Mechanical PropertiesSession 07-14

Subject : S1014 / MECHANICS of MATERIALSYear : 2008

Bina Nusantara

Mechanical Properties

Bina Nusantara

What is Stress ?

Much Work with limited time

High Stress

Bina Nusantara

What is Stress ?

Less Work with long

time Low Stress

Bina Nusantara

What is Stress ?

stress is according to strength and failure of solids. The stress field is

the distribution of internal "tractions" that balance a given set of external tractions and body forces

Bina Nusantara

Stress

stress is according to strength and failure of solids. The stress field is

the distribution of internal "tractions" that balance a given set of external tractions and body forces

Bina Nusantara

Stress

Look at the external traction T that represents the force per unit area acting at a given location on the body's surface.

Bina Nusantara

Stress

Traction T is a bound vector, which means T cannot slide along its line of action or translate to another location and keep the same meaning. In other words, a traction vector cannot be fully described unless both the force and the surface where the force acts on has been specified. Given both DF and Ds, the traction T can be defined as

                                          

Bina Nusantara

Stress

The internal traction within a solid, or stress, can be defined in a similar manner.

Bina Nusantara

Stress Suppose an arbitrary slice is made across the solid shown in the above figure, leading to the free body diagram shown at right.

Bina Nusantara

Stress

Surface tractions would appear on the exposed surface, similar in form to the external tractions applied to the body's exterior surface.

Bina Nusantara

Stress

The stress at point P can be defined using the same equation as was used for T.

Bina Nusantara

Stress

Stress therefore can be interpreted as internal tractions that act on a defined internal datum plane. One cannot measure the stress without first specifying the datum plane.

Bina Nusantara

Stress

Surface tractions, or stresses acting on an internal datum plane, are typically

decomposed into three mutually orthogonal components. One component is normal to the surface and

represents direct stress. The other two components are tangential to the

surface and represent shear stresses.

Bina Nusantara

Stress

What is the distinction between normal and tangential tractions, or equivalently, direct and shear stresses?

Direct stresses tend to change the volume of the material (e.g. hydrostatic pressure) and are resisted by the body's bulk modulus (which depends on the Young's modulus and Poisson ratio).

Bina Nusantara

Stress

What is the distinction between normal and tangential tractions, or equivalently, direct and shear stresses?

Shear stresses tend to deform the material without changing its volume, and are resisted by the body's shear modulus.

Bina Nusantara

Stress

These nine components can be organized into the matrix:

Bina Nusantara

Stress

where shear stresses across the diagonal are identical (xy = yx, yz = zy, and zx = xz) as a result of static equilibrium (no net moment).

Bina Nusantara

Stress

This grouping of the nine stress components is known as the stress tensor (or stress matrix).

Bina Nusantara

Stress

The subscript notation used for the nine stress components have the following meaning:

Bina Nusantara

What is Strain?

A propotional

dimensional change ( intensity or degree of distortion )

Bina Nusantara

What is Strain measure?

a total elongation per unit length of material due to some applied stress.

oL

Bina Nusantara

What are the types of strain ?

1.Elastic Strain2.Plastic Deformatio

n

Bina Nusantara

Elastic Strain

Transitory dimensional change that exists only while the initiating stress is applied and dissapears immediately upon removal of the stress.

Bina Nusantara

Elastic Strain

The applied stresses cause the atom are displaced the same amount and still maintain their relative geometic. When streesses are removed, all the atom return to their original positions and no permanent deformation occurs

Bina Nusantara

Plastic Deformation

a dimentional change that does not dissapear when the initiating stress is removed. It is usually accompanied by some elastic strain.

Bina Nusantara

Elastic Strain & Plastic Deformation

The phenomenon of elastic strain & plastic Deformation in a material

are called elasticity & Plasticity respectively

Bina Nusantara

Elastic Strain & Plastic Deformation

Most of Metal material

At room temperature they have some elasticity, which manifests itself as soon as the slightest stress is applied. Usually, they are also posses some plasticity , but this may not become apparent until the stress has been raised appreciablty.

Bina Nusantara

Elastic Strain & Plastic Deformation

Most of Metal material

The magnitude of Plastic strain, when it does appear , is likely to be much greater than that of the

elastic strain for a given stress increment

Bina Nusantara

Constitutive

Solid material by force, F, at a point, as shown in the figure.

F

Bina Nusantara

Constitutive

Let the deformation at the the point be infinitesimal and be represented by vector d, as shown.

The work done = F .d

F

d

Bina Nusantara

Constitutive

For the general case:

W = Fx dxi.e., only the force in the direction of the deformation does work.

F

d

xz

y

Bina Nusantara

Amount of Work done

Constant ForceIf the Force is constant, the work is simply the product of the force and the displacement,

W = FxF

Displacementx

Bina Nusantara

Amount of Work done

Linear Force: If the force is proportional to the displacement, the work is

ooxFW2

1

F

Displacementx

Fo

xo

Bina Nusantara

Strain Energy

Fx

A simple spring system, subjected to a Force is

proportional to displacement x; F=kx.Now determine the work done when F= Fo, from before:

ooxFW2

1

This energy (work) is stored in the spring and is released when the force is returned to zero

Bina Nusantara

Hooke’s LawFor systems that obey Hooke's law, the extension produced is directly proportional to the load:

F=kx• where:

– x = the distance that the spring has been stretched or compressed away from the equilibrium position, which is the position where the spring would naturally come to rest (usually in meters),

– F = the restoring force exerted by the material (usually in newtons), and

– K = force constant (or spring constant). The constant has units of force per unit length (usually in newtons per meter).

Bina Nusantara

Hooke’s Law

Bina Nusantara

Hooke’s Law

Bina Nusantara

Strain Energy Density

y

xa

a

a

Consider a cube of material acted upon by a force, Fx,

creating stress sx=Fx/a2

xFW2

1

Bina Nusantara

Strain Energy Density

causing an elastic displacement, d in the x direction, and strain ex=d/a

y

xa

Fx

d

xFW2

1

32

2

1

2

1aeaeaU xxxx

Where U is called the Strain Energy, and u is the Strain Energy Density.

xxxx eaaeV

Uu

2

1/

2

1 33

Bina Nusantara

(a) For a linear elastic material

0.0100.0080.0060.0040.0020.0000

100

200

300

400

500

CONTINUED

Str

ess (

MP

a)

Strain

u=1/2(300)(0.0015) N.mm/mm3

=0.225 N.mm/mm3

Bina Nusantara

(b) Consider elastic-perfectly plastic

0.0100.0080.0060.0040.0020.0000

100

200

300

400

500

CONTINUED

Str

ess (

MP

a)

Strain

u=1/2(350)(0.0018) +350(0.0022)

=1.085 N.mm/mm3

Bina Nusantara

Shear Strain Energy

Consider a cube of material acted upon by a shear stress,xy

causing an elastic shear strain xy

3

2

1aU yxy

xyxyxyxy aau 2

1/

2

1 33

y

xaa

a

y

x

xya

xy

xy

Bina Nusantara

Total Strain Energy for a Generalized State of Stress

xzxzyzyzxyxyzzyyxxu 2

1

zxzx

yzyz

xyxy

yyxxzzzz

xxzzyyyy

zzyyxxxx

E

E

E

E

E

E

)1(2

)1(2

)1(2

(1

(1

(1

Bina Nusantara

Strain Energy for axially loaded bar

AE

LFFU

AE

FL

A

F

22

1

;;

2

axial

F= Axial Force (Newtons, N)A = Cross-Sectional Area Perpendicular to “F” (mm2)E = Young’s Modulus of Material, MPaL = Original Length of Bar, mm

F

A

L

Bina Nusantara

Comparison of Energy Stored in Straight and Stepped bars

F

a

A

L

AE

LFU

2

2

(a)

Bina Nusantara

Comparison of Energy Stored in Straight and Stepped bars b

FA

L/2

nA

L/2

n

n

AE

LF

nAE

LF

AE

LFU

2

1

2

2

2/

2

2/

2

22

(b)

Bina Nusantara

What is Torsion ?

an external torque is applied and an internal torque, shear stress, and deformation (twist) develops in response to the externally applied torque.

Bina Nusantara

What is Torsion ?

For solid and hollow circular shafts, in which assume the material is homogeneous and isotropic , that the stress which develop remain within the elastic limits, and that plane sections of the shaft remain plane under the applied torque.

Bina Nusantara

What is isotropic ?

is properties of the materials are the same in all directions in the material

Bina Nusantara

Torsion of shafts

Shafts are members with length greater than the largest cross sectional dimension used in transmitting torque from one plane to another

Bina Nusantara

Internal Torque

Bina Nusantara

Consider circular shaft AC subjected to equal and opposite torques T and T’. A cutting plane is passed through the shaft at B.

The FBD for section BCmust include the applied torqueand elementary shearing forcesdF. These forces are perpendicular to radius of the shaft and must balance to maintain equilibrium.The axis of the shaft is denoted

as r.

Bina Nusantara

0:0 ' dFTM

dFTTT :'

Taking moments about the Axis of the shaft results in

dF is related to the shearing stress: dF = dA

So the applied torque can be related to the shearing stress as

dAT

Bina Nusantara

dAT

Equation is independent of material model as it represents static equivalency between shear stress and internal torque on a cross section

Bina Nusantara

Shear stress can’t exist on one plane only.

The applied torque

produces a shear stress to the axis of the shaft. The equilibrium require equal stresses on the faces

Bina Nusantara

What is Torsion ?

To obtain a formula for the relative rotation f2-f1 in terms of the internal torque T.To obtain a formula for the shear stress txq in terms of the internal torque T.

- angle of twist

Bina Nusantara

Shearing Strain

Bina Nusantara

Assume: Material is linearly elastic and isotropic

dAT

dAJ 2 Polar moment of inertia for the cross section

Bina Nusantara

Torsion formula

Circular hollow shaft with outer radius R, inner radius r

44

2rRJ

Bina Nusantara

Sign Convention

Bina Nusantara

Relative rotation in terms of the internal torque T.

Bina Nusantara

Shear stress x in terms of the internal torque T.

Maximum occurs at shaft’s outer radius

Bina Nusantara

Direction of Shearing

Bina Nusantara