stress/strain relationship for solids

Definition of normal stress (axial stress)

AF

Definition of normal strain

0LL

Poisson’s ratio

Definition of shear stress

0AF

Definition of shear strain

lx

tan

Tensile Testing

Stress-Strain Curves

Stress-Strain Curves

http://www.uoregon.edu/~struct/courseware/461/461_lectures/461_lecture24/461_lecture24.html

Stress-Strain Curve (ductile material)

http://www.shodor.org/~jingersoll/weave/tutorial/node4.html

Stress-Strain Curve (brittle material)

Example: stress-strain curve for low-carbon steel

•1 - Ultimate Strength •2 - Yield Strength •3 - Rupture •4 - Strain hardening region •5 - Necking region

Hooke's law is only valid for the portion of the curve between the origin and the yield point.

http://en.wikipedia.org/wiki/Hooke's_law

σPL ⇒ Proportional Limit - Stress above which stress is not longer proportional to strain.

σEL ⇒ Elastic Limit - The maximum stress that can be applied without resulting in permanent deformation when unloaded.

σYP ⇒ Yield Point - Stress at which there are large increases in strain with little or no increase in stress. Among common structural materials, only steel exhibits this type of response.

σYS ⇒ Yield Strength - The maximum stress that can be applied without exceeding a specified value of permanent strain (typically .2% = .002 in/in).

OPTI 222 Mechanical Design in Optical Engineering 21

σU ⇒ Ultimate Strength - The maximum stress the material can withstand (based on the original area)

True stress and true strain

True stress and true strain are based upon instantaneous values of cross sectional

area and gage length

The Region of Stress-Strain Curve

Stress Strain Curve

• Similar to Pressure-Volume Curve• Area = Work

Volume

Pressure

Volume

Uni-axial Stress State Elastic analysis

Stress-Strain Relationship

EE -- Young’s modulus

GG -- shear modulus

Hooke’s Law:

Stresses on Inclined Planes

Thermal Strain

Straincaused by temperature changes. α is a material characteristic called the coefficient of thermal expansion.

Strains caused by temperature changes and strains caused by applied loads are essentially independent. Therefore, the total amount of strain may be expressed as follows:

Bi-axial state elastic analysis

(1) Plane stress

• State of plane stress occurs in a thin plate subjected to forces acting in the mid-plane of the plate• State of plane stress also occurs on the free surface of a structural element or machine component, i.e., at any point of the surface not subjected to an external force.

Transformation of Plane Stress

Mohr’s Circle (Plane Stress)

http://www.tecgraf.puc-rio.br/etools/mohr/mohreng.html

Mohr’s Circle (Plane Stress)

Instruction to draw Mohr’s Circle 1. Determine the point on the body in which the principal stresses are to be

determined.2. Treating the load cases independently and calculated the stresses for the point

chosen.3. Choose a set of x-y reference axes and draw a square element centered on the

axes.4. Identify the stresses σx, σy, and τxy = τyx and list them with the proper sign.5. Draw a set of σ - τ coordinate axes with σ being positive to the right and τ being

positive in theupward direction. Choose an appropriate scale for the each axis.6. Using the rules on the previous page, plot the stresses on the x face of the element

in this coordinate system (point V). Repeat the process for the y face (point H).7. Draw a line between the two point V and H. The point where this line crosses the

σ axis establishes the center of the circle.8. Draw the complete circle.9. The line from the center of the circle to point V identifies the x axis or reference

axis for angle measurements (i.e. θ = 0).Note: The angle between the reference axis and the σ axis is equal to 2θp.

Mohr’s Circle (Plane Stress)

http://www.egr.msu.edu/classes/me423/aloos/lecture_notes/lecture_4.pdf

Principal Stresses

Maximum shear stress

http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm

Stress-Strain Relationship(Plane stress)

))((1yxz E

xy

y

x

xy

y

x E

2100

0101

1 2

(2) Plane strain

Coordinate Transformation

The transformation of strains with respect to the {x,y,z} coordinates to the strains with respect to {x',y',z'} is performed via the equations

Mohr's Circle (Plane Strain)

(εxx' - εavg)2 + ( γx'y' / 2 )2 = R2

εavg = εxx + εyy

2http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/calc_principal_strain.cfm

Principal Strain

Maximum shear strain

Stress-Strain Relationship (Plane strain)

)(

211 yxzE

z

y

x

z

y

x E

)1(22100

011

01

1

)21)(1()1(

Tri-axial stress state elastic analysis

3D stress at a point

three (3) normal stresses may act on faces of the cube, as well as, six (6) components of shear stress

Stress and strain components

The stress on a inclined plane

))(()2

()2

( 312122322232

lnn

))(()2

()2

( 123222132213

mnn

))(()2

()2

( 231322212221

nnn

y

x

z

(l, m, n)

p2

3

1n

n

3-D Mohr’s Circle

* The 3 circles expressed by the 3 equations intersect in point D, and the value of coordinates of D is the stresses of the inclined plane

D

Stress-Strain Relationship

For isotropic materials

Generalized Hooke’s Law:

000111

21

22100000

02210000

00221000

000100010001

)21)(1(

TEE

zx

yz

xy

z

y

x

zx

yz

xy

z

y

x