

´ ´ ´ ´

´ ´ ´ ´ ´

x x y x z r r r

x y y y z r

x z y z z r

2 sen dV r dr d d

Mohr’s Circle is a two-dimensional graphical representation of the state of stress at a point.

For a 3D state of stress:

Summary

1 2 , 02

2 2 2C R

Equation for Mohr’s circle (2D) sin 2

Mohr’s circle is a two-dimensional graphical representation of the state of stress at a point. For a 3D state of stress:

Summary

1 2 , 02

2 2 2C R

Equation for Mohr’s circle (2D)

C1 12 2 3

C2 12 1 3

C3 12 1 2

sin 2R

2D Mohr’s circle can be used to solve: Direct problem

Inverse problem

Summary

x y x yxy

1 2 1 2

cos 22 2

sin 22

1sin 211

tg 2 2

principal stresses principal stress directions

Any straight line drawn from the pole or origin of planes will intersect the Mohr circle at a point that represents the state of stress on a plane parallel in space to that line. The inverse is also true.

Particular cases:

Summary

Hydrostatic state of stress

A tensor and its deviator

Pure shear state of stress

Sign criterion

The sign criterion used in soil mechanics, is the inverse of the one used in continuum mechanics, except the criterion for angles, which remains the same (positive angles are measured counterclockwise).

Summary

t n<0 compressive stress>0 tensile stress

ij aortensile stresscompressive stress

positive (+)negative (−)

positive direction of the b-axisnegative direction of the b-axis

continuum mechanics

soil mechanics

continuum mechanics

soil mechanics

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