Final Report Summary - MEMIC (Fracture mechanics of microstructured composites incorporating intrinsic length-scales)
Description of the project objectives:
- modelling the propagation of a crack in microstructured interphases, incorporating the microstructural properties (length-scales) into the definition of fracture toughness and crack propagation stability;
- modelling the propagation of a crack in discrete interphases and lattice structures joining dissimilar materials; the formulation of the corresponding dynamic problem and crack-waves interactions in the heterogeneous structure;
- derivation of nonlocal interface models from homogenization of discrete structures and interfaces, showing that internal characteristic length can be deduced from the consideration of the microstructure through homogenization techniques;
- crack nucleation from, and interaction with, material micro-instabilities (such as shear bands and flutter)
Description of the work performed since the beginning of the project and main results:
1. General transmission conditions for thin elasto-plastic pressure-dependent interphase between dissimilar materials
In this work, a thin soft adhesive interphase between dissimilar elastic media is considered. The material of the intermediate layer is modelled by elasto-plastic pressure-sensitive constitutive law. An asymptotic procedure, together with a novel formulation of the deformation theory of plasticity for pressure-sensitive materials, is used in order to derive nonlinear transmission conditions for the corresponding imperfect zero-thickness interface. A FEM analysis of the original three-phase structure is performed to validate the transmission conditions for the simplified bimaterial structure. This work has been published in:
- Mishuris, G., Miszuris, W., Ochsner, A. and Piccolroaz, A. (2013). Transmission conditions for thin elasto-plastic pressure-dependent interphases. In: Plasticity of Pressure-Sensitive Materials. Altenbach, H., Ochsner, A., Eds., Springer-Verlag, Berlin.
- Sonato, M., Piccolroaz, A., Miszuris, W., Mishuris, G. (2015). General transmission conditions for thin elasto-plastic pressure-dependent interphase between dissimilar materials. International Journal of Solids and Structures, 64-65, 9-21. http://dx.doi.org/10.1016/j.ijsolstr.2015.03.009
2. Dispersion and localisation in structured Rayleigh beams
This part of the work brings a comparative analysis between dynamic models of couple-stress elastic materials and structured Rayleigh beams on a Winkler foundation. Although physical phenomena have different physical origins, the underlying equations appear to be similar, and hence mathematical models have a lot in common. In the present work, the main focus is on the analysis of dispersive waves, band-gaps and localised waveforms in structured Rayleigh beams. The Rayleigh beam theory includes the effects of rotational inertia which are neglected in the Euler–Bernoulli beam theory. This makes the approach applicable to higher frequency regimes. Special attention is given to waves in pre-stressed Rayleigh beams on elastic foundations. This work has been published in:
- Piccolroaz, A., Movchan, A.B. (2014). Dispersion and localization in structured Rayleigh beams. International Journal of Solids and Structures, 51, 4452-4461. http://dx.doi.org/10.1016/j.ijsolstr.2014.09.016
3. Dispersion degeneracies in flexural waves supported by Rayleigh beam structures
The paper presents a novel analysis of Floquet-Bloch flexural waves in a periodic lattice-like structure consisting of flexural beam ligaments. A special feature of this structure is in the presence of the rotational inertia, which is commonly neglected in conventional models of the Euler-Bernoulli type. The dispersion properties of the Rayleigh beam structure with rotational inertia include degeneracies linked to Dirac cones on the dispersion diagrams as well as directional anisotropy and special refraction properties. Steering of Dirac cones is described for rectangular flexural structures with a rotational inertia. This work has been published in:
- Piccolroaz, A., Movchan, A.B. Cabras, L. (2017) Dispersion degeneracies and standing modes in flexural waves supported by Rayleigh beam structures. International Journal of Solids and Structures, 109, 152-165. http://dx.doi.org/10.1016/j.ijsolstr.2017.01.017
4. Strain localization and shear band propagation in ductile materials
A model of a shear band as a zero-thickness non-linear interface is proposed and tested using finite element simulations. An imperfection approach is used in this model where a shear band that is assumed to lie in a ductile matrix material (obeying von Mises plasticity with linear hardening), is present from the beginning of loading and is considered to be a zone in which yielding occurs before the rest of the matrix. This approach is contrasted with a perturbative approach, developed for a J2-deformation theory material, in which the shear band is modelled to emerge at a certain stage of a uniform deformation. Both approaches concur in showing that the shear bands (differently from cracks) propagate rectilinearly under shear loading and that a strong stress concentration should be expected to be present at the tip of the shear band, two key features in the understanding of failure mechanisms of ductile materials. This work has been published in:
- Bordignon, N., Piccolroaz, A., Dal Corso, F., Bigoni, D. (2015). Strain localization and shear band propagation in ductile materials. Front. Mater. 2:22. http://dx.doi.org/10.3389/fmats.2015.00022
- Bordignon, N., Piccolroaz, A. (2017) Cohesive modelling of thin elasto-plastic pressure-dependent adhesive joints. Proceedings of the 21st International Conference on Composite Materials, ICCM-21.
5. Multiscale asymptotic homogenization analysis of thermo-diffusive composite materials
In this work, an asymptotic homogenization method for the analysis of composite materials with periodic microstructure in presence of thermodiffusion is described. Appropriate down-scaling relations correlating the microscopic fields to the macroscopic displacements, temperature and mass concentration are introduced. The effects of the material inhomogeneities are described by perturbation functions derived from the solution of recursive cell problems. Exact expressions for the overall elastic and thermodiffusive constants of the equivalent first-order thermodiffusive continuum are derived. The proposed approach is applied to the case of a two-dimensional bi-phase orthotropic layered material, where the effective elastic and thermodiffusive properties can be determined analytically. Considering this illustrative example and assuming periodic body forces, heat and mass sources acting on the medium, the solution performed by the first order homogenization approach is compared with the numerical results obtained by the heterogeneous model. This work has been published in:
- Bacigalupo, A., Morini, L., Piccolroaz, A. (2016) Multiscale asymptotic homogenization analysis of thermo-diffusive composite materials. International Journal of Solids and Structures, 85-86, 15-33. http://dx.doi.org/10.1016/j.ijsolstr.2016.01.016
6. Overall thermomechanical properties of layered materials
This work is concerned with the analysis of effective thermomechanical properties of multi-layered materials of interest for solid oxide fuel cells (SOFC) and lithium ions batteries fabrication. The recently developed asymptotic homogenization procedure is applied in order to express the overall thermoelastic constants of the first order equivalent continuum in terms of microfluctuations functions, and these functions are obtained by the solution of the corresponding recursive cell problems. The effects of thermal stresses on periodic multi-layered thermoelastic composite reproducing the characteristics of solid oxide fuel cells (SOFC-like) are studied assuming periodic body forces and heat sources, and the solution derived by means of the asymptotic homogenization approach is compared with the results obtained by finite element analysis of the associate heterogeneous material. This work has been published in:
- Bacigalupo, A., Morini, L., Piccolroaz, A. (2016) Overall thermomechanical properties of layered materials for energy devices applications. Composite Structures, 157, 366-385. http://dx.doi.org/10.1016/j.compstruct.2016.07.048
7. Crack dynamics in a bimaterial lattice
This works addresses the propagation of a crack along a structured interface (lattice) joining two dissimilar materials. The problem is reduced to the functional equation of the Wiener–Hopf type, which is solved analytically The existence of interface waves along the structured layer is theoretically proved. The dispersion properties of such waves are analytically derived. The load–crack speed dependence is obtained, which also has implications for the stability analysis for the crack propagating along the structured layer. In particular, we address the evaluation of the dissipation rate and the energy release rate, which is found to be strongly dependent on the crack speed, the mismatch between the two materials and the properties of the discrete interface. A paper on this work is in preparation.
- Piccolroaz, A., Gorbushin, N., Mishuris, G. Crack dynamics in a bimaterial lattice, in preparation.
8. Energy release rate in hydraulic fracture
A novel hydraulic fracture (HF) formulation is introduced which accounts for the hydraulically induced shear stress at the crack faces. It utilizes a general form of the elasticity operator alongside a revised fracture propagation condition based on the critical value of the energy release rate. It is shown that the revised formulation describes the underlying physics of HF in a more accurate way and is in agreement with the asymptotic behaviour of the linear elastic fracture mechanics. A number of numerical simulations by means of the universal HF algorithm previously developed in [Wrobel M., Mishuris G. (2015) Hydraulic fracture revisited: Particle velocity based simulation. International Journal of Engineering Science, 94: 23–58] are performed in order to: (i) compare the modified HF formulation with its classic counterpart and (ii) investigate the peculiarities of the former. Computational advantages of the revised HF model are demonstrated. Asymptotic estimations of the main solution elements are provided for the cases of small and large toughness. The modified formulation opens new ways to analyse the physical phenomenon of HF and also improves the reliability and efficiency of its numerical simulations. This work has been published in:
- Wrobel, M., Mishuris, G., Piccolroaz, A. (2016) Energy release rate in hydraulic fracture: Can we neglect an impact of the hydraulically induced shear stress?. International Journal of Engineering Science, 111, 28-51. http://dx.doi.org/10.1016/j.ijengsci.2016.09.013
9. Redirection of a crack driven by viscous fluid
As shown by Wrobel et al. (2017), the hydraulically induced tangential traction on fracture walls changes local displacement and stress fields. This resulted in the formulation of a new hydraulic fracture (HF) propagation condition based on the critical value of the energy release rate that accounts for the hydraulically-induced shear stress. Therefore it is clear that the crack direction criteria, which depend on the tip distributions of the stress and strain fields, need to be changed. We analyse the two commonly used criteria, one based on the maximum circumferential stress (MCS) and another - on the minimum strain energy density (MSED). We show that the impact of the hydraulically induced shear stress on the direction of the crack propagation is negligible in the case of large material resistance to fracture, while for small toughness the effect is significant. Moreover, values of the redirection angles, corresponding to the so-called viscosity dominated regime, depend dramatically on the ratios of the stress intensity factors. This work has been published in:
- Perkowska, M., Piccolroaz, A., Wrobel, M., Mishuris, G. (2017) Redirection of a crack driven by viscous fluid. International Journal of Engineering Science, in press.
10. Floquet-Bloch waves in periodic networks of the Rayleigh beams: honeycomb systems, dispersion degeneracies and structured interfaces.
This work addresses novel dispersion properties of elastic flexural waves in periodic structures which possess rotational inertia. The structure is represented as a lattice, whose elementary links are formally defined as the Rayleigh beams. Although in the quasi-static regime such beams respond similarly to the classical Euler-Bernoulli beams, as the frequency increases the dispersion of flexural waves possesses new interesting features. For a doubly periodic lattice, we give a special attention to degeneracies associated with so-called Dirac cones on the dispersion surfaces as well as directional anisotropy. Comparative analysis for Floquet-Bloch waves in periodic flexural lattices of different geometries is presented and accompanied by numerical simulations. This work has been published in:
- Cabras, L., Movchan, A.B. Piccolroaz, A. (2017) Floquet-Bloch waves in periodic networks of the Rayleigh beams: honeycomb systems, dispersion degeneracies and structured interfaces. Mechanics of Solids, A Journal of Russian Academy of Sciences, 5, 93-108.
The final result is the development of a consistent theoretical and computational framework for the analysis of crack and failure propagation in materials with complex heterogeneous structure. This will have a potential impact on the design of new composite high-performance materials. The address of the project website is http://www.ing.unitn.it/dims/memic/
- modelling the propagation of a crack in microstructured interphases, incorporating the microstructural properties (length-scales) into the definition of fracture toughness and crack propagation stability;
- modelling the propagation of a crack in discrete interphases and lattice structures joining dissimilar materials; the formulation of the corresponding dynamic problem and crack-waves interactions in the heterogeneous structure;
- derivation of nonlocal interface models from homogenization of discrete structures and interfaces, showing that internal characteristic length can be deduced from the consideration of the microstructure through homogenization techniques;
- crack nucleation from, and interaction with, material micro-instabilities (such as shear bands and flutter)
Description of the work performed since the beginning of the project and main results:
1. General transmission conditions for thin elasto-plastic pressure-dependent interphase between dissimilar materials
In this work, a thin soft adhesive interphase between dissimilar elastic media is considered. The material of the intermediate layer is modelled by elasto-plastic pressure-sensitive constitutive law. An asymptotic procedure, together with a novel formulation of the deformation theory of plasticity for pressure-sensitive materials, is used in order to derive nonlinear transmission conditions for the corresponding imperfect zero-thickness interface. A FEM analysis of the original three-phase structure is performed to validate the transmission conditions for the simplified bimaterial structure. This work has been published in:
- Mishuris, G., Miszuris, W., Ochsner, A. and Piccolroaz, A. (2013). Transmission conditions for thin elasto-plastic pressure-dependent interphases. In: Plasticity of Pressure-Sensitive Materials. Altenbach, H., Ochsner, A., Eds., Springer-Verlag, Berlin.
- Sonato, M., Piccolroaz, A., Miszuris, W., Mishuris, G. (2015). General transmission conditions for thin elasto-plastic pressure-dependent interphase between dissimilar materials. International Journal of Solids and Structures, 64-65, 9-21. http://dx.doi.org/10.1016/j.ijsolstr.2015.03.009
2. Dispersion and localisation in structured Rayleigh beams
This part of the work brings a comparative analysis between dynamic models of couple-stress elastic materials and structured Rayleigh beams on a Winkler foundation. Although physical phenomena have different physical origins, the underlying equations appear to be similar, and hence mathematical models have a lot in common. In the present work, the main focus is on the analysis of dispersive waves, band-gaps and localised waveforms in structured Rayleigh beams. The Rayleigh beam theory includes the effects of rotational inertia which are neglected in the Euler–Bernoulli beam theory. This makes the approach applicable to higher frequency regimes. Special attention is given to waves in pre-stressed Rayleigh beams on elastic foundations. This work has been published in:
- Piccolroaz, A., Movchan, A.B. (2014). Dispersion and localization in structured Rayleigh beams. International Journal of Solids and Structures, 51, 4452-4461. http://dx.doi.org/10.1016/j.ijsolstr.2014.09.016
3. Dispersion degeneracies in flexural waves supported by Rayleigh beam structures
The paper presents a novel analysis of Floquet-Bloch flexural waves in a periodic lattice-like structure consisting of flexural beam ligaments. A special feature of this structure is in the presence of the rotational inertia, which is commonly neglected in conventional models of the Euler-Bernoulli type. The dispersion properties of the Rayleigh beam structure with rotational inertia include degeneracies linked to Dirac cones on the dispersion diagrams as well as directional anisotropy and special refraction properties. Steering of Dirac cones is described for rectangular flexural structures with a rotational inertia. This work has been published in:
- Piccolroaz, A., Movchan, A.B. Cabras, L. (2017) Dispersion degeneracies and standing modes in flexural waves supported by Rayleigh beam structures. International Journal of Solids and Structures, 109, 152-165. http://dx.doi.org/10.1016/j.ijsolstr.2017.01.017
4. Strain localization and shear band propagation in ductile materials
A model of a shear band as a zero-thickness non-linear interface is proposed and tested using finite element simulations. An imperfection approach is used in this model where a shear band that is assumed to lie in a ductile matrix material (obeying von Mises plasticity with linear hardening), is present from the beginning of loading and is considered to be a zone in which yielding occurs before the rest of the matrix. This approach is contrasted with a perturbative approach, developed for a J2-deformation theory material, in which the shear band is modelled to emerge at a certain stage of a uniform deformation. Both approaches concur in showing that the shear bands (differently from cracks) propagate rectilinearly under shear loading and that a strong stress concentration should be expected to be present at the tip of the shear band, two key features in the understanding of failure mechanisms of ductile materials. This work has been published in:
- Bordignon, N., Piccolroaz, A., Dal Corso, F., Bigoni, D. (2015). Strain localization and shear band propagation in ductile materials. Front. Mater. 2:22. http://dx.doi.org/10.3389/fmats.2015.00022
- Bordignon, N., Piccolroaz, A. (2017) Cohesive modelling of thin elasto-plastic pressure-dependent adhesive joints. Proceedings of the 21st International Conference on Composite Materials, ICCM-21.
5. Multiscale asymptotic homogenization analysis of thermo-diffusive composite materials
In this work, an asymptotic homogenization method for the analysis of composite materials with periodic microstructure in presence of thermodiffusion is described. Appropriate down-scaling relations correlating the microscopic fields to the macroscopic displacements, temperature and mass concentration are introduced. The effects of the material inhomogeneities are described by perturbation functions derived from the solution of recursive cell problems. Exact expressions for the overall elastic and thermodiffusive constants of the equivalent first-order thermodiffusive continuum are derived. The proposed approach is applied to the case of a two-dimensional bi-phase orthotropic layered material, where the effective elastic and thermodiffusive properties can be determined analytically. Considering this illustrative example and assuming periodic body forces, heat and mass sources acting on the medium, the solution performed by the first order homogenization approach is compared with the numerical results obtained by the heterogeneous model. This work has been published in:
- Bacigalupo, A., Morini, L., Piccolroaz, A. (2016) Multiscale asymptotic homogenization analysis of thermo-diffusive composite materials. International Journal of Solids and Structures, 85-86, 15-33. http://dx.doi.org/10.1016/j.ijsolstr.2016.01.016
6. Overall thermomechanical properties of layered materials
This work is concerned with the analysis of effective thermomechanical properties of multi-layered materials of interest for solid oxide fuel cells (SOFC) and lithium ions batteries fabrication. The recently developed asymptotic homogenization procedure is applied in order to express the overall thermoelastic constants of the first order equivalent continuum in terms of microfluctuations functions, and these functions are obtained by the solution of the corresponding recursive cell problems. The effects of thermal stresses on periodic multi-layered thermoelastic composite reproducing the characteristics of solid oxide fuel cells (SOFC-like) are studied assuming periodic body forces and heat sources, and the solution derived by means of the asymptotic homogenization approach is compared with the results obtained by finite element analysis of the associate heterogeneous material. This work has been published in:
- Bacigalupo, A., Morini, L., Piccolroaz, A. (2016) Overall thermomechanical properties of layered materials for energy devices applications. Composite Structures, 157, 366-385. http://dx.doi.org/10.1016/j.compstruct.2016.07.048
7. Crack dynamics in a bimaterial lattice
This works addresses the propagation of a crack along a structured interface (lattice) joining two dissimilar materials. The problem is reduced to the functional equation of the Wiener–Hopf type, which is solved analytically The existence of interface waves along the structured layer is theoretically proved. The dispersion properties of such waves are analytically derived. The load–crack speed dependence is obtained, which also has implications for the stability analysis for the crack propagating along the structured layer. In particular, we address the evaluation of the dissipation rate and the energy release rate, which is found to be strongly dependent on the crack speed, the mismatch between the two materials and the properties of the discrete interface. A paper on this work is in preparation.
- Piccolroaz, A., Gorbushin, N., Mishuris, G. Crack dynamics in a bimaterial lattice, in preparation.
8. Energy release rate in hydraulic fracture
A novel hydraulic fracture (HF) formulation is introduced which accounts for the hydraulically induced shear stress at the crack faces. It utilizes a general form of the elasticity operator alongside a revised fracture propagation condition based on the critical value of the energy release rate. It is shown that the revised formulation describes the underlying physics of HF in a more accurate way and is in agreement with the asymptotic behaviour of the linear elastic fracture mechanics. A number of numerical simulations by means of the universal HF algorithm previously developed in [Wrobel M., Mishuris G. (2015) Hydraulic fracture revisited: Particle velocity based simulation. International Journal of Engineering Science, 94: 23–58] are performed in order to: (i) compare the modified HF formulation with its classic counterpart and (ii) investigate the peculiarities of the former. Computational advantages of the revised HF model are demonstrated. Asymptotic estimations of the main solution elements are provided for the cases of small and large toughness. The modified formulation opens new ways to analyse the physical phenomenon of HF and also improves the reliability and efficiency of its numerical simulations. This work has been published in:
- Wrobel, M., Mishuris, G., Piccolroaz, A. (2016) Energy release rate in hydraulic fracture: Can we neglect an impact of the hydraulically induced shear stress?. International Journal of Engineering Science, 111, 28-51. http://dx.doi.org/10.1016/j.ijengsci.2016.09.013
9. Redirection of a crack driven by viscous fluid
As shown by Wrobel et al. (2017), the hydraulically induced tangential traction on fracture walls changes local displacement and stress fields. This resulted in the formulation of a new hydraulic fracture (HF) propagation condition based on the critical value of the energy release rate that accounts for the hydraulically-induced shear stress. Therefore it is clear that the crack direction criteria, which depend on the tip distributions of the stress and strain fields, need to be changed. We analyse the two commonly used criteria, one based on the maximum circumferential stress (MCS) and another - on the minimum strain energy density (MSED). We show that the impact of the hydraulically induced shear stress on the direction of the crack propagation is negligible in the case of large material resistance to fracture, while for small toughness the effect is significant. Moreover, values of the redirection angles, corresponding to the so-called viscosity dominated regime, depend dramatically on the ratios of the stress intensity factors. This work has been published in:
- Perkowska, M., Piccolroaz, A., Wrobel, M., Mishuris, G. (2017) Redirection of a crack driven by viscous fluid. International Journal of Engineering Science, in press.
10. Floquet-Bloch waves in periodic networks of the Rayleigh beams: honeycomb systems, dispersion degeneracies and structured interfaces.
This work addresses novel dispersion properties of elastic flexural waves in periodic structures which possess rotational inertia. The structure is represented as a lattice, whose elementary links are formally defined as the Rayleigh beams. Although in the quasi-static regime such beams respond similarly to the classical Euler-Bernoulli beams, as the frequency increases the dispersion of flexural waves possesses new interesting features. For a doubly periodic lattice, we give a special attention to degeneracies associated with so-called Dirac cones on the dispersion surfaces as well as directional anisotropy. Comparative analysis for Floquet-Bloch waves in periodic flexural lattices of different geometries is presented and accompanied by numerical simulations. This work has been published in:
- Cabras, L., Movchan, A.B. Piccolroaz, A. (2017) Floquet-Bloch waves in periodic networks of the Rayleigh beams: honeycomb systems, dispersion degeneracies and structured interfaces. Mechanics of Solids, A Journal of Russian Academy of Sciences, 5, 93-108.
The final result is the development of a consistent theoretical and computational framework for the analysis of crack and failure propagation in materials with complex heterogeneous structure. This will have a potential impact on the design of new composite high-performance materials. The address of the project website is http://www.ing.unitn.it/dims/memic/