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Massachusetts Institute of Technology Biological Engineering Division Department of Mechanical Engineering Department of Electrical Engineering and Computer Science 2.797/20.310/3.0536.024, Fall 2006 MOLECULAR, CELLULAR, & TISSUE BIOMECHANICS Problem Set #8 Issued: 11/17/06 Due: 12/7/06 You only need to turn in Problems 2 and 3. Solutions will be distributed for the others. True-False 1. A specimen that continues to creep (deform) under a constant stress, as long as the stress is applied, could be modeled as a Kelvin (or Standard Linear Solid) material. 2. If the applied stress is held constant with time, a purely elastic material will exhibit a constant, steady strain. 3. For a poroelastic material subjected to an oscillatory load at a single frequency, the strain is in phase with the applied load. 4. When subjected to confined compression, a tissue specimen is reduced in volume due to the expulsion of liquid through the boundaries. 5. For unconfined compression of a poroelastic material, the following constitutive law applies: tot 11 = 2G11 + ( 11 + 22 + 33 )  p True-False 1. A specimen that continues to creep (deform) under a constant stress, as long as the stress is applied, could be modeled as a Kelvin (or Standard Linear Solid) material. False. A Kelvin material will creep to a certain extent, but will eventually reach a new equilibrium. 2. If the applied stress is held constant with time, a purely elastic material will exhibit a constant, steady strain. True. A purely elastic material will deform immediately upon the application of stress and will change its deformation only when the applied stress changes.3. For a poroelastic material subjected to an oscillatory load at a single frequency, the strain is in phase with the applied load. False. A poroelastic material, like a viscoelastic material, is dissipative so that the load will lead the deformation. 4. When subjected to confined compression, a tissue specimen is reduced in volume due to the expulsion of liquid through the boundaries. True. In confined compression, the upper boundary is usually porous and interstitial fluid will leak out. 5. For unconfined compression of a poroelastic material, the following constitutive law applies: 11 = 2G11 + ( 11 + 22 + 33 )  p True. The only difference between this expression and that for an elastic material is the addition of the fluid pressure, p. Problem 1: Linear, isotropic, homogeneous, poroelastic material Consider a Poroelastic tissue specimen subjected to confined compression. In class we demonstrated that the displacement u1(x1,t) is described by a partial differential equation having the form of a diffusion equation with equivalent “diffusivity” equal to Hk, the product of the confined compression modulus H = (2G + ) and the hydraulic permeability k. (a) Derive an analogous diffusion equation that describes the spatial and temporal dependence of the fluid pressure p. What is the equivalent “diffusivity”? tot 11 x1 = 0 x1 = L x1(b) A step in displacement is applied at x1 = 0 having amplitude u0. State the boundary conditions on u1(x1=0,t) and u1(x1=L,t) and the initial condition u1(x1,t=0) that would be used to solve for the displacement u1(x1,t) occurring during this “stress relaxation”. (Do not solve.) (c) A step in stress is applied at x1 = 0 of amplitude 0. State the boundary conditions on the displacement (or its slope) and the initial condition on u1(x1,t=0) that would be used to solve for the creep displacement u1(x1,t). (Do not solve.) (d) For the stress relaxation example of part (b), the solution below was provided in one of the slides from class. Use that solution to show (1) that the higher frequency components of the solution decay more rapidly with time, and (2) that the displacement is a linear function of x1 as t . What is the expression for the slowest (“n=1”) decay time; i.e., the stress relaxation time, in terms of material and geometric constants? t n x1x1u1(x1,t ) = u0 1   Asin nexpL L n n L2 = n 2n 2 Hk Problem 2: Measuring H and k You wish to perform a simple set of experiments on a sample of cartilage to obtain values for the hydraulic permeability k and confined compression modulus H using an apparatus of the type shown in the sketch. The sample is placed into a compression chamber with rigid, non-permeable sides and bottom. On top of the sample is placed a permeable but rigid platen to which a vertical force can be applied. For this problem, design an appropriate experiment or set of experiments that will allow you to compute individual values for k and H. You may specify either a time-varying (or static) force or displacement for the upper platen. Assume that all displacements are purely uni-directional (in the x1-direction), that the sample has homogeneous properties, and that it satisfies the following expressions derived in class:tot  p tot 11 1) U1 = k 2) 11 = H11 + p 3) = 0 z x1 =4) U1 = u1 5) 11 u1 t x1tot 11 x1 = 0 x1 = L x1 where U is the x1-component of velocity, u1 is displacement of the cartilage matrix, p is1hydrostatic pressure, and 1 tot is total stress. There is no single correct answer to this problem; there are a variety of schemes that will work. All you need to do is describe one experimental procedure, and then provide the appropriate analysis indicating how H and k are to be computed from the experimental measurements. Problem 3: Arterial wall poroelasticity As the wall of an artery expands and contracts due to arterial pressure variations, there is a tendency for fluid to be periodically drawn into and expelled from the tissue comprising the wall. In this problem, you will model the arterial wall as a poroelastic material and analyze this fluid motion. This is of special interest in the context of arterial disease since this represents one method by which lipids normally found in the blood plasma might enter the arterial wall where they could react with extracellular matrix proteins and form the nucleus for lipid aggregation (see e.g., Yin et al., A model for the initiation and growth of extracellular lipid liposomes in arterial intima. Am J Physiol. 1997 Feb;272(2 Pt 2):H1033-46.). Assume here that: • the wall is thin compared to the radius of the


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MIT 2 797J - MOLECULAR, CELLULAR AND TISSUE BIOMECHANICS

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