lecture-3 (definition, notation, stress symmetry, deviator stress, equilibrium equations)

7/27/2019 Lecture-3 (Definition, Notation, Stress Symmetry, Deviator Stress, Equilibrium Equations)

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CE 251 (Solid Mechanics)

Lecture-3Definition, notation, stress symmetry,

deviator stress, equilibrium equations

Anil Mandariya 1


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Fundamentals of Tensor Analysis

Elasticity theory is formulated in terms of many differenttypes of variables that are either specified or sought atspatial points in the body under study .

These variables are scalar, vector and matrix quantities.

Scalar quantities, : A physical quantity that can becompletely described by a real number. Representing asingle magnitude at each point in space. Exa. , E etc.

Vector quantities, a: A physical quantity that has bothdirection and length. Expressible in 2-D & 3-D coordinatesystem. Exa. diplacement + rotation of material pointsAnil Mandariya 2

Concepts of Scalar, Vector, and Tensor:


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Tensor A: More then 3-components requires to quantify.Exa. Stress, strain etc.

Defines an 2 nd order operation that transforms a vector to another vector.

Scalar: Zero order tensor Vector: 1 st order tensor

Matrix: 2nd

order tensor

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A 2 nd order tensor is a linear operator that transforms avector a into another vector b through a dot product.


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Consider a general body subjected to forces acting on

its surface.Pass a fictitious plane Q through the body, cutting thebody along surface A

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Definition of stress at a point:


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Let the force that is transmitted through an incrementalarea A of A by the part on the positive side Q be denotedby F.

The force F may be resolved into components FN andF s , along unit normal N and unit tangent S, respectively, tothe plane Q.

FN = normal (perpendicular) force on area A andF s = shear (tangential) force on A.

The forces F, FN, and F s depend on the area A andthe orientation of plane Q.

F/ A = average stress; FN / A = average normalstress; F s / A = average shear stress acting on area A.

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Stress Notation :

Traction is a vector, whose direction and magnitudedepend on how the surface S is obtained, or thedirection of S.

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Stress Tensor : Traction


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i represent the normal to the plane where stressacting and j represent the direction of stress.


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Unit Dyads


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As each part of deformable body remains inequilibrium so

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Symmetry of Stress Components:


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If the nine stress components relative to rectangular coordinate axes (x, y, z) may be tabulated in array formas follows:

Summation of moments leads to the result:

Only six components of stress are required to describethe state of stress at a point in a member.

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The stress vectors x, y, and z on planes that areperpendicular, respectively to the x, y, and z axes are:

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Mean stress ( m ) = ( x + y + z)/3

Imagine hydrostatic type of stress having all thenormal stresses equal to m and all the shear stresseszero.

Stress tensor = Hydrostatic stress (spherical stress)+ Deviatorial stress tensor

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Deviator stress:

+

m m

m

00

0 00

0

- m

- m

- m


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The deviatorial part of stress produces all the changesof shape in the body and finally causes failure.

The spherical part is rather harmless, produces onlyuniform volume changes without any change of shape,and does not cause failure.

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Consider a general deformable body

Choose a differential volume element at point o in thebody.

We choose rectangular coordinateaxes (x, y, z) whosedirections are parallel to

the edges of the volumeelement.

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Differential equations of motion of a deformablebody and Equations of equilibrium:


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To write the differential equations of motion, eachstress component must be multiplied by the area onwhich it acts and each body force must be multiplied bythe volume of the element since (B x, B y, B z) havedimensions of force per unit volume.

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Summation of forces in the x direction gives:

Summation of forces in the y and z directions yieldssimilar results. The three equations of motion are thus:

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lecture-3 (definition, notation, stress symmetry, deviator stress, equilibrium equations)

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