LS50 - Essential Mechanics Lecture 35: Elasticity

Home , Bulk modulus, Elastic modulus

March 23, 2016

1 Basic modes of deformations

2 Tensile and compression stress and strain

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 constant 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 √ L p 1 end-to-end distance ∼ ξp N = ξp ∼ ξp ∼ √ . 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 √ L p 1 end-to-end distance ∼ ξp N = ξp ∼ ξp ∼ √ . 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 √ L p 1 end-to-end distance ∼ ξp N = ξp ∼ ξp ∼ √ . 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