lecture 8 – viscoelasticity and deformation 2/4/2011 bae2023 physical properties of biological...
Post on 19-Dec-2015
219 views
TRANSCRIPT
BAE2023 Physical Properties of Biological Materials
1
Lecture 8 – Viscoelasticity andDeformation
2/4/2011
Deformation due to applied forces varies widely among different biomaterials
Depends on many factors• Rate of applied force• Previous loading• Moisture content• Biomaterial composition
BAE2023 Physical Properties of Biological Materials
22/4/2011
Lecture 8 – Viscoelasticity andDeformation
Normal stress: Force per unit area applied perpendicular to the planeNormal strain: Change in length per unit of length in the direction of the applied normal stress
BAE2023 Physical Properties of Biological Materials
32/4/2011
Modulus of elasticityLinear region of stress strain curve E = σ/εFor biomaterials: apparent E = σ/ε at any given point (secant method)
Tangent method: slope of stress/strain curve at any point
Lecture 8 – Viscoelasticity andDeformation
BAE2023 Physical Properties of Biological Materials
4
Lecture 8 – Viscoelasticity andDeformation
Poisson’s Ratio, μ• When a material is compressed in one direction, it usually
tends to expand in the other two directions perpendicular to the direction of compression
• The Poisson ratio is the ratio of the fraction (or percent) of expansion divided by the fraction (or percent) of compression, for small values of these changes.
2/4/2011
BAE2023 Physical Properties of Biological Materials
5
Poisson’s Ratio
• Ratio of the strain in the direction perpendicular to the applied force to the strain in the direction of the applied force.
• For uniaxial compression in Z direction:εz = σz/Eεy = -μ·εz εx = -μ·εz
2/4/2011
BAE2023 Physical Properties of Biological Materials
6
Poisson’s RatioMulti-axial Compression
See equations in 4.2 page 117Maximum Poisson’s = 0.5 for incompressible materials to 0.0 for easily compressed materialsExamples:• Gelatin gel – 0.50• Soft rubber – 0.49• Cork – 0.0• Potato flesh – 0.45 – 0.49• Apple flesh - 0.21 – 0.29• Wood – 0.3 to 0.5More porous means smaller Poisson’s Ratio2/4/2011
BAE2023 Physical Properties of Biological Materials
7
Lecture 8 – Viscoelasticity andDeformation
Shearing StressesShear Stress: Force per unit area acting in the direction parallel to the surface of the plane, τShear Strain: Change in the angle formed between two planes that are orthogonal prior to deformation that results from application of sheer stress, γ
2/4/2011
BAE2023 Physical Properties of Biological Materials
8
Lecture 8 – Viscoelasticity andDeformation
Shear Modulus:Ratio of shear stress to shear strain G = τ/γ
Measured with parallel plate shear test
(pg. 119)
2/4/2011
BAE2023 Physical Properties of Biological Materials
9
Lecture 8 – Viscoelasticity andDeformation
Example Problem
The bottom surface (8 cm x 12 cm) of a rectangular blockof cheese (8 cm wide, 12 cm long, 3 cm thick) is clampedin a cheese grater.• The grating mechanism moving across the top surface ofthe cheese applies a lateral force of 20N.• The shear modulus, G, of the cheese is 3.7kPa.• Assuming the grater applies the force uniformly to theupper surface, estimate the lateral movement of the uppersurface w/respect to the lower surface
2/4/2011
BAE2023 Physical Properties of Biological Materials
10
Lecture 8 – Viscoelasticity andDeformation
Stresses and Strains: Described as Deviatoric or Dilitational
Dilitational: Causes change in volume
Deviatoric: Causes change in shape but negligiblechanges in volume
Bulk Modulus, K: describes response of solid todilitational stresses2/4/2011
BAE2023 Physical Properties of Biological Materials
11
Lecture 8 – Viscoelasticity andDeformation
Dilatation: (Vf – V0)/V0
Δ V = Vf – V0
K = ΔP/(Δ V / V0)ΔP = Average normal stress, uniform hydrostatic gaugepressureK = average normal stress/dilatationV is negative, so K is negative
•Example of importance: K (Soybean oil) > K (diesel)•Will effect the timing in an engine burning biodiesel2/4/2011
BAE2023 Physical Properties of Biological Materials
12
Lecture 8 – Viscoelasticity andDeformation
Apples compress easier than potatoes so they have a smaller bulk modulus, K (pg. 120) but larger bulk compressibility
K-1 =bulk compressibility
Strain Energy Density: Area under the loading curve of stress-strain diagram
• Sharp drop in curve = failure
2/4/2011
BAE2023 Physical Properties of Biological Materials
13
Stress strain curve for uniaxial compression of cylindrical sample of food product
2/4/2011
BAE2023 Physical Properties of Biological Materials
14
Lecture 8 – Viscoelasticity andDeformation
Stress-Strain Diagram, pg. 122Toughness: Area under curve until it fails Bio yield point: Failure point Resilience: Area under the unloading curve• Resilient materials “spring back”…all energy isrecovered upon unloadingHysteresis: strain density – resilienceFigure 4.6, page 124Figure 4.7, page 1252/4/2011
BAE2023 Physical Properties of Biological Materials
15
Lecture 8 – Viscoelasticity andDeformation
Factors Affecting Force-Deformation Behavior• Moisture Content, Fig. 4.6b• Water Potential, Fig. 4.8• Strain Rate: More stress required for higher strain
rate, Fig. 4.8
• Repeated Loading, Fig. 4.9
2/4/2011
BAE2023 Physical Properties of Biological Materials
16
Effect of water potential and strain rate on stress-strain curve of cylindrical Ida Red apple tissue
2/4/2011
BAE2023 Physical Properties of Biological Materials
17
Lecture 8 – Viscoelasticity andDeformation
Stress Relaxation: Figure 4.10 pg 129Material is deformed to a fixed strain and strain is held constant…stress required to hold strain constant decreases with time.
2/4/2011
BAE2023 Physical Properties of Biological Materials
18
Lecture 8 – Viscoelasticity andDeformation
Creep: Figure 4.11 pg. 130
A continual increase in deformation (strain) with time with constant load
2/4/2011
BAE2023 Physical Properties of Biological Materials
19
Lecture 8 – Viscoelasticity andDeformation
Tensile Testing• Not as common as compression testing• Harder to do
See figure 4.12 page 132
2/4/2011
BAE2023 Physical Properties of Biological Materials
20
Lecture 8 – Viscoelasticity andDeformation
Bending E=modulus of elasticityD=deflection F=forceI = moment of inertia E=L3(48DI)-1
I=bh3/12
2/4/2011
BAE2023 Physical Properties of Biological Materials
21
Lecture 8 – Viscoelasticity andDeformation
Bending• Can be used for testing
critical tensile stress at failure
• Max tensile stress occurs at bottom surface of beam
σmax=3FL/(2bh2)
2/4/2011
BAE2023 Physical Properties of Biological Materials
22
Lecture 8 – Viscoelasticity andDeformation
Contact Stresses (handout from Mohsenin book)
Hertz Problem of Contact StressesImportance:“In ag. products the Hertz method can beused to determine the contact forces anddisplacements of individual units”
2/4/2011
BAE2023 Physical Properties of Biological Materials
23
Lecture 8 – Viscoelasticity andDeformation
Contact Stresses Assumptions:• Material is homogeneous• Loads applied are static• Hooke’s law holds• Contacting stresses vanish at the opposite ends• Radii of curvature of contacting solid are very large• compared to radius of contact surface• Contact surface is smooth
2/4/2011
BAE2023 Physical Properties of Biological Materials
24
Lecture 8 – Viscoelasticity andDeformation
Contact Stresses
Maximum contact stress occurs at the center of the surface of contact
a and b are the major and minor semi axes the elliptic contact area
For ag. Products, consider bottom 2 figures in Figure 6.1
2/4/2011
BAE2023 Physical Properties of Biological Materials
25
Lecture 8 – Viscoelasticity andDeformation
2/4/2011
BAE2023 Physical Properties of Biological Materials
26
Lecture 8 – Viscoelasticity andDeformation
2/4/2011
BAE2023 Physical Properties of Biological Materials
27
Lecture 8 – Viscoelasticity andDeformation
2/4/2011
BAE2023 Physical Properties of Biological Materials
28
HW Assignment Due 2/11
Problem 1:An apple is cut in a cylindrical shape 28.7 mm indiameter and 22.3 mm in height. Using an InstronUniversal Testing Machine, the apple cylinder iscompressed. The travel distance of the compression head of the Instron is 3.9 mm. The load cell records a force of 425.5 N. Calculate the stress εz , and strain σz on the apple cylinder.
2/4/2011
BAE2023 Physical Properties of Biological Materials
29
HW Assignment Due 2/11
Problem 2:A sample of freshly harvested miscanthus is shaped into a beam with a square cross section of 6.1 mm by 6.1 mm. Two supports placed 0.7 mm apart support the miscanthus sample and a load is applied halfway between the support points in order to test the Force required to fracture the sample. If ultimate tensile strength is 890 MPa, what would be the force F (newtons) required to cause this sample to fail?
2/4/2011
BAE2023 Physical Properties of Biological Materials
30
HW Assignment Due 2/11
Problem 3:Ham is to be sliced for a deli tray. A prepared block of the ham has a bottom surface of 10 cm x 7 cm. The block is held securely in a meat slicing machine. A slicing blade moves across the top surface of the ham with a uniform lateral force of 27 N and slices a thin portion of meat from the block. The shear modulus, G, of the ham is 32.3 kPa. Estimate the deflection of the top surface with respect to the bottom surface of the block during slicing.
2/4/2011
BAE2023 Physical Properties of Biological Materials
31
HW Assignment Due 2/11
Problem 4:•Sam, the strawberry producer, has had complaints from the produce companythat his strawberries are damaged during transit. Sam would like to know theforce required to damage the strawberries if they are stacked three deep in theircontainer.•The damage occurs on the bottom layer at the interface with the parallel surfaceof the container and also at the point of contact between the layers ofstrawberries.•An hydrostatic bulk compression test on a sample of Sam’s strawberries indicatesan average bulk modulus of 225 psi. Testing of specimens from Sam’sstrawberry crop shows a compression modulus E of 200 psi. The averagestrawberry diameter is 1.25 inches and the axial deformation due to the damagein transit averages 0.23 inches. The modulus of elasticity for Sam’s variety ofstrawberries is reported to be 130 psi.•Estimate the force Sam’s strawberries may be encountering during transit. (Hertzmethod)
2/4/2011