MECH 5312 – Solid Mechanics IIDr. Calvin M. Stewart

Department of Mechanical Engineering

The University of Texas at El Paso

Table of Contents

• Principal Strains

• Strain Invariants

• Volume Changes and Dilatation

• Strain Deviator

Principal Strains

• Principal stresses exists on an Arbitrary plane, n in a body where only normal stresses appear. Find two complementary planes that are orthogonal to n. We arrive at the principal stress tensor.

• What form does the corresponding principal strain tensor take?

1

2

3

0 0

0 0

0 0

σ

1

2

3

n

n

n

1 2 3n n n

Principal Strain Tensor

• Since, the principal stress tensor contains only normal stresses, the principal strain tensor will contain only normal strains.

• Our procedure for finding the principal strains follows

1

2

3

0 0

0 0

0 0

ε

1

2

3

n

n

n

1 2 3n n n

Principal Strain

• The principal directions coincide with Arbitrary plane, n through the body where the normal strain, εn is maximized and shear strain disappears, γ =0

0

0

0

xx n xy xz x

yx yy n yz y

zzx zy zz n

n

n

n

n ε I n 0

2 2 2 1x y zn n n

Strain Invariants

• The exact solution to the determinant

• resolves into a quadratic equation

• where J1, J2, and J3 are the strain Invariants.

det 0n ε I

3 2

1 2 3 0n n nJ J J

1

2 2 2

2

3 det

xx yy zz

xx yy yy zz zz xx xy xz yz

J tr

J

J

ε

ε

ExampleMathCAD

Volume Change and Dilatation

• A differential element is subject to normal strain in the x direction

• The length of the side increases by

• The new length is now

xx

xxdx

1 xx dx

Example, for dx

dx dx xxdx

1 xx dx

Volume Change and Dilatation

• If the differential element is subject to normal strains

• The lengths of the sides increase by

• The new lengths are now

, , xx yy zz

, , xx yy zzdx dy dz

1 , 1 , 1xx yy zzdx dy dz

dx

1 1 1def xx yy zzV dxdydz

The Deformed Volume, Vdef

V dxdydz

The Initial Volume, V

dy

dz

Volume Change and Dilatation

• Assuming finite displacement, the change in volume becomes

• The change in volume per unit volume is

• where is the dilatation and

0def xx yy zzV V V dxdydz

xx yy zz

V

V

1J

V dxdydz

Second order terms are neglected!

Strain Deviator

• Strain can be decomposed into volumetric and deviatoric parts

• The deviatoric strain can be found by rearranging

1

3d

ε I ε

1

3d

ε ε I

ExampleMathCAD

Calvin M. StewartAssistant ProfessorDepartment of Mechanical EngineeringThe University of Texas at El Paso500 W. University Blvd.Suite A126El Paso, Texas 79968-0717

Email: cmstewart@utep.eduURL: http://me.utep.edu/cmstewart/Phone: 915-747-6179Fax: 915-747-5019

Contact Information

mailto:cmstewart@utep.eduhttp://me.utep.edu/cmstewart/