Final Report Summary - CERMAT2 (New ceramic technologies and novel multifunctional ceramic devices and structures)
Description of the project objectives:
The CERMAT2 project aims to train young researchers in understanding the modelling of Solid Mechanics problems applied to the process and design of advanced ceramics in a synergic collaboration between academia and industry. The objective will be pursued by a training-through-research methodology focused on:
i) improvement of processes related to ceramic powder compaction and design of ceramic pieces;
ii) fundamental understanding of the behaviour of granular and composite materials;
iii) development of multifunctional ceramic materials and structures.
Description of the work performed since the beginning of the project and main results:
Constitutive modelling of compaction of ceramic powders: Cold densification of a ceramic or a metallic powder is a mechanical process in which a loose material is progressively transformed into a fully dense solid, the so-called ‘green body’, during a process which does not involve phase transformations, thermal effects, chemical or compositional changes. The description of this transformation in terms of a constitutive model is a scientific challenge, since it involves the transition between two quite different states of matter: the granular state (in which the material can, as a fluid, conform in shape to fill a container, but at the same time can carry significant shear stress before plastically flowing) and the dense state (in which the material has lost its particulate nature and can only sustain limited stresses before plastic strain and subsequent failure occur). The following tasks were achieved:
Multiscale approach and constitutive modelling A new coupled elastoplastic constitutive framework for cold powder densification was introduced in which the nonlinear and anisotropic elastic response of the granulate is modelled through a hyper-elastic potential (in which stiffness and directional characteristics strongly depend on plastic strain). This modelling eliminates problems (generation of spurious permanent strain) related to the hypo-elastic behaviour that is usually employed for describing the behaviour of granular matter within its elastic range.
Optimal forming and sintering of ceramic compounds During the heating process the powder particles grow together through diffusion process and become a single body. This process is called sintering and technical ceramics are usually produced by this process. During this process the dimensions of the body changes drastically, and thus the end shape of a part is not predictable in a simple straight-forward fashion. A model has been developed to predict the shape-change and thus to be able to produce parts that are closer to the desired geometry (``near net-shape''). The model of sintering is also valid for non-isothermal processes. This model is able to describe the mechanical state during and also after the sintering process, thus predicting dimensions, porosity and residual stresses.
Numerical strategies and implementation. The constitutive modelling of granular, porous and quasi-brittle materials is based on yield (or damage) functions, which may exhibit features (for instance, lack of convexity, or branches where the values go to infinity, or false elastic domains) preventing the use of efficient return-mapping integration schemes. This problem was solved by proposing a general construction strategy to define an implicitly defined convex yield function starting from any convex yield surface. Based on this implicit definition of the yield function, a return-mapping integration scheme is implemented and tested for elastic-plastic (or -damaging) rate equations. The scheme is general and, although it introduces a numerical cost when compared to situations where the scheme is not needed, is demonstrated to perform correctly and accurately.
The material model for sintering was implemented as a thermomechanical model into a Finite Element Method routine that extends the classic continuum description to non-isothermal states. The FEM environment of AceGen/AceFem was used. This allows for an easy conversion to a material routine for the commercial solver Abaqus.
Mechanical testing and experimental processing characterization. Experimental testing is crucial for the calibration of material parameters appearing in the constitutive models. Forming of alumina powder in hedometric conditions were performed. The objective is to calibrate and validate the model on ad-hoc experiments. As the current industrial praxis usually destructive tests on compacted green bodies are employed to assess material parameters – a task that requires significant amount of time and material. One of the project objectives was to develop more accurate and more effective calibration. The novel procedure relies only on data collected from compaction tests, thus avoiding any further experimentation on green bodies. Methodology adopted to reach this objective is based on inverse analysis, centered on the minimization of a suitable objective function which quantifies the difference between experimentally measured and computed quantities. A testing protocol was formulated defining green body geometries, measurable quantities, and algorithms to solve the resulting minimization problem. The proposed calibration procedure offers important advantages for industrial routine use since being more precise and significantly faster. The performances of the model proved to be excellent, so that the model can be considered an available tool for the design of pieces to be obtained through cold forming.
Enhancing performances of ceramics:
Micro-structural fields and asymptotic models for near defect fields. The research focused on the mathematical modeling of performance of ceramic materials under extreme mechanical loading conditions leading to fracture. The developed mathematical models can be applied to the analysis of crack propagation both in ceramics and bi-materials having ceramics as one of the phase, especially for the investigation of interfacial cracks. Bi-materials themselves are formed from two separate bulk materials joined together in order to benefit the overall material properties. These materials are common in engineering application, aircraft constructions, etc. At the same time, under production stage cracks appear along the interface between joined materials and can later grow resulting in failure of the whole structure. The developed models explains the mechanisms that lead to the crack appearance and their propagation.
Numerical simulations. An effective implementation of the boundary element method in analyzing piezoelectric ceramics has been developed. The method is based on constitutive equations, Stroh formulations and numerical modeling. This method can be used to get a better knowledge of damage mechanism in a crack analysis by discretising the integral equation in the ceramic materials. The influences of inclusion shape (cylindrical, rectangular) and kinked crack on behaviour of piezoelectric media have been investigated.
Modelling of ceramic structures with imperfect interfaces:
Evaluation of transmission conditions and solutions for ceramics with imperfect interfaces.
An important area of interest in which there is the potential wide application of ceramics is the bone replacement operations. A central point from a mechanical point of view in articular replacements is the understanding of the features of the contact problem between the bones at the joint which is facilitated in nature by the presence of cartilage layers. These layers show frictionless surfaces and the capability of distribute the stresses in order not to damage the underlying bone. The developed model allows the mechanical analysis of a fully three-dimensional contact and permits to observe the influence of the material anisotropy and inhomogeneity on human articular mechanics.
Interaction between cracks and composite structures.
During recent decades, application of piezoelectric ceramics such as actuator, sensors and transducers have been increased due to their efficiency in converting mechanical loads to electrical ones and viceversa. However, the most important problem is that piezoelectric ceramics are fragile due to their special microstructural characteristics. The developed multi-scale approach is able to model the properties of materials simultaneously at different scales.
Dynamic response of ceramic composite structures:
Dynamic response of ceramic composite structures for elastic wave filters. Mathematical models of Bloch-Floquet waves in piezoelectric metamaterials were developed. Analysis of dispersion properties and localisation id dynamic periodic Rayleigh beams on elastic foundations was performed successfully. Further work was carried out on dispersion of Floquet flexural waves in two-dimensional structured Rayleigh plates; the novel properties involving Dirac cone regimes are identified and analysed.
Localization phenomena in ceramic structures for resonators. Several results concerning the modelling of elastic and electromagnetic wave propagation through one- and two-dimensional layered piezoelectric structures were achieved. For 1D structures whose unit cell is made of two different layers of piezoelectric materials the dispersion diagrams were obtained. The reflection and transmission coefficients associated with the problem of scattering of an elastic wave by a finite number of piezoelectric layers (piezoelectric stack) were analysed. The width and the position of band gap of periodic 1D structures and the occurrence of transmission resonances in finite stacks depend on piezoelectric effect. The analysis was extended to a 2D periodic piezoelectric structure. The unit cell of this structure coincides with the one of a periodic checkerboard, where the white and black rectangles will be different piezoelectric materials. In this 2D case, the dispersion surfaces for Bloch waves have been constructed and analysed. From a mathematical and physical point of view, particular attention was given to the dynamic anisotropy and localisation phenomena.
Multilayer piezoelectric actuators. We performed the analysis of the properties of multilayer inhomogeneous ceramics with parasitic residual stresses or specifically pre-stressed during the production process. The importance of this research can be highlighted by the fact that the famous Prince Rupert Drop (PRD) phenomenon, very well understood qualitatively, has not been convincingly modelled numerically. We start with a model problem where the layers are concentric cylinders having different material properties and pre-stressed when the multilayer cylinder is contracted. We showed that thermoelastic response of such a structure demonstrates some peculiarities appearing for the PRD.
Ceramic seals. In order to develop suitable sealants, operating at about 950°C, several glass-ceramics with different compositional additives and ratios in BaO–Al2O3–SiO2 (BAS) glass matrix have been prepared by using ceramic method. The thermal properties of all these glass-ceramics can be tailored by proper selection of composition, particularly by taking suitable amount of metal oxides.
The CERMAT2 project aims to train young researchers in understanding the modelling of Solid Mechanics problems applied to the process and design of advanced ceramics in a synergic collaboration between academia and industry. The objective will be pursued by a training-through-research methodology focused on:
i) improvement of processes related to ceramic powder compaction and design of ceramic pieces;
ii) fundamental understanding of the behaviour of granular and composite materials;
iii) development of multifunctional ceramic materials and structures.
Description of the work performed since the beginning of the project and main results:
Constitutive modelling of compaction of ceramic powders: Cold densification of a ceramic or a metallic powder is a mechanical process in which a loose material is progressively transformed into a fully dense solid, the so-called ‘green body’, during a process which does not involve phase transformations, thermal effects, chemical or compositional changes. The description of this transformation in terms of a constitutive model is a scientific challenge, since it involves the transition between two quite different states of matter: the granular state (in which the material can, as a fluid, conform in shape to fill a container, but at the same time can carry significant shear stress before plastically flowing) and the dense state (in which the material has lost its particulate nature and can only sustain limited stresses before plastic strain and subsequent failure occur). The following tasks were achieved:
Multiscale approach and constitutive modelling A new coupled elastoplastic constitutive framework for cold powder densification was introduced in which the nonlinear and anisotropic elastic response of the granulate is modelled through a hyper-elastic potential (in which stiffness and directional characteristics strongly depend on plastic strain). This modelling eliminates problems (generation of spurious permanent strain) related to the hypo-elastic behaviour that is usually employed for describing the behaviour of granular matter within its elastic range.
Optimal forming and sintering of ceramic compounds During the heating process the powder particles grow together through diffusion process and become a single body. This process is called sintering and technical ceramics are usually produced by this process. During this process the dimensions of the body changes drastically, and thus the end shape of a part is not predictable in a simple straight-forward fashion. A model has been developed to predict the shape-change and thus to be able to produce parts that are closer to the desired geometry (``near net-shape''). The model of sintering is also valid for non-isothermal processes. This model is able to describe the mechanical state during and also after the sintering process, thus predicting dimensions, porosity and residual stresses.
Numerical strategies and implementation. The constitutive modelling of granular, porous and quasi-brittle materials is based on yield (or damage) functions, which may exhibit features (for instance, lack of convexity, or branches where the values go to infinity, or false elastic domains) preventing the use of efficient return-mapping integration schemes. This problem was solved by proposing a general construction strategy to define an implicitly defined convex yield function starting from any convex yield surface. Based on this implicit definition of the yield function, a return-mapping integration scheme is implemented and tested for elastic-plastic (or -damaging) rate equations. The scheme is general and, although it introduces a numerical cost when compared to situations where the scheme is not needed, is demonstrated to perform correctly and accurately.
The material model for sintering was implemented as a thermomechanical model into a Finite Element Method routine that extends the classic continuum description to non-isothermal states. The FEM environment of AceGen/AceFem was used. This allows for an easy conversion to a material routine for the commercial solver Abaqus.
Mechanical testing and experimental processing characterization. Experimental testing is crucial for the calibration of material parameters appearing in the constitutive models. Forming of alumina powder in hedometric conditions were performed. The objective is to calibrate and validate the model on ad-hoc experiments. As the current industrial praxis usually destructive tests on compacted green bodies are employed to assess material parameters – a task that requires significant amount of time and material. One of the project objectives was to develop more accurate and more effective calibration. The novel procedure relies only on data collected from compaction tests, thus avoiding any further experimentation on green bodies. Methodology adopted to reach this objective is based on inverse analysis, centered on the minimization of a suitable objective function which quantifies the difference between experimentally measured and computed quantities. A testing protocol was formulated defining green body geometries, measurable quantities, and algorithms to solve the resulting minimization problem. The proposed calibration procedure offers important advantages for industrial routine use since being more precise and significantly faster. The performances of the model proved to be excellent, so that the model can be considered an available tool for the design of pieces to be obtained through cold forming.
Enhancing performances of ceramics:
Micro-structural fields and asymptotic models for near defect fields. The research focused on the mathematical modeling of performance of ceramic materials under extreme mechanical loading conditions leading to fracture. The developed mathematical models can be applied to the analysis of crack propagation both in ceramics and bi-materials having ceramics as one of the phase, especially for the investigation of interfacial cracks. Bi-materials themselves are formed from two separate bulk materials joined together in order to benefit the overall material properties. These materials are common in engineering application, aircraft constructions, etc. At the same time, under production stage cracks appear along the interface between joined materials and can later grow resulting in failure of the whole structure. The developed models explains the mechanisms that lead to the crack appearance and their propagation.
Numerical simulations. An effective implementation of the boundary element method in analyzing piezoelectric ceramics has been developed. The method is based on constitutive equations, Stroh formulations and numerical modeling. This method can be used to get a better knowledge of damage mechanism in a crack analysis by discretising the integral equation in the ceramic materials. The influences of inclusion shape (cylindrical, rectangular) and kinked crack on behaviour of piezoelectric media have been investigated.
Modelling of ceramic structures with imperfect interfaces:
Evaluation of transmission conditions and solutions for ceramics with imperfect interfaces.
An important area of interest in which there is the potential wide application of ceramics is the bone replacement operations. A central point from a mechanical point of view in articular replacements is the understanding of the features of the contact problem between the bones at the joint which is facilitated in nature by the presence of cartilage layers. These layers show frictionless surfaces and the capability of distribute the stresses in order not to damage the underlying bone. The developed model allows the mechanical analysis of a fully three-dimensional contact and permits to observe the influence of the material anisotropy and inhomogeneity on human articular mechanics.
Interaction between cracks and composite structures.
During recent decades, application of piezoelectric ceramics such as actuator, sensors and transducers have been increased due to their efficiency in converting mechanical loads to electrical ones and viceversa. However, the most important problem is that piezoelectric ceramics are fragile due to their special microstructural characteristics. The developed multi-scale approach is able to model the properties of materials simultaneously at different scales.
Dynamic response of ceramic composite structures:
Dynamic response of ceramic composite structures for elastic wave filters. Mathematical models of Bloch-Floquet waves in piezoelectric metamaterials were developed. Analysis of dispersion properties and localisation id dynamic periodic Rayleigh beams on elastic foundations was performed successfully. Further work was carried out on dispersion of Floquet flexural waves in two-dimensional structured Rayleigh plates; the novel properties involving Dirac cone regimes are identified and analysed.
Localization phenomena in ceramic structures for resonators. Several results concerning the modelling of elastic and electromagnetic wave propagation through one- and two-dimensional layered piezoelectric structures were achieved. For 1D structures whose unit cell is made of two different layers of piezoelectric materials the dispersion diagrams were obtained. The reflection and transmission coefficients associated with the problem of scattering of an elastic wave by a finite number of piezoelectric layers (piezoelectric stack) were analysed. The width and the position of band gap of periodic 1D structures and the occurrence of transmission resonances in finite stacks depend on piezoelectric effect. The analysis was extended to a 2D periodic piezoelectric structure. The unit cell of this structure coincides with the one of a periodic checkerboard, where the white and black rectangles will be different piezoelectric materials. In this 2D case, the dispersion surfaces for Bloch waves have been constructed and analysed. From a mathematical and physical point of view, particular attention was given to the dynamic anisotropy and localisation phenomena.
Multilayer piezoelectric actuators. We performed the analysis of the properties of multilayer inhomogeneous ceramics with parasitic residual stresses or specifically pre-stressed during the production process. The importance of this research can be highlighted by the fact that the famous Prince Rupert Drop (PRD) phenomenon, very well understood qualitatively, has not been convincingly modelled numerically. We start with a model problem where the layers are concentric cylinders having different material properties and pre-stressed when the multilayer cylinder is contracted. We showed that thermoelastic response of such a structure demonstrates some peculiarities appearing for the PRD.
Ceramic seals. In order to develop suitable sealants, operating at about 950°C, several glass-ceramics with different compositional additives and ratios in BaO–Al2O3–SiO2 (BAS) glass matrix have been prepared by using ceramic method. The thermal properties of all these glass-ceramics can be tailored by proper selection of composition, particularly by taking suitable amount of metal oxides.