186
7 Nano-Mechanical Properties of Solid Surfaces Obtained …
The deviation from the Hertzian behavior in Fig. 7.6 identifies the onset of plastic
deformation. In some cases, this deviation is slight, but still feasible to identify, while
in many cases, as shown in Fig. 7.7, an abrupt discontinuity appears in a load-depth
curve, which identifies the onset of dislocation activity. This abrupt discontinuity
is named “pop-in” or “excursion”. The dislocation activity continues until the load
rises again. The mean applied stress σ p normal to the sample surface at the initiation
of dislocation activity may be expressed as follows [6]:
σ p =
L p
π Rh p
,
(7.10)
where the subscript p indicates the value at the plastic threshold. The plastic deformation results from shear stresses. It is necessary to estimate the shear stresses beneath
the indenter in order to obtain a critical shear stress at which dislocations are activated. For a Hertzian stress distribution for a material with Poisson’s ratio of ν = 0.3,
the maximum shear stress arises along the axis of symmetry at a depth of 0.48 a,
where a is the radius of the contact area [5]. At this depth, the shear stress τ is given
by [5]:
τ = 0.31
3L
2π Rh
.
(7.11)
Consequently, the maximum shear stress τ p at the plastic threshold load L p can be
estimated from the following relationship [6, 7]:
τ p = 0.465
L p
π Rh p
= 0.465σ p ,
(7.12)
Fig. 7.6 Schematic
illustration of a load-depth
curve (solid line) and the
Hertzian behavior (dotted
line)
Load, L
Depth, h
Hertzian
Plastic threshold
7 Nano-Mechanical Properties of Solid Surfaces Obtained …
The deviation from the Hertzian behavior in Fig. 7.6 identifies the onset of plastic
deformation. In some cases, this deviation is slight, but still feasible to identify, while
in many cases, as shown in Fig. 7.7, an abrupt discontinuity appears in a load-depth
curve, which identifies the onset of dislocation activity. This abrupt discontinuity
is named “pop-in” or “excursion”. The dislocation activity continues until the load
rises again. The mean applied stress σ p normal to the sample surface at the initiation
of dislocation activity may be expressed as follows [6]:
σ p =
L p
π Rh p
,
(7.10)
where the subscript p indicates the value at the plastic threshold. The plastic deformation results from shear stresses. It is necessary to estimate the shear stresses beneath
the indenter in order to obtain a critical shear stress at which dislocations are activated. For a Hertzian stress distribution for a material with Poisson’s ratio of ν = 0.3,
the maximum shear stress arises along the axis of symmetry at a depth of 0.48 a,
where a is the radius of the contact area [5]. At this depth, the shear stress τ is given
by [5]:
τ = 0.31
3L
2π Rh
.
(7.11)
Consequently, the maximum shear stress τ p at the plastic threshold load L p can be
estimated from the following relationship [6, 7]:
τ p = 0.465
L p
π Rh p
= 0.465σ p ,
(7.12)
Fig. 7.6 Schematic
illustration of a load-depth
curve (solid line) and the
Hertzian behavior (dotted
line)
Load, L
Depth, h
Hertzian
Plastic threshold