Matter: properties have an orientation dependence and must conform to material's symmetry (must obey Neumann's principle)

Field: relates quantities, can have any orientation with respect to material, can exist in isotropic bodies, need not conform to mateiral's symmetry (ie conductivity/stiffness)

A property must have at least the symmetry of the underlying material but could have more

aka the symmetry elements of a material property must include the symmetry elements of the point group of the material

• infinite symmetry
• macroscopic continuum
• amorphous solids at atomic scale

strain depends only on stress state not orientation

prinicpal axes for stress and strain will coincide

Hooke's law can be  used to related stress and strain

• recoverable: remove load and it returns to initial state
• metals and ceramics are elastic at SMALL strains
• polymers are elastic at small and large strains
• can get elastic failure of a material (buckling)

can not simply find the inverse of individual terms; must find the inverse of the entire tensor

1/ρ_ij≠1/σ_ij

e₁₂=e₂₁ "pure shear"

e₁₂=-e₂₁ "rotation"

e₁₂=c, e₂₁=0 "simple shear"

e_ij=ε_ij+ω_ij

ε_ij-symmetric/strain/shear tensor=(e_ij+e_ji)/2

ω_ij-antisymmetric/rotation tensor=(e_ij-e_ji)/2

• permanent - irrecoverable
• strains may be large
• important in metal working (forging/extrusion)
• microscopic scale (deformation at crack tip)
• often the desireable failure mode (absorbs energy (impace/crumble zone)

• catastrophic propagation of crack
• no macroscopic deformation

• finding principal axes/stresses/strains
• knowing stresses for one orietnation and calculate for any other
• calculate maximum shear stress

hydrostatic: volume change

deviatoric: shape change

plastic flow only depends on deviatoric and this is true even to very high hydrostatic

dilation based on hydrostatic bit

metals ~ 0.25-0.33

rubber ~0.5

Bulk Modulus, K: found by relating sum of strains (dilation) and sum of stresses (3*hydrostatic) via Hooke's law

Lame's constants: Used to get stresses from strains

two indepent elastic statnts for an isotropic material can specify many elastic properties

when atoms lie at centers of symmetry and interatomic forces are entirely along the line joining the centers (C12=C44)

roughly ture for simply ionic crystals

does not work for metals bbecause bonding is not localized

isotropic: 2 independent elastic constants and 2 deformation modes: dilation/shear

cubic: 3 independent cosntants and 3 deformation modes: dilation, shear on cube face, shear at 45 degrees to cube axis

3 independent terms

C11=C22=C33, C12=C23=C31, C44=C55=C66

devces which exploit changes in resisitance as a function of change in length

based on R=ρL/A

to measure strain, we fix wire secuely to object.  once fixed we allow deformation to occur.  we measure changes in R and infer strain however strains are usually very small so it is desirable to have a small gauge length but large actual length; done by folding

strain guage contacts are connected along a wheatstone bridge

these gauges are sensitive only to normal strains

Photoelasticity

birefringent coatings

applying a stress to a transparent material results in optical anisotropy

light will propagate with two different velocities and two mutually perpendicular wave fronts

n1-n2=c(σ1-σ2) [c-stress optical coefficient]

if wavelength dependent: isochromatic fringes

if wavelengeth independent: isoclinic fringes

birefringent coatings are placed on surface under stress then birefringence is viewed.  materials with high stress optical coefficient are used (PMMA, epoxy, etc.)

Birefringent coatings

brittle coatings

XRD

Ultrasonics

Residual stresses

Stress strain analysis

used to determine teh directions of principal axes on surface

apply a thin coating of a brittle mateiral to unstressed body, cracks appear once critical stress is reached

first set of cracks appear perpendicular to sigma 1 then second appear later

if principal stresses are the same, "crazed" surface appears as direction of cracks is indeterminate

σ_specimen=E_specimenε_coating

Advantages:

can be used on real structure, obtain actual stresses, useful (determine directions and instrument with strain gauges)

Disadvantages:

need ceramic coating for high T, failure strain of coating must be calibrated

measure strain by changes in lattice parameter (thin films)

non destructive/no contact

useful only for surface strains (penetration less than a micron)

applicable to larger objects than XRD

real materials are non linearly elastic so modulus varies with stress/strain

optical methods are attractive: non invasive, sensitive

lookign at small movements on surface rather than penetrating

retained internal stresses not a result of external force

must be retained by interlocking

origins: thermal stress/differential thermal expansion, plastic flow/bending due to shot peening or machining, volume changes (phase transformations, carbides, nitriding)

thin films

there is no force on the free surface and in most cases σ1=σ2=σ [σ3=0]

these stresses are due to thermal expansion differences, deposition stresses, epitaxial stresses (difference in lattice parameters)

if hte forces acting on a small part of the surface of a body are replaced by statically equivalent forces, the stress state is negligibly changed at large distance

upon mathematical treatment, it is evident that the decay of the effect of this small load is irrelevant at distances larger than the dimensions of that effect

(Venant's principle states that high order momentum of mechanical load ( momentum with order higher than torque) decays so fast that they never need to be considered for regions far from the short boundary. Therefore, the Saint-Venant's principle can be regarded as a statement on the asymptotic behavior of the Green's function by a point-load.)

The most generic relationship between stress and strain based on three principles. used for discontinuities such as cracks/holes

1. stress equilibrium (the sum of all forces must equal zero though force can change through a body)
2. strain compatibility  (displacements must vary smoothly)
3. stress/strain relationships

direct piezoelectricity: polarization derived directly from application of stress

converse piezoelectricity:strains derived from application of electric field

can not include centrosymmetric terms

this process can exist if indivdiual grains are themselves piezoelectric and there is some texture

important in dynamic response of materials/structures

important in mechanical testing at high strain rates

in general for bulk isotropic materials there are two wave modes which correspond to the two deformation modes

if polarization of ferroelectric material is temperature sensitive

primary pyroelectricity: volume of teh sample is held constant

secondary pyroelectricity: volume change exists

non linear piezoelectric effect

could include centrosymmetric crystals

Representation surface of second rank tensors

Optical indicatrix

the radius length in any direction is the property to the ^-1/2 power

the normal to the representation surface at the point where the radius intersects is parallel to the resultant vector

Optical indicatrix: n=(K)^1/2=(B)^-1/2

J_i=λ_iσ_iE_i

σ_ijx_ix_j=1

x₁=rλ₁

r²σ_ijx_ix_j=1

r²σ=1

r=(σ)^-1/2