elastic properties of materials. elastic properties of foods many food systems are solids or display...
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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.