Read: pg 130 – 168 (rest of Chpt. 4)Lecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 81•Poisson’s Ratio, µ (pg. 115)–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:Lecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 82–For uniaxial compression:–εz= σz/E, εy= -µ·εzand εx = -µ·εy•Poisson’s Ratio–For multi-axial compression–See equations in 4.2 page 117–Maximum Poisson’s = 0.5 for incompressible materials to 0.0 for easily compressed materials–Examples: gelatin gel –0.50Lecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 83–Examples: gelatin gel –0.50Soft rubber – 0.49Cork – 0.0Potato flesh – 0.45 – 0.49Apple flesh - 0.21 – 0.29 Wood – 0.3 to 0.5More porous means smaller Poisson’s•In addition to Normal stresses: Shearing Stresses–Shear stress: force per unit area acting in the direction parallel to the surface of the plane,τ–Shear strain: change in the angle formed between Lecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 84–Shear strain: change in the angle formed between two planes that are orthogonal prior to deformation that results from application of sheer stress, γ•Shear modulus: ratio of shear stress to shear strain, G = τ/γ•Measured with parallel plate shear test (pg. 119)Lecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 85(pg. 119)Lecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 86Lecture 8 – Viscoelasticity and DeformationExample Problem• The bottom surface (8 cm x 12 cm) of a rectangular block of cheese (8 cm wide, 12 cm long, 3 cm thick) is clamped in a cheese grater.• The grating mechanism moving across the top surface of the cheese applies a lateral force of 20N. 2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 87the 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 the upper surface, estimate the latera movement of the upper surface w/respect to the lower surface.•Stresses and Strains: described as deviatoric or dilitational•Dilitational: causes change in volume•Deviatoric: causes change in shape but negligible changes in volumeLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 88changes in volume•Bulk Modulus, K: describes response of solid to dilitational stresses•K = average normal stress/dilatation•Dilatation: (Vf– V0)/V0•K = average normal stress/dilatation•Dilatation: (Vf– V0)/V0•Average normal stress = ∆P, uniform hydrostatic gauge pressure•∆V = Vf–V0Lecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 89•∆V = Vf–V0•So: K = ∆P/(∆V / V0)•∆ V is negative, so K is negative•Example of importance: K (Soybean oil) > K (diesel) •Will effect the timing in an engine burning biodiesel• Apples compress easier than potatoes so they have a smaller bulk modulus, K (pg. 120) but larger bulk compressibility•K-1=bulk compressibilityLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 810•K=bulk compressibility• Strain energy density: area under the loading curve of stress-strain diagram• Sharp drop in curve = failure• Stress-Strain Diagram, pg. 122• Area under curve until it fails = toughness• Failure point = bioyield point•Resilience: area under the unloading curveLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 811•Resilience: area under the unloading curve• Resilient materials “spring back”…all energy is recovered upon unloading• Hysteresis = strain density – resilience• Figure 4.6, page 124• Figure 4.7, page 125•Factors Affecting Force-Deformation Behavior–Moisture Content, Fig. 4.6bLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 812–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•Stress Relaxation: Figure 4.10 pg 129–Material is deformed to a fixed strain and strain is held constant…stress required to Lecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 813strain is held constant…stress required to hold strain constant decreases with time.•Creep: Figure 4.11 pg. 130–A continual increase in deformation (strain) with time with constant load•Tensile testing–Not as common as compression testing–Harder to doLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 814–Harder to do–See figure 4.12 page 132•Tensile testingLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 815•Bending: E=modulus of elasticityD=deflection, F=force, I = moment of inertiaLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 816•E=L3(48DI)-1•I=bh3/12•Can be used for testing critical tensile stress at failure•Max tensile stress occurs at bottom surface of beamLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 817surface of beam•σmax=3FL/(2bh2)•Contact Stresses (handout from Mohsenin book)–Hertz Problem of Contact StressesLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials Lecture 818–Hertz Problem of Contact StressesImportance: “In ag products the Hertz method can be used to determine the contact forces and displacements of individual units”•Assumptions:–Material is homogeneous–Loads applied are staticLecture 8 – Viscoelasticity and Deformation2/8/2010 BAE2023 Physical Properties of Biological Materials
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