11.2 Mechanical Properties of Bulk Nanocrystalline Materials 251
diagram. The type, labeled as rubber, is typical of highly elastic materials, where
Hooke’s law is not applicable. Also for the type labeled collagen, Hooke’s law is not
valid. This material shows a behavior that is typical of biological materials. Certainly, in experimental reality, one observes stress–strain diagrams that could be
described as a superposition of these three basic types.
11.2
Mechanical Properties of Bulk Nanocrystalline Materials
It is well-known that hardness and yield stress increase with decreasing grain size.
The Hall–Petch relation describes this experimental correlation between yield
stress σ y ; and grain size d mathematically
σ σ κ
y
d
= +
−
0
0 5
. .
(11.4)
In Eq. (11.4) σ 0 and κ are constant parameters, which are specific for the materials.
The grain size d is related to the smallest units, which may be limited by smallangle grain boundaries and not necessarily defined by wide-angle grain boundaries. The exponent −0.5 is experimentally and theoretically well established;
however, experiments sometimes lead to exponents in the range from −0.3 to −0.7.
Experiments with nanomaterials sometimes lead to results deviating significantly from the Hall–Petch relation. Figure 11.3 displays a comparison of a stress–
strain diagram of palladium with different grain sizes.
The comparison depicted in Figure 11.3 contains a lot of information. One sees
the higher yield stress of 259 MPa for the nanocrystalline material with a grain
size of 14 nm, and a significantly lower value of 52 MPa for the coarse-grained
Figure 11.2 Most important types of stress–
strain diagrams: the most common one is
the one for metals, furthermore, diagrams
found with rubber and collagen are depicted.
It is important to note that the absolute
values of stresses and strains are entirely
different for these three materials; just for
demonstration of the types, they were plotted
in comparable ranges of stresses and strains.
0
3
6
9
12
15
18
strain
0
4
8
12
stress
Material
Metal
Rubber
Collagen
diagram. The type, labeled as rubber, is typical of highly elastic materials, where
Hooke’s law is not applicable. Also for the type labeled collagen, Hooke’s law is not
valid. This material shows a behavior that is typical of biological materials. Certainly, in experimental reality, one observes stress–strain diagrams that could be
described as a superposition of these three basic types.
11.2
Mechanical Properties of Bulk Nanocrystalline Materials
It is well-known that hardness and yield stress increase with decreasing grain size.
The Hall–Petch relation describes this experimental correlation between yield
stress σ y ; and grain size d mathematically
σ σ κ
y
d
= +
−
0
0 5
. .
(11.4)
In Eq. (11.4) σ 0 and κ are constant parameters, which are specific for the materials.
The grain size d is related to the smallest units, which may be limited by smallangle grain boundaries and not necessarily defined by wide-angle grain boundaries. The exponent −0.5 is experimentally and theoretically well established;
however, experiments sometimes lead to exponents in the range from −0.3 to −0.7.
Experiments with nanomaterials sometimes lead to results deviating significantly from the Hall–Petch relation. Figure 11.3 displays a comparison of a stress–
strain diagram of palladium with different grain sizes.
The comparison depicted in Figure 11.3 contains a lot of information. One sees
the higher yield stress of 259 MPa for the nanocrystalline material with a grain
size of 14 nm, and a significantly lower value of 52 MPa for the coarse-grained
Figure 11.2 Most important types of stress–
strain diagrams: the most common one is
the one for metals, furthermore, diagrams
found with rubber and collagen are depicted.
It is important to note that the absolute
values of stresses and strains are entirely
different for these three materials; just for
demonstration of the types, they were plotted
in comparable ranges of stresses and strains.
0
3
6
9
12
15
18
strain
0
4
8
12
stress
Material
Metal
Rubber
Collagen