In order to investigate the anisotropic micromechanical properties of singlecrystal nickelbased superalloy DD99 of four crystallographic orientations, (001), (215), (405), and (605), microindentation test (MIT) was conducted with different loads and loading velocities by a sharp Berkovich indenter. Some material parameters reflecting the micromechanical behavior of DD99, such as microhardness
In recent years, singlecrystal nickelbased superalloys are widely used as blade of modern gas turbine aeroengines, as they significantly raise the operation temperature and efficiency due to excellent mechanical properties in service [
Conventional tensile tests are difficult to conduct at nanoand microscales to determine orientation dependent behavior. Microindentation can be an alternative approach to tension or compression to probe the micromechanical properties, such as elastic modulus and hardness. Due to its precise measurement and advantages of celerity, accuracy, and nondestructiveness [
FEM may be used as a supplement to solve complex 3D problems and more information can be extracted from microindentation simulation [
In this paper, MIT were conducted on crystal planes of nickelbased singlecrystal DD99 to investigate its micromechanical properties. Subsequently, a FEM model based on the results of MIT was implemented in 3D ABAQUS/Explicit to prove the accuracy of this FEM model.
The loaddisplacement response obtained by microindentation contains information about the elastic and plastic deformation of the indented materials. Therefore, it is often regarded as “fingerprint” of materials’ properties under identification [
Powerlaw elasticplastic, true stressstrain behavior [
When
Thus,
In order to describe the mechanical properties of a powerlaw material, Young’s modulus
The typical
The typical
According to the model proposed by Oliver and Pharr [
The relationship between the apparent modulus
According to King [
The next task is to find the yield stress
According to
Similarly, the unloading slope can be described as follows at
Finally, after plugging value of
The ultimate tensile strength characterizes the resistance of largest uniform plastic deformation. The tensile strength is usually determined by uniaxial tensile test: the highest point of the stressstrain curve is the tensile strength. However, it can also be calculated by MIT.
In the stage of uniform plastic deformation stage, load
The result of differential calculation of
When nonuniform deformation such as necking occurs on certain part of materials,
It has the following relationship, according to
According to
Its formula is
According to
Hardness is an ability of resistance to permanent (plastic) deformation. It represents the overall mechanical properties of materials. For example, hardness is related to other mechanics parameters, such as Young’s modulus, yield stress, and strain hardening component [
Shim et al. [
The nominal chemical compositions of asreceived singlecrystal nickelbased superalloy DD99 (5 mm × 5 mm × 3 mm rectangle block) which has its crystal plane (001) marked in advance are shown in Table
Chemical composition of DD99 superalloy (mass fraction, %).
C  Cr  Co  W  Al  Ti  Ta  Ni 

0.016  8.5  5.0  9.5  5.5  2.2  2.8  Bal. 
Although nickelbased single crystal has the optimum performance along crystal orientation [001] [
Schematic diagram of specimens with different cutting angles: (a) 30°; (b) 45°; (c) 60°.
The cutting specimens along with the original one with crystal plane (001) were carefully ground with sand paper. Then, they were polished with 1.5
MIT condition.
Max load (mN)  500  1000  1500  2000  2500  3000  3500  4000  4500 

Velocity (mN/s)  17.2842  34.5684  51.8527  103.7053 
In the researches of Li et al. [
In terms of the shape of indenter, the Berkovich indenter of 2D model is represented by a straight with an angle of 70.3° to axis of symmetry, indicating the indenter itself as a whole is conical. In contrast, the real shape of Berkovich indenter is triangular pyramid. Considering the boundary conditions, circumferential displacement in 2D model is constrained and only the displacement in radial direction is permitted. By comparison, constrained circumferential displacement just appears on the symmetry plane and deformation can be expanded in both circumferential and radial direction within matrix in 3D model. Considering above variations and the fact that 2D model cannot be utilized to discuss the orientation dependent properties due to its rotational symmetry, a 3D simulation seems preferable, even though computational time is considerably higher.
The Berkovich indenter is a triangularbased pyramid having a threefold symmetry. The load is applied along the axis of the indenter; thus the load symmetry is the same as the geometric one. For these considerations, a threedimensional model is defined only by onethird of the entire system. The 3D model setup is shown in Figure
Microindentation model setup.
The loaddisplacement curves (
Based on
Microhardness
It can be seen from Figure
According to Figure
Moreover, Nix and Gao proposed a new model of
According to
As for each crystal plane,
The statistically stored dislocations density on four crystal planes can be calculated through
Density of dislocation density
Of the various unique mechanical properties of materials, Young’s modulus, which is a measure of elasticity, has attracted particular attention. Figure
Young’s modulus
It can be seen from Figure
Figure
Compared with
For each crystal plane,
According to
According to theory of metallic plasticity, the crystal
For the same material,
For crystal nickelbased DD99, value of
According to equations in Section
Values of
As shown in Figure
It is well known that
In Figure
Schematic diagram of uniaxial tension model.
This is called Schmid’s law.
Crystal indices 




(001)  0.3449  0.8248  0.408 
(215)  0.2529  1.6701  0.435 
(405)  0.3338  0.9667  0.448 
(605)  0.3242  1.0585  0.442 
For (001) crystal having eight equivalent
Based on
Elasticplastic properties parameters.
Crystal indices 





(001)  3.8941  0.3449  0.8248  1.9110 
(215)  3.3090  0.2529  1.6701  1.8150 
(405)  3.9661  0.3338  0.9667  1.9694 
(605)  3.8123  0.3242  1.0585  1.9133 
Mechanical property parameters,
Simulation calculations have been performed by using the commercial finite element software ABAQUS. The experimental and FEM results of 500 mN, 2500 mN, and 4500 mN on crystal plane (405) were shown in Figure
Experimental versus computed
Although FEM results deviate from experimental results slightly, it is of great importance to focus on their difference. In order to illustrate cause of deviation to achieve better simulations in the future, experimental and FEM results of 2500 mN on different crystal planes were shown in Figure
Experimental and FEM results of 2500 mN on different crystal planes: (a) (001); (b) (215); (c) (405); (d) (605).
As shown in Figure
There is a relationship between
According to
Therefore, Kick’s law seems to be reasonable. However, microindentation test is not an ideal plastic deformation process. ISE exists in the process of microindentation as discussed above, resulting in decrease of
Experimental and FEM results of (405) with different loads: (a) 500 mN; (b) 2500 mN.
Also the decrease of
With regard to unloading process, the slope of unloading curve is related with
In order to interpret the matching degree of
According to literature [
Error analysis of four different crystal planes.
Crystal indices  Correlation coefficient 
Average absolute relative error (AARE %) 

(001)  0.9884  15% 
(215)  0.9914  14% 
(405)  0.99024  14.3% 
(605)  0.992515  12.8% 
Microindentation measurements using a sharp Berkovich indenter on singlecrystal nickelbased superalloy DD99 of four crystallographic orientations, that is, (001), (215), (405), and (605), were made to determine the loaddisplacement relations. Some material parameters reflecting the micromechanical behavior of DD99, such as microhardness
Although FEM results deviate slightly from experimental results, they can be used as sufficient evidences indicating the accuracy of 3D FEM model and material elasticplastic model founded from MIT.
Crosssectional area, mm^{2}
Crosssectional area corresponding to
Lattice constant
Fitting coefficients
Burgers vector
A variable related to material properties as well as indenter geometry
Constraint factor
Elastic modulus, MPa
Effective elastic modulus of damaged material, MPa
Microhardness, kg
microhardness regardless of strain gradient plasticity, kg
Indenter displacement,
Residual depth after unloading,
Miller indices
Schmid factor
Strain hardening exponent
Indenter load, mN
Strength coefficient
Independent elastic compliance constant.
True strain
Corresponding strain to initial yield stress
Strain corresponding to
Initial yield stress, MPa
Representative stress, MPa
Ultimate tensile strength, MPa
Flow stress, MPa
Poisson’s ratio
Shear modulus, MPa
Density of statistically stored dislocations.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to express their sincere thanks for the research grants supported by the National Natural Science Foundation of China (Grant no. 51275414), the Aeronautical Science Foundation of China (Grant no. 2011ZE53059), and the Graduate Starting Seed Fund of Northwestern Polytechnic University (Grant no. Z2014007).