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Introduction
Chapter 1
Introduction
1
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 microstrucural 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.
2
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.
However, the specific, 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 undertanding of the microstructural characteristics of materials and reasonably clear
pictures of the atomic constitution of most engineering materials are now available.
2.1
Classification of Materials
Engineering Materials can be classified 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 defined 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.
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2.2
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.
2.3
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 different structures, the most
common being zinc blende, wurzite, perovskite, fluorite, sodium chloride, cesium chloride,
spinel, corundum and crystobalite.
2.4
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,
branched, cross linked or ladder type. Also, depending on the pattern along the chain one
has homopolymers and copolymers. Organic polymers can also be classified 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 fiber 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.
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The Theoretical Strength of a Crystal
Orowan first 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
2
crystal plane. Using a simple approximation for the cohesive strength of interatomic bonds
he derived the following expression for the maximum theoretical strength
σmax ≈
E
π
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
τmax ≈
G
5
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 influence in the determination of the actual strength
of a given piece of material.
4
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
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 significant engineering usefulness, it is
important to have a good understanding of classical constitutive models of material behavior.
4.1
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=
n
X
i=1
ai Pi
where the coefficients ai are independent of Pi . A simple picture of linear elastic behavior is
that of a spring.
A linear elastic material returns to the undeformed state once the loads are removed and
the effects 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
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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
= ui
∂Pi
and the associated principle of virtual work
∂U
= Pi
∂ui
4.2
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 fluid). 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
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
F = µu + η
du
dt
where µ is the spring constant and η the viscosity, and the displacement function resulting
from a unit step force is a monotonic continuous function of time.
4.3
Plasticity
If the load applied to a piece of metal is increased from zero, the metal first 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 instantaneoulsy and it does not return to the undeformed state
when unloaded. Another characteristic frequently found is that the material strain hardens.
4.4
Creep/Viscoplasticity
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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5
Thermodynamics of Materials
Thermodynamic principles lead to the formal statement of energy conservation and they also
place restrictions on the constitutive behavior of materials.
5.1
The First Law of Thermodynamics
The first 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 first law is expressed in rate form it becomes the principle of energy conservation.
5.2
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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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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