ExpPhys I Lect17

8/13/2019 ExpPhys I Lect17

1/35

The Annual Ks Lab X-Mas Bash 2013

When: December 14th 7pmWhere: Feuerbachstr 4 LeipzigPlease, bring a christmas ornament and clothing for the homeless!


2/35

Dr. M arei ke Zi nk /

P ro f . Dr. Jo sef A. Ks

Experimental Physics IWinter 2013/14


3/35

Elastic modulus

An elastic modulus, is the description of an object or substance's tendency to

be deformed elastically (i.e., non-permanently) when a force is applied.

The elastic modulus is defined as the slope of its stress-strain curve in the

elastic deformation region:

where (lambda) is the elastic modulus;

stressis the force causing the deformation divided by the area to which the force

is applied; and

strainis the ratio of the change caused by the stress to the original state of the

object

If stress is measured in pascals, since strain is a unitless ratio, then the units

ofare pascals as well.


4/35

Young's modulus

measure of the stiffness of an isotropic elastic material

ratio of the uniaxial stress over the uniaxial strain in the range of stressin which Hooke's Law holds

Young's modulus,E, can be calculated by dividing the tensile stress by

the tensile strain:

whereEis the Young's modulus,

Fis the force applied to the object,A0 is the original cross-sectional area through which the force isapplied,Lis the amount by which the length of the object changes,

L0is the original length of the object.


5/35


6/35

Elastic Limit


7/35

True elastic limit (1): The lowest stress at which dislocations move. This

definition is rarely used, since dislocations move at very low stresses, anddetecting such movement is very difficult.

Proportionality limit (2):Up to this amount of stress, stress is proportionalto strain (Hooke's law), so the stress-strain graph is a straight line, and thegradient will be equal to the elastic modulus of the material. Elastic limit (yieldstrength) Beyond the elastic limit, permanent deformation will occur. Thelowest stress at which permanent deformation can be measured.

Elastic Limit (3)

Yield point (4): The point in the stress-strain curve at which the curve levelsoff and plastic deformation begins to occur. Offset yield point (proofstress) When a yield point is not easily defined based on the shape of the stress-

strain curve an offset yield point is arbitrarily defined. The value for this iscommonly set at 0.1 or 0.2% of the strain.


8/35

Linear elasticity is an approximation for the potential

energy near the minimum:

Minimum for potential energy

Taylor Expansion:

Harmonic potential

Hookes Law


9/35


10/35

Shear modulus


11/35

Bulk modulus

the bulk modulus Kcan be formally defined by the equation:

where Pis pressure,

Vis volume, andP/Vdenotes the partial derivative of pressure with respect to

volume

inverse of the bulk modulus gives a substance's compressibility.


12/35

Relation among elastic constants

For homogeneous isotropic materials simple relations exist betweenelastic constants (Young's modulusE, shear modulusG, bulk modulus

K, and Poisson's ratio ) that allow calculating them all as long as twoare known:


13/35

Poisson's ratio

when a material is compressed in one direction, it usually tends toexpand in the other two directions perpendicular to the compression=Poisson effect

Poisson's ratio = a measure of the Poisson effect

the ratio of the fraction of expansion divided by the fraction (or percent)of compression, for small values of these changes

in certain rare cases, a material will actually shrink in the transversedirection when compressed (or expand when stretched) which will yielda negative value of the Poisson ratio.

the Poisson's ratio of a stable, isotropic, linear elastic material cannot beless than 1.0 nor greater than 0.5 due to the requirement that Young'smodulus, the shear modulus and bulk modulus have positive values

most materials have Poisson's ratio values ranging between 0.0 and 0.5

incompressible material have a Poisson's ratio of exactly 0.5


14/35

A cube with sides of lengthLof an isotropic linearly elastic material subject totension along the x axis, with a Poisson's ratio of 0.5. The green cube isunstressed, the red is expanded in the xdirection by L due to tension, andcontracted in theyandzdirections byL'


15/35

Poisson number


16/35

Pressure (the symbol:P) is the force per unit area applied in a directionperpendicular to the surface of an object

where:

Pis the pressure,Fis the normal force,Ais the area of the surface area oncontact

Body under compressive pressure:


17/35

Compression modul K


18/35

Bending

EulerBernoulli beam theory= a simplification of the linear theory ofelasticity which

provides a means of calculating the load-carrying and deflectioncharacteristics of beams

covers the case for small deflections of a beam which is subjected to lateral

loads only

pure bending (of positive sign) willcause zero stress at the neutral axis,positive (tensile) stress at the "top"

of the beam, and negative(compressive) stress at the bottomof the beam


19/35

Rectangular beam

Small piece

Radius curvature at dashed line r

Stretching above: Compression below:


20/35

Bending force:

Torque:


21/35


22/35

Bending moment:


23/35

Solid mechanics

amount of departure from rest shape is called

deformation, the proportion of deformation to original sizeis called strain

if the applied stress is low (or the imposed strain is small),

solid materials behave in such a way that the strain isdirectly proportional to the stress, linearly elastic


24/35

three types how a solid responds to an applied stress:

Elastically When an applied stress is removed, thematerial returns to its undeformed state

Viscoelastically materials that behave elastically, butalso have damping, implies that the material response hastime-dependence

Plastically Materials that behave elastically generallydo so when the applied stress is less than a yield value.

When the stress is greater than the yield stress, the material

behaves plastically and does not return to its previous state.That is, deformation that occurs after yield is permanent.


25/35

Th e Op t i ca l St r et ch er


26/35

Th e Op t i ca l St r et ch er

Gedankenexperiment:

Momentum of light:


27/35

E f f ect i v e Cel l Com p l i a n ce

a s Cel l M a r k er

stress: strain

Jo: steady state compliance


28/35

Worm-like chain


29/35


30/35

Bending stiffness:

in the case of small undulations around a straight shape the total bending energyHbendof the filament can be expressed by

for a normal-mode analysis of the thermal bending excitations a Fourierdecomposition of the tangential angle:

kcdenotes the bending modulus


31/35

using the equipartition theorem we obtained for the mean bending energy ofeach mode

by solving this equation we derive the following equation for the mean square

amplitudes:


32/35


33/35

breast cervix

Benign

Tumor

-> F. Wetzel et al, Cancer Cell, submitted-> A. Fritsch et al, Nature Physics, 2010

Soft cells in solid tumors


34/35

Cancer cell contractility and optical stretcher

-0.04 -0.03 -0.02 -0.01 0.000

20

40

60

80

100

no.

ofcells(absolutvalues)

relaxation

breast tumor G3+

breast tumor G3-

FA (1+2)

0 1 2 3 4 5

0.000

0.005

0.010

0.015

0.020

0.025

0.030breast tumor G3+

breast tumor G3-

FA2

FA1

re

lative

deform

ation

time [s]

B

relaxation

-> A. Fritsch et al, Nature Physics, 2010

Benign

Tumor

Metastatic


35/35

Metastatic cells

Non-metastatic metastatic

Documents

ExpPhys I Lect17