ELASTIC PROPERTIES OF MATERIALS

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

Elastic Properties and Texture Food texture is evaluated by application of forces to the food The perceived texture of a food is a combination of its mechanical properties and structure Measurement of elastic properties well defined; measurement of “texture” more tenuous

Solid Foods Solid behavior is characterized by elastic properties Examples of elastic solid foods: egg shells macaroni noodles hard candies

Strength 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 Relations Solids described by the strain produced by an applied stress Stress: force per unit area that causes a strain Strain: 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 Stress Normal Stress: acts perpendicular to a surface area Area A Force

Shear Stress: acts parallel to the area Force

Stress and Strain 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

Usually expressed as a fraction of change per length of material 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 l Area A F ∆l F

Hooke’s law Stress = Constant X Strain

Area A Force F Force F Force 2 F Force 2 F Area 2A

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

Types of Stress Three types of stress are possible Tension stress Compression stress Shear stress Other stresses (twisting, bending) are derived from these

Tension Stress Tension stress is the force per unit area that produces a small elongation of a material (l) l Area A F ∆l F F

Compressive Stress Compression stress is the force per unit area that produces a reduction in length Area A ∆l F F F l

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

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

ELASTIC MODULI The rheological properties of solids are described by elastic moduli which relate the amount of deformation caused by a given stress Assumptions: 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:

Shear Modulus:

Bulk Modulus:

Poisson’s Ratio Usually, when you stretch a sample in one direction, it contracts in the other direction Defined by Poisson’s ratio µ

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

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

For a small displacements

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

The 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:

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

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.

The problem is similar to bending but we consider a polar moment of inertia The torque T is related to the deformation a

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

Linear Region: Hooke’s law obeyed Stress proportional to strain Linear limit reached at point A

A to B: material still elastic and returns to orignal state when force removed Stress not proportional to strain Point B: elastic limit

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

D: fracture point Curve from B-D: plastic deformation Area under curve up to D is work required to break the material

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

Brittle

Ductile

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

Ductile fracture Completely ductile fracture

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

Brittle material Stress Ductile material Strain

Ductile Brittle

There are two principal stages of the fracture process: 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.