1. Analysis of Elastic Strains
Ref: L.D.Landau, E.M.Lifshitz, “Theory of Elasticity”, Pergamon Press (59/86)
Continuum approximation: good for λ > 30A.
Description of deformation (Cartesian coordinates):
Displacement vector field u(r).
Material point
Nearby point
= strain tensor
= linear strain tensor

2. Dilation
uik is symmetric → diagonalizable →  principal axes such that
(no summation over i ) → 
Fractional volume change 
 Trace of uik

3. Stress Total force acting on a volume element inside solid 
f  force density
Newton’s 3rd law → internal forces cancel each other
→ only forces on surface contribute
This is guaranteed if
σ  stress tensor
so that
σik  ith component of force acting on the surface element normal to the xk axis.
Moment on volume element 
Only forces on surface contribute →
(σ is symmetric)

4. Elastic Compliance & Stiffness Constants
σ and u are symmetric → they have at most 6 independent components
Compact index notations (i , j) → α :
(1,1) → 1, (2,2) → 2, (3,3) → 3, (1,2) = (2,1) → 4, (2,3) = (3,2) → 5, (3,1) = (1,3) → 6
Elastic energy density:
i , j , k, l = 1,2,3
α , β = 1,2,…,6 21
where
 elastic stiffness constants
 elastic modulus tensor
uik & uki treated as independent
Stress:
S α β  elastic compliance constants

5. Elastic Stiffness Constants for Cubic Crystals
Invariance under reflections xi → –xi  C with odd numbers of like indices vanishes
Invariance under C3 , i.e.,
 All C i j k l = 0 except for (summation notation suspended):

6. where 

7. Bulk Modulus & Compressibility
Uniform dilation:
δ = Tr uik = fractional volume change
B = Bulk modulus
= 1/κ
κ = compressibility
See table 3 for values of B & κ .

8. Elastic Waves in Cubic Crystals
Newton’s 2nd law:
don’t confuse ui with uα → 
Similarly

9. Dispersion Equation →
dispersion equation

10. Waves in the [100] direction→
Longitudinal
Transverse, degenerate

11. Waves in the [110] direction→
Lonitudinal
Transverse
Transverse

12. Prob 3.10

15. T stress tensor is defined by:
where the dFi are the components of the resultant force vector acting on a small area dA which can be represented by a vector dAj perpendicular to the area element, facing outwards and with length equal to the area of the element. In elementary mechanics, the subscripts are often denoted x,y,z rather than 1,2,3.
Stress tensor is symmetric, otherwise the volume element would rotate (to seet this look at zy and yz component in figure)
Hookes Law
(Voigt) notation
1 = 11, 2 = 22 3 = 33 4 = 23 5 = 31 6 = 12

16. Normal : load perpendicular to area
Types of Stress
Normal : load perpendicular to area

17. Types of Stress
a) Normal
b) Shear : load parallel to area

18. Types of Stress
a) Normal
b) Shear
c) Hydrostatic (uniform pressure)

19. Normal : deformation in the the direction of length
Types of Strain
Normal : deformation in the the direction of length
Longitudinal Strain
e = u / l
Transverse Strain
e = v / l

20. Types of Strain
Normal
Shear : deformation normal to length

21. Types of Strain
Normal
Shear
Dilatational (volume change)

22. Relating Stress and Strain
Called a “Constitutive Model”
Simple Linear Model:
“Hooke’s Law”

23. Relating Stress and Strain
Normal
S = E e
Stress = Young’s modulus x strain

24. Relating Stress and Strain Shear
Hooke’s Law
 = G 
Shear Modulus
G = E / [2 (1+  )]
 = shear stress
G = shear modulus
 = shear strain

25. Relating Stress and Strain Dilatational (hydrostatic)
Hooke’s Law
P = K 
K= E/3(1-2)
P = pressure
K = bulk modulus
 = V/ Vo

26. Real Material Behavior
-Hookean to a limiting strain
-Proportional limit : Deviation from linear
-After PL
*Linear
*Flat
*Non linear

27. ELASTIC PROPERTIES OF MATERIALS
When a tensile stress is imposed on a metal specimen, an elastic elongation and accompanying strain result in the direction of the applied stress (z).
As a result:
constrictions in the lateral (x and y) directions perpendicular to the tension stress
compressive strains
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28. <TK091355>

29. ELASTIC PROPERTIES OF MATERIALS
Many materials are elastically anisotropic;
the elastic behavior (the magnitude of E) varies with crystallographic direction (see Table 3.3).
For these materials the elastic properties are completely characterized only by the specification of several elastic constants, which depend on the crystal structure.
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30. ELASTIC PROPERTIES OF MATERIALS
Even for isotropic materials, for complete characterization of the elastic properties, at least two constants must be given.
Since the grain orientation is random in most polycrystalline materials, these may be considered to be isotropic; inorganic ceramic glasses are also isotropic.
The discussion of mechanical behavior assumes isotropy and polycrystallinity because such is the character of most engineering materials.
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31. Isotropic Stress-Strain Relations
Other elastic constants
Bulk Modulus K
where p is the average normal stress.
Lamé’s constant 
Note: 0 ≤  ≤ 1/2
E=2G(1+)

32. Relations Among the Elastic Constants
R. Reismann and P.S. Pawlik, Elasticity: Theory and Practice, Robert E. Krieger Publishing Company , 1991

33. Uniaxial Loading x y
is the biaxial modulus
Biaxial Loading

34. Anisotropic Materials
where [C] is the 6x6 stiffness matrix
where [S] = [C]-1 is the 6x6 compliance matrix

35. Cubic Material (Single Crystal Silicon)
There are 3 independent elastic constants C11, C12, C44
For silicon C11= 166 GPa, C12= 64 GPa, C44 = 80 GPa
Appendix C
where l1, l2, l3 are the direction cosines of the angle between the direction of interest and the x,y,z directions respectively, e.g. l1=cos(x,dir).

36. Elasticity:
The one-dimensional stress-strain relationship may be written as:
where C is an elastic constant.
Note that:
The response is instantaneous.
Here the strain is the infinitesimal strain.

37. The material is said to be linear elastic if n=1.
Hooke’s law:
In three dimensions, Hooke’s law is written as:
where Cijkl is a matrix whose entries are the stiffness coefficients.

38. It thus seems that one needs 81(
It thus seems that one needs 81(!) constants in order to describe the stress strain relations.

39. Thanks to the symmetry of the stress tensor, the number of independent elastic constants is reduced to 54.

40. Thanks to the symmetry of the strain tensor, the number of independent elastic constants is further reduced to 36.

41. The following formalism is convenient for problems in which the strains components are known and the stress components are the dependent variables:
In cases where the strain components are the dependent parameters, it is more convenient to use the following formalism:
Where Sijkl is a matrix whose entries are the compliance coefficients.

42. The case of isotropic materials:
A material is said to be isotropic if its properties are independent of direction.
In that case, the number of non-zero stiffnesses (or compliances) is reduced to 12, all are a function of only 2 elastic constants.
Young modulus:

43. Shear modulus (rigidity):
Bulk modulus (compressibility):

44. Poisson’s ratio:
Poisson’s ratio of incompressible isotropic materials equals 0.5:

45. All elastic constants can be expressed as a function of only 2 elastic constants. Here is a conversion table:

46. Generalized Hooke’s law, i.e. Hooke’s law for isotropic materials:
Normal stresses along X, Y and Z directions do not cause shear strains along these directions.
Tensions along one direction cause shortening along perpendicular directions.
Shear stresses at X, Y and Z directions do not cause strains at perpendicular directions.
Shear stresses at one direction do not cause shear strains at perpendicular directions.

47. Elasticity
Elastic Limit -> Maximum amount of stress up to which the deformation is absolutely temporary
Proportionality Limit -> Maximum stress up to which the relationship between stress & strain is linear.
Hooke’s Law -> Within elastic limit, the strain produced in a body is directly proportional to the stress applied.
σ = E ε
0. This curve is for a ductile material
Last -> The proportionality constant connecting stress and strain is known as the elastic modulus and may
be either (a) the elastic or Young’s modulus, E, (b) the rigidity or shear modulus μ, or (c) the bulk
modulus K, depending on whether the strain is tensile, shear or hydrostatic compressive, respectively.

48. Important Terms
Young’s Modulus of elasticity -> the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds
Elasticity -> the tendency of a body to return to its original shape after it has been stretched or compressed
Yield Point -> the stress at which a material begins to deform plastically
It is a measure of STIFFNESS for a material. E = Stress/Strain; Units for “E” are same as that for Stress. .
For pure metals it is yield point and for alloys it is yield strength (obtained by intersection of a line at 0.2% strain offset to the original stress-strain curve). Yield strength is also called as yield stress. Yield point marks the transition to the plastic region and start of plastic deformation

49. Important Terms
Plasticity -> the deformation of a material undergoing non-reversible changes of shape in response to applied forces
Ultimate Strength -> It is the maxima of the stress-strain curve. It is the point at which necking will start.
Necking -> A mode of tensile deformation where relatively large amounts of strain localize disproportionately in a small region of the material
Plastic range of a ductile material is the area of interest for manufacturing ppl. The region is not governed by the Hooke’s Law and the deformation magnitude is exceedingly large as compared to that in elastic range for the same increment in the applied stress .
During necking, the stress begins to decline and a localized elongation sets in the test specimen. Instead of continuing strain uniformly, straining becomes concentrated in one small region of the specimen.

50. Important Terms
Fracture Point -> The stress calculated immediately before the fracture.
Ductility -> The amount of strain a material can endure before failure.
Ductility is measured by percentage elongation or area reduction
Fracture stress is less than ultimate strength. The increase in stress from the initial yield up to the TS indicates that the specimen hardens during deformation (i.e. work hardens). On straining beyond the TS the metal still continues to work-harden, but at a rate too small to compensate for the reduction in cross-sectional area of the test piece
Ductility is the ability of the material to plastically strain without failure. Ductility refers to workability of any material

51. Poisson’s Ratio
It is the ratio of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load).
The Poisson ratio is the ratio of the fraction (or percent) of expansion divided by the fraction (or percent) of compression, for small values of these changes
Most materials have Poisson's ratio values ranging between 0.0 and 0.5
When a material is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the direction of compression. This phenomenon is called the Poisson effect. Poisson's ratio ν (nu) is a measure of the Poisson effect

52. Poisson’s Ratio – Length Change
The infinitesimal strains are given by:
Integrating the definition of Poisson's ratio:
Solving and exponentiating, the relationship between ΔL and ΔL' is found to be:
0. For a cube stretched in the x-direction with a length increase of ΔL in the x direction, and a length decrease of ΔL' in the y and z directions, :

53. Poisson’s Ratio – Length Change
For very small values of ΔL and ΔL', the first-order approximation yields:

54. Poisson’s Ratio – Volumetric Change
Using V = L3 and V + ΔV = (L + ΔL)(L − ΔL')2:
Using the derived relationship between ΔL and ΔL'
and for very small values of ΔL and ΔL', the first-order approximation yields:
0. The relative change of volume ΔV/V due to the stretch of the material can now be calculated.

55. Elastic Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions.
It can be defined as a difference, when measured along different axes, in a material's physical property (absorbance, refractive index, density, etc.)
Hooke’s law for anisotropic materials can be expressed in terms of compliances, sijmn, which relate the response of individual strain components to individual stress components
Compliance is the inverse of stiffness

56. -> σmn = σnm and γij = γji = 2emn = 2enm,
Elastic Anisotropy
Every strain component depends linearly on the stress components
eij = sijmnσmn
The compliances, sijmn, form a fourth-order tensor
Simplification of the Hooke’s Law relations
-> σmn = σnm and γij = γji = 2emn = 2enm,
Hooke’s law becomes:
Sijmn i,j,m,n X i,j,m,n (fourth-order)
Equations -> Note the subscripts on the strain and stress terms refer to the crystallographic axes

57. (Compliance in 2nd row, 1st column is incorrect)
Elastic Anisotropy
Because it can be shown that sij = sji, further simplification results in
(Compliance in 2nd row, 1st column is incorrect)
So there are at most 21 independent compliances.
In the majority of cases, symmetry reduces the number of constants
3. … Further reduction of the tensor is not going to be studied here (although available in the book)

58. Equations of Elasticity
Full equations of nonlinear elastodynamics
Nonlinearities due to
geometry (large deformation; rotation of local coord frame)
material (nonlinear stress-strain curve; volume preservation)
Simplification for small-strain (“linear geometry”)
Dynamic and quasistatic cases useful in different contexts
Very stiff almost rigid objects
Haptics
Animation style

59. Stress Tensor Describes forces acting inside an object n w
dA (tiny area)

60. Newton’s 2nd Law of Motion
Simple (finite volume) discretization… w dV

61. 27.1 Stress
The stress in a material is the ratio of the force acting through the material divided by the cross section area through which the force is carried.
The metric unit of stress is the pascal (Pa).
One pascal is equal to one newton of force per square meter of area (1 N/m2).
Force (N)
s = F A
Stress (N/m2)
Area (m2)

62. 27.1 Properties of Solids

63. 26.1 Properties of Solids
A thicker wire can support more force at the same stress as a thinner wire because the cross section area is increased.
Even thought the stress is the same, the tensile strength of the thicker wire is higher
314 N/0.8 mm2 = 400N/mm2
1256N/3.1 mm2 = 400N/mm2

64. 26.1 Tensile strength
The tensile strength is the stress at which a material breaks under a tension force.
The tensile strength also describes how materials break in bending.

65. 27.1 Tensile strength

66. 27.1 Strain
The Greek letter epsilon (ε) is usually used to represent strain.
Change in
length (m)
e = Dl l
Strain
Original length (m)

67. 27.1 Properties of solids
The modulus of elasticity plays the role of the spring constant for solids.
A material is elastic when it can take a large amount of strain before breaking.
A brittle material breaks at a very low value of strain.

68. 27.1 Modulus of Elasticity

69. 27.1 Stress for solids s = -E e
Calculating stress for solids is similar to using Hooke's law for springs.
Stress and strain take the place of force and distance in the formula:
Modulus of
elasticity (pa)
s = -E e
Stress (Mpa)
Strain