An estimation is found for the energy release due to fragmentation of a brittle inhomogeneity of ... more An estimation is found for the energy release due to fragmentation of a brittle inhomogeneity of ellipsoidal shape embedded in a ductile matrix under remote static loading. In the state of completed fragmentation the inhomogeneity is replaced by a void with zero stiffness. Thus, the problem of estimating the energy release reduces to the eigenstrain problem solved by Eshelby. The energy release calculated for prolate spheroidal inhomogeneities is used in the balance of energy to determine the crack density. The application to the geological system of garnet inhomogeneities embedded in a quartz matrix is considered.
Variational based effective models for inelastic materials
PAMM, 2021
Mechanical systems with inelastic materials and given boundary conditions can be generally define... more Mechanical systems with inelastic materials and given boundary conditions can be generally defined and solved. However, for large systems, this problem becomes non trivial and computationally costly, especially when microstructures occure. Therefore, multi‐scale modeling is used to capture the behavior of the system.In this study, we present a methodology to provide such models for inelastic materials starting from the variational scheme. Using the variational principle we define the material strain energy and dissipation potential at the coarse scale (macro scale) starting from their counterparts on the fine scale, through the homogenization scheme. Then we solve the material model at the macro level instead of solving the fine scale model or the whole combination of scales. This multi‐scale model offers a reduced number of degrees of freedom and can generally provide the behavior of the system. Moreover, the macro scale model preserves the mathematical structure of the micro scale...
Predicting the Geological Structure Ahead of a Tunnel Using Full Waveform Inversion - A Blind Test
78th EAGE Conference and Exhibition 2016, 2016
Looking ahead of a tunnel is an important issue in order to have a smooth and safe construction p... more Looking ahead of a tunnel is an important issue in order to have a smooth and safe construction process. Seismic methods are used to locate the geological anomalies ahead of a tunnel; the waves reflected from the geological changes carry information about the location and mechanical properties of the objects. Different migration techniques and first arrivals of the reflected waves are commonly used to achieve the goal. Full waveforms carry more information about the geological structure of a domain than the first traveltimes of the reflected waves. This work emerges from the curiosity to see whether full waveforms can successfully image the velocity field in tunnel applications. There are two parties in this study. The first party chooses a synthetic model and provides the second party with the precisely calculated waveforms in the time domain. The second party has no prior information about the synthetic model except for the waveforms and the inversion is carried out in the frequency domain as a blind test. The aim is to see whether the velocity field can be successfully imaged with the full waveform inversion.
Journal of the Mechanics and Physics of Solids, 2021
When they are damaged or injured, soft biological tissues are able to self-repair and heal. Mecha... more When they are damaged or injured, soft biological tissues are able to self-repair and heal. Mechanics is critical during the healing process, as the damaged extracellular matrix (ECM) tends to be replaced with a new undamaged ECM supporting homeostatic stresses. Computational modeling has been commonly used to simulate the healing process. However, there is a pressing need to have a unified thermodynamics theory for healing. From the viewpoint of continuum damage mechanics, some key parameters related to healing processes, for instance, the volume fraction of newly grown soft tissue and the growth deformation, can be regarded as internal variables and have related evolution equations. This paper is aiming to establish this unified framework inspired by thermodynamics for continuum damage models for the healing of soft biological tissues. The significant advantage of the proposed model is that no ad hoc equations are required for describing the healing process. Therefore, this new model is more concise and offers a universal approach to simulate the healing process. Three numerical examples are provided to demonstrate the effectiveness of the proposed model, which is in good agreement with the existing works, including an application for balloon angioplasty in an arteriosclerotic artery with a fiber cap.
Seismic exploration in tunneling using full waveform inversion with a frequency domain model
PAMM, 2021
With the knowledge of the geology in front of a tunnel, the excavation process can be optimized t... more With the knowledge of the geology in front of a tunnel, the excavation process can be optimized to avoid damage at the tunnel boring machine and settlements on the surface. Therefore, dwell times can be decreased and additional expenses can be avoided. Transmitted seismic waves will be spread, reflected and refracted due to geological changes. By utilizing geophones, the seismic waves will be captured and information about the geological structure in front of the tunnel can be extracted from the measured seismograms using e.g. the concept of full waveform inversion. A frequency domain model has been employed to demonstrate the potential of full waveform inversion for seismic reconnaissance in a tunnel environment. The success of the inversion procedure depends strongly on the positions of the utilized sender and receiver stations, the chosen initial material parameter distribution, and on the selected frequency groups for the inversion. Further challenges are an accurate representat...
International Journal for Multiscale Computational Engineering, 2021
Modeling and simulation have quickly become equivalent pillars of research along with traditional... more Modeling and simulation have quickly become equivalent pillars of research along with traditional theory and experimentation. The growing realization that most complex phenomena of interest span many orders of spatial and temporal scales has led to an exponential rise in the development and application of multiscale modeling and simulation over the past two decades. In this perspective, the associate editors of the International Journal for Multiscale Computational Engineering and their co-workers illustrate current applications in their respective fields spanning biomolecular structure and dynamics, civil engineering and materials science, computational mechanics, aerospace and mechanical engineering, and more. Such applications are highly tailored, exploit the latest and ever-evolving advances in both computer hardware and software, and contribute significantly to science, technology, and medical challenges in the 21st century.
Acoustic waveform inversion in frequency domain: Application to a tunnel environment
Underground Space, 2021
Abstract Waveform inversion is an approach used to find an optimal model for the velocity field o... more Abstract Waveform inversion is an approach used to find an optimal model for the velocity field of a ground structure such that the dynamic response is close enough to the given seismic data. First, a suitable numerical approach is employed to establish a realistic forward computer model. The forward problem is solved in the frequency domain using higher-order finite elements. The velocity field is inverted over a specific number of discrete frequencies; thereby, reducing the computational cost of the forward calculation and the nonlinearity of the inverse problem. The results are presented for different frequency sets and with different source and receiver locations for a two-dimensional model. The influence of attenuation effects is also investigated. The results of two blind tests are presented where only the seismic records of an unknown synthetic model with an inhomogeneous material parameter distribution are provided to mimic a more realistic case. Finally, the result of the inversion in a three-dimensional space is illustrated.
An Efficient Treatment of the Laplacian in a Gradient‐Enhanced Damage Model
PAMM, 2019
As is well‐known, softening effects that are characteristic for damage models are accompanied by ... more As is well‐known, softening effects that are characteristic for damage models are accompanied by ill‐posed boundary value problems. This arises from non‐convex and non‐coercive energies and results in mesh‐dependent finite element results. For that reason, regularization strategies, which somehow take into account the non‐local behavior, have to be applied in order to prevent ill‐posedness and to achieve mesh‐independence. Hereto, most commonly gradient‐enhanced formulations [1, 2] are considered, but also integral‐type [3,4] and viscous [5,6] regularization are well‐known.Gradient‐enhanced damage models such as [1,2], to what group our new model [7] basically belongs, come along with a field function acting on the non‐local level. Two variational equations are resulting and, however, usually the number of nodal unknowns is increased and consequently the numerical effort is increased as well.In contrast, we present an improved algorithm for brittle damage [7] combining finite elemen...
By transmitting seismic waves in the ground the spread, reflected and refracted waves can be reco... more By transmitting seismic waves in the ground the spread, reflected and refracted waves can be recorded at different positions. The arising seismograms contain information of the geology, which can be evaluated with the Full Waveform Inversion (FWI). For the application of FWI for reconnaissance in tunneling extensive information has to be provided to achieve an appropriate model of the geology in front of the tunnel face. Such information are the position and geometry of reflecting boundaries, exact position of sources and receivers, a proper initial model of the distribution of the material parameters. The signal, which is transmitted by the senders, has also an influence on the inversion. Even if the intended source function is known, the effective signal depends on the transfer functions between sender-soil and soil-receiver and its identification is challenging. To avoid cumbersome investigations on the source function in advance we applied a simultaneous identification of the source signature during the FWI in the frequency domain. For synthetic models of a 2D tunnel configuration with disturbances, we will show how the calculated source signature of a frequency converges to the real source signature during the iterations of the FWI. The influence of different sender and receiver configurations will be analyzed. Furthermore, the influence of the simultaneous evaluation of the source signature on the results of the FWI will be evaluated. In a post-processing step of the FWI, the source signature of the whole frequency spectrum will be determined.
The prediction of failure mechanism in structures are always an important topic in the field of c... more The prediction of failure mechanism in structures are always an important topic in the field of computational mechanics. Finite element computations of an inelastic material involving softening behavior (e.g. softening plasticity or damage) can suffer from strongly mesh‐dependent results. Therefore, such continuum models should be equipped with a regularization (localization limiter) strategy to overcome the above‐mentioned problem.In this study, we present a framework for gradient‐enhancement for coupled damage‐plasticity material model derived by means of Hamilton's principle for non‐conservative continua. This model is applied for the numerical investigation of wear processes as they occur, e.g. in the case of mechanized tunneling. These investigations require a fine resolution of the involved constituents (cut sheet and abrasive particles in the soil). Consequently, a numerical strategy for the damage‐plasticity model is demanded that allows for time‐efficient simulations.In...
, phone +49 234 322 6025, 1 For the definition of j surface flux, j flux along the triple line, µ... more , phone +49 234 322 6025, 1 For the definition of j surface flux, j flux along the triple line, µ i chemical potential see [1]
Material models, including softening effects due to, for example, damage and localizations, share... more Material models, including softening effects due to, for example, damage and localizations, share the problem of ill-posed boundary value problems that yield mesh-dependent finite element results. It is thus necessary to apply regularization techniques that couple local behavior described, for example, by internal variables, at a spatial level. This can take account of the gradient of the internal variable to yield mesh-independent finite element results. In this paper, we present a new approach to damage modeling that does not use common field functions, inclusion of gradients or complex integration techniques: Appropriate modifications of the relaxed (condensed) energy hold the same advantage as other methods, but with much less numerical effort. We start with the theoretical derivation and then discuss the numerical treatment. Finally, we present finite element results that prove empirically how the new approach works. Communicated by Andreas Öchsner.
International Journal for Numerical Methods in Engineering, 2016
This study presents a polytree-based adaptive methodology for multi-material topology optimizatio... more This study presents a polytree-based adaptive methodology for multi-material topology optimization (MP-TOP). Polytree data structure is introduced as a general recursive multi-level mesh that is automatically refined in processing based on error analysis. In order to resolve hanging nodes in element edges, the Wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons. A new definition of filter radius is also proposed to improve the efficiency of filters and optimized results. The combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases (including void phase), but also decreases the computing time of the overall process in comparison to using the regular fine meshes. Several benchmark and practical problems are considered to show distinct features of the proposed method.
International Journal for Multiscale Computational Engineering, 2016
Our previous work proposes a micromechanical model for dissolution-precipitation creep, an elasto... more Our previous work proposes a micromechanical model for dissolution-precipitation creep, an elasto-viscoplastic process supposed to be one of the main reasons for the tectonic motion of earth plates in the subduction zone. While the model in its original form enables the simulation of polycrystals with a limited number of crystals, the topic of the present contribution is its extension to simulating structures on a much larger spatial scale. For this purpose, a homogenization technique known as the multiscale finite element method is used. Here, the behavior of a heterogeneous body is simulated by solving two boundary value problems: one related to the structural level and one related to the representative volume element. The coupling of scales is established by introducing the Hill macrohomogeneity condition requiring the equality of the macropower with the volume average of the micropower. The method allows the simulating of various tasks at both levels. The examples concerned with simulating the tension tests of a macroscopic plate with different types of the microstructure are presented.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015
A micromechanical model for finite single crystal plasticity was introduced by Kochmann & Hackl (... more A micromechanical model for finite single crystal plasticity was introduced by Kochmann & Hackl (2011 Contin. Mech. Thermodyn. 23, 63–85 ( doi:10.1007/s00161-010-0714-5 )). This model is based on thermodynamic variational principles and leads to a non-convex variational problem. Based on the Lagrange functional, an incremental strategy was outlined to model the time-continuous evolution of a first-order laminate microstructure. Although this model provides interesting results on the material point level, owing to the global minimization in the evolution equations, the calculation time and numerical instabilities may cause problems when applying this model to macroscopic specimens. In this paper, a smooth transition zone between the laminates is introduced to avoid global minimization, which makes the numerical calculations cumbersome compared with the model in Kochmann & Hackl. By introducing a smooth viscous transition zone, the dissipation potential and its numerical treatment hav...
An estimation is found for the energy release due to fragmentation of a brittle inhomogeneity of ... more An estimation is found for the energy release due to fragmentation of a brittle inhomogeneity of ellipsoidal shape embedded in a ductile matrix under remote static loading. In the state of completed fragmentation the inhomogeneity is replaced by a void with zero stiffness. Thus, the problem of estimating the energy release reduces to the eigenstrain problem solved by Eshelby. The energy release calculated for prolate spheroidal inhomogeneities is used in the balance of energy to determine the crack density. The application to the geological system of garnet inhomogeneities embedded in a quartz matrix is considered.
Variational based effective models for inelastic materials
PAMM, 2021
Mechanical systems with inelastic materials and given boundary conditions can be generally define... more Mechanical systems with inelastic materials and given boundary conditions can be generally defined and solved. However, for large systems, this problem becomes non trivial and computationally costly, especially when microstructures occure. Therefore, multi‐scale modeling is used to capture the behavior of the system.In this study, we present a methodology to provide such models for inelastic materials starting from the variational scheme. Using the variational principle we define the material strain energy and dissipation potential at the coarse scale (macro scale) starting from their counterparts on the fine scale, through the homogenization scheme. Then we solve the material model at the macro level instead of solving the fine scale model or the whole combination of scales. This multi‐scale model offers a reduced number of degrees of freedom and can generally provide the behavior of the system. Moreover, the macro scale model preserves the mathematical structure of the micro scale...
Predicting the Geological Structure Ahead of a Tunnel Using Full Waveform Inversion - A Blind Test
78th EAGE Conference and Exhibition 2016, 2016
Looking ahead of a tunnel is an important issue in order to have a smooth and safe construction p... more Looking ahead of a tunnel is an important issue in order to have a smooth and safe construction process. Seismic methods are used to locate the geological anomalies ahead of a tunnel; the waves reflected from the geological changes carry information about the location and mechanical properties of the objects. Different migration techniques and first arrivals of the reflected waves are commonly used to achieve the goal. Full waveforms carry more information about the geological structure of a domain than the first traveltimes of the reflected waves. This work emerges from the curiosity to see whether full waveforms can successfully image the velocity field in tunnel applications. There are two parties in this study. The first party chooses a synthetic model and provides the second party with the precisely calculated waveforms in the time domain. The second party has no prior information about the synthetic model except for the waveforms and the inversion is carried out in the frequency domain as a blind test. The aim is to see whether the velocity field can be successfully imaged with the full waveform inversion.
Journal of the Mechanics and Physics of Solids, 2021
When they are damaged or injured, soft biological tissues are able to self-repair and heal. Mecha... more When they are damaged or injured, soft biological tissues are able to self-repair and heal. Mechanics is critical during the healing process, as the damaged extracellular matrix (ECM) tends to be replaced with a new undamaged ECM supporting homeostatic stresses. Computational modeling has been commonly used to simulate the healing process. However, there is a pressing need to have a unified thermodynamics theory for healing. From the viewpoint of continuum damage mechanics, some key parameters related to healing processes, for instance, the volume fraction of newly grown soft tissue and the growth deformation, can be regarded as internal variables and have related evolution equations. This paper is aiming to establish this unified framework inspired by thermodynamics for continuum damage models for the healing of soft biological tissues. The significant advantage of the proposed model is that no ad hoc equations are required for describing the healing process. Therefore, this new model is more concise and offers a universal approach to simulate the healing process. Three numerical examples are provided to demonstrate the effectiveness of the proposed model, which is in good agreement with the existing works, including an application for balloon angioplasty in an arteriosclerotic artery with a fiber cap.
Seismic exploration in tunneling using full waveform inversion with a frequency domain model
PAMM, 2021
With the knowledge of the geology in front of a tunnel, the excavation process can be optimized t... more With the knowledge of the geology in front of a tunnel, the excavation process can be optimized to avoid damage at the tunnel boring machine and settlements on the surface. Therefore, dwell times can be decreased and additional expenses can be avoided. Transmitted seismic waves will be spread, reflected and refracted due to geological changes. By utilizing geophones, the seismic waves will be captured and information about the geological structure in front of the tunnel can be extracted from the measured seismograms using e.g. the concept of full waveform inversion. A frequency domain model has been employed to demonstrate the potential of full waveform inversion for seismic reconnaissance in a tunnel environment. The success of the inversion procedure depends strongly on the positions of the utilized sender and receiver stations, the chosen initial material parameter distribution, and on the selected frequency groups for the inversion. Further challenges are an accurate representat...
International Journal for Multiscale Computational Engineering, 2021
Modeling and simulation have quickly become equivalent pillars of research along with traditional... more Modeling and simulation have quickly become equivalent pillars of research along with traditional theory and experimentation. The growing realization that most complex phenomena of interest span many orders of spatial and temporal scales has led to an exponential rise in the development and application of multiscale modeling and simulation over the past two decades. In this perspective, the associate editors of the International Journal for Multiscale Computational Engineering and their co-workers illustrate current applications in their respective fields spanning biomolecular structure and dynamics, civil engineering and materials science, computational mechanics, aerospace and mechanical engineering, and more. Such applications are highly tailored, exploit the latest and ever-evolving advances in both computer hardware and software, and contribute significantly to science, technology, and medical challenges in the 21st century.
Acoustic waveform inversion in frequency domain: Application to a tunnel environment
Underground Space, 2021
Abstract Waveform inversion is an approach used to find an optimal model for the velocity field o... more Abstract Waveform inversion is an approach used to find an optimal model for the velocity field of a ground structure such that the dynamic response is close enough to the given seismic data. First, a suitable numerical approach is employed to establish a realistic forward computer model. The forward problem is solved in the frequency domain using higher-order finite elements. The velocity field is inverted over a specific number of discrete frequencies; thereby, reducing the computational cost of the forward calculation and the nonlinearity of the inverse problem. The results are presented for different frequency sets and with different source and receiver locations for a two-dimensional model. The influence of attenuation effects is also investigated. The results of two blind tests are presented where only the seismic records of an unknown synthetic model with an inhomogeneous material parameter distribution are provided to mimic a more realistic case. Finally, the result of the inversion in a three-dimensional space is illustrated.
An Efficient Treatment of the Laplacian in a Gradient‐Enhanced Damage Model
PAMM, 2019
As is well‐known, softening effects that are characteristic for damage models are accompanied by ... more As is well‐known, softening effects that are characteristic for damage models are accompanied by ill‐posed boundary value problems. This arises from non‐convex and non‐coercive energies and results in mesh‐dependent finite element results. For that reason, regularization strategies, which somehow take into account the non‐local behavior, have to be applied in order to prevent ill‐posedness and to achieve mesh‐independence. Hereto, most commonly gradient‐enhanced formulations [1, 2] are considered, but also integral‐type [3,4] and viscous [5,6] regularization are well‐known.Gradient‐enhanced damage models such as [1,2], to what group our new model [7] basically belongs, come along with a field function acting on the non‐local level. Two variational equations are resulting and, however, usually the number of nodal unknowns is increased and consequently the numerical effort is increased as well.In contrast, we present an improved algorithm for brittle damage [7] combining finite elemen...
By transmitting seismic waves in the ground the spread, reflected and refracted waves can be reco... more By transmitting seismic waves in the ground the spread, reflected and refracted waves can be recorded at different positions. The arising seismograms contain information of the geology, which can be evaluated with the Full Waveform Inversion (FWI). For the application of FWI for reconnaissance in tunneling extensive information has to be provided to achieve an appropriate model of the geology in front of the tunnel face. Such information are the position and geometry of reflecting boundaries, exact position of sources and receivers, a proper initial model of the distribution of the material parameters. The signal, which is transmitted by the senders, has also an influence on the inversion. Even if the intended source function is known, the effective signal depends on the transfer functions between sender-soil and soil-receiver and its identification is challenging. To avoid cumbersome investigations on the source function in advance we applied a simultaneous identification of the source signature during the FWI in the frequency domain. For synthetic models of a 2D tunnel configuration with disturbances, we will show how the calculated source signature of a frequency converges to the real source signature during the iterations of the FWI. The influence of different sender and receiver configurations will be analyzed. Furthermore, the influence of the simultaneous evaluation of the source signature on the results of the FWI will be evaluated. In a post-processing step of the FWI, the source signature of the whole frequency spectrum will be determined.
The prediction of failure mechanism in structures are always an important topic in the field of c... more The prediction of failure mechanism in structures are always an important topic in the field of computational mechanics. Finite element computations of an inelastic material involving softening behavior (e.g. softening plasticity or damage) can suffer from strongly mesh‐dependent results. Therefore, such continuum models should be equipped with a regularization (localization limiter) strategy to overcome the above‐mentioned problem.In this study, we present a framework for gradient‐enhancement for coupled damage‐plasticity material model derived by means of Hamilton's principle for non‐conservative continua. This model is applied for the numerical investigation of wear processes as they occur, e.g. in the case of mechanized tunneling. These investigations require a fine resolution of the involved constituents (cut sheet and abrasive particles in the soil). Consequently, a numerical strategy for the damage‐plasticity model is demanded that allows for time‐efficient simulations.In...
, phone +49 234 322 6025, 1 For the definition of j surface flux, j flux along the triple line, µ... more , phone +49 234 322 6025, 1 For the definition of j surface flux, j flux along the triple line, µ i chemical potential see [1]
Material models, including softening effects due to, for example, damage and localizations, share... more Material models, including softening effects due to, for example, damage and localizations, share the problem of ill-posed boundary value problems that yield mesh-dependent finite element results. It is thus necessary to apply regularization techniques that couple local behavior described, for example, by internal variables, at a spatial level. This can take account of the gradient of the internal variable to yield mesh-independent finite element results. In this paper, we present a new approach to damage modeling that does not use common field functions, inclusion of gradients or complex integration techniques: Appropriate modifications of the relaxed (condensed) energy hold the same advantage as other methods, but with much less numerical effort. We start with the theoretical derivation and then discuss the numerical treatment. Finally, we present finite element results that prove empirically how the new approach works. Communicated by Andreas Öchsner.
International Journal for Numerical Methods in Engineering, 2016
This study presents a polytree-based adaptive methodology for multi-material topology optimizatio... more This study presents a polytree-based adaptive methodology for multi-material topology optimization (MP-TOP). Polytree data structure is introduced as a general recursive multi-level mesh that is automatically refined in processing based on error analysis. In order to resolve hanging nodes in element edges, the Wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons. A new definition of filter radius is also proposed to improve the efficiency of filters and optimized results. The combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases (including void phase), but also decreases the computing time of the overall process in comparison to using the regular fine meshes. Several benchmark and practical problems are considered to show distinct features of the proposed method.
International Journal for Multiscale Computational Engineering, 2016
Our previous work proposes a micromechanical model for dissolution-precipitation creep, an elasto... more Our previous work proposes a micromechanical model for dissolution-precipitation creep, an elasto-viscoplastic process supposed to be one of the main reasons for the tectonic motion of earth plates in the subduction zone. While the model in its original form enables the simulation of polycrystals with a limited number of crystals, the topic of the present contribution is its extension to simulating structures on a much larger spatial scale. For this purpose, a homogenization technique known as the multiscale finite element method is used. Here, the behavior of a heterogeneous body is simulated by solving two boundary value problems: one related to the structural level and one related to the representative volume element. The coupling of scales is established by introducing the Hill macrohomogeneity condition requiring the equality of the macropower with the volume average of the micropower. The method allows the simulating of various tasks at both levels. The examples concerned with simulating the tension tests of a macroscopic plate with different types of the microstructure are presented.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015
A micromechanical model for finite single crystal plasticity was introduced by Kochmann & Hackl (... more A micromechanical model for finite single crystal plasticity was introduced by Kochmann & Hackl (2011 Contin. Mech. Thermodyn. 23, 63–85 ( doi:10.1007/s00161-010-0714-5 )). This model is based on thermodynamic variational principles and leads to a non-convex variational problem. Based on the Lagrange functional, an incremental strategy was outlined to model the time-continuous evolution of a first-order laminate microstructure. Although this model provides interesting results on the material point level, owing to the global minimization in the evolution equations, the calculation time and numerical instabilities may cause problems when applying this model to macroscopic specimens. In this paper, a smooth transition zone between the laminates is introduced to avoid global minimization, which makes the numerical calculations cumbersome compared with the model in Kochmann & Hackl. By introducing a smooth viscous transition zone, the dissipation potential and its numerical treatment hav...
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