Toughening

In polycrystalline ceramics, the crack can propagate in an intergranular way. The associated irreversible work per unit area is 2γ-γ_gb, where γ is the surface energy of material and γ_gb is the grain boundary energy. Though the irreversible work is decreased because of grain boundary energy, the fracture area is increased in intergranular crack propagation. Moreover, Mode II crack can be caused by deflection from normal fracture plane during intergranular crack propagation, which furtherly improves the toughness of ceramics. As a result, the ceramics with intergranular fracture shows a higher toughness than that with transgranular fracture. In SiC, the fracture toughness is ~2-3 ${\displaystyle MPa\cdot m^{0.5}}$if it fractures transgranularly and the fracture toughness is improved to 10 ${\displaystyle MPa\cdot m^{0.5}}$when it fractures intergranularly.[8]

Microcrack toughening means that the formation of microcracks before the main crack can toughen the ceramic. Additional microcracks will cause stress to concentrate in front of the main crack. This leads to additional irreversible work required for crack propagation. In addition, these microcracks can cause crack branches, and one crack can form multiple cracks. Because of the formation of these cracks, irreversible work is increased. The increment of toughness ${\displaystyle G_{C}}$ due to microcrack toughening can be expressed by:[1]

${\displaystyle \Delta G_{C}\cong 2r_{c}\sigma _{R}(\Delta \alpha )(\Delta T)V_{f}}$

where ${\displaystyle r_{c}}$ is the distance between microcracks and fracture plane, ${\displaystyle \sigma _{R}}$is residual stress, ${\displaystyle \Delta \alpha }$ is the difference of thermal expansion coefficient between adjacent grains, ${\displaystyle \Delta T}$ is the temperature difference causing thermal strain, and ${\displaystyle V_{f}}$ is the fraction of grains that is related to microcracks in an affected volume. In this equation, it has been assumed that residual stress is dominant in nucleating microcracks and formation of microcracks is caused by elastic work. In order to retard crack propagation, these microcracks must form during crack propagation. The grain size should be smaller than a critical grain size to avoid spontaneous formation of microcracks. The distance between microcrack and fracture plane should be larger than grain size to have a toughening effect.

Schematic of transformation toughening in partially stabilized zirconia

The TRIP effect is found in partially stabilized zirconia. Partially stabilized zirconia is composed of tetragonal phase at high temperature and monoclinic phase and cubic phase at lower temperature in equilibrium. In some components, the onset temperature of tetragonal monoclinic martensite transformation is lower than room temperature. The stress field near the crack tip triggers the martensitic transformation at velocities hypothesized to approach that of sound in the material.[9] The martensitic transformation causes volume expansion (volumetric/ dilatational strain) and shear strains of about 4% and 16% respectively. It applies compressive stress at the crack tip to prevent crack propagation as well as closure tractions at the crack wake.[10] From another point of view, the work associated to this phase transformation contributes to the improvement of toughness. The increment of toughness caused by transformation toughening can be expressed by:[1]

${\displaystyle \Delta G_{C}\cong 2r_{c}\sigma _{M}\epsilon _{M}V_{f}}$

where ${\displaystyle r_{c}}$is the distance between boundary of transformed region with fracture plane, ${\displaystyle \sigma _{M}}$is the stress triggering martensite transformation, ${\displaystyle \epsilon _{M}}$ is the strain of martensite transformation, and ${\displaystyle V_{f}}$is the fraction of tetragonal grains that is related to microcracks in an affected volume. The tetragonal particle size should be controlled properly. It is due to that too large particle size leads to spontaneous transformation and too small particle size leads to a very small toughening effect.

When a crack propagates in an irregular path, some grains of each side of main crack may protrude into other side. This leads to additional work for a complete fracture. This irreversible work is related to residual stress, which is about . The increment of toughness can be expressed by:[1]

${\displaystyle \Delta G_{C}\cong {1 \over 8}\mu \sigma _{R}V_{f}d}$

where ${\displaystyle \mu }$is the coefficient of friction, ${\displaystyle \sigma _{R}}$is residual stress, ${\displaystyle d}$ is the edge length of grain, and ${\displaystyle V_{f}}$is the fraction of grains associated with crack bridging.

There are some other approaches to improve the toughness of ceramics through crack bridging. The phenomenon of abnormal grain growth, or AGG, can be harnessed to impart a crack bridging microstructure within a single phase ceramic material. The presence of abnormally long grains serves to bridge crack-wakes and hinders their opening.[11] This has been demonstrated in silicon carbide and silicon nitride. Abnormally large grains may also serve to toughen ceramics through crack deflection mechanisms. Formation of a textured internal structure within ceramics can be used as a toughening approach.[12] silicon carbide materials have toughened by this approach.[13] Because the interfacial surface area is increased due to the internal structure, the irreversible fracture work is increased in this material.

Schematic fiber-reinforced composites

When fiber is ductile, the work from plastic deformation mainly contributes to the improvement of toughens. The additional toughness contributed by plastic deformation can be expressed by:[1]

${\displaystyle \Delta G_{C}=C\sigma _{y}\epsilon _{f}V_{f}l_{d}}$

where ${\displaystyle C}$ is a constant between 1.5-6, ${\displaystyle \sigma _{y}}$is the flow stress of fibers, ${\displaystyle \epsilon _{y}}$is the fracture strain of fibers, ${\displaystyle V_{f}}$is the fraction of fibers, and ${\displaystyle l_{d}}$is the debond length. From the equation, it can be found that higher flow stress and longer debond length can improve the toughening. However, longer debond length usually lead to a decrease of flow stress because of loss of constraint for plastic deformation.

The toughness in a composite with ductile phase toughening can also be shown using stress intensity factor, ${\displaystyle K_{c}}$ by linear superposition of the matrix and crack bridging based on solutions by Tada.[15] This model can predict behavior for small-scale bridging (bridge length << crack length) under monotonic loading conditions, but not large scale bridging.[16][17]

${\displaystyle K_{c}=K_{m}+\Delta K_{b}=K_{m}+{\sqrt {\frac {2}{\pi }}}\alpha V_{f}\int _{0}^{L}{\frac {\sigma _{y}}{\sqrt {x}}}dx}$

Several toughening mechanism in epoxies

where ${\displaystyle K_{m}}$ is the fracture toughness of the matrix, ${\displaystyle \Delta K_{b}}$ is the toughening due to crack bridging, ${\displaystyle L}$ is the bridge length, ${\displaystyle x}$ is the distance behind the crack tip, ${\displaystyle \sigma _{y}}$is the uniaxial yield stress, and ${\displaystyle \alpha }$ is a constraint/ triaxiality factor.