BMES 345 CH03 Problem Set 20240402(1).docx

BMES 345 CHAPTER 3 PROBLEMS: LINEAR VISCOELASTICITY Contents Problem 1: Spring-Dashpot Models ............................................................................................................................. 2 Problem 2: Spring-Dashpot Models ............................................................................................................................. 3 Problem 3: Viscoelasticity ............................................................................................................................................. 4 Problem 4: Viscoelasticity ............................................................................................................................................. 5 Problem 5: Spring-Dashpot Models ............................................................................................................................. 6 Problem 6: Spring-Dashpot Models ............................................................................................................................. 7 Problem 7: Spring-Dashpot Models ............................................................................................................................. 8 Problem 8: Spring-Dashpot Models ............................................................................................................................. 9 Problem 9: Spring-Dashpot Models ........................................................................................................................... 10 Problem 10: Spring-Dashpot Models ......................................................................................................................... 11 [Solution] Problem 1 ...................................................................................................................................................... 12 [Solution] Problem 2 ...................................................................................................................................................... 13 [Solution] Problem 3 ...................................................................................................................................................... 16 [Solution] Problem 4 ...................................................................................................................................................... 18 [Solution] Problem 5 ...................................................................................................................................................... 20 [Solution] Problem 6 ...................................................................................................................................................... 22 [Solution] Problem 7 ...................................................................................................................................................... 23 [Solution] Problem 8 ...................................................................................................................................................... 25 [Solution] Problem 9 ...................................................................................................................................................... 27 [Solution] Problem 10 .................................................................................................................................................... 28 1

Problem 1: Spring-Dashpot Models Match the creep curve with the corresponding spring-dashpot model. 2

Problem 2: Spring-Dashpot Models In a recent study by Rubiano et al ( J Mech Behav Biomed Mater 2016), researchers injected adipose- derived stem cells into the damaged hearts of hypertensive rats, and observed significant improvements in mechanical properties and clinical outcomes. One of the ways they measured improvements was via viscoelastic testing of the heart tissue. In this study, they modeled the viscoelasticity of the heart using the standard linear solid. Rubiano et al determined that the standard linear solid parameters for the normal, healthy heart tissue were E 1 = 13 kPa, E 2 = 15 kPa, and η = 2,380 kPa-s. Using this information, sketch the following (make sure the curves and axes are clearly and properly labeled): a) A creep curve for the heart tissue, using an applied stress of σ 0 = 2 kPa. b) A stress relaxation curve for the heart tissue, using an applied strain of ε 0 = 0.05. 3

Problem 3: Viscoelasticity Consider the following stress relaxation curve for an osteoblast (bone-producing cell) subjected to a constant strain ε 0 = 0.03: 0 50 100 150 200 250 300 350 400 0 5 10 15 20 25 30 35 40 45 50 Stress (Pa) Time (s) Based on the stress vs. time graph, determine the following: a) Instantaneous modulus b) Equilibrium modulus c) Relaxation time constant d) If the same osteoblast was subjected to a creep test at a constant stress σ 0 = 600 Pa, what would the instantaneous and equilibrium (asymptotic) strains be? Sketch the creep curve with appropriately labeled and scaled axes. 4

Problem 4: Viscoelasticity In a study of how botox-induced unloading of the Achilles tendon altered the biomechanical properties of the tissue (Khayyeri et al, Scientific Reports 2017), researchers measured the viscoelastic properties of rat Achilles tendons using a creep test. Using a constant applied stress of σ 0 = 16 MPa , the researchers obtained the following creep curve: 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0 30 60 90 120 150 Normal Strain Time (s) Find the following parameters (remember to show your work): a) Instantaneous modulus E 0 b) Equilibrium modulus E ∞ c) Creep time constant τ c 5

Problem 5: Spring-Dashpot Models Download the Excel file containing stress relaxation data (stress_relax.xlsx) from BB Learn. In that file, you have the stress relaxation response of a breast cancer cell subjected to a normal strain ε = 0.05. You also have a curve fit (which matches the general form of the stress relaxation response of a standard linear solid): σ ( t )= E 2 ε 0 exp ( − t τ ) + E 1 ε 0 Determine the parameters E 1 , E 2 , and τ of the breast cancer cell using the standard linear solid model. a) Using the equation above and the definitions of instantaneous modulus, equilibrium modulus, and the characteristic time constant, estimate the parameters. b) Now use the Excel Solver add-in to minimize the sum of squared error term to come up with a “best fit” curve. 6

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