Improvement of the operating properties of ceramic components is possible on the basis of a profound study of processes occurring within a surface layer [1–3]. It is particularly important to determine features of material inhomogeneous deformation, under action of external loads [4, 5]. However, these studies are significantly complicated by a requirement for considering boundaries between ceramic interacting structural elements having different properties [6, 7].

Analysis of the stress-strained state of structural inhomogeneous bodies makes it possible to determine features of their failure, and to formulate effective recommendations for a reduction in the intensity of a process [8, 9]. An important aspect of analysis is a study of local stresses at boundaries of interacting structural elements of inhomogeneous material and evaluation of the volumetric stress distribution in a surface layer of a component of this material with different external action [10, 11]. Solution of this complex task makes it possible to realize the idea of computer design of structurally inhomogeneous materials [12]. This approach may also be used for automation of a search for optimum sintered material compositions for manufacture of innovative objects intended for specific operating conditions.

The aim of this work is to evaluate stress inhomogeneity in a surface layer of zirconium oxide ceramic under action of a complex mechanical load. The effect of thermal and combined loading is also studied; results of these studies will be provided future articles.

Research Procedure

Solution of this problem is based on numerical modelling of deformation processes using methodology of control by ceramic component operating properties [5, 9], and a set of models [2, 6]. A design scheme is formulated on their basis for carrying out numerical experiments in a SolidWorks simulation software module (Fig. 1).

Fig. 1
figure 1

Design layout.

Full size image

The design layout is presented in the from of a structure with a size of 38 × 18 × 4 μm, consisting of fragments of components D1 and D2 made of ceramic and copper respectively. These components are in close contact, and the form of the interface takes account of real microgeometry of a ceramic component, formed after sintering or machining. A fragment of component D1 consists of three grains Z1, Z2, Z3 of elliptical shape and sizes a = 5 μm and b = 4 μm, which are fixed in a matrix through intergranular phase of thickness δ f  = 1 μm. The major axis of grain Z2 coincides with axis x, and major axes of grains Z1 and Z3 are arranged at angle α = 15° to axis x, and the distance between the axes of neighboring grains is 11 μm. Junctions of intergranular phase at grain boundaries have radii of contact curves R 1 = R 2 = 1 μm and R 3 = R 4 = 4 μm. At the outer side of a grains, intergranular phase, and matrix there is a fragment of component D2 of thickness δ f  = 2 μm. The grain and matrix of component D1 is made of zirconium dioxide (density ρ = 5.68 g/cm3, elasticity modulus E = 180 GPa, Poisson’s ratio μ = 0.2); the intergranular phase is made of MgO (ρ = 3.4 g/cm3, E = 315 GPa, μ = 0.18), and D2 is made of copper (ρ = 8.9 g/cm3, E = 110 GPa, μ = 0.37). In future this structure will be called “ceramic system ZrO2–MgO–ZrO2–Cu.”

Two forms of force were applied to the free surface of a component in the design layout (see Fig. 1); distributed of forces P 1, P 2, P 3, and P 4, were applied over the whole contact length; concentrated force F was applied to a section of contact with grain Z2, and this force operated over a direction towards the grain center. The specific combination of these loads was considered by a force complex, each if which is characterized by values of load provided in Table 1.

Table 1 Combination of Load in Force Complexes, N
Full size table

In order to analyze results of numerical experiments a control point (CP) method was used, by which we understand fixed finite structural elements. The number and location of CP was determined taking account of the purpose oft his study. Schematic arrangement and numbering of CP, and also a fragment of an actual finite element network are shown in Fig. 2. All of the CP selected for component D1 are located in surface layers of grains Z1 (CP1-CP25), Z2 (CP26-CP51), Z3 (CP52-CP76), and intergranular phase, adjacent to grains (CP77-CP137) and to the matrix (CP138-CP182), and also matrix surface layer adjacent to intergranular phase (CP183-CP221). Stresses in component D1 were not studied and CP in this structural element were not separated. After carrying out calculations, values of stress intensity σ i in each CP were obtained.

Fig. 2
figure 2

CP location in object D1 surface layer of ceramic of the system ZrO2–MgO–ZrO2–Cu.

Full size image

Results of calculations are presented in the form of graphical image of the stress intensity field and curves connecting numerical values of stress intensity at CP. Stress inhomogeneity coefficient K Δ is used for quantitative evaluation of stress inhomogeneity in a surface layer of a working channel of a ceramic fiber, i.e., a dimensionless index determining the level of change in stresses in a surface layer of a ceramic structural element, and expressed numerically by the ratio of the greatest value of stress intensity to the least value. Values of K Δ were analyzed in a surface layer of each grain, layers of intergranular phase adjacent to a grain and to a matrix, and in layer of matrix adjacent to intergranular phase. Use of this procedure made it possible to study the effect in detail of a set of force loads on the stress-strained state of ceramic tools taking account of material structure and to supplement knowledge obtained in [13, 14].

Results and Discussion

An example is given in Fig. 3 of a graphical image of a stress intensity field in the surface layer of ceramic of the system ZrO2–MgO–ZrO2–Cu under action of a complex mechanical load. A qualitative estimate of the stress field shows that the stress-strained state of a surface layer of this ceramic differs significantly in different areas. Values of stress intensity σ i in structural elements of ceramic of the system ZrO2–MgO–ZrO2–Cu under action of force complex No. 1 (see Table 1) vary from 42 to 1600 MPa. The greatest stresses arise in the central grain Z2, and in intergranular phase and matrix around it.

Fig. 3
figure 3

Diagram of stress-strained state of structural elements of ceramic of the system ZrO2–MgO–ZrO2–Cu under action of force complex No. 1.

Full size image

For comparison of a more detailed picture of the effect of force loads on stress-strained state in a surface layer we analyze results of calculating stress intensity in each structural element of ceramic of the system ZrO2–MgO–ZrO2–Cu.

The stress-strained state is shown in Fig. 4 for three grains under the action of different force complexes. The nature of change in local stresses in the surface layer of grain Z1 under action of a complex force load has a quite simple form (see Fig. 4 a). The gradual increase in stress intensity σ i recorded from CP1 to CP18, and then from CP18 to CP25 reveals a reduction in σ i , i.e., the greatest stresses σ i form at CP18.

Fig. 4
figure 4

Effect of complex force load on stress intensity σ i at CP of surface layer of grain Z1 (a), Z2 (b), and Z3 (c) in ceramic of the system ZrO2–MgO–ZrO2–Cu. Force complex number shown on curves.

Full size image

An increase in force load leads to a marked increase in stress intensity σ i at all CP of a surface layer of grain Z1. For example, under action of force complex No. 1 the range of change in stress intensity σ i is 82 – 256 MPa, with load No. 2 it is 164 – 512 MPa, and with load No. 3 it is 329 – 1024 MPa. The stressed state of a surface layer of grain Z 1 is characterized by inhomogeneity coefficient K Δz1 = 3.12.

The nature of change in local stresses in the surface layer of grain Z2 under action of complex force loads is different, and it has a more complex form (see Fig. 4 b). In the area from CP26 to CP31 there is a reduction in σ i . In the section from CP31 to CP38 there is a marked increase in stress, and then in section CP38 to CP46 a reduction in stress is recorded. In the section from CP46 to CP51 there is a sharp increase in local stresses. The greatest values of σ i are recorded at CP51, and their values are 1125, 250, and 4500 MPa under action of force complexes Nos. 1 – 3 respectively. It is seen that the greatest local stresses in grain Z2 form at CP26 and CP51, which are most closely placed to the area concentrated force F i application. Analysis of the dependences obtained showed that K Δz2 in a surface layer of grain Z2 is 4.83, which is significantly higher compared with the value in the first case.

A change in stresses in a surface layer for grain Z3 under action of a complex load is shown in Fig. 4 c. It is seen that curves almost “mirror” image the dependence typical for grain Z1, and the least value of σ i is recorded at CP59. With force load No. 1 the range of change in σ i is 79 – 245 MPa, with load No. 2 it is 159 – 491 MPa, and with load No. 3 it is 319 – 982 MPa. The stressed state of a surface layer of grain Z3 is characterized by K Δz3 = 3.1.

The effect of complex force load on local stresses in layers of intergranular phase adjacent to grains and matrix is shown in Fig. 5. It is established that the nature of change in σ i in the layer of intergranular phase under action of complex force load has a complex form with sharp changes in stress values, and curves are almost symmetrical with respect to central grain Z1. The range of change in σ i in layers of intergranular phase adjacent to grains and matrix is almost the same, and under action of force complex No. 1 it is 110 – 424 MPa, for No. 2 it is 221 – 848 MPa, and for No. 3 it is 443 – 1697 MPa.

Fig. 5
figure 5

Effect of complex force load on stress intensity σ i at CP of intergranular phase surface layer, adjacent to grains (a) and matrix (b) in ceramic of the system ZrO2–MgO–ZrO2–Cu. Force complex number shown on curves.

Full size image

Average values of stress inhomogeneity coefficients K Δip1 and K Δip2 in layers of intergranular phase adjacent to grains and matrix comprise 3.85 and 3.62 respectively. Attention is drawn to the extremely unstable change in stresses in a layer of intergranular phase adjacent to grain Z2. However, in the section from CP99 to CP115 (see Fig. 5 a) K Δip1 = 1.6, which is lower by almost a factor of three than the average value of this coefficient calculated for the whole surface. It is apparent that in order to estimate instability it is necessary to use a coefficient capable of considering presence of structural stress concentration.

The nature of change in σ i in a layer of matrix adjacent to intergranular phase under the action of a complex mechanical load is presented in Fig. 6. All curves are symmetrical with respect to CP202. Values of σ i in the surface layer of matrix under action of force complex No. 1 change from 73 to 334 MPa, complex No. 2 from 146 to 668 MPa, and No. 3 from 293 to 1337 MPa. The greatest stresses form in a matrix surface layer in the area of grain Z2, for which there is the greatest effect of concentrated force F i . The stressed sate of a matrix surface layer has K Δm  = 4.6.

Fig. 6
figure 6

Effect of complex force load on stress intensity σ i at CP of intergranular phase surface layer, adjacent to intergranular phase of ceramic of the system ZrO2–MgO–ZrO2–Cu. Force complex number shown on curves.

Full size image

Conclusion

By summarizing results of numerical experiments it may be noted that the stress-strained state of a surface layer of ceramic based on zirconium dioxide under action of a complex mechanical load is characterized by high inhomogeneity. The greatest local stresses under action of a complex force load form within grain Z2 and intergranular phase around it, which is determined by action of a concentrated force. The inhomogeneity coefficient K Δz for stresses in a surface layers of grains Z1, Z2, Z3 equals 3.12, 4.83, and 3.1 respectively. The stress inhomogeneity coefficient in layers of intergranular phase adjacent to grains comprises 3.85 and 3.62 respectively. The stressed state of a surface layer of matrix adjacent to intergranular phase is characterized by K Δz1 = 4.6.

Results of this study also point to a requirement for considering stress inhomogeneity forming a surface layer of a ceramic component based on zirconium dioxide under action of a complex mechanical load, in describing their wear and failure mechanisms, and also in designing components for prescribed operating conditions.