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Introduction
Chapter 1
Introduction
1 Overview of the Course
This is a course about the mechanical response of structural materials to the application
of external loads. The atomic and microstructural features of materials that are directly
associated with the various forms of mechanical behavior are described. Using continuum
models, the quantitative aspects of mechanical response are also fully examined. Analysis
is possible by introducing conservation laws and suitable constitutive equations of material
behavior. Conservation laws invoke well established physical principles of mass, momentum
and energy conservation while constitutive equations are empirically obtained relationships
relating responses to stimulae. An integrated picture of mechanical behavior of materials is
developed by combining concepts of materials science, mechanical engineering and computational mechanics. All major manifestations of mechanical response of structural materials are
examined including: linear elasticity, thermo elasticity, viscoelasticity, plasticity, creep, viscoplasticity and damage mechanics including fracture and fatigue. On successful completion,
students will be ready to undertake advanced analytical work on classical and computational
mechanics of structural materials.
2 Mechanical Engineering Materials
Mechanical Engineering systems and structures are composed of various types of materials.
Reliability and life of such structures depends on the internal microstructural characteristics
of the materials employed as well as on the structural design parameters. This course investigates the fundamental microstructural characteristics of engineering materials and the
principles of structural mechanics employed in the design of mechanical engineering systems.
3 Structure of Materials
Ultimately, all matter is made of atomic size particles held together by interatomic forces.
The mechanical response of matter to external loads is directly related to atomic cohesion.
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However, the specic, quantitative form of the relationship is complicated and has yet to be
elucidated.
The disciplines of metallurgy and science of materials have achieved substantial progress
in our understanding of the microstructural characteristics of materials and reasonably clear
pictures of the atomic constitution of most engineering materials are now available.
3.1 Classication of Materials
Engineering Materials can be classied according to various criteria. If one considers the
nature of atomic arrangements in the material, two main groups emerge; crystalline materials
and amorphous materials. Crystalline materials are characterized by atomic arrangements of
great regularity with long range order. Atoms occupy well dened positions in a geometrically
regular crystal lattice characterized by a high degree of pattern repetitiveness. In contrast,
atoms in amorphous materials are located on rather more random locations and exhibit a
lack of long range order. Examples of crystalline materials include metals and ceramics while
glass and polymers are examples of amorphous materials. However, it is possible to produce
metals with amorphous structures and polymers with crystalline arrangements as well as
materials with various degrees of crystallinity.
3.2 Crystal Structures
A crystal structure is a regular three dimensional pattern of reticular locations. The basic
unit of repetition of the crystalline pattern is called the unit cell. The unit cells of all known
crystals belong to one of the 14 Bravais space lattices.
Crystallographic directions are indicated using the vectors associated with them with
the components enclosed in square brackets. Crystallographic planes are indicated by Miller
indices which are produced by listing the reciprocal intersections of the plane with the
coordinate axes in round parenthesis.
3.3 Metals and Ceramics
Most elements of periodic table are metals. Metals appear mostly in three structures: face
centered cubic (FCC), hexagonal close packed (HCP) and body centered cubic (BCC).
Ceramics are mainly ionic compounds and exists in many dierent structures, the most
common being zinc blende, wurzite, perovskite, uorite, sodium chloride, cesium chloride,
spinel, corundum and crystobalite.
3.4 Polymers, Glasses, Composite and Porous Materials
Polymers are constituted by assemblies of large chain molecules with a distribution of molecular weights. Organic polymers are based on hydrocarbon chains. The chains can be linear,
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branched, cross linked or ladder type. Also, depending on the pattern along the chain one
has homopolymers and copolymers. Organic polymers can also be classied according to
their response to heat treatment as thermoplastics or thermosets. Further, by alignment of
polymer chains polymers with various degrees of crystallinity can be produced.
Glasses are materials lacking long range order. Their structure rather resembles that of
a liquid. A common method for the production of glasses is by cooling from the melt while
preventing crystallization.
Composite materials consists of dispersions of multiple phases in intimate contact. Most
engineering materials are composites but the term is commonly used to refer to ber reinforced materials. The properties of composites depend in a complex manner on the characteristics of the constituent phases and their interfaces.
Many structural materials have porous or cellular structures. Wood, bone and space
shuttle tiles are good examples. The mechanical properties of these materials are complex
functions of pore structure and characteristics.
4 The Theoretical Strength of a Crystal
Orowan rst proposed a model to estimate the intrinsic mechanical strength of a crystal. He
envisioned the crystal loaded in tension and failing at a certain critical load along a single
crystal plane. Using a simple approximation for the cohesive strength of interatomic bonds
he derived the following expression for the maximum theoretical strength
max
E
where E is the elastic modulus of the material.
A similar simple model was produced to estimate the maximum strength under shear
loading. The corresponding expression is
max
G5
where G is the shear modulus of the material. The above formulae represent truly maximum
values and are approximated in practice only under extreme circumstances as all sorts of
microstructural defects exert a powerful inuence in the determination of the actual strength
of a given piece of material.
5 Solid Mechanics of Materials
Solid mechanics was originally conceived and used as a continuum theory for the estimation
of the mechanical response of materials subjected to loads. In continuum theory, the details
of the atomistic structure of the material are neglected. Material response is expressed in
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terms of empirically determined constitutive equations of behavior. Much progress has been
achieved in structural mechanics despite this assumption. Nowadays, it is recognized that
some knowledge of materials science aspects is a helpful tool for understanding the mechanical response of materials. However, because of their signicant engineering usefulness, it is
important to have a good understanding of classical constitutive models of material behavior.
5.1 Linear Elasticity
Linear elasticity is the material deformation behavior described by Hooke's law which states
that displacement is linearly proportional to the applied load, i.e. for a point inside a material
subjected to external loads P1; P2; :::; ; Pn , the displacement can be expressed as
u
=
X
n
i=1
ai Pi
where the coeÆcients ai are independent of Pi. A simple picture of linear elastic behavior is
that of a spring.
Linear elastic behavior is well described by Hooke's law
=
E
where is the strain, is the stress and E is the modulus of elasticity (Young's modulus).
A linear elastic material returns to the undeformed state once the loads are removed and
the eects of multiple load systems can be computed by simple linear superposition. Moreover, the work done by the forces is calculated by multiplying the loads by the displacements
and the Maxwell and Betty reciprocity relation are valid. A linear elastic material under
load accumulates elastic strain energy U and one has Castigliano's theorem
@U
@Pi
= ui
and the associated principle of virtual work
@U
@ui
= Pi
5.2 Viscoelasticity
A linear elastic solid that remembers its deformation history is called a viscoelastic material.
Viscoelastic behavior can be represented by combinations of springs and dashpots (pistons
that move inside a viscous uid). While linear springs instantaneously produce deformation
proportional to the load, a dashpot produces a velocity proportional to the load at each
instant. If a spring and a dashpot are placed in parallel one obtains Maxwell's viscoelastic
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model. If they are in arranged in series, one has Voigt's model. Finally, a series/parallel
arrangement yields Kelvin's model.
As an example, for the Voigt's model, the relationship among load F , displacement u
and velocity du=dt is
= u + du
dt
where is the spring constant and the viscosity, and the displacement function resulting
from a unit step force is a monotonic continuous function of time.
F
5.3 Plasticity
If the load applied to a piece of metal is increased from zero, the metal rst deforms elastically
according to Hooke's law but at some critical threshold load, it yields and continues deforming
at stresses that are much smaller than those that would be required for continued elastic
deformation. This behavior is known as plasticity. Two characteristic features of plasticity
are that the material deforms instantaneously and it does not return to the undeformed state
when unloaded. Another characteristic frequently found is that the material strain hardens.
From a quantitative standpoint, at least two pieces of information are required to investigate
plastic behavior. One rst needs a yield criterion to specify the level of stress intensity at
which the material ceases behaving as a linear elastic solid and residual strain remains upon
unloading. The yield criterion is typically stated by saying that plastic ow begins once the
eective stress
p
2 [( )2 + ( )2 + [( )2]1=2 = ( 3 )1=2
ef f =
2
3
3
1
2 1 2
2 ij ij
rst exceeds a critical value 0, obtained from a simple uni-axial tensile test of a standardized
specimen.
The second item required is a ow rule to describe the specic strain response of the material to the applied stress. One commonly used empirical form of the stress-strain relationship
used to describe plastic ow is the so-called power law
n
ef f = Kef f
where ef f is the eective strain and the material parameters K and n must also be determined from a standard tensile test.
0
5.4 Creep/Viscoplasticity
0
Metals at high temperatures exhibit time dependent deformation under constant load. Even
very small loads may produce deformation and the body remains deformed after unloading.
This behavior is known as creep or viscoplasticity.
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Creep and viscoplasticity account for the combined eects of temperature, time and stress
on the deformation response. A widely used, empirical relationship for the representation of
the stress-strain relationship under creep/viscoelastic deformation is the Bailey-Norton law,
sometimes called power law creep law. This is given by
n
m
ef f = Aef f t
where ef f is the eective creep strain, ef f is the eective stress, t is time, A is a temperature
dependent material parameter, as are also n and m, all of which must be determined from
standardized creep tests.
5.5 Damage Mechanics, Fracture and Fatigue
Material damage is the gradual process of mechanical deterioration that ultimately results in
component failure. Fracture, fatigue and creep rupture are all instances of material damage.
Damage mechanics is the study of material damage based on the introduction of damage
variables and their evolution. Fracture Mechanics and Fatigue Mechanics provide guidance
towards understanding and controlling catastrophic structural breakdown of materials.
Fracture studies start by considering the stress concentration factor S , dened as
max
S =
where max and are, respectively, the maximum (localized) and nominal stresses in the
structural component. Fracture criteria specify the critical condition in a component that
leads to rapid and catastrophic crack growth and propagation. Following the original idea of
GriÆth that assumed that all engineering materials always contain pre-existing cracks, Irwin
and Orowan produced the following expression for the critical applied stress c required for
structural breakdown
s 2EG
c =
a(1
2)
where E is the elastic modulus, is Poisson's ratio, a is the original crack size and G is the
strain energy release rate and is a physical quantity associated with the energy involved in
plastic deformation at the crack tip and with the energy required to form new free surfaces
at that same location.
Fatigue studies usually involve analysis of the number of loading cycles at a given stress
level that are required for rapid and catastrophic fatigue crack growth and propagation.
Paris et al, rst proposed the following empirical relationship between the crack growth rate
per cycle and the stress intensity factor range K = Kmax Kmin,
m
=
C (K )
dN
da
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Here C and m are material parameters that are determined from standardized tests.
Damage mechanics aims to quantitatively represent the accrual of mechanical deterioration of a material or component subjected to certain loading. This is done by introducing
a damage variable. By considering a small, representative volume element of material, with
cross sectional area dS , the damage is dened as
D
D
= dS
dS
where dSD be the amount of area inside an area dS that is occupied by material discontinuities characterizing damage such as cracks or voids. The damage variable D is a number
2 [0; 1]. The value D = 0 describes undamaged material and D = 1 represents the ruptured
component. Damage mechanics studies then focus on the quantitative determination of the
response of the damage variable to specic loading conditions.
6 Thermodynamics of Materials
Thermodynamic principles lead to the formal statement of energy conservation and they also
place restrictions on the constitutive behavior of materials.
6.1 The First Law of Thermodynamics
The rst law of thermodynamics states that the increase of internal energy of a material
system is equal to the amount of heat absorbed by it minus the amount of work done, i.e.
U = Q W
when the rst law is expressed in rate form it becomes the principle of energy conservation.
6.2 The Second Law of Thermodynamics
The second law of thermodynamics requires the concepts of absolute temperature and entropy. Absolute temperature is an intensive quantity and a positive number associated with
the notion of hotness. Entropy is an extensive property of the system that changes as a
result of interaction with the external environment and also as a result of internal processes
in the system. The second law states that the change in entropy resulting from internal
processes is never negative, i.e.
dSi 0
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7 Continuum Thermomechanics of Materials
Continuum thermomechanics investigates the deformation behavior of materials under mechanical and thermal loads. The discipline is founded on well established conservation principles of universal applicability. The conservation principles used in continuum thermomechanics are:
Principle of Mass Conservation (Equation of Continuity)
Principle of Conservation of Linear Momentum (Equation of Motion)
Principle of Conservation of Angular Momentum (Equation of Moment of Momentum)
Principle of Conservation of Energy (Energy Equation)
Principle of Entropy Production (Clausius-Duhem Equation)
Ultimate predictions of material deformation require incorporating constitutive equations
into the formulation. While the conservation principles apply to any material, constitutive
relations specify individual material responses.
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