ELASTIC PROPERTIES OF MATERIALS

ELASTIC PROPERTIES OF MATERIALS

Elastic Properties of FoodsElastic Properties of FoodsElastic Properties of FoodsElastic Properties of Foods

Many food systems are solids or display partial solid behaviorKnowledge of solid behavior important to understanding solids, semi-solids, and visco-elastic foodsTo understand food texture, we need to understand how foods respond when we apply forces to them

Elastic Properties and TextureElastic Properties and Texture

Food texture is evaluated by application of forces to the foodThe perceived texture of a food is a combination of its mechanical properties and structureMeasurement of elastic properties well defined; measurement of “texture” more tenuous

Solid FoodsSolid Foods

Solid behavior is characterized by elastic properties

Examples of elastic solid foods:egg shells

macaroni noodles

hard candies

Strength of MaterialsStrength of Materials

The study of the elastic properties of materials usually falls under “strength of materials: how do bridges, concrete, steel bolts respond to small deformations

Food texture concerned with weakness of materials- how forces cause large deformations in the food that it breaks or disintegrates

Stress/Strain RelationsStress/Strain Relations

Solids described by the strain produced by an applied stressStress: force per unit area that causes a strainStrain: some fractional change in the dimensions of a material due to stress. The type of strain produced depends on the way in which the stress is applied

Normal vs Shear StressNormal vs Shear StressNormal Stress: acts perpendicular to a surface area

Area A

Force

•Shear Stress: acts parallel to the area

Force

If a force acts on an eraser, it will stretch

If the cross-section of the eraser is twice as large it will take twice the force to stretch it the same amount. The stress is defined as the force per area

Stress and StrainStress and Strain

σ =F

A

The strain is a measure of how much the material deforms when subject to a stress

Usually expressed as a fraction of change per length of material

ε =strain =

Δl

l

l

Area A

F F

∆l

Hooke’s lawHooke’s law

Stress = Constant X Strain

Area A

Area 2A

Force F

Force 2 F

Force F

Force 2 F

The stress is opposed by intermolecular forces within the material. The more the material, the greater the internal force resisting the stress.

Types of StressTypes of Stress

Three types of stress are possible Tension stress Compression stress Shear stress

Other stresses (twisting, bending) are derived from these

Tension StressTension Stress

Tension stress is the force per unit area that produces a small elongation of a material (l)

ε =strain =

Δl

l

l

Area A

F F F

∆l

Compressive StressCompressive Stress

Compression stress is the force per unit area that produces a reduction in length

ε =strain =

Δl

l

l

Area A

F F

∆l

F

Shear StressShear Stress

Shear stress acts tangent to a surface and moves the surface out of line with layers underneath

h

δ

α

εs = shear strain =δ

h= tanα

FF

Hydrostatic PressureHydrostatic Pressure

Hydrostatic pressure is a variation of compression in which the stress acts inward in all directions

F

Pistonarea A

σ =stress =F

A≡ P (hydrostatic pressure)

ε = strain =ΔVV

ELASTIC MODULIELASTIC MODULI

The rheological properties of solids are described by elastic moduli which relate the amount of deformation caused by a given stressAssumptions: elements are elastic: complete recovery occurs

when stress is removed small strains are applied (1-3%) material is continuous, homogeneous

There are 4 elastic moduli for solids, all of which are variations of Hooke’s law

Stress = Constant X Strain

Young’s Modulus:

E =stressstrain

=σε=

FAll

Longitudinal compression or stretching

Shear Modulus:

G =stressstrain

=σs

εs

=

Ft

Aδh

=

Ft

Atanα

Shearing

Bulk Modulus:

K =stressstrain

=σε=

Pε=

FAVV

Volume compression

Poisson’s RatioPoisson’s Ratio

Usually, when you stretch a sample in one direction, it contracts in the other direction

Defined by Poisson’s ratio µ

w

w=

Δh

h= −μ

Δl

l

Fh

wl ∆ l

∆ w

∆h

The elastic moduli and Poisson’s ratio are sufficient information to describe the elastic properties of a material

Superposition Principle

In the simple case, stress is linearly proportional to the strain produced

The resulting displacements of more than one stress is the sum of the displacements

Example: Volume Compression

For a block in a tank of water, we could consider linear compression along each direction

F

Pistonarea A

w

w= −

P

E

Δh

h= −

P

E

Δl

l= −

P

E

Force in any one direction is countered by a force due to squeezing of the other sides

Thus:

€

w

w=

Δh

h=

Δl

l= −

P

E+ μ

P

E+ μ

P

E

= −P

E(1− 2μ)

For a small displacements

V

V=

Δl

l+

Δw

w+

Δh

h

= −3P

E(1− 2μ )

σ = P = - KΔV

V

Bending

An objects resistance to bending depends on both material properties and its shape (not just cross sectional area)

Bending is a combination of compression and tension

L

ab

F

F/ 2 F/ 2R

The forces form a couple that tend to rotate the barThe forces form a couple that tend to rotate the bar

The upper half of the bar is compressed; the lower half is under tension

Upper and lower surfaces are distorted the most and experience the greatest compression and tension forces

The beam bends with radius R. The torque is given by:

Γ =internal torque = EIA

R

where

IA = moment of inertia =a3b

12

Cross section IA

Rectangle IA=

Solid cylinder IA=

Hollow cylinder IA=

I beam IA=

a

b

r

a

b

a

b

t

a3b12

πr3

4

π(a4 -b4 )4

a2bt2

a3 t1 2

Buckling

Failure often occurs due to large torques rather than simple linear compression or tensionLarge diameter-thin wall tructures tend to fail by bucklingIf the center of gravity of a hollow cylinder is off-center, the weight will exert a force about a point

w

P P

Twisting

If a cylinder is fixed at one end, and coupled forces are applied at the other, a torque is produced that twists the object.

F

- F

Γ

α

l

The problem is similar to bending but we consider a polar moment of inertia

The torque T is related to the deformation α

Γ =GI p

α

l

F

- F

Γ

α

l

Ip =πr4

2

Large Deformations

As more and more force is applied over an area, the strain increases

After a certain point, Hooke’s law may no longer apply

A typical stress-strain curve

Strain, ε

Stress, σ

AB

CD

Linearlimit

Elasticlimit

Ultimatetensionstrength Fracture

point

Linear Region: Hooke’s law obeyed

Stress proportional to strain

Linear limit reached at point A

Strain, ε

Stress, σ

AB

CD

Linearlimit

Elasticlimit

Ultimatetensionstrength Fracture

point

A to B: material still elastic and returns to orignal state when force removed

Stress not proportional to strain

Point B: elastic limit Strain, ε

Stress, σ

AB

CD

Linearlimit

Elasticlimit

Ultimatetensionstrength Fracture

point

B to C: further stress causes rapid increase in strain

If force removed object does not return to original dimensions

Point C: ultimate tension strength. Even smaller force will cause deformation

Strain, ε

Stress, σ

AB

CD

Linearlimit

Elasticlimit

Ultimatetensionstrength Fracture

point

D: fracture point

Curve from B-D: plastic deformation

Area under curve up to D is work required to break the material

Strain, ε

Stress, σ

AB

CD

Linearlimit

Elasticlimit

Ultimatetensionstrength Fracture

point

Strain, ε

Stress, σ

AB

CD

Linearlimit

Elasticlimit

Ultimatetensionstrength Fracture

point

•B-D is “plastic deformation”B-D is “plastic deformation”•Brittle materials: C and D are close Brittle materials: C and D are close togethertogether•Ductile materials: C and D are far apartDuctile materials: C and D are far apart•Area under curve up to point D is energy Area under curve up to point D is energy needed to break the materialneeded to break the material

BrittleBrittle

DuctileDuctile

Malleability: a material's ability to deform under compressive stress; this is often characterized by the material's ability to form a thin sheet by hammering or rolling.

Ductility: mechanical property used to describe the extent to which materials can be deformed plastically without fracture

(a)(a) Ductile fractureDuctile fracture(b)(b) Ductile fractureDuctile fracture(c)(c) Completely ductile fractureCompletely ductile fracture

Ductile materials deform quite a bit (through plastic deformation) before they break

Brittle materials deform very little before they break

StrainStrain

Str

ess

Str

ess

Brittle materialBrittle material

Ductile materialDuctile material

Ductile BrittleDuctile Brittle

Fracture is a process of breaking a solid into pieces as a result of stress.

There are two principal stages of the fracture process:

Crack formation

Crack propagation

Ductile fracture

Ductile materials undergo plastic deformation and absorb significant energy before fracture.

A crack, formed as a result of the ductile fracture, propagates slowly and when the stress is increased.

Permanent deformation at the tip of the advancing crack that leaves distinct patterns in SEM images.

Fractures are perpendicular to the principal tensile stress, although other components of stress can be factors.

The fracture surface is dull and fibrous.There has to be a lot of energy available to extend the crack.

Brittle Fracture

Very low plastic deformation and low energy absorption prior to breaking.

A crack, formed as a result of the brittle fracture, propagates fast and without increase of the stress applied to the material.

The brittle crack is perpendicular to the stress direction.

There is no gross, permanent deformation of the material.

Characteristic crack advance markings frequently point to where the fracture originated.The path the crack follows depends on the material's structure. In metals, transgranular and intergranular cleavage are important. Cleavage shows up clearly in the SEM.