Mechanical modelling of material under loads

Mechanical modelling of material under loads

1. ME729: Modelling of Mechanical Properties of
Materials
Sumit Basu, Associate Professor, Department of
Mechanical Engineering, IIT Kanpur, Kanpur 208016, UP.
Email: [email protected], Ph: [phone],

2. Course Objectives
Learn to model the constitutive behaviourof materials
i.e devise laws describing how a particular material
deforms under application of loads which can be body
forces (eg. gravity), surface tractions (contact with other
bodies), thermal (heating and consequent expansion) or
electrical loads or chemical action (eg. material in a
corrosive environment) on the material.

3. What is a constitutive equation?
Deformation Rate Temperature
Other internal variablesthat characterise physical
quantities like the microstructure of the material,
history of prior deformation etc
}

5. Method I: Based on Phenomenology
Test under
uniaxial
conditions
Guess all parameters
that can have an
influence on the stress-
strain response. Vary
them one by one and
get curves.
Curve fit:
Construct a
function F( ) for
multi-
dimensional
situations, that,
when applied to
the uniaxial
case, explains all
the experimental
results.

6. Phenomenological models contd
Biaxial stretching tests on polyethylene
terephthalate (PET) conducted at ESA Lab,
IIT Kanpur.

7. From www.grantadesign.com
Extensive testing has led to material
property handbooks used routinely
by practising engineers.

8. Such information is used routinely for
design of components
From: http://www-materials.eng.cam.ac.uk/

9. Bottom up routes to F( )
Macroscopic structure deformed
under load
Deformations at the
macro-level are smoothly
varying
zoom
At a macro-point, the material is
homogeneous
A micro-
structure
underlies every
macro-point.
Properties are
determined by
the
microstructure
F(..) at this point
obtained through
micromechanics at the
level of the
microstructure

10. Suggested Reading:
1.Mechanics of Solids and Materials, Asaro and Lubarda, Cambridge Univ
Press, 2006.
2.Crystals, Defects and Microstructures, Philips, Cambridge Univ Press,
2001
3.Foundations and Applications of Mechanics, vol 1, Jog, Narosa, 2005
4.Introduction to the Mechanics of a continuous Medium, Malvern, Prentice-
Hall
5.Nonlinear Solid Mecahnics, A continuum approach for engineering,
Holzapfel, Wiley, 2000.
6.The Physics of Polymers, Strobl, Springer, 1996
7.Fluid Mechanics of Viscoelasticity, Huilgol and Phan-Thien, Elsevier,
1997
8.Mechanics of the Cell, Boal, Cambridge Univ Press, 2002.
9.Research papers

11. Evaluation procedure:
1.2 quizzes, after first and second mid semester
periods respectively. Open book, open notes, use of
laptops and internet allowed. (30%)
2.About 6 assignments. You do not need to submit
them.
3.1 one hour mid semester exam, open book, open
notes, use of laptops and internet allowed. (30%)
4.3 hour end semester exam, open book, open notes,
use of computers and internet allowed. (40%)

12. A test generally determines properties under restricted
conditions
A UTM determines uniaxial
properties
Strain gauges
determine surface
strains

13. Real life is seldom uniaxial!
Loading and geometry renders a real life
problem non-uniaxial!

14. Maity et al. (2007a,b), IEEE
TDEI
Any observable,
macroscopic property P is
given by
Average over a
statistically
representative periodic
volume
On top of that, materials are
inhomogeneous!
Macroscopic properties are homogenised forms of
microscopic properties.

15. Homogenisation scheme applied to a typical
heterogeneous single crystal superalloy: (a) RVE and
(b) predicted contours of accumulated inelastic strain
under [010] uniaxial loading
The micro picture is always inhomogeneous, while the macro
may be homogeneous.
From Busso, 2004

16. Micro may be big!
Bandopadhyay et al Constr. Bldg
Mater, 2006
Maitra et al, ASCE J Civil Engng
Mater, submitted
aggregates
Air voids
Asphalt binder

17. Main crack
Intense localisation of
deformation, porous
material.
Micro may be local
Basu and Narasimhan,
JMPS, 1999,2000

18. Interfaces: properties are difficult to test
Substrate
Coating
From Hohlfelder
(1998)
.
Pressure (MPa)
Volume(micro-l)
Experimental data from Hohlfelder et al., 1998
Blister tests are used to assess
integrity of thin films on hard
substrates, eg in electronic
packaging
Funded by ISRO

19. Simulating interfaces
Kulmi and Basu, Model Simul Mater Sc
Engng, 2006
Sudarkodi and Basu, JMPS, submitted.
Funded by DST

20. 500 mm
crack
Shear bands
craze
Bulk
Bulk
Fibrils
~750nm ~50nm
crazes
Micro may be local and small!
Basu and van der Giessen, IJP,
2002
Basu et al., Polymer, 2005

21. Experiments:
simple loading,
macroscopic data.
Model:
Generalised but
should explain
experimental
observations under
restricted
conditions.
Material
inhomogeneities
interfaces
Damage
Size effects

22. 1. Concept of strain and deformation (Jog) (2 lectures)
i.Lagrangian and Eulerian descriptions of motion, deformation gradient tensor
ii.Transformation of length elements
iii.Transformation of area and volume elements
iv.Velocity, acceleration and rates of deformation
v.Examples of simple motions
2. Conservation laws and stress measures (Jog) (5 lectures)
i.Transport theorems, balance of mass, momentum and angular momentum
ii.Cauchy theorem and problems
iii.Momentum balance in reference configuration
iv.Variational theorems
v.Thermodynamics of deformation (first and second laws) and problems
vi.Example of modeling of growing mass (Asaro+papers)
vii.Assignment
Topics to be covered

23. 1. General theories of constitutive behaviour of materials (4 lectures, Asaro)
i.Introduction to constitutive laws
ii.Applications of the laws of thermodynamics, concept of thermodynamic potentials.
iii.Example of linear thermoelasticity
iv.Irreversible thermodynamics and internal variables
v.Assignment
vi.Concept of frame indifference and objectivity (cover Jaumann and other rates)
vii.Assignment
2. Specific constitutive theories and applications-1, elasticity (12 lectures) (Asaro)
i.Non-linear and linear elasticity (2)
ii.Example of non linear elasticity: rubber elasticity (1)(Strobl)
iii.Biological membranes + surface elasticity
iv.Simple problems: hole in an infinite plate (1)
v.Green’s function solutions: indentation problems (1)
vi.Inclusion problem: Eshelby formalism (1) (Asaro+Mura)
vii.Effective properties of composites (3)
viii.Fields due to dislocations (2)
ix.Misc problems involving dislocations (near interfaces, dislocation arrays etc.) (2)
Assignment

24. 1. Specific constitutive theories and applications-II, plasticity (11 lectures) Asaro+papers
i.Phenomenological theory of plasticity, pressure dependent yield, kinematic hardening. (4)
ii.Discussion of porous plasticity (1)
iii.Assignment
iv.Micromechanics of crystallographic slip (1)
v.Crystal plasticity, example of shear localisation (2)
vi.Strain gradient plasticity (1)
vii.Discrete dislocation plasticity (1)
viii.Open issues in metal plasticity esp. size effects (1)
ix.Computer assignment
2. Phenomenological viscoelasticity (1)
3. Microscopic viscoelasticity: Rouse, dumbbell and reptation models (1) (Huilgol)
4. Assignments
5. Length scale bridging issues and open questions (5 lectures) (Phillips+papers)
i.Problems with Multiple time and length scales
ii.Historic examples of multiscale modeling
iii.Multiscale models in fracture, cohesive models
iv.Dislocation dynamics, Molecular dynamics
v.Molecular dynamics of metals, quasicontinuum approaches
Molecular dynamics of amorphous materials

25. A quick introduction to Continuum Mechanics

27. Kinematics of deformation
P
p
Motion:
Notes

28. Example

29. Example

31. An example

32. Animation of the
motion described in
the previous slide . The
arrows denote
velocities at the
corners of the
rectangle in the
deformed
configuration.

33. Notes
Material Derivative

34. Velocity profile of the problem in Slide 31 at
times t=0.5 (blue) and 0.8 (green)

35. Example on the application of the
material derivative

36. Deformation Gradient tensor
The deformation gradient tensor maps an
infinitesimal line element tangent to a
curve on the reference configuration to an
infinitesimal line element to a curve on the
deformed/current configuration
P
p
Notes

37. Deformation Gradient: in terms of the
displacement gradient

39. P
p
Concept of strain

43. Physical significance of the Green Lagrange strain tensor

45. Example

47. Mapping surfaces and volumes
Proof

48. P
p

49. P
p

50. Example:

51. Polar decomposition
Definition
Proof

52. Uembodies only stretch. Polar decomposition splits the
motion into a stretch and a rotation

53. Spectral decomposition of C

54. Example

57. Example: finding Uin 2-d

58. Exercise

59. Rates of deformation

62. Example: Lin terms of U

63. Some more rates

67. Balance Laws and Stress Measures

68. Reynolds Transport theorem
}}}

69. See explanation
See explanation

71. Mass conservation

73. Example

74. Example

75. Balance of momenta
Notes

78. Stress in the reference configuration:
First Piola Kirchoff stress

80. Equation of motion in terms of the 1
st
PK stress

82. Second Piola Kirchoff stress tensor

83. Cooking up stresses!

84. Balance of angular momentum

86. Principal stresses and directions

87. Example: find the maximum shear stress directions at a
point

89. Example: The meaning of the first Piola Kirchoff stress

91. Reference configuration
Current configuration
Note that the surface with a cross in the reference configuration
goes to the surface with a cross in the deformed configuration.
This means that the motion is a combination of a rotation about
the X
3axis followed by a stretch in the X
1direction and a
contraction in the X
2direction.

95. Balance Laws: The two laws of thermodynamics

96. Balance of energy: first
law of thermodynamics

98. P
p
Rate of thermal work

102. Second law of thermodynamics

104. Clausius Duhem inequality in the reference
configuration
Entropy due to heat
conductionRate of stress work

106. Example:Moving discontinuity surfaces:
Application of balance laws

109. Internal variables
Internal variables quantify the missing dissipation!!

110. Types of thermomechanical processes

111. Free energy

112. Example: Second law and constitutive
equations

115. Example: Thermodynamic restrictions on elastic
solids

117. Objectivity and Frame Indifference

118. Objectivity and Frame Indifference

119. Proof
Proof

120. Example: Is velocity objective?

121. Example: What about acceleration?
Coriolis acceleration
Centrifugal
acceleration

122. Some tensors
Proof
Proof

123. Rate of Cauchy stress

124. Objective and material response

128. The requirement that constitutive equations be ‘Material
Frame Indifferent’ poses restrictions on the nature of these
equations!

129. Example: Jaumann rate

132. Putting it all together: Case Study I:
Mechanics of a growing mass.
A very important problem in biomechanics. Growth and remodelling of tissues takes
place during normal developmental growth, healing processes and pathological
conditions
Objectives
•Formulate the basic balance equations for the problem
•Assume a constitutive framework consistent with the
Clausius Duhem inequality and the principle of
objectivity.
•Solve a simple problem

137. Momentum principles

138. Energy principle: 1
st
law of thermodynamics for a
continuum with growing mass

139. 2
nd
law for a continuum with growing mass

140. Building a constitutive framework

142. Example: circumferential growth of a blood vessel

153. Reversible materials: hyperelasticity

156. Isotropic Green elasticity

158. Usual assumptions…
This is the usual linear elastic case….

159. Alternate forms of expressing the constitutive equation for
Green elastic materials

163. Incompressible hyperelasticity

164. Further forms of hyperelastic
constitutive laws

166. Examples: Mooney Rivlin rubber

170. Inflating a spherical balloon

172. Snap through
Snap back
For more read:Needleman (1977), Inflation of spherical balloons, Int J Solids
Structures, v13, p409.

173. How do we get the free energy?
Case study 1: a linearly elastic crystal

178. Linking micro to macro: introduction to statistical
mechanics

179. subsystem
reservoir
A short review of statistical mechanics

184. This equation links the micro to the macro world!

185. Polymer chain as a random walk

194. Compare this with the Mooney Rivlin material

195. Linear elastic isotropic materials

196. Navier equation
Potential Methods for problems in Linear Elasticity:
Papkovich Neuber potentials

197. See, http://www.engin.brown.edu/courses/en224/for details

198. Green’s function

199. Some Fourier transforms

200. Useful Fourier transforms

203. Concept of averaging over space
X
Y
x
y
V
Macrostress
Microstress
Representative
Volume Element
(RVE)

206. B
H
Example:

208. Phase averages
RVE

209. Basic idea of homogenisation
techniques

210. Properties of the localisation tensor

212. To find the stiffness Cor the compliance D, knowledge of the localisation
tensors are sufficient.

213. The simplest assumption: Voigt
model
Strain everywhere is same as
the average strain
Maybe valid for low contrast
cases
Rule of mixtures

214. Reuss model

215. Mean field approaches: dilute inclusions
Low volume fraction
Each particle sees an infinite
matrix around it.

216. A non-mathematical overview of the dilute
particle theory
Composite subjected to macrostrain
Assumption:if volume fraction f<0.25
All points in the matrix also has strain
since the matrix does not feelthe particles
Inside the particles, according to Eshelby’s
theory it can be shown that strains are
constant.
The strain in the particles can be found by
using the Eshelby tensor.
Thus the localisation tensor in the matrix is the identity tensor and that in the
particles can be found (constant). The total localisation tensor can thus be
derived.

217. Concept of eigenstrains

218. Unconstrained
expansion
Eigenstrains
Eigenstrains are non-stress strains.
Stress is caused only by the elastic
strain.

220. The Eshelby procedure
I
II
III
IV
VVI

222. Disturbance strain due to the
particle.
Main result of Eshelby’s (1957) analysis

224. Eshelby tensor
Ref: Micromechanics of defects in solids, T. Mura, 1982

225. Estimates of average moduli from Eshelby’s concept of the
equivalent inclusion.

227. A digression: how do we invert fourth order
tensors?

228. Alternate method: E bases

229. Beyond dilute distributions: estimating matrix
strains
Dilute distribution assumes that matrix strains ~
At higher volume fractions, matrix strains are non-uniform

230. A particle is ‘embedded’ in a matrix
with strains
The localisation tensor is same as
in the case when the matrix had
strains

231. Self consistent schemes
When it is not clear which is the matrix and which are particulates i.e
phases are not distinct.
Assume:the particles are embedded in a medium with the overall
elastic modulus
This approximately accounts for the inclusion-inclusion interaction
Since both sides are functions of , iterative solution of the above
equation is necessary

233. How good are the estimates: bounds
Ref: Hill, J Mechanics Physics Solids (1963)

234. Tighter bounds: Hashin Shtrikman variational principle
Ref: Hashin, Shtrikman, J. Mech Phys Solids, 1963

236. Generalisation of the H-S approach is found in Willis, J Mech Phys Solids,
1977

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