Thermal effects on the fracture mechanics of ferroelectric ceramics

Thermal effects on the fracture mechanics of ferroelectric ceramics

Project Supervisor

Prof. Dr.rer.nat.habil. Meinhard Kuna, i.R.

Responsible Research Assistant

M.Sc. Omar El Khatib


Ferroelectric ceramics possess special electromechanical properties allowing them to be an appropriate choice for various applications, such as actuator, sensor and energy convertors. The use of ferroelectric materials in such applications usually involves strong mechanical and electrical loading conditions. These severe conditions can alter the functional properties of ferroelectrics and raise the questions of their reliability, strength and service lifetime, which are of major importance during the design and production of adaptive structures. Moreover, thermal effects, which have been largely ignored, have a significant influence on the behavior of ferroelectrics. Besides external loading, internal heat generation occur by dissipation of the inelastic work resulting from ferroelectric hysteresis. Furthermore, at extreme field concentrations like crack, ferroelectric devices may fail by brittle fracture or fatigue.


This project aims toward  a more realistic modeling of the ferroelectric materials. For that purpose, the ignored thermal effects should be investigated, on the one hand, on the constitutive modeling of the non-linear domain switching and on the fracture mechanical behavior on the other hand. 


The well established micromechanical material model for ferroelectric domain switching is enhanced to represent the fully coupled thermo-electro-mechanical behavior. The coupling considers the pyroelectric and thermal strains effects. The internal heat production, which leads to transient temperature fields , is taken into account, as well as the temperature dependency of material parameters. A three-dimensional finite element code is implemented with the help of Abaqus user defined element. The finite element formulation is based on the principle of virtual displacements, electric potential and temperature to obey the balance laws for forces, electric charges and heat energy. 

The temperature influence on the ferroelectric domain switching behavior is shown by means of the strain and polarization hysteresis loops.

The thermo-electro-mechanical fields at the crack tip are analyzed using a boundary layer approach for small scale switching conditions. The temperature increase around the crack tip is determined at first assuming an adiabatic approach. However, a fully transient heat conduction problem is later considered, which emphasize the effect of the driving frequencies on heat generation. The impact of all external and internal factors on the crack driving energy is described by means of the configurational forces and the thermo-electro-mechanical extension of the J-integral.


Non-linear thermo-electro-mechanical ferroelectric material model

In order to predict the coupled thermo-electro-mechanical behavior of a ferroelectric polycrystalline material at the macro-scale, a homogenization process of the micromechanical model must be conducted. For this purpose, a cubic representative volume element was chosen, which represent a polycrystal with randomly oriented grains.

Figure 1: Representative sketches of the RVE with 125 random oriented grains and the used eight-noded hexahedral element with its 8 integration pointsThe RVE consists of 125 eight-noded hexahedral elements, where each element represents a randomly oriented grain (see Fig. 1). A cyclic electric field E in z-direction is applied as an external loading.

In Fig. 2, a domain switching event is shown (with the help of ABAQUSER) under an alternated bipolar uniaxial electric field. At the beginning, the spontaneous polarization of each domain is randomly oriented and the applied electric field is relatively small. As the electric field increases and reaches a threshold value, the switching processes of the domain are initiated and the domains start to align with the direction of the electric field. Further increase of the electric field will lead to a saturated state where most of the domains are aligned with the electric field and no additional switching can occur. During removal of the electric field, there is slight depolarization of the domains, yet they do not return to their original states after totally removing the electric field. If the direction of the electric field is reversed, the domains start to depolarize before the threshold value is attained again. Increasing further the electric field in the opposite direction, the domain switchng processes are initiated again and domains start to align with the new direction of the electric field until the saturation state is reached again. As the cycle continues, the hysteresis is completed symmetrically. The area of the hysteresis loop corresponds to the dissipated energy in the course of a load cycle. Switching is an irreversible process, whereby electromechanical energy is dissipated and transformed into heat inside the material.

Figure 3: Temperature increase due to self heatingFig. 3 shows the temperature change due to domain switching with respect to the number of cycles. Figures 4 and 5 show the strain and polarization hysteresis loops at different load cycles. One can clearly observe the influence of the temperature increase on the behavior of ferroelectric materials. From the strain hysteresis, one can see that the maximum strains are decreasing with the increase of temperature each cycle. On the other hand the minimum and remanent strains increase with the temperature. In the case of polarization, a decrease in the remanent and maximum polarization is observed with the increase of temperature. Finally, from both hysteresis, one can notice the decrease of the coercive field strength.

Figure 4: Strain hysteresis at different cycles

Figure 5: Polarization hysteresis at different cycles

In general, the temperature increase due to domain switching leads to the degradation of the ferroelectric hysteresis properties. This is represented by the decrease of the hysteresis loops areas of both strains and polarization with the number of cycles.

Heat generation in cracked ferroelectric ceramics

In order to investigate the effects of applied electric fields and mechanical stresses on the fracture behavior under small scale switching conditions, the so-called boundary layer approach is adopted. The asymptotic problem of the Griffith crack in a ferroelectric ceramic under combined electrical and mechanical loading is considered, whereby the analytical solution for the near crack tip fields in a linear piezoelectric material is applied as boundary conditions on a circular domain including the crack tip in a ferroelectric material. The crack can be subjected to either external monotonic mechanical loading (KI controlled loading) or cyclic electric loading (KIV controlled loading). Figures 6 and 7 show the switching zones around the crack tip of a Griffith crack under a constant mechanical Mode-I load and bipolar cyclic electric KIV  loading, respectively. For both cases the initial poling direction is taken perpendicular to the crack faces. 

Figure 6: Switching zone around the crack tip under KI controlled loading.

Switching zone at the crack tip

On the first hand, different switching events occur at the crack tip under Mode-I (figure 6), the +90° and -90°  as well as 180°. Along the crack faces, and due to the initial poling direction which is perpendicular to the crack faces and the traction and charge free boundary conditions, only +/-90° switching can occur, which is necessary to fulfill these boundary conditions. Additionally, switching of some 180° domains is also observed, in the region in front of the crack tip. Given that, a 180° event is always caused by electric field, means that the mechanical stresses applied are generating through the piezoelectric effect some electric field concentration at the crack tip causing these 180° switching. The 180° switching region in front of the crack (along the ligament) is relatively small compared to the one normal to the crack faces , and to the +/-90° switching regions.

Figure 7: Switching zone under positive and negative applied electric field

On the other hand,  for the Mode-IV case (figure 7), the applied electric far field acts at the beginning of the load cycle in the same direction as the initial polarization. The external load creates inhomogeneous field concentrations around the crack tip causing the domains to switch. Since most of the domains have initially a similar orientation as the local electric field, the applied electric field can only induce 90° domain switching in the area along the crack faces, which is necessary to fulfil the charge free boundary conditions. During the next part of the load cycle the electric field goes from its maximum value to zero whereby no further switching events occur. Next, the electric field is reversed and it is now in the opposite direction of the initial polarization. During this part, the reverse switching (180°) of the domains is initiated ahead of the crack tip creating a larger switching zone area, which depends on the load magnitude. Also, a larger  90° domain switching area is created along the crack faces due to the applied reversed electric field.

Heat generation around the crack tip

Through one load cycle heat generation occurs only during domain switching which leads to a step wise temperature increase. Additionally, the switching phenomenon is confined to the switching zone area around the crack tip. Assuming an adiabatic process, heat generation is then limited to the switching zone area.

Figure 8: Temperature field around the crack tip

Fig. 8 shows the temperature field generated around the crack tip for the case where the load magnitude is E = 0.6Ec and after 100 load cycles. Higher temperature increase can be observed closer to the crack tip as higher field concentrations can be found in this area leading to complete switching of all domains and consequently more electromechanical energy dissipation arises, resulting in higher heat generation. Closer to the switching zone boundary only partially switched domains exist meaning less energy dissipation followed by less temperature increases.

Fracture mechanical behavior

For the investigation of the fracture mechanical behavior, the concept of configurational forces [2,3] is used. The J-integral at the crack tip node, which is considered to be a relevant fracture parameter, is calculated for the different load cases. The load magnitude, self heating due to domain switching as well as the initial temperature effect on the J-integral are investigated.

Figure 9: J-integral at different cycles for the load case E = 0.3Ec

Fig. 9 shows the J-integeral variation for different load cycles. More load cycles means higher temperature increase is accumulated around the crack tip and stronger material degradation can be found. It can be seen in Fig. 9, that the J-integral values during load application are increasing at higher cycles. During the application of the load, a shielding effect linked to the domain switching around the crack tip is observed. An increase in J-integral means that this shielding effect is weakened and the crack is becoming more vulnerable to external fields. So, with more heat generation and temperature increase after many cycles, the ferroelectric material properties are degrading and consequently the hysteresis loops are degenerating. This leads to the weakening of the shielding effect associated with the domain switching around the crack tip during load application. This indicates that the switched domains effectiveness is diminishing as the temperature increases, despite the probable increase in the switching zone are around the crack tip and the higer possibility of domain switching events at higher temperatures.

Figure 10: J-integral values at different uniform temperatures for E = 0.6Ec

Similar to the heat generation due to domain switching, the homogeneous initial temperature would also affect the fracture behavior of ferroelectric materials. Fig. 10 show the J-integral values at different initial temperatures.  It can be seen that the values are increasing with the increase of the temperature. Since at higher temperatures the shielding effect diminishes because of the loss in the ferroelectric properties and degeneration in the hysteresis loops, this increase of the J-integral values was expected.

Transient thermal fracture analysis

The previously assumed adiabatic heat generation could be enough for some caes, where for example the heat generation is fast inside the material and it does not have enough time to flow out, allowing higher temperature increase in the regions of heat generation. However, for other cases, heat may have enough time to flow and spread in the body, deeming the adiabatic assumption insufficient. Additional to the dependency of the heat generation on the load magnitude and temperature, the frequency of the applied load plays an equal role. 

Figure 11: Temperature change with time at different locations in the structure. E = 1Ec and f = 1Hz.In Fig. 11 the temperature change with time is shown at different locations in the structure for a load magnitude of E=Ec at a driving frequency of 1Hz. Four different points were chosen to show the temperature change: the crack tip node (R = 0), a point inside of the switching zone(R = 0.05RK), a point outside the switching zone (R = 0.2RK) and a point at the boundary of the model (R = RK). At the crack tip high field concentrations are found and the switching will start first followed by the other points inside the switching zone. This leads to kind of delay in temperature change between these two points. It can also be seen that the temperature increase per switching event is higher for the point inside the switching zone than at the crack tip, this is mainly due to the heat flow from the crack tip to the surrounding nodes as heat is generated there before, which allows the other nodes to have additional heat to their respective generated heat, thus, see a higher temperature increase.  At this low frequency the application of the load is slow, which gives the heat generated in the switching zone plenty of time to flow in the structure and allows the temperature to reach a steady state at all points even at the boundary, before the switching starts again, and the same behavior is repeated for different load cycles.

Figure 12: Temperature change with time at different locations in the structure. E = 0.3Ec and f = 1kHz.

At a higher frequency a slightly different behavior is observed, especially with respect to heat conduction in the structure. Figure 12 shows the temperature increase for E = 0.3Ec at a driving frequency of f = 1kHz at different locations. Here, the temperature increase at the crack tip is higher than at other locations, as heat flow is slower than the frequency of applied load, which does not give the heat generated an opportunity to flow to the neighboring locations like for the case of lower driving frequency. And the heat generated is almost restricted in the switching zone similar to the adiabatic case, and the outer boundary can barely get any heat and no temperature change will be observed.

Financial Support

Internal funding Prof. Kuna


  • O. El Khatib: Thermal effects on the fracture mechanics of ferroelectric ceramics, Dissertation, TU Bergakademie Freiberg, 2020, ISBN: 978-3-86012-646-2


  1. El Khatib, O., Kozinov, S., & Kuna, M. (2019). A micro–macro scale approach for thermal effects in ferroelectrics. Continuum Mechanics and Thermodynamics31(5), 1439-1452.
  2. El Khatib, O., Kuna, M., & Kozinov, S. (2020). Switching induced heating at the crack tip in ferroelectric ceramics. International Journal of Fracture221(2), 141-154.
  3. El Khatib, O., Kuna, M., & Kozinov, S. (2021). Transient thermal fracture analysis of ferroelectric ceramics under electromechanical loading. Smart Materials and Structures30(8), 085033.