A family of hyperelastic models for human brain tissue - Mihai, L. Angela; Budday, Silvia; Holzapfel, Gerhard A.; Kuhl, Ellen; Goriely, Alain (Journal of the Mechanics and Physics of Solids, 2017)
Experiments on brain samples under multiaxial loading have shown that human brain tissue is both extremely soft when compared to other biological tissues and characterized by a peculiar elastic response under combined shear and compression/tension: there is a significant increase in shear stress with increasing axial compression compared to a moderate increase with increasing axial tension. Recent studies have revealed that many widely used constitutive models for soft biological tissues fail to capture this characteristic response. Here, guided by experiments of human brain tissue, we develop a family of modeling approaches that capture the elasticity of brain tissue under varying simple shear superposed on varying axial stretch by exploiting key observations about the behavior of the nonlinear shear modulus, which can be obtained directly from the experimental data.
Keywords: Brain mechanics; Hyperelasticity; Large deformations; Combined loading; Constitutive modeling

A finite element method for cohesive crack modelling - Mergheim, Julia; Steinmann, Paul; Kuhl, Ellen (Proceedings GAMM'05, 2005)
UnivIS-Import:2015-04-16:Pub.2005.tech.FT.FT-TM.afinit

A Finite Element Method for the Computational Modelling of Cohesive Cracks - Mergheim, Julia; Kuhl, Ellen; Steinmann, Paul (International Journal for Numerical Methods in Engineering, 2005)
The present contribution is concerned with the computational modelling of cohesive cracks in quasibrittle materials, whereby the discontinuity is not limited to interelement boundaries, but is allowed to propagate freely through the elements. In the elements, which are intersected by the discontinuity, additional displacement degrees of freedom are introduced at the existing nodes. Therefore, two independent copies of the standard basis functions are used. One set is put to zero on one side of the discontinuity, while it takes its usual values on the opposite side, and vice versa for the other set. To model inelastic material behaviour, a discrete damage-type constitutive model is applied, formulated in terms of displacements and tractions at the surface. Some details on the numerical implementation are given, concer ning the failure criterion, the determination of the direction of the discontinuity and the integration scheme. Finally, numerical examples show the performance of the method. Copyright © 2005 John Wiley & Sons, Ltd.
UnivIS-Import:2015-03-09:Pub.2005.tech.FT.FT-TM.afinit_6
Keywords: Cohesive cracks; Discontinuous elements; Propagating discontinuities

A Hybrid Discontinuous Galerkin Interface Method for the Computational Modelling of Failure - Mergheim, Julia; Kuhl, Ellen; Steinmann, Paul (Communications in Numerical Methods in Engineering, 2004)
The present contribution is concerned with the computational modelling of failure along well-defined surfaces, which occur for example in the case of light-weight composite materials. A hybrid method will be introduced which makes use of the discontinuous Galerkin method in combination with a finite element interface approach. As a natural choice interface elements are introduced along the known failure surface. The discontinuous Galerkin method is applied in the pre-failure regime to avoid the unphysical use of penalty terms and instead to enforce the continuity of the solution along the interface weakly. Once a particular failure criterion is fulfilled, the behaviour of the interface is determined constitutively, depending on the displacement jump. The applicability of the proposed method is illustrated by means of two computational model problems. © 2004 John Wiley and Sons, Ltd.
UnivIS-Import:2015-03-09:Pub.2004.tech.FT.FT-TM.ahybri
Keywords: Discontinuous Galerkin method; Failure; FE-technology; Interface approach

A mechanical approach to explain cortical folding phenomena in healthy and diseased brains - Budday, Silvia; Steinmann, Paul; Kuhl, Ellen (PAMM, 2014)
The development of the human brain with its characteristically folded surface morphology remains an intensively discussed topic. Impressive advancements in different fields of research have enhanced the understanding of the brain. However, the mechanism that underlies the folding process in healthy and diseased brains remains undetermined. Here, we hypothesize that growth induced mechanical instabilities drive folding. Using the nonlinear field theories of continuum mechanics supplemented by the theory of finite growth [1], we model the human brain as a bi-material with the cerebral cortex, a morphogenetically growing outer layer of gray matter, and the subcortex, a strain-driven growing inner core of white matter [2]. This approach integrates the two popular but competing hypotheses that cortical folding is either driven by differential growth or by axon elongation. Through systematic sensitivity analyses, we identify the critical process parameters of cortical folding and quantify their impact on brain morphology. We further simulate phenomena causing malformations like lissencephaly and polymicrogyria [3], which are associated with neurological disorders, including severe retardation, epilepsy, schizophrenia, and autism. Understanding the mechanisms of cortical folding during brain development might facilitate the diagnostics and treatment of malformed brains. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
UnivIS-Import:2015-07-08:Pub.2014.tech.FT.FT-TM.amecha_7

A mechanical model predicts morphological abnormalities in the developing human brain - Budday, Silvia; Kuhl, Ellen; Raybaud, Charles (Scientific Reports, 2014)
The developing human brain remains one of the few unsolved mysteries of science. Advancements in developmental biology, neuroscience, and medical imaging have brought us closer than ever to understand brain development in health and disease. However, the precise role of mechanics throughout this process remains underestimated and poorly understood. Here we show that mechanical stretch plays a crucial role in brain development. Using the nonlinear field theories of mechanics supplemented by the theory of finite growth, we model the human brain as a living system with a morphogenetically growing outer surface and a stretch-driven growing inner core. This approach seamlessly integrates the two popular but competing hypotheses for cortical folding: axonal tension and differential growth. We calibrate our model using magnetic resonance images from very preterm neonates. Our model predicts that deviations in cortical growth and thickness induce morphological abnormalities. Using the gyrification index, the ratio between the total and exposed surface area, we demonstrate that these abnormalities agree with the classical pathologies of lissencephaly and polymicrogyria. Understanding the mechanisms of cortical folding in the developing human brain has direct implications in the diagnostics and treatment of neurological disorders, including epilepsy, schizophrenia, and autism.
UnivIS-Import:2015-03-09:Pub.2014.tech.FT.FT-TM.amecha

A new approach for the FE modelling of cohesive cracks - Mergheim, Julia; Steinmann, Paul; Kuhl, Ellen (Proceedings of 21st International Congress of Theoretical and Applied Mechanics, 2004)
UnivIS-Import:2015-04-16:Pub.2004.tech.FT.FT-TM.anewap_9

A novel strategy to identify the critical conditions for growth-induced instabilities - Javili, Ali; Steinmann, Paul; Kuhl, Ellen (Journal of the Mechanical Behavior of Biomedical Materials, 2014)
Geometric instabilities in living structures can be critical for healthy biological function, and abnormal buckling, folding, or wrinkling patterns are often important indicators of disease. Mathematical models typically attribute these instabilities to differential growth, and characterize them using the concept of fictitious configurations. This kinematic approach toward growth-induced instabilities is based on the multiplicative decomposition of the total deformation gradient into a reversible elastic part and an irreversible growth part. While this generic concept is generally accepted and well established today, the critical conditions for the formation of growth-induced instabilities remain elusive and poorly understood. Here we propose a novel strategy for the stability analysis of growing structures motivated by the idea of replacing growth by prestress. Conceptually speaking, we kinematically map the stress-free grown configuration onto a prestressed initial configuration. This allows us to adopt a classical infinitesimal stability analysis to identify critical material parameter ranges beyond which growth-induced instabilities may occur. We illustrate the proposed concept by a series of numerical examples using the finite element method. Understanding the critical conditions for growth-induced instabilities may have immediate applications in plastic and reconstructive surgery, asthma, obstructive sleep apnoea, and brain development. © 2013 Elsevier Ltd.
UnivIS-Import:2015-03-09:Pub.2014.tech.FT.FT-TM.anovel
Keywords: Growth; Instabilities; Morphogenesis; Prestress; Residual stress; Stability analysis

Automated model discovery for human cardiac tissue - Martonová, Denisa; Peirlinck, Mathias; Linka, Kevin; Holzapfel, Gerhard A.; Leyendecker, Sigrid; Kuhl, Ellen (conference, Netzwerktagung der Alexander von Humboldt Stiftung, 2024)

Automated model discovery for human cardiac tissue: Discovering the best model and parameters - Martonová, Denisa; Peirlinck, Mathias; Linka, Kevin; Holzapfel, Gerhard A.; Leyendecker, Sigrid; Kuhl, Ellen (Computer Methods in Applied Mechanics and Engineering, 2024)
For more than half a century, scientists have developed mathematical models to understand the behavior of the human heart. Today, we have dozens of heart tissue models to choose from, but selecting the best model is limited to expert professionals, prone to user bias, and vulnerable to human error. Here we take the human out of the loop and automate the process of model discovery. Towards this goal, we establish a novel incompressible orthotropic constitutive neural network to simultaneously discover both, model and parameters, that best explain human cardiac tissue. Notably, our network features 32 individual terms, 8 isotropic and 24 anisotropic, and fully autonomously selects the best model, out of more than 4 billion possible combinations of terms. We demonstrate that we can successfully train the network with triaxial shear and biaxial extension tests and systematically sparsify the parameter vector with L1" role="presentation" style="display: inline-block; line-height: normal; font-size: 14.4px; font-size-adjust: none; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: none; position: relative;">-regularization. Strikingly, we robustly discover a four-term model that features a quadratic term in the second invariant I2" role="presentation" style="display: inline-block; line-height: normal; font-size: 14.4px; font-size-adjust: none; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: none; position: relative;">, and exponential quadratic terms in the fourth and eighth invariants I4f" role="presentation" style="display: inline-block; line-height: normal; font-size: 14.4px; font-size-adjust: none; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: none; position: relative;">, I4n" role="presentation" style="display: inline-block; line-height: normal; font-size: 14.4px; font-size-adjust: none; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: none; position: relative;">, and I8fs" role="presentation" style="display: inline-block; line-height: normal; font-size: 14.4px; font-size-adjust: none; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: none; position: relative;">. Importantly, our discovered model is interpretable by design and has parameters with well-defined physical units. We show that it outperforms popular existing myocardium models and generalizes well, from homogeneous laboratory tests to heterogeneous whole heart simulations. This is made possible by a new universal material subroutine that directly takes the discovered network weights as input. Automating the process of model discovery has the potential to democratize cardiac modeling, broaden participation in scientific discovery, and accelerate the development of innovative treatments for cardiovascular disease. Our source code, data, and examples are available at https://github.com/LivingMatterLab/CANN.<br

Brain stiffness increases with myelin content - Weickenmeier, Johannes; de Rooij, Rijk; Budday, Silvia; Steinmann, Paul; Ovaert, Timothy; Kuhl, Ellen (Acta Biomaterialia, 2016)
Brain stiffness plays an important role in neuronal development and disease, but reported stiffness values vary significantly for different species, for different brains, and even for different regions within the same brain. Despite extensive research throughout the past decade, the mechanistic origin of these stiffness variations remains elusive. Here we show that brain tissue stiffness is correlated to the underlying tissue microstructure and directly proportional to the local myelin content. In 116 indentation tests of six freshly harvested bovine brains, we found that the cerebral stiffnesses of 1.33 ± 0.63 kPa in white matter and 0.68 ± 0.20 kPa in gray matter were significantly different (p < 0.01). Strikingly, while the inter- specimen variation was rather moderate, the minimum and maximum cerebral white matter stiffnesses of 0.59 ± 0.19 kPa and 2.36 ± 0.64 kPa in each brain varied by a factor of four on average. To provide a mechanistic interpretation for this variation, we performed a histological characterization of the tested brain regions. We stained the samples with hematoxylin and eosin and luxol fast blue and quantified the local myelin content using image analysis. Interestingly, we found that the cerebral white matter stiffness increased with increasing myelin content, from 0.72kPa at a myelin content of 64–2.45kPa at a myelin content of 89%, with a Pearson correlation coefficient of 0.91 (p < 0.01). This direct correla- tion could have significant neurological implications. During development, our results could help explain why immature, incompletely myelinated brains are softer than mature, myelinated brains and more vul- nerable to mechanical insult as evident, for example, in shaken baby syndrome. During demyelinating disease, our findings suggest to use stiffness alterations as clinical markers for demyelination to quantify the onset of disease progression, for example, in multiple sclerosis. Taken together, our study indicates that myelin might play a more important function than previously thought: It not only insulates signal propagation and improves electrical function of single axons, it also provides structural support and mechanical stiffness to the brain as a whole.
Keywords: Brain; Mechanical testing; Indentation; Soft matter; Stiffness; Myelin

Challenges and perspectives in brain tissue testing and modeling - Budday, Silvia; Holzapfel, Gerhard A.; Steinmann, Paul; Kuhl, Ellen (Proceedings in Applied Mathematics and Mechanics, 2019)
While long underestimated, more and more evidence confirms that mechanics play a critical role for brain function and dysfunction. Therefore, computational simulations based on the field equations of nonlinear continuum mechanics can provide important insights into the underlying mechanisms of brain injury and disease. Realistic numerical predictions, however, require models capable of capturing the complex and unique mechanical behavior of this ultrasoft, highly heterogeneous and adaptive tissue. In recent years, contradictory experimental results have retarded progress in this field. Here, we summarize the key characteristics of brain tissue behavior on different length and time scales and propose application-specific modeling approaches, which are as complex as necessary but as simple as possible. The presented considerations will, on the one hand, facilitate well-designed future experiments and, on the other hand, help to choose the appropriate constitutive law for a specific application.

Chapter two-neuromechanics: From neurons to brain - Goriely, Alain; Budday, Silvia; Kuhl, Ellen (Advances in Applied Mechanics, 2015)
Arguably, the brain is the most complex organ in the human body, and, at the same time, the least well understood. Today, more than ever before, the human brain has become a subject of narcissistic study and fascination. The fields of neuroscience, neu- rology, neurosurgery, and neuroradiology have seen tremendous progress over the past two decades; yet, the field of neuromechanics remains underappreciated and poorly understood. Here, we show that mechanical stretch, strain, stress, and force play a critical role in modulating the structure and function of the brain. We discuss the role of neuromechanics across the scales, from individual neurons via neuronal tissue to the whole brain. We review current research highlights and discuss challenges and potential future directions. Using the nonlinear field theories of mechanics, we illustrate three phenomena which are tightly regulated by mechanical factors: neuroelasticity, the extremely soft behavior of the brain independent of time; neurodevelopment, the evo- lution of the brain at extremely long time scales; and neurodamage, the degradation of the brain at extremely short time scales. We hope that this review will become a starting point for a multidisciplinary approach to the mechanics of the brain with potential impact in preventing, diagnosing, and treating neurological disorders.

Computational Modelling of Failure with the Discontinuous Galerkin Method - Mergheim, Julia; Steinmann, Paul; Kuhl, Ellen (Proceedings of the 1st International LUXFEM Conference, 2003)
UnivIS-Import:2015-04-16:Pub.2003.tech.FT.FT-TM.comput_4

Constitutive neural networks for model discovery of myocardial tissue - Martonová, Denisa; Peirlinck, Mathias; Linka, Kevin; Holzapfel, Gerhard A.; Leyendecker, Sigrid; Kuhl, Ellen (conference, CMBBE 2024 SYMPOSIUM, 19th International Symposium on Computer Methods in Biomechanics and Biomedical Engineering, 2024)

Convex neural networks learn generalized standard material models - Flaschel, Moritz; Steinmann, Paul; De Lorenzis, Laura; Kuhl, Ellen (Journal of the Mechanics and Physics of Solids, 2025)
We propose Generalized Standard Material Networks, a machine learning framework based on convex neural networks for learning the mechanical behavior of generalized standard materials. The theory of these materials postulates the existence of two thermodynamic potentials, the Helmholtz free energy density and the dissipation rate density potential, which alone determine the constitutive material response with guaranteed thermodynamic consistency. We parameterize the two potentials with two artificial neural networks and, due to a specifically designed network architecture, we satisfy by construction all the needed properties of the two potentials. Using automatic differentiation, an implicit time integration scheme and the Newton-Raphson method, we can thus describe a multitude of different material behaviors within a single unified overarching framework, including elastic, viscoelastic, plastic, and viscoplastic material responses with hardening. By probing our framework on the synthetic data generated by five benchmark material models, we demonstrate satisfactory prediction accuracy to unseen data and a high robustness to noise. In this context, we observe a non-uniqueness of thermodynamic potentials and discuss how this affects the results of the training process. Finally, we show that a carefully chosen number of internal variables strikes a balance between fitting accuracy and model complexity.
CRIS-Team Scopus Importer:2025-03-28
Keywords: Automatic differentiation; Generalized standard materials; Input convex neural networks; Internal variables; Thermodynamic consistency

Discovering uncertainty: Bayesian constitutive artificial neural networks - Linka, Kevin; Holzapfel, Gerhard A.; Kuhl, Ellen (Computer Methods in Applied Mechanics and Engineering, 2025)
Understanding uncertainty is critical, especially when data are sparse and variations are large. Bayesian neural networks offer a powerful strategy to build predictable models from sparse data, and inherently quantify both, aleatoric uncertainties of the data and epistemic uncertainties of the model. Yet, classical Bayesian neural networks ignore the fundamental laws of physics, they are non-interpretable, and their parameters have no physical meaning. Here we integrate concepts of Bayesian learning and constitutive neural networks to discover interpretable models, parameters, and uncertainties that best explain soft matter systems. Instead of training an individual constitutive neural network and learning point values of the network weights, we train an ensemble of networks and learn probability distributions of the weights, along with their means, standard deviations, and credible intervals. We use variational Bayesian inference and adopt an efficient backpropagation-compatible algorithm that approximates the true probability distributions by simpler distributions and minimizes their divergence through variational learning. When trained on synthetic data, our Bayesian constitutive neural network successfully rediscovers the initial model, even in the presence of noise, and robustly discovers uncertainties, even from incomplete data. When trained on real data from healthy and aneurysmal human arteries, our network discovers a model with more stretch stiffening, more anisotropy, and more uncertainty for diseased than for healthy arteries. Our results demonstrate that Bayesian constitutive neural networks can successfully discriminate between healthy and diseased arteries, robustly discover interpretable models and parameters for both, and efficiently quantify uncertainties in model discovery. We anticipate our approach to generalize to other soft biomedical systems for which real-world data are rare and inter-personal variations are large. Ultimately, our calculated uncertainties will help enhance model robustness, promote personalized predictions, enable informed decision-making, and build confidence in automated model discovery and simulation. Our source code, data, and examples are available at https://github.com/LivingMatterLab/CANN.
CRIS-Team Scopus Importer:2024-11-15
Keywords: Automated model discovery; Bayesian inference; Bayesian neural networks; Constitutive artificial neural networks; Uncertainty

Fifty Shades of Brain: A Review on the Mechanical Testing and Modeling of Brain Tissue - Budday, Silvia; Ovaert, Timothy; Holzapfel, Gerhard A.; Steinmann, Paul; Kuhl, Ellen (Archives of Computational Methods in Engineering, 2020)
Brain tissue is not only one the most important but also the most complex and compliant tissue in the human body. While long underestimated, increasing evidence confirms that mechanics plays a critical role in modulating brain function and dysfunction. Computational simulations–based on the field equations of nonlinear continuum mechanics–can provide important insights into the underlying mechanisms of brain injury and disease that go beyond the possibilities of traditional diagnostic tools. Realistic numerical predictions, however, require mechanical models that are capable of capturing the complex and unique characteristics of this ultrasoft, heterogeneous, and active tissue. In recent years, contradictory experimental results have caused confusion and hindered rapid progress. In this review, we carefully assess the challenges associated with brain tissue testing and modeling, and work out the most important characteristics of brain tissue behavior on different length and time scales. Depending on the application of interest, we propose appropriate mechanical modeling approaches that are as complex as necessary but as simple as possible. This comprehensive review will, on the one hand, stimulate the design of new experiments and, on the other hand, guide the selection of appropriate constitutive models for specific applications. Mechanical models that capture the complex behavior of nervous tissues and are accurately calibrated with reliable and comprehensive experimental data are key to performing reliable predictive simulations. Ultimately, mathematical modeling and computational simulations of the brain are useful for both biomedical and clinical communities, and cover a wide range of applications ranging from predicting disease progression and estimating injury risk to planning surgical procedures.<br

Folding drives cortical thickness variations - Holland, Maria A.; Budday, Silvia; Li, Gang; Shen, Dinggang; Goriely, Alain; Kuhl, Ellen (European Physical Journal - Special Topics, 2020)
The cortical thickness is a characteristic biomarker for a wide variety of neurological disorders. While the structural organization of the cerebral cortex is tightly regulated and evolutionarily preserved, its thickness varies widely between 1.5 and 4.5 mm across the healthy adult human brain. It remains unclear whether these thickness variations are a cause or consequence of cortical development. Recent studies suggest that cortical thickness variations are primarily a result of genetic effects. Previous studies showed that a simple homogeneous bilayered system with a growing layer on an elastic substrate undergoes a unique symmetry breaking into a spatially heterogeneous system with discrete gyri and sulci. Here, we expand on that work to explore the evolution of cortical thickness variations over time to support our finding that cortical pattern formation and thickness variations can be explained – at least in part – by the physical forces that emerge during cortical folding. Strikingly, as growth progresses, the developing gyri universally thicken and the sulci thin, even in the complete absence of regional information. Using magnetic resonance images, we demonstrate that these naturally emerging thickness variations agree with the cortical folding pattern in n = 9 healthy adult human brains, in n = 564 healthy human brains ages 7–64, and in n = 73 infant brains scanned at birth, and at ages one and two. Additionally, we show that cortical organoids develop similar patterns throughout their growth. Our results suggest that genetic, geometric, and physical events during brain development are closely interrelated. Understanding regional and temporal variations in cortical thickness can provide insight into the evolution and causative factors of neurological disorders, inform the diagnosis of neurological conditions, and assess the efficacy of treatment options.
CRIS-Team Scopus Importer:2020-11-20

Mechanical characterization of human brain tissue - Budday, Silvia; Sommer, Gerhard; Birkl, Christoph; Langkammer, Christian; Haybaeck, Johannes; Kohnert, Julius; Bauer, Melanie; Paulsen, Friedrich; Steinmann, Paul; Kuhl, Ellen; Holzapfel, Gerhard A. (Acta Biomaterialia, 2017)
Mechanics are increasingly recognized to play an important role in modulating brain form and function. Computational simulations are a powerful tool to predict the mechanical behavior of the human brain in health and disease. The success of these simulations depends critically on the underlying constitutive model and on the reliable identification of its material parameters. Thus, there is an urgent need to thor- oughly characterize the mechanical behavior of brain tissue and to identify mathematical models that cap- ture the tissue response under arbitrary loading conditions. However, most constitutive models have only been calibrated for a single loading mode. Here, we perform a sequence of multiple loading modes on the same human brain specimen – simple shear in two orthogonal directions, compression, and tension – and characterize the loading-mode specific regional and directional behavior. We complement these three indi- vidual tests by combined multiaxial compression/tension-shear tests and discuss effects of conditioning and hysteresis. To explore to which extent the macrostructural response is a result of the underlying microstructural architecture, we supplement our biomechanical tests with diffusion tensor imaging and histology. We show that the heterogeneous microstructure leads to a regional but not directional depen- dence of the mechanical properties. Our experiments confirm that human brain tissue is nonlinear and vis- coelastic, with a pronounced compression-tension asymmetry. Using our measurements, we compare the performance of five common constitutive models, neo-Hookean, Mooney-Rivlin, Demiray, Gent, and Ogden, and show that only the isotropic modified one-term Ogden model is capable of representing the hyperelastic behavior under combined shear, compression, and tension loadings: with a shear modulus of 0.4–1.4 kPa and a negative nonlinearity parameter it captures the compression-tension asymmetry and the increase in shear stress under superimposed compression but not tension. Our results demonstrate that material parameters identified for a single loading mode fail to predict the response under arbitrary loading conditions. Our systematic characterization of human brain tissue will lead to more accurate com- putational simulations, which will allow us to determine criteria for injury, to develop smart protection sys- tems, and to predict brain development and disease progression.
Keywords: Human brain; Mechanical testing; Hyperelastic modeling; Mooney Rivlin model; Ogden model

Mechanical properties of gray and white matter brain tissue by indentation - Budday, Silvia; Nay, Richard; de Rooij, Rijk; Steinmann, Paul; Wyrobek, Thomas; Ovaert, Timothy; Kuhl, Ellen (Journal of the Mechanical Behavior of Biomedical Materials, 2015)
UnivIS-Import:2015-07-08:Pub.2015.tech.FT.FT-TM.mechan

Modeling and simulation of viscous electro-active polymers - Vogel, Franziska Anna; Göktepe, Serdar; Steinmann, Paul; Kuhl, Ellen (European Journal of Mechanics A-Solids, 2014)
Electro-active materials are capable of undergoing large deformation when stimulated by an electric field. They can be divided into electronic and ionic electro-active polymers (EAPs) depending on their actuation mechanism based on their composition. We consider electronic EAPs, for which attractive Coulomb forces or local re-orientation of polar groups cause a bulk deformation. Many of these materials exhibit pronounced visco-elastic behavior. Here we show the development and implementation of a constitutive model, which captures the influence of the electric field on the visco-elastic response within a geometrically non-linear finite element framework. The electric field affects not only the equilibrium part of the strain energy function, but also the viscous part. To adopt the familiar additive split of the strain from the small strain setting, we formulate the governing equations in the logarithmic strain space and additively decompose the logarithmic strain into elastic and viscous parts. We show that the incorporation of the electric field in the viscous response significantly alters the relaxation and hysteresis behavior of the model. Our parametric study demonstrates that the model is sensitive to the choice of the electro-viscous coupling parameters. We simulate several actuator structures to illustrate the performance of the method in typical relaxation and creep scenarios. Our model could serve as a design tool for micro-electro-mechanical systems, microfluidic devices, and stimuli-responsive gels such as artificial skin, tactile displays, or artificial muscle. © 2014 Elsevier Masson SAS. All rights reserved.
UnivIS-Import:2015-03-09:Pub.2014.tech.FT.FT-TM.modeli
Keywords: Electroactivity; Electroelasticity; Electrostatics; Viscoelasticity

Modeling the life cycle of the human brain - Budday, Silvia; Kuhl, Ellen (Current Opinion in Biomedical Engineering, 2020)
In recent years, computational mechanics have become a powerful tool to study and predict the behavior of the human brain. Yet, an important challenge that remains unaddressed is that brain tissue is not a constant uniform material. Throughout its life time, our brain’s microstructure, mechanical properties, and macroscopic shape keep changing in close relation to brain function. In this review, we summarize the evolution of various microscopic and macroscopic features during brain development and aging. We discuss how to use mechanical models to translate processes on the microscopic scale into the evolution of mechanical properties and brain shape on the macroscopic scale. In addition, we propose how to incorporate additional coupling effects that arise from the ‘mechanosensitivity’ of brain cells. Considering the entire life cycle of the human brain will be critical to refine existing brain models towards personalized simulations of brain injury, brain damage, and brain protectio
Keywords: brain development; brain aging; nonlinear continuum mechanics; mechanosensing; growth; coupled problems

Modelling and Computation of 3D Discontinuities in Solids - Mergheim, Julia; Steinmann, Paul; Kuhl, Ellen; Jäger, Philippe (Proceedings GAMM'06, 2007)
UnivIS-Import:2015-04-16:Pub.2007.tech.FT.FT-TM.modell

Modelling of failure with a hybrid dG/ interface approach - Mergheim, Julia; Kuhl, Ellen; Steinmann, Paul (Proceedings GAMM'03, 2003)
UnivIS-Import:2015-04-16:Pub.2003.tech.FT.FT-TM.comput_6

Natural element analysis of the Cahn-Hilliard phase-field model - Rajagopal, Amirtham; Fischer, Peter; Kuhl, Ellen; Steinmann, Paul (Computational Mechanics, 2010)
We present a natural element method to treat higher-order spatial derivatives in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear partial differential equation that allows to model phase separation in binary mixtures. Standard classical C0-continuous finite element solutions are not suitable because primal variational formulations of fourth-order operators are well-defined and integrable only if the finite element basis functions are piecewise smooth and globally C1-continuous. To ensure C1-continuity, we develop a natural-element-based spatial discretization scheme. The C1-continuous natural element shape functions are achieved by a transformation of the classical Farin interpolant, which is basically obtained by embedding Sibsons natural element coordinates in a Bernstein–Bézier surface representation of a cubic simplex. For the temporal discretization, we apply the (second-order accurate) trapezoidal time integration scheme supplemented with an adaptively adjustable time step size. Numerical examples are presented to demonstrate the efficiency of the computational algorithm in two dimensions. Both periodic Dirichlet and homogeneous Neumann boundary conditions are applied. Also constant and degenerate mobilities are considered. We demonstrate that the use of C1-continuous natural element shape functions enables the computation of topologically correct solutions on arbitrarily shaped domains.
UnivIS-Import:2015-04-14:Pub.2010.tech.FT.FT-TM.natura

On the Application of Hansbo's Method for Interface Problems - Kuhl, Ellen; Jäger, Philippe; Mergheim, Julia; Steinmann, Paul (Proceedings of IUTAM Symposium on Discretization Methods for Evolving Discontinuities, 2006)
UnivIS-Import:2015-04-16:Pub.2006.tech.FT.FT-TM.onthea

On the mechanics of continua with boundary energies and growing surfaces - Steinmann, Paul; Papastavrou, Areti; Kuhl, Ellen (Journal of the Mechanics and Physics of Solids, 2013)
Many biological systems are coated by thin films for protection, selective absorption, or transmembrane transport. A typical example is the mucous membrane covering the airways, the esophagus, and the intestine. Biological surfaces typically display a distinct mechanical behavior from the bulk; in particular, they may grow at different rates. Growth, morphological instabilities, and buckling of biological surfaces have been studied intensely by approximating the surface as a layer of finite thickness; however, growth has never been attributed to the surface itself. Here, we establish a theory of continua with boundary energies and growing surfaces of zero thickness in which the surface is equipped with its own potential energy and is allowed to grow independently of the bulk. In complete analogy to the kinematic equations, the balance equations, and the constitutive equations of a growing solid body, we derive the governing equations for a growing surface. We illustrate their spatial discretization using the finite element method, and discuss their consistent algorithmic linearization. To demonstrate the conceptual differences between volume and surface growth, we simulate the constrained growth of the inner layer of a cylindrical tube. Our novel approach toward continua with growing surfaces is capable of predicting extreme growth of the inner cylindrical surface, which more than doubles its initial area. The underlying algorithmic framework is robust and stable; it allows to predict morphological changes due to surface growth during the onset of buckling and beyond. The modeling of surface growth has immediate biomedical applications in the diagnosis and treatment of asthma, gastritis, obstructive sleep apnoea, and tumor invasion. Beyond biomedical applications, the scientific understanding of growth-induced morphological instabilities and surface wrinkling has important implications in material sciences, manufacturing, and microfabrication, with applications in soft lithography, metrology, and flexible electronics. © 2013 Elsevier Ltd.
UnivIS-Import:2015-03-09:Pub.2013.tech.FT.FT-TM.onthem_8
Keywords: Airway wall remodeling; Boundary energy; Finite elements; Growth; Thin films

Period-doubling and period-tripling in growing bilayered systems - Budday, Silvia; Kuhl, Ellen; Hutchinson, John W. (Philosophical Magazine, 2015)
Growing layers on elastic substrates are capable of creating a wide variety of surface morphologies. Moderate growth generates a regular pattern of sinusoidal wrinkles with a homogeneous energy distribution. While the critical conditions for periodic wrinkling have been extensively studied, the rich pattern formation beyond this first instability point remains poorly understood. Here, we show that upon continuing growth, the energy progressively localizes and new complex morphologies emerge. Previous studies have often overlooked these secondary bifurcations; they have focused on large stiffness ratios between layer and substrate, where primary instabilities occur early, long before secondary instabilities emerge. We demonstrate that secondary bifurcations are particularly critical in the low stiffness ratio regime, where the critical conditions for primary and secondary instabilities move closer together. Amongst all possible secondary bifurcations, the mode of period-doubling plays a central role – it is energetically favourable over all other modes. Yet, we can numerically suppress period-doubling, by choosing boundary conditions, which favour alternative higher order modes. Our results suggest that in the low stiffness regime, pattern formation is highly sensitive to small imperfections: surface morphologies emerge rapidly, change spontaneously and quickly become immensely complex. This is a common paradigm in developmental biology. Our results have significant applications in the morphogenesis of living systems where growth is progressive and stiffness ratios are low.
UnivIS-Import:2015-07-08:Pub.2015.tech.FT.FT-TM.period
Keywords: bifurcation; growth; instabilities; period-doubling; period-tripling; thin films

Physical biology of human brain development - Budday, Silvia; Steinmann, Paul; Kuhl, Ellen (Frontiers in Cellular Neuroscience, 2015)
Neurodevelopment is a complex, dynamic process that involves a precisely orchestrated sequence of genetic, environmental, biochemical, and physical events. Developmental biology and genetics have shaped our understanding of the molecular and cellular mechanisms during neurodevelopment. Recent studies suggest that physical forces play a central role in translating these cellular mechanisms into the complex surface morphology of the human brain. However, the precise impact of neuronal differentiation, migration, and connection on the physical forces during cortical folding remains unknown. Here we review the cellular mechanisms of neurodevelopment with a view toward surface morphogenesis, pattern selection, and evolution of shape. We revisit cortical folding as the instability problem of constrained differential growth in a multi-layered system. To identify the contributing factors of differential growth, we map out the timeline of neurodevelopment in humans and highlight the cellular events associated with extreme radial and tangential expansion. We demonstrate how computational modeling of differential growth can bridge the scales-from phenomena on the cellular level toward form and function on the organ level-to make quantitative, personalized predictions. Physics-based models can quantify cortical stresses, identify critical folding conditions, rationalize pattern selection, and predict gyral wavelengths and gyrification indices. We illustrate that physical forces can explain cortical malformations as emergent properties of developmental disorders. Combining biology and physics holds promise to advance our understanding of human brain development and enable early diagnostics of cortical malformations with the ultimate goal to improve treatment of neurodevelopmental disorders including epilepsy, autism spectrum disorders, and schizophrenia.
UnivIS-Import:2015-10-26:Pub.2015.tech.FT.FT-TM.physic
Keywords: Connectivity; Cortical folding; Differential growth; Lissencephaly; Neurodevelopment; Polymicrogyria; Synaptogenesis

Primary and secondary instabilities in soft bilayered systems - Budday, Silvia; Steinmann, Paul; Kuhl, Ellen; Andres, Sebastian (PAMM, 2015)
Wrinkling phenomena emerging from mechanical instabilities in inhomogeneously compressed soft bilayered systems can evoke a wide variety of surface morphologies. Applications range from undesired instabilities in engineering structures such as sandwich panels, via fabricating surfaces with controlled buckling patterns of unique properties, to wrinkling phenomena in living matter such as lungs, mucosas, and brain convolutions. While moderate compression evokes periodic sinusoidal wrinkles, higher compression induces secondary instabilities - the surface bifurcates into increasingly complex morphologies. Periodic wrinkling has already been extensively studied, but the rich pattern formation in the highly nonlinear post-buckling regime remains poorly understood. Here, we establish a computational model of differential growth to explore the evolving buckling pattern of a growing layer bonded to a non-growing substrate. Our model provides a mechanistic understanding of growth-induced primary and secondary instabilities. We show that amongst all possible secondary bifurcations, the mode of period-doubling is energetically favorable. We experimentally validate our numerical results by examining buckling of a compressed polymer film on a soft foundation. Our computational studies have broad applications in the microfabrication of distinct surface patterns as well as in the morphogenesis of living systems, where growth is progressive and the formation of structural instabilities is critical to biological function.

Region‐ and loading‐specific finite viscoelasticity of human brain tissue - Budday, Silvia; Sommer, Gerhard; Paulsen, Friedrich; Holzapfel, Gerhard A.; Steinmann, Paul; Kuhl, Ellen (Proceedings in Applied Mathematics and Mechanics, 2018)
Computational simulations are a powerful tool to understand the mechanical behavior of our brain in health and disease, with the ultimate goal to prevent pathological conditions. Accurate numerical predictions, however, require the development of appropriate constitutive models and, equally important, the careful calibration of the corresponding constitutive parameters. This has been exceptionally challenging due to the ultrasoft and heterogeneous nature of brain tissue, resulting in a distinctly nonlinear, rate‐dependent, compression‐tension asymmetric, and region‐dependent behavior. Previous constitutive models have been deduced from a single loading mode, but fail to predict the behavior under various loading conditions. Here, we developed a large strain, nonlinear, viscoelastic constitutive model for brain tissue on the basis of cyclic and relaxation experiments under multiple loading modes, simple shear, compression, and tension. We carefully calibrated individual parameter sets for four different regions of the human brain, the cortex, the basal ganglia, the corona radiata, and the corpus callosum. The model captures effects such as nonlinearity and compression‐tension asymmetry, but also time‐dependent effects with substantial pre‐conditionning during the first loading cycle, only minor conditioning effects during subsequent cycles, and successive softening when the applied strain is stepwise increased. With close consideration of the underlying microstructure, we evaluate the physical meaning of viscoelastic material parameters with rate‐dependent regional trends. Our results help to improve the accuracy of human brain simulations during development and disease or to predict outcomes of neurosurgical procedure

Rheological characterization of human brain tissue - Budday, Silvia; Sommer, Gerhard; Haybaeck, Johannes; Steinmann, Paul; Holzapfel, Gerhard A.; Kuhl, Ellen (Acta Biomaterialia, 2017)
The rheology of ultrasoft materials like the human brain is highly sensitive to regional and temporal variations and to the type of loading. While recent experiments have shaped our understanding of the time-independent, hyperelastic response of human brain tissue, its time-dependent behavior under various loading conditions remains insufficiently understood. Here we combine cyclic and relaxation testing under multiple loading conditions, shear, compression, and tension, to understand the rheology of four different regions of the human brain, the cortex, the basal ganglia, the corona radiata, and the corpus callosum. We establish a family of finite viscoelastic Ogden-type models and calibrate their parameters simultaneously for all loading conditions. We show that the model with only one viscoelastic mode and a constant viscosity captures the essential features of brain tissue: nonlinearity, pre-conditioning, hysteresis, and tension-compression asymmetry. With stiffnesses and time constants of μ∞=0.7 kPa, μ1=2.0 kPa, and τ1=9.7 s in the gray matter cortex and μ∞=0.3 kPa, μ1=0.9 kPa and τ1=14.9 s in the white matter corona radiata combined with negative parameters α∞ and α1, this five-parameter model naturally accounts for pre-conditioning and tissue softening. Increasing the number of viscoelastic modes improves the agreement between model and experiment, especially across the entire relaxation regime. Strikingly, two cycles of pre-conditioning decrease the gray matter stiffness by up to a factor three, while the white matter stiffness remains almost identical. These new insights allow us to better understand the rheology of different brain regions under mixed loading conditions. Our family of finite viscoelastic Ogden-type models for human brain tissue is simple to integrate into standard nonlinear finite element packages. Our simultaneous parameter identification of multiple loading modes can inform computational simulations under physiological conditions, especially at low to moderate strain rates. Understanding the rheology of the human brain will allow us to more accurately model the behavior of the brain during development and disease and predict outcomes of neurosurgical procedures.
Keywords: Human brain; Rheological testing; Finite viscoelasticity; Ogden model; Parameter identification

Secondary instabilities modulate cortical complexity in the mammalian brain - Budday, Silvia; Steinmann, Paul; Kuhl, Ellen (Philosophical Magazine, 2015)
UnivIS-Import:2015-07-08:Pub.2015.tech.FT.FT-TM.second

Size and curvature regulate pattern selection in the mammalian brain - Budday, Silvia; Steinmann, Paul; Goriely, Alain; Kuhl, Ellen (Extreme Mechanics Letters, 2015)
Mammalian brains display a wide variety of shapes and surface morphologies. Their characteristically folded surface is closely correlated to neuronal activity and serves as a clinical indicator for physiological and pathological conditions. Yet, the regulators of pattern formation in evolution and development remain poorly understood. Here we show how brain size and curvature affect the folding pattern in the developing mammalian brain. We model cortical folding as the instability problem of a bilayered system subjected to growth-induced compression. Using analytical estimates and continuum models for finite growth, we systematically explore the effects of geometric factors on the evolution of surface shape. We demonstrate that extrinsic geometric features-including brain size, cortical thickness, and cortical curvature-tightly regulate pattern selection: The mammalian brain is extremely soft and even small environmental changes can create extremely large alterations in its surface morphology. Our simulations explain why gyrification increases with brain size and why longer brains tend to fold more longitudinally than radially. Our results suggest that brain folding is driven, at least in part, by extreme mechanics, rather than by phylogeny alone.
UnivIS-Import:2015-10-26:Pub.2015.tech.FT.FT-TM.sizean
Keywords: Brain mechanics; Cortical folding; Growth; Instabilities; Living matter; Soft matter; Thin films

Symmetry Breaking in Wrinkling Patterns: Gyri Are Universally Thicker than Sulci - Holland, Maria; Goriely, Alain; Budday, Silvia; Kuhl, Ellen (Physical Review Letters, 2018)
Wrinkling instabilities appear in soft materials when a flat elastic layer on an elastic substrate is sufficiently stressed that it buckles with a wavy pattern to minimize the energy of the system. This instability is known to play an important role in engineering, but it also appears in many biological systems. In these systems, the stresses responsible for the wrinkling instability are often created through differential growth of the two layers. Beyond the instability, the upper and lower sides of the elastic layer are subject to different forces. This difference in forces leads to an interesting symmetry breaking whereby the thickness becomes larger at ridges than at valleys. Here we carry out an extensive analysis of this phenomenon by combining analytical, computational, and simple polymer experiments to show that symmetry breaking is a generic property of such systems. We apply our idea to the cortical folding of the brain for which it has been known for over a century that there is a thickness difference between gyri and sulci. An extensive analysis of hundreds of human brains reveals a systematic region-dependent thickness variation. Our results suggest that the evolving thickness patterns during brain development, similar to our polymer experiments, follow simple physics-based laws: Gyri are universally thicker than sulci.

The mechanical and sensory signature of plant-based and animal meat - St. Pierre, Skyler R.; Darwin, Ethan C.; Adil, Divya; Aviles, Magaly C.; Date, Archer; Dunne, Reese A.; Lall, Yanav; Parra Vallecillo, María; Perez Medina, Valerie A.; Linka, Kevin; Levenston, Marc E.; Kuhl, Ellen (npj Science of Food, 2024)
Eating less meat is associated with a healthier body and planet. Yet, we remain reluctant to switch to a plant-based diet, largely due to the sensory experience of plant-based meat. Food scientists characterize meat using a double compression test, which only probes one-dimensional behavior. Here we use tension, compression, and shear tests–combined with constitutive neural networks–to automatically discover the behavior of eight plant-based and animal meats across the entire three-dimensional spectrum. We find that plant-based sausage and hotdog, with stiffnesses of 95.9 ± 14.1 kPa and 38.7 ± 3.0 kPa, successfully mimic their animal counterparts, with 63.5 ± 45.7 kPa and 44.3 ± 13.2 kPa, while tofurky is twice as stiff, and tofu is twice as soft. Strikingly, a complementary food tasting survey produces in nearly identical stiffness rankings for all eight products (ρ = 0.833, p = 0.015). Probing the fully three-dimensional signature of meats is critical to understand subtle differences in texture that may result in a different perception of taste. Our data and code are freely available at https://github.com/LivingMatterLab/CANN
CRIS-Team Scopus Importer:2024-11-29

The mechanical importance of myelination in the central nervous system - Weickenmeier, Johannes; de Rooij, Rijk; Budday, Silvia; Ovaert, Timothy; Kuhl, Ellen (Journal of the Mechanical Behavior of Biomedical Materials, 2017)
Neurons in the central nervous system are surrounded and cross-linked by myelin, a fatty white substance that wraps around axons to create an electrically insulating layer. The electrical function of myelin is widely recognized; yet, its mechanical importance remains underestimated. Here we combined nanoindentation testing and histological staining to correlate brain stiffness to the degree of myelination in immature, pre-natal brains and mature, post-natal brains. We found that both gray and white matter tissue stiffened significantly (p ≪ 0.001) upon maturation: the gray matter stiffness doubled from 0.31 ± 0.20 kPa pre-natally to 0.68 ± 0.20 kPa post-natally; the white matter stiffness tripled from 0.45 ± 0.18 kPa pre-natally to 1.33±0.64kPa post-natally. At the same time, the white matter myelin content increased significantly (p ≪ 0.001) from 58 ± 2% to 74 ± 9%. White matter stiffness and myelin content were correlated with a Pearson correlation coefficient of ρ=0.92 (p ≪ 0.001). Our study suggests that myelin is not only important to ensure smooth electrical signal propagation in neurons, but also to protect neurons against physical forces and provide a strong microstructural network that stiffens the white matter tissue as a whole. Our results suggest that brain tissue stiffness could serve as a biomarker for multiple sclerosis and other forms of demyelinating disorders. Understanding how tissue maturation translates into changes in mechanical properties and knowing the precise brain stiffness at different stages of life has important medical implications in development, aging, and neurodegeneration.
Keywords: Brain; Stiffness; Indentation; White matter; Myelin; Myelination

The role of mechanics during brain development - Budday, Silvia; Steinmann, Paul; Kuhl, Ellen (Journal of the Mechanics and Physics of Solids, 2014)
Convolutions are a classical hallmark of most mammalian brains. Brain surface morphology is often associated with intelligence and closely correlated with neurological dysfunction. Yet, we know surprisingly little about the underlying mechanisms of cortical folding. Here we identify the role of the key anatomic players during the folding process: cortical thickness, stiffness, and growth. To establish estimates for the critical time, pressure, and the wavelength at the onset of folding, we derive an analytical model using the Föppl-von Kármán theory. Analytical modeling provides a quick first insight into the critical conditions at the onset of folding, yet it fails to predict the evolution of complex instability patterns in the post-critical regime. To predict realistic surface morphologies, we establish a computational model using the continuum theory of finite growth. Computational modeling not only confirms our analytical estimates, but is also capable of predicting the formation of complex surface morphologies with asymmetric patterns and secondary folds. Taken together, our analytical and computational models explain why larger mammalian brains tend to be more convoluted than smaller brains. Both models provide mechanistic interpretations of the classical malformations of lissencephaly and polymicrogyria. Understanding the process of cortical folding in the mammalian brain has direct implications on the diagnostics of neurological disorders including severe retardation, epilepsy, schizophrenia, and autism.
UnivIS-Import:2015-03-09:Pub.2014.tech.FT.FT-TM.therol
Keywords: Brain development; Cortical folding; Growth; Instabilities; Thin films

Towards microstructure-informed material models for human brain tissue - Budday, Silvia; Sarem, Melika; Starck, Laurent; Sommer, Gerhard; Pfefferle, Jule; Phunchago, Nichapa; Kuhl, Ellen; Paulsen, Friedrich; Steinmann, Paul; Shastri, V. Prasad; Holzapfel, Gerhard A. (Acta Biomaterialia, 2020)
Emerging evidence suggests that the mechanical behavior of the brain plays a critical role in development, disease, and aging. Recent studies have begun to characterize the mechanical behavior of gray and white matter tissue and to identify sets of material models that best reproduce the stress-strain behavior of different brain regions. Yet, these models are mainly phenomenological in nature, their parameters often lack clear physical interpretation, and they fail to correlate the mechanical behavior to the underlying microstructural composition. Here we make a first attempt towards identifying general relations between microstructure and mechanics with the ultimate goal to develop microstructurally motivated constitutive equations for human brain tissue. Using histological staining, we analyze the microstructure of brain specimens from different anatomical regions, the cortex, basal ganglia, corona radiata, and corpus callosum, and identify the regional stiffness and viscosity under multiple loading conditions, simple shear, compression, and tension. Strikingly, our study reveals a negative correlation between cell count and stiffness, a positive cor- relation between myelin content and stiffness, and a negative correlation between proteoglycan content and stiffness. Additionally, our analysis shows a positive correlation between lipid and proteoglycan content and viscosity. We demonstrate how understanding the microstructural origin of the macroscopic behavior of the brain can help us design microstructure-informed material models for human brain tissue that inherently capture regional heterogeneities. This study represents an important step towards using brain tissue stiffness and viscosity as early diagnostic markers for clinical conditions including chronic traumatic encephalopathy, Alzheimer’s and Parkinson’s disease, or multiple sclerosis.
Keywords: material modeling; biomechanical testing; hyperelasticity; Ogden model; human brain tissue; microstructure

Towards the Algorithmic Treatment of 3D Strong Discontinuities - Mergheim, Julia; Kuhl, Ellen; Steinmann, Paul (Communications in Numerical Methods in Engineering, 2007)
A geometrically non-linear finite element framework for the modelling of propagating discontinuities in three-dimensional continua is presented. By doubling the degrees of freedom in the discontinuous elements, the algorithm allows for arbitrary discontinuities which are not restricted to inter-element boundaries. The deformation field is interpolated independently on both sides of the discontinuity. In contrast to the X-FEM, the suggested approach thus relies exclusively on displacement degrees of freedom. On the discontinuity surface, the jump in the deformation is related to the cohesive tractions to account for smooth crack opening. Computational difficulties characteristic of three-dimensional crack propagation are addressed. The performance of the method is elaborated by means of a homogeneous three-dimensional tension problem and by means of the classical peel test. Copyright © 2006 John Wiley & Sons, Ltd.
UnivIS-Import:2015-03-09:Pub.2007.tech.FT.FT-TM.toward
Keywords: Cohesive surfaces; Discontinuous elements; Nitsche's method; Propagating discontinuities

Towards the treatment of boundary conditions for global crack path tracking in three-dimensional brittle fracture - Jäger, Paul; Steinmann, Paul; Kuhl, Ellen (Computational Mechanics, 2009)
The aim of the present paper is a systematic elaboration of a three dimensional finite element analysis tool for discontinuous fracture in brittle solids. Brittle or quasi-brittle fracture usually occurs when a material reaches the limit of its strength and no plastic deformation has been observed prior to failure. In the present approach, this kind of failure is characterized by three sets of governing equations: (i) the elastic bulk problem and (ii) the cohesive interface problem regarding the solid deformation field and (iii) the crack tracking problem concerning the crack kinematics. This manuscript describes a unique modular tool set for the coupled set of nonlinear equations. We focus in particular on the boundary conditions for the crack tracking problem of this analysis tool. We critically discuss important implementation details: (i) the choice of crack onset boundary conditions for the additional global field, (ii) the numerical integration, (iii) the modeling of geometrically exact crack tips for cohesive fracture, and (iv) the post-processing procedure for the discontinuity visualization. The potential of the method to simulate brittle fracture is demonstrated by qualitative and quantitative comparisons with experiments from the literature as well as by common benchmark problems. © 2009 Springer-Verlag.
UnivIS-Import:2015-03-09:Pub.2009.tech.FT.FT-TM.toward_2
Keywords: Brittle fracture; Cohesive crack concept; Concrete modeling; Discontinuous failure; Interface; Visualization

Viscoelastic parameter identification of human brain tissue - Budday, Silvia; Sommer, Gerhard; Holzapfel, Gerhard A.; Steinmann, Paul; Kuhl, Ellen (Journal of the Mechanical Behavior of Biomedical Materials, 2017)
Understanding the constitutive behavior of the human brain is critical to interpret the physical environment during neurodevelopment, neurosurgery, and neurodegeneration. A wide variety of constitutive models has been proposed to characterize the brain at different temporal and spatial scales. Yet, their model parameters are typically calibrated with a single loading mode and fail to predict the behavior under arbitrary loading condi- tions. Here we used a finite viscoelastic Ogden model with six material parameters–an elastic stiffness, two viscoelastic stiffnesses, a nonlinearity parameter, and two viscous time constants–to model the characteristic nonlinearity, conditioning, hysteresis and tension-compression asymmetry of the human brain. We calibrated the model under shear, shear relaxation, compression, compression relaxation, and tension for four different regions of the human brain, the cortex, basal ganglia, corona radiata, and corpus callosum. Strikingly, un- conditioned gray matter with 0.36 kPa and white matter with 0.35 kPa were equally stiff, whereas conditioned gray matter with 0.52 kPa was three times stiffer than white matter with 0.18 kPa. While both unconditioned viscous time constants were larger in gray than in white matter, both conditioned constants were smaller. These rheological differences suggest a different porosity between both tissues and explain–at least in part–the ongoing controversy between reported stiffness differences in gray and white matter. Our unconditioned and conditioned parameter sets are readily available for finite element simulations with commercial software packages that feature Ogden type models at finite deformations. As such, our results have direct implications on improving the accuracy of human brain simulations in health and disease.
Keywords: Human brain; Rheological testing; Finite viscoelasticity; Ogden model; Parameter identification

Wrinkling instabilities in soft bi-layered systems - Budday, Silvia; Andres, Sebastian; Walter, Bastian; Steinmann, Paul; Kuhl, Ellen (Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 2017)
Wrinkling phenomena control the surface morphology of many technical and biological systems. While primary wrinkling has been extensively studied, experimentally, analytically and computationally, higher-order instabilities remain insufficiently understood, especially in systems with stiffness contrasts well below 100. Here, we use the model system of an elastomeric bilayer to experimentally characterize primary and secondary wrinkling at moderate stiffness contrasts. We systematically vary the film thickness and substrate prestretch to explore which parameters modulate the emergence of secondary instabilities, including period-doubling, period-tripling and wrinkle-to-fold transitions. Our experiments suggest that period-doubling is the favourable secondary instability mode and that period-tripling can emerge under disturbed boundary conditions. High substrate prestretch can suppress period-doubling and primary wrinkles immediately transform into folds. We combine analytical models with computational simulations to predict the onset of primary wrinkling, the post-buckling behaviour, secondary bifurcations and the wrinkle-to-fold transition. Understanding the mechanisms of pattern selection and identifying the critical control parameters of wrinkling will allow us to fabricate smart surfaces with tunable properties and to control undesired surface patterns like in the asthmatic airway.
Keywords: thin film; instability; bifurcation; period-doubling; period-tripling