Young's Modulus - Tensile and Yield Strength for Materials

introduction :

A force exerted on a body can cause a change in either the shape or the motion of the body. The unit of force  is the newton*, N. No solid body is perfectly rigid and when forces are applied to it, changes in dimensions occur. Such changes are not always perceptible to the human eye since they are
so small. For example, the span of a bridge will sag under the weight of a vehicle and a spanner will bend slightly when tightening a nut. It is important for engineers and designers to appreciate the effects of forces on materials, together with their mechanical properties.

The three main types of mechanical force that can act on a body are:
(i) tensile (ii) compressive and (iii) shear 

Tension is a force that tends to stretch a material, as shown in

(i) the rope or cable of a crane carrying a load is in Tension 

(ii) rubber bands, when stretched, are in tension

(iii) when a nut is tightened, a bolt is under tension

A tensile force, i.e. one producing tension, increases the length of the material on which it acts.

Compression is a force that tends to squeeze or crush a material, as shown in

(i) a pillar supporting a bridge is in compression 

(ii) the sole of a shoe is in compression 

(iii) the jib of a crane is in compression 

A compressive force, i.e. one producing compression, will decrease the length of the material on which it acts Shear is a force that tends to slide one face of the material over an adjacent face. For example, 

(i) a rivet holding two plates together is in shear if  a tensile force is applied between the plates

(ii) a guillotine cutting sheet metal, or garden shears, each provide a shear force 

(iii) a horizontal beam is subject to shear force 

(iv) transmission joints on cars are subject to shear Forces

DIRECT STRESS σ 

When a force F acts directly on an area A as shown in, the resulting direct stress is the force per unit area Stress is force per unit area and can be expressed as 

σ = F / A 

where 

σ = stress (N/m², lb/in², psi) 

F = applied force (N, lb) 

A = stress area of object (m², in²)

· tensile stress - stress that tends to stretch or lengthen the material - acts normal to the stressed area
· compressible stress - stress that tends to compress or shorten the material - acts normal to the             stressed area
· shearing stress - stress that tends to shear the material - acts in plane to the stressed area at right-       angles to compressible or tensile stress

where F is the force in newtons and A is the cross-sectional area in square metres. For tensile and compressive forces, the cross-sectional area is that which is at right angles to the direction of the force. 

For a shear force the shear stress is equal to F/A 

where the cross sectional area A is that which is parallel to the direction of the shear force. The symbol used for shear stress is the Greek letter tau, τ

Since 1 Pa is a small unit kPa , MPa and GPa are commonly used.

If the force pulls on the area so that the material is stretched then it is a tensile force and stress. This is regarded as positive.If the force pushes on the surface so that the material is compressed, then the force and stress is compressive and negative

DIRECT STRAIN ε

Consider a piece of material of length L as shown above. The direct stress produces a change in length dL. The direct strain produced is ε (epsilon) defined as

ε = dL / L 

where

ε = strain (m/m, in/in)

dL = elongation or compression (offset) of object (m, in)

L = length of object (m, in)

The units of change in length and original length must be the same and the strain has no units. Strains are normally very small so often to indicate a strain of 10⁻ ⁶  we use the name micro strain and write it as με

Tensile strain is positive and compressive strain is negative.

For a shear force, strain is denoted by the symbol  γ (Greek letter gamma) and, is given by:

γ = x/L

EXAMPLE No.1
A metal wire is 2.5 mm diameter and 2 m long. A force of 12 N is applied to it and it stretches 0.3 mm. Assume the material is elastic. Determine the following.
i. The stress in the wire σ.
ii. The strain in the wire ε.
SOLUTION
A=πd²∕4=π×2.5²=4.909 mm²
σ=F/A = 12/4.909=2.44 N/mm²
Answer (i) is hence 2.44 MPa
ε=x/L=0.3/2000=0.00015 or 150 με

Elasticity

Elasticity is the ability of a material to return to its original shape and size on the removal of external forces.  Plasticity is the property of a material of being permanently deformed by a force without breaking. Thus if a material does not return to the original shape, it is said to be plastic.

MODULUS OF ELASTICITY E 

Many materials are elastic up to a point. This means that if they are deformed in any way, they will spring back to their original shape and size when the force is released It has been established that so long as the material remains elastic, the stress and strain are related by the simple formula

Young's modulus can be expressed as
E = stress / strain
    = σ / ε
    = (F / A) / (dL / L) 
where
E = Young's Modulus of Elasticity (N/m², lb/in², psi)

named after the 18th-century English physician and physicist Thomas Young E is called the MODULUS OF ELASTICITY. The units are the same as those of stress.

Hooke’s law states: 

note above MODULUS OF ELASTICITY

Within the limit of proportionality, the extension of a material is proportional to the applied force 

Within the limit of proportionality of a material, the strain produced is directly proportional to the stress producing it Young’s modulus of elasticity 

Within the limit of proportionality, stress α strain, hence stress = (a constant) × strain 

This constant of proportionality is called Young’s modulus of elasticity* and is given the symbol E. 

The value of E may be determined from the gradient of the straight line portion of the stress/strain graph. The dimensions of E are pascals (the same as for stress, since strain is dimension-less). 

E = σ / ε 

Some typical values for Young’s modulus of elasticity, 

Aluminium alloy 70 GPa (i.e. 70 × 10⁹ Pa), brass 90 GPa, copper 96 GPa, titanium alloy 110 GPa,
 mild steel 210 GPa, lead 18 GPa, tungsten 410 GPa, cast iron 110 GPa, zinc 85 GPa, glass
fibre 72 GPa, carbon fibre 300 GPa. 

A spring is an example of an elastic object - when stretched, it exerts a restoring force which tends to bring it back to its original length. This restoring force is in general proportional to the stretch described by Hooke's Law. 

It takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon the stretching force is called Hooke's law and can be expressed as

Fs = -k dL 

Fs = force in the spring (N) 

k = spring constant (N/m) 

dL = elongation of the spring (m) 

Note that Hooke's Law can also be applied to materials undergoing three dimensional stress (triaxial loading).

Stiffness
A material having a large value of Young’s modulus is said to have a high value of material stiffness, where stiffness is defined as:
Stiffness = force F / extension X

yield strength - σy
Yield strength is defined in engineering as the amount of stress (Yield point) that a material can undergo before moving from elastic deformation into plastic deformation. Yielding - a material deforms permanently
The Yield Point is in mild- or medium-carbon steel the stress at which a marked increase in deformation occurs without increase in load. In other steels and in nonferrous metals this phenomenon is not observed.

ULTIMATE TENSILE STRESS σu
If a material is stretched until it breaks, the tensile stress has reached the absolute limit and this stress level is called the ultimate tensile stress. Values for different materials may be found in various sources

Young's Modulus or Tensile Modulus alt. Modulus of Elasticity - and Ultimate Tensile and Yield Strength for steel, glass, wood and other common materials

Material	Tensile Modulus (Young's Modulus, Modulus of Elasticity) - E -		Ultimate Tensile Strength - σu - (MPa)	Yield Strength - σy - (MPa)
	(106 psi)	(GPa)
ABS plastics		1.4 - 3.1	40
A53 Seamless and Welded Standard Steel Pipe - Grade A			331	207
A53 Seamless and Welded Standard Steel Pipe - Grade B			414	241
A106 Seamless Carbon Steel Pipe - Grade A			400	248
A106 Seamless Carbon Steel Pipe - Grade B			483	345
A106 Seamless Carbon Steel Pipe - Grade C			483	276
A252 Piling Steel Pipe - Grade 1			345	207
A252 Piling Steel Pipe - Grade 2			414	241
A252 Piling Steel Pipe - Grade 3			455	310
A501 Hot Formed Carbon Steel Structural Tubing - Grade A			400	248
A501 Hot Formed Carbon Steel Structural Tubing - Grade B			483	345
A523 Cable Circuit Steel Piping - Grade A			331	207
A523 Cable Circuit Steel Piping - Grade B			414	241
A618 Hot-Formed High-Strength Low-Alloy Structural Tubing - Grade Ia & Ib			483	345
A618 Hot-Formed High-Strength Low-Alloy Structural Tubing - Grade II			414	345
A618 Hot-Formed High-Strength Low-Alloy Structural Tubing - Grade III			448	345
API 5L Line Pipe			310 - 1145	175 - 1048
Acetals		2.8	65
Acrylic		3.2	70
Aluminum Bronze		120
Aluminum	10.0	69	110	95
Aluminum Alloys	10.2
Antimony	11.3
Aramid		70 - 112
Beryllium (Be)	42	287
Beryllium Copper	18.0
Bismuth	4.6
Bone, compact		18	170 (compression)
Bone, spongy		76
Boron				3100
Brass		102 - 125	250
Brass, Naval		100
Bronze		96 - 120
CAB		0.8
Cadmium	4.6
Carbon Fiber Reinforced Plastic		150
Carbon nanotube, single-walled		1000+
Cast Iron 4.5% C, ASTM A-48			170
Cellulose,  cotton, wood pulp and regenerated			80 - 240
Cellulose acetate, molded			12 - 58
Cellulose acetate, sheet			30 - 52
Cellulose nitrate, celluloid			50
Chlorinated polyether		1.1	39
Chlorinated PVC (CPVC)		2.9
Chromium	36
Cobalt	30
Concrete		17
Concrete, High Strength (compression)		30	40 (compression)
Copper	17	117	220	70
Diamond (C)		1220
Douglas fir Wood		13	50 (compression)
Epoxy resins		3-2	26 - 85
Fiberboard, Medium Density		4
Flax fiber		58
Glass		50 - 90	50 (compression)
Glass reinforced polyester matrix		17
Gold	10.8	74
Granite		52
Graphene		1000
Grey Cast Iron		130
Hemp fiber		35
Inconel	31
Iridium	75
Iron	28.5	210
Lead	2.0
Magnesium metal (Mg)	6.4	45
Manganese	23
Marble			15
MDF - Medium-density fiberboard		4
Mercury
Molybdenum (Mo)	40	329
Monel Metal	26
Nickel	31	170
Nickel Silver	18.5
Nickel Steel	29
Niobium (Columbium)	15
Nylon-6		2 - 4	45 - 90	45
Nylon-66			60 - 80
Oak Wood (along grain)		11
Osmium (Os)	80	550
Phenolic cast resins			33 - 59
Phenol-formaldehyde molding compounds			45 - 52
Phosphor Bronze		116
Pine Wood (along grain)		9	40
Platinum	21.3
Plutonium	14	97
Polyacrylonitrile, fibers			200
Polybenzoxazole		3.5
Polycarbonates		2.6	52 - 62
Polyethylene HDPE (high density)		0.8	15
Polyethylene Terephthalate, PET		2 - 2.7	55
Polyamide		2.5	85
Polyisoprene, hard rubber			39
Polymethylmethacrylate (PMMA)		2.4 - 3.4
Polyimide aromatics		3.1	68
Polypropylene, PP		1.5 - 2	28 - 36
Polystyrene, PS		3 - 3.5	30 - 100
Polyethylene, LDPE (low density)		0.11 - 0.45
Polytetrafluoroethylene (PTFE)		0.4
Polyurethane cast liquid			10 - 20
Polyurethane elastomer			29  - 55
Polyvinylchloride (PVC)		2.4 - 4.1
Potassium
Rhodium	42
Rubber, small strain		0.01 - 0.1
Sapphire		435
Selenium	8.4
Silicon	16	130 - 185
Silicon Carbide		450		3440
Silver	10.5
Sodium
Steel, High Strength Alloy ASTM A-514			760	690
Steel, stainless AISI 302		180	860	502
Steel, Structural ASTM-A36	29	200	400	250
Tantalum	27
Polytetrafluoroethylene (PTFE)		0.5
Thorium	8.5
Tin		47
Titanium	16
Titanium Alloy		105 - 120	900	730
Tooth enamel		83
Tungsten (W)		400 - 410
Tungsten Carbide (WC)		450 - 650
Uranium	24	170
Vanadium	19
Wrought Iron		190 - 210
Zinc	12

THEORIES OF ELASTIC FAILURE

Modern CADD systems allow the engineer to calculate stress levels in a component using finite stress analysis linked to the model. The reasons why a given material fails however, is not something a computer can predict without the results of research being added to its data bank. In some cases it fails because the maximum tensile stress has been reached and in others because the maximum shear stress has been reached. The exact combination of loads that makes a component fail depends very much on the properties of the material such as ductility, grain pattern and so on. This section is about some of the theories used to predict whether a complex stress situation is safe or not. There are many theories about this and we shall examine three. First we should consider what we regard as failure. Failure could be regarded as when the material breaks or when the material yields. If a simple tensile test is conducted on a ductile material, the stress strain curve may look like this.

The maximum allowable stress in a material is σmax. This might be regarded as the stress at fracture (ultimate tensile stress), the stress at the yield point or the stress at the limit of proportionality (often the same as the yield point). The Modulus of elasticity is defined as

 E = stress/strain = σ/ε and this is only true up to the limit of proportionality. 

Note that some materials do not have a proportional relationship at all. The maximum allowable stress may be determined with a simple tensile test. 

There is only one direct stress in a tensile test (σ = F/A) so it follows that σmax = σ1 and it will have a corresponding strain εmax = ε1. Complex stress theory tells us that there will be a shear stress τ and strain γ that has a maximum value on a plane at 45⁰ to the principal plane. It is of interest to note that in a simple tensile test on a ductile material, at the point of failure, a cup and cone is formed with the sides at 45⁰  to the axis. Brittle materials often fail with no narrowing (necking) but with a flat fail plane at 45⁰  to he axis. This suggests that these materials fail due to the 

maximum shear stress being reached

THE GREATEST PRINCIPAL STRESS THEORY (RANKINE) 

This simply states that in a complex stress situation, the material fails when the greatest principal stress equals the maximum allowable value. σ₁ = σmax 

σmax could be the stress at yield or at fracture depending on the definition of failure. 

If σ₁ is less than σmax then the material is safe. 

Safety Factor = σmax  /σ₁

EXAMPLE No.2 

A certain material fractured in a simple tensile test at a stress level of 800 MPa. The same material when used as part of a structure must have a safety factor of 3. Calculate the greatest principal stress that should be allowed to occur in it based on Rankine’s theory. 

SOLUTION 

S.F. = 3 = σmax /σ1= 800/σ1 

σ1 = 800/3 = 266.7 MPa

THE GREATEST PRINCIPAL STRAIN THEORY (St. VENANT)

This states that in a complex stress situation, the material fails when the greatest principal strain reaches the maximum allowable strain determined in a simple tensile test.
ε1 = εmax εmax  is the value determined in a simple tensile test. If the maximum allowable stress is taken as the value at the limit of proportionality, we may further develop the theory using the modulus of elasticity. εmax =  σmax/E 

THE MAXIMUM SHEAR STRESS THEORY (GUEST and COULOMB) 

This states that in a complex stress situation, the material fails when the greatest shear strain in the material equals the value determined in a simple tensile test. Applying complex stress theory to a tensile test gives this as 

τmax = ½ σmax 

In a simple tensile test, σmax could be what ever stress is regarded as the maximum allowable. 

In a 3 dimensional complex stress situation the maximum shear strain is τ = ½ (σ1 - σ3 ) 

If this is less than τmax then the material is safe. 

The allowable stress or allowable strength is the maximum stress (tensile, compressive or bending) that is allowed to be applied on a structural material. The allowable stresses are generally defined by building codes, and for steel, and aluminum is a fraction of their yield stress (strength) 

safety factor is generally defined by the building codes based on particular condition under consideration.

Typical overall Factors of Safety

Typical overall Factors of Safety:

Equipment	Factor of Safety - FOS -
Aircraft components	1.5 - 2.5
Boilers	3.5 - 6
Bolts	8.5
Cast-iron wheels	20
Engine components	6 - 8
Heavy duty shafting	10 - 12
Lifting equipment - hooks ..	8 - 9
Pressure vessels	3.5 - 6
Turbine components - static	6 - 8
Turbine components - rotating	2 - 3
Spring, large heavy-duty	4.5
Structural steel work in buildings	4 - 6
Structural steel work in bridges	5 - 7
Wire ropes	8 - 9