1I. Fundamentals of Mechanical Properties of Materials
Four types of mechanical properties of thin films are of interest. The first type is the elastic
constants, in which the typical property is Young’s modulus. The other three types are strengths
including yield strength, σy , ultimate tensile strength, σu, fracture strength, σf, ductility, i.e.,
elongation, δ, and reduction of area, ψ or RA, and toughness, i.e., fracture toughness, KIC or GC,
and impact energy, which correspond to stress, strain and energy, respectively.
1. Stress-strain Curve
Two definitions of stresses and strains are widely used in academia and industry, namely,
engineering (or nominal) stresses and engineering (or nominal) strains, and true stresses and true
strains. The basic test to characterize mechanical properties of materials is the tensile test. Figure
2-1 shows a general stress-strain curve obtained from a uniaxial tensile test. In the first stage,
stress increases linearly with strain with the slope of Young’s modulus. For brittle-ductile
materials, if the stress-strain curve is smooth, yield strength (σy) is determined by an engineering
standard and it is the stress under an offset strain, which is usually taken as 0.2 %. Thus, the
yield strength is marked as σ0.2. The ultimate tensile strength represents the ability of a sample
to resist uniform deformation under tension, which is calculated from the maximum load (Pmax)
and the original cross section area (A0), i.e., σu =Pmax/A0. The ultimate tensile strength is the
critical point, at which necking occurs and uniform deformation is ended in the tested sample.
Referring to Fig. 1, the conventional tensile test gives the following mechanical properties
defined as follows:
Ultimate tensile strength: σu = Pmax/A0, (1)
Yield strength: σy = Py /A0, (2)
Elongation: δ = (Lf -L0)/L0, (3)
Reduction of area: RA = ψ = (A0 -Af)/A0, (4)
where A0 and L0 are respectively the original cross sectional area and gauge length, Af and Lf
are respectively the final cross sectional area and gauge length at fracture, and δ and ψ are the
parameters to evaluate ductility. Since the cross sectional area (A) and the gauge length (L)
change during tensile testing, true stress (σ) and true strain (ε) are defined as:
σ = P/A (5)
)/ln(/
0
0LLLdLL
L==∫ε. (6)
Nominal (engineering) stress and nominal (engineering) strain are defined as:
σ’=P/A0, (7)
ε’=ΔL/L0=(L-L0)/L0=(L/L0)-1. (8)
Combining Eqs. (6) and (8) yields
ε=ln(1+ ε’). (9)
2It is a widely accepted assumption that during plastic deformation, the sample volume is
conserved:
V=A0L0
=AL
=constant. (10)
If plastic deformation is much more severe than elastic deformation, elastic strain can be
ignored. Then, combining Eqs. (10) and (6) leads to
ε= ln(L/L0)=ln(A0/A). (11)
From Eqs. (5) and (7), we have:
σ=(P/A0)(A0/A)=σ’ (A0/A). (12)
Substituting Eq. (9) into Eq. (10) gives
σ=σ’(1 + ε’). (13)
During a tensile test, necking (local narrowing of the cross section) starts at the maximum load
(Pmax). At the maximum load, we have:
dP = σdA + Adσ =
0. (14)
The constant volume approximation gives
dL/L = -dA/A. (15)
Combining Eqs. (15) and (14) leads to:
dL/L = dσ/σ = dε. (16)
Equation (16) is the important condition for necking.
The true stress-true strain (σ - ε) curve for plastic deformation is usually expressed as:
σ = kεn, (17)
where k is a constant, which equals the true stress at a unity strain, and n is called strain
hardening exponent, the slope of a log(σ)-log(ε) plot. It can be proved that n is numerically
equal to the limit of uniform plastic strain. Substituting (17) into (16) results in dσ/d ε = nkεn-
1=σ =kεn. So
n=ε, (18)
i.e., when the true strain is equal to the strain hardening exponent, the necking starts.
The true strain at fracture is calculated from Eq. (9) as follows:
εf = ln(Lf/L0) = ln(A0/Af). (19) 3
Combining Eq. (19) with Eq. (4), we obtain the relationship between ψ and εf,
⎟⎟
⎠⎞
⎜⎜
⎝⎛
−=
ψε
11
ln
f. (20)
It should be emphasized that σu corresponds to the maximum load at which necking starts. After
necking, the cross section of the specimen reduces, so the fracture load (Pf) is smaller than the
maximum load (Pmax). However, the fracture strength σf in true stress-strain curve could be
greater than σu. The strains corresponding to Pmax and Pf are called the maximum uniform
elongation (εu) and total elongation (εt) respectively. Thus:
εt = εu + εnu, (21)
where εnu is the maximum non-uniform elongation after necking.
2. Theoretical Strength
2-1. Fracture Strength
When a piece of solid is under stress, its atoms are displaced from their equilibrium
positions. The displacement is governed by the interactomic potential. An applied tensile force
against the internal force tends to lengthen the solid and thus to increase the interatomic
distance. At equilibrium, an applied force has the same magnitude as the internal force but with
an opposite sign. Figure 2(a) shows the applied force versus the interatomic distance, where
maxF is the maximum force which corresponds to the dissociative distance
Dr.
maxF is the
maximum tensile force needed to pull the solid apart, because the force needed to increase the
interatomic distance beyond
Dr is less than
maxF. We can regard
maxF as the theoretical fracture
strength of the solid, which can be estimated from the interatomic potential of a solid.
In simple cases, interatomic potential energy can be, in general, represented by
⎥⎥
⎦⎤
⎢⎢
⎣⎡
⎟
⎠⎞
⎜
⎝⎛−⎟
⎠⎞
⎜
⎝⎛
−=qp
bra
qra
pqppq
r0011
)(εφ, (22)
where p and q are numbers whose values depend on the shape of the potential. The first term in
the right side of Eq. (22) represents the repulsive interaction, which is due to the Pauli exclusion
principle, and the repulsive interaction occurs only within a very short distance. The second term
is the attractive interaction, which binds atoms together. At the equilibrium position,
0ar=, the
attraction is balanced by the repulsion, the potential is at its minimum,
baεφ−=)(
0, and the
force F=0.
For short-range interactions, such as solid Ar, a frozen inert gas, the Lennard-Jones potential
(p=12 and q=6) applies, which takes the form:
