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LS50 - Essential Mechanics Lecture 35: Elasticity

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LS50 - Essential Mechanics Lecture 35: Elasticity LS50 - Essential Mechanics Lecture 35: Elasticity March 23, 2016 1 Basic modes of deformations 2 Tensile and compression stress and strain F Tensile stress = ? A units of the tensile stress are P a. ∆` Tensile strain = `0 When the two are sufficiently small can define Young's Modulus as Tensile stress F =A F ` Y = = ? = 0 : Tensile strain ∆`=`0 A ∆` Stress and strain Stress is the strength of the deforming force per unit area Strain is the resulting deformation Model intermolecular interactions as springs =) Linear theory (Hooke's law) Stress = Elastic modulus × Strain . 3 Stress and strain Stress is the strength of the deforming force per unit area Strain is the resulting deformation Model intermolecular interactions as springs =) Linear theory (Hooke's law) Stress = Elastic modulus × Strain . Tensile and compression stress and strain F Tensile stress = ? A units of the tensile stress are P a. ∆` Tensile strain = `0 When the two are sufficiently small can define Young's Modulus as Tensile stress F =A F ` Y = = ? = 0 : Tensile strain ∆`=`0 A ∆` 3 Concept pause: the effective spring con- stant of an elastic deformation What is the effective spring constant k of an individual pili, whose cross-sectional area is A, length L, and Young's modulus Y ? What would be the frequency of oscillations of a small DNA molecule of mass 10−5pg bound to a pili of length 1µm, if Y × A = 154pN. Young's modulus Tensile stress F ` Y = = 0 : Tensile strain A ∆` Material Y (in Pa) Blood cell 500 − 103 Soft tissue 6×104 Skin 4×105 Rubber 1×106 Microtubule 1×109 Silk 4×109 Teeth 1.5×1010 Bone 1.7×1010 Steel 2×1011 4 Young's modulus Concept pause: the effective spring con- Tensile stress F ` Y = = 0 : stant of an elastic deformation Tensile strain A ∆` What is the effective spring constant k of an individual pili, whose cross-sectional area is Material Y (in Pa) A, length L, and Young's modulus Y ? What Blood cell 500 − 103 would be the frequency of oscillations of a Soft tissue 6×104 small DNA molecule of mass 10−5pg bound Skin 4×105 to a pili of length 1µm, if Y × A = 154pN. Rubber 1×106 Microtubule 1×109 Silk 4×109 Teeth 1.5×1010 Bone 1.7×1010 Steel 2×1011 4 Compression strain and compressive stress Very similar to tension... in most materials, the same Young's modulus. In some cases, parts of an object are under tension, and others under compression. (recall: bending is NOT a basic mode of deformation.) The energy required for compression or tension is, 1 YA U = (∆`)2 : 2 `0 1 2 This is analogous to U = 2 kx for a simple spring. 5 Bulk stress and strain, compressibility When the strain is applied uniformly by the surrounding fluid. Bulk stress ∆p Bulk modulus = B = = − Bulk strain ∆V=V0 The inverse of the Bulk modulus is called compressibility. Bulk strain 1 ∆V Compressibility = k = = − Bulk stress V ∆p −1 Material k (in Pa ) k=kw 1 @V 1 9 Ideal gas − V @p = p 2:2 × 10 Playdough 10−6 2174 Ethyl alcohol 1.1×10−9 2.4 Proteins 2-10×10−9 4.3-21.7 Lipid bilayer 5.6×10−10 1.2 Water 4.6×10−10 1 White blood cells 4×10−10 0.87 6 Shear stress and strain We define F Shear stress = k A (units of the shear stress are P a, just like the tensile stress). ∆x Tensile strain = h Shear stress F =A Shear modulus = S = = − k Shear strain ∆x=h Shear modulus applies only to solid materials. For a homogeneous material, the Shear modulus is about 1/3 of Young's modulus. 7 The persistence length Thermal fluctuations randomize the orientation of a polymer. The persistence length ξp is the length-scale over which the polymer is roughly rigid. To estimate it, set L = R = ξp: Y Iξp YI kB T = 2 =) ξp = 2ξp 2kB T Entropic spring Think about a long polymer as a random walker. If its length is L, it makes N = L/ξp independent steps, so its s p L p 1 end-to-end distance ∼ ξp N = ξp ∼ ξp ∼ p : ξp T Thus \entropic spring" shrinks when the temperature increases (!) Bending 1 The energy of bending a beam of length L to a curvature κ = R is YIL U = bending 2R2 P 2 where I is the moment of inertia I = i miRi (remember?). 8 Entropic spring Think about a long polymer as a random walker. If its length is L, it makes N = L/ξp independent steps, so its s p L p 1 end-to-end distance ∼ ξp N = ξp ∼ ξp ∼ p : ξp T Thus \entropic spring" shrinks when the temperature increases (!) Bending 1 The energy of bending a beam of length L to a curvature κ = R is YIL U = bending 2R2 P 2 where I is the moment of inertia I = i miRi (remember?). The persistence length Thermal fluctuations randomize the orientation of a polymer. The persistence length ξp is the length-scale over which the polymer is roughly rigid. To estimate it, set L = R = ξp: Y Iξp YI kB T = 2 =) ξp = 2ξp 2kB T 8 Bending 1 The energy of bending a beam of length L to a curvature κ = R is YIL U = bending 2R2 P 2 where I is the moment of inertia I = i miRi (remember?). The persistence length Thermal fluctuations randomize the orientation of a polymer. The persistence length ξp is the length-scale over which the polymer is roughly rigid. To estimate it, set L = R = ξp: Y Iξp YI kB T = 2 =) ξp = 2ξp 2kB T Entropic spring Think about a long polymer as a random walker. If its length is L, it makes N = L/ξp independent steps, so its s p L p 1 end-to-end distance ∼ ξp N = ξp ∼ ξp ∼ p : ξp T Thus \entropic spring" shrinks when the temperature increases (!) 8 Example: DNA looping is an important mechanism in compaction of the DNA in vivo and in regulation of gene expression. How much energy does it take to loop DNA? YI Using ξp = we find kB T U 2 loop πξp 2π ξp 3000 = = ' ; kB T R Nbp a Nbp where a is the length of a single base, a ' 0:34nm, and ξp = 50nm for dsDNA. Looping YIL If Ubending = 2R2 , how much energy does it take to make a loop? When we make a full loop we have L = 2πR and then πY I U = : loop R 9 Looping YIL If Ubending = 2R2 , how much energy does it take to make a loop? When we make a full loop we have L = 2πR and then πY I U = : loop R Example: DNA looping is an important mechanism in compaction of the DNA in vivo and in regulation of gene expression. How much energy does it take to loop DNA? YI Using ξp = we find kB T U 2 loop πξp 2π ξp 3000 = = ' ; kB T R Nbp a Nbp where a is the length of a single base, a ' 0:34nm, and ξp = 50nm for dsDNA. 9 Elasticity and plasticity A material shows elastic behavior when it returns to its original form upon removal of the stress (purple range), and a plastic behavior when it doesn't. Near the transition (between a and b) Hooke's law is not a good approximation. When the stress is removed from a point in this range, the material will return to its original shape but may take a new path (displaying elastic hysteresis). The behavior between b and d is called a plastic deformation. A ductile material can undergo significant plastic deformation before it breaks. A brittle material breaks soon beyond its yield stress. 10 Elasticity and plasticity A material shows elastic behavior when it returns to its original form upon removal of the stress (purple range), and a plastic behavior when it doesn't. Near the transition (between a and b) Hooke's law is not a good approximation. When the stress is removed from a point in this range, the material will return to its original shape but may take a new path (displaying elastic hysteresis). The behavior between b and d is called a plastic deformation. A ductile material can undergo significant plastic deformation before it breaks. A brittle material breaks soon beyond its yield stress. 10 Elasticity and plasticity A material shows elastic behavior when it returns to its original form upon removal of the stress (purple range), and a plastic behavior when it doesn't. Near the transition (between a and b) Hooke's law is not a good approximation. When the stress is removed from a point in this range, the material will return to its original shape but may take a new path (displaying elastic hysteresis). The behavior between b and d is called a plastic deformation. A ductile material can undergo significant plastic deformation before it breaks. A brittle material breaks soon beyond its yield stress. 10 Elasticity and plasticity A material shows elastic behavior when it returns to its original form upon removal of the stress (purple range), and a plastic behavior when it doesn't. Near the transition (between a and b) Hooke's law is not a good approximation. When the stress is removed from a point in this range, the material will return to its original shape but may take a new path (displaying elastic hysteresis). The behavior between b and d is called a plastic deformation.
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