Nonlinear Field Theories Of Mechanics Research Articles

A new family of Constitutive Artificial Neural Networks towards automated model discovery

For more than 100 years, chemical, physical, and material scientists have proposed competing constitutive models to best characterize the behavior of natural and man-made materials in response to mechanical loading. Now, computer science offers a universal solution: Neural Networks. Neural Networks are powerful function approximators that can learn constitutive relations from large data without any knowledge of the underlying physics. However, classical Neural Networks entirely ignore a century of research in constitutive modeling, violate thermodynamic considerations, and fail to predict the behavior outside the training regime. Here we design a new family of Constitutive Artificial Neural Networks that inherently satisfy common kinematical, thermodynamical, and physical constraints and, at the same time, constrain the design space of admissible functions to create robust approximators, even in the presence of sparse data. Towards this goal we revisit the non-linear field theories of mechanics and reverse-engineer the network input to account for material objectivity, material symmetry and incompressibility; the network output to enforce thermodynamic consistency; the activation functions to implement physically reasonable restrictions; and the network architecture to ensure polyconvexity. We demonstrate that this new class of models is a generalization of the classical neo Hooke, Blatz Ko, Mooney Rivlin, Yeoh, and Demiray models and that the network weights have a clear physical interpretation in the form of shear moduli, stiffness-like parameters, and exponential coefficients. When trained with classical benchmark data for rubber under uniaxial tension, biaxial extension, and pure shear, our network autonomously selects the best constitutive model and learns its set of parameters. Our findings suggest that Constitutive Artificial Neural Networks have the potential to induce a paradigm shift in constitutive modeling, from user-defined model selection to automated model discovery. Our source code, data, and examples is available at https://github.com/LivingMatterLab/CANN.

Bulging brains.

Brain swelling is a serious condition associated with an accumulation of fluid inside the brain that can be caused by trauma, stroke, infection, or tumors. It increases the pressure inside the skull and reduces blood and oxygen supply. To relieve the intracranial pressure, neurosurgeons remove part of the skull and allow the swollen brain to bulge outward, a procedure known as decompressive craniectomy. Decompressive craniectomy has been preformed for more than a century; yet, its effects on the swollen brain remain poorly understood. Here we characterize the deformation, strain, and stretch in bulging brains using the nonlinear field theories of mechanics. Our study shows that even small swelling volumes of 28 to 56 ml induce maximum principal strains in excess of 30%. For radially outward-pointing axons, we observe maximal normal stretches of 1.3 deep inside the bulge and maximal tangential stretches of 1.3 around the craniectomy edge. While the stretch magnitude varies with opening site and swelling region, our study suggests that the locations of maximum stretch are universally shared amongst all bulging brains. Our model has the potential to inform neurosurgeons and rationalize the shape and position of the skull opening, with the ultimate goal to reduce brain damage and improve the structural and functional outcomes of decompressive craniectomy in trauma patients.

Secondary instabilities modulate cortical complexity in the mammalian brain

Disclosing the origin of convolutions in the mammalian brain remains a scientific challenge. Primary folds form before we are born: they are static, well defined and highly preserved across individuals. Secondary folds occur and disappear throughout our entire lifetime: they are dynamic, irregular and highly variable among individuals. While extensive research has improved our understanding of primary folding in the mammalian brain, secondary folding remains understudied and poorly understood. Here, we show that secondary instabilities can explain the increasing complexity of our brain surface as we age. Using the nonlinear field theories of mechanics supplemented by the theory of finite growth, we explore the critical conditions for secondary instabilities. We show that with continuing growth, our brain surface continues to bifurcate into increasingly complex morphologies. Our results suggest that even small geometric variations can have a significant impact on surface morphogenesis. Secondary bifurcations, and with them morphological changes during childhood and adolescence, are closely associated with the formation and loss of neuronal connections. Understanding the correlation between neuronal connectivity, cortical thickness, surface morphology and ultimately behaviour, could have important implications on the diagnostics, classification and treatment of neurological disorders.

Patient-Specific Airway Wall Remodeling in Chronic Lung Disease.

Chronic lung disease affects more than a quarter of the adult population; yet, the mechanics of the airways are poorly understood. The pathophysiology of chronic lung disease is commonly characterized by mucosal growth and smooth muscle contraction of the airways, which initiate an inward folding of the mucosal layer and progressive airflow obstruction. Since the degree of obstruction is closely correlated with the number of folds, mucosal folding has been extensively studied in idealized circular cross sections. However, airflow obstruction has never been studied in real airway geometries; the behavior of imperfect, non-cylindrical, continuously branching airways remains unknown. Here we model the effects of chronic lung disease using the nonlinear field theories of mechanics supplemented by the theory of finite growth. We perform finite element analysis of patient-specific Y-branch segments created from magnetic resonance images. We demonstrate that the mucosal folding pattern is insensitive to the specific airway geometry, but that it critically depends on the mucosal and submucosal stiffness, thickness, and loading mechanism. Our results suggests that patient-specific airway models with inherent geometric imperfections are more sensitive to obstruction than idealized circular models. Our models help to explain the pathophysiology of airway obstruction in chronic lung disease and hold promise to improve the diagnostics and treatment of asthma, bronchitis, chronic obstructive pulmonary disease, and respiratory failure.

Morphoelastic control of gastro-intestinal organogenesis: Theoretical predictions and numerical insights

With nine meters in length, the gastrointestinal tract is not only our longest, but also our structurally most diverse organ. During embryonic development, it evolves as a bilayered tube with an inner endodermal lining and an outer mesodermal layer. Its inner surface displays a wide variety of morphological patterns, which are closely correlated to digestive function. However, the evolution of these intestinal patterns remains poorly understood. Here we show that geometric and mechanical factors can explain intestinal pattern formation. Using the nonlinear field theories of mechanics, we model surface morphogenesis as the instability problem of constrained differential growth. To allow for internal and external expansion, we model the gastrointestinal tract with homogeneous Neumann boundary conditions. To establish estimates for the folding pattern at the onset of folding, we perform a linear stability analysis supplemented by the perturbation theory. To predict pattern evolution in the post-buckling regime, we perform a series of nonlinear finite element simulations. Our model explains why longitudinal folds emerge in the esophagus with a thick and stiff outer layer, whereas circumferential folds emerge in the jejunum with a thinner and softer outer layer. In intermediate regions like the feline esophagus, longitudinal and circumferential folds emerge simultaneously. Our model could serve as a valuable tool to explain and predict alterations in esophageal morphology as a result of developmental disorders or certain digestive pathologies including food allergies.

A mechanical model predicts morphological abnormalities in the developing human brain.

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.

Some Indentities for the Gradient of the Principal Invariants, Traces and Determinants via Grassmann Calculus

Let A be a second order tensor in a finite dimensional space. In this work we determine the gradient of the principal invariants of A and obtain some trace and determinant identities using only some standard rigorous statements concerning Grassmann calculus. We recover some of the results of Dui et al. (J. Elast. 75:193–196, 2004) and of Truesdell and Noll (The Non-linear Field Theories of Mechanics, Springer, Berlin, 2002) and solve an old problem proposed in SIAM Review concerning a determinant identity from a new perspective in a concise and simple manner.

Evolution of Surfaces and the Kinematics of Membranes

The polar decomposition theorem for a two dimensional continuum (a membrane) is used to produce a set of equations that describe the evolution of the geometrical quantities of a moving surface, i.e., the metric, the unit normal, the shape operator, the second fundamental form, the mean and the Gauss curvature. A link to the kinematical quantities of the continuum is also given. The version of the polar decomposition theorem for membranes we use was proved by Chi-Sing Man and H. Cohen (J. Elast. 16:97–104, 1986). Both the geometric and the kinematical framework are coordinate-free, in an attempt to contribute to a coordinate-free description for the kinematics of membranes in analogy to the kinematics of three dimensional continuum bodies as it emerges from the classical works of Noll (Arch. Rat. Mech. Anal. 2:197–226, 1958), Truesdell and Noll (The Non-linear Field Theories of Mechanics, 3rd edn., Springer, Berlin, 2004), Truesdell (A First Course in Rational Continuum Mechanics, vol. 1, Academic Press, San Diego, 1977).

Combined axial shearing, extension, and straightening of elastic annular cylindrical sectors

The combined axial shearing, extension, and straightening of an annular cylindrical sector is a deformation that, following Truesdell & Noll (1965, The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics III/3) and Hill (1973, Z. Angew. Math. Phys., 24, 609-618), we describe in terms of two prescribed constants and two unknown functions that depend only on the radial material coordinate. Under the assumption that the material is elastic, compressible, and isotropic we show that for equilibrium in the absence of body forces the unknown functions must satisfy a system of first-order nonlinear ordinary differential equations. The system of differential equations can be decoupled for certain material classes, one of which is the class of Hadamard-Green materials. Thus, several new exact solutions are obtained and, under the assumption that the annular cylindrical sector is composed of a Hadamard-Green material that is strongly elliptic, the existence and uniqueness of solutions for two types of boundary conditions is established.

The Genesis of Truesdell's Nonlinear Field Theories of Mechanics

The Genesis of Truesdell's Nonlinear Field Theories of Mechanics

Constitutive Modelling of Geomaterials

Constitutive Modelling of Geomaterials

On the validity of the Clausius-Duhem inequality

The author discusses the applicability of various theories for describing processes in continuous matter and deals in some detail with the entropy-free thermodynamics of irreversible processes, basing his discourse on the work of J. Meixner in this field. Today there are essentially three different phenomenological theories available for describing processes in continuous matter: (1) The classical Thermodynamics of Irreversible Processes (TIP)1. (2) The Non-linear Field-Theories of Mechanics and their thermodynamic extensions (NFT)2. (3) The Entropy-free Thermodynamics of Processes (ETIP)3' . These theories have in common the conservation laws of continuous matter and the laws of thermostatics. They differ essentially in the arguments on which the constitutive equations (CE) are based. Classical TIP starts from a generalized Clausius—Duhem Inequality (CDI) for the well-known (specific) thermostatic entropy s p5+divJ8O (1) The quantity is well defined by the second law of thermodynamics and is a function = s5(u, p) for fluids and a function = s(u, F) for solid systems. Here u means the (specific) internal energy, p... density, Ff1... deformation gradient, substantial time derivative of and J... current of s. In the case of thermodynamic systems without diffusion one has

The Non-Linear Field Theories of Mechanics

The Non-Linear Field Theories of Mechanics

Nonlinear Viscoelastic Solids

1. Introduction. The classical theories of the mechanics of deformable materials are, for the most part, linear theories and ones in which the stress depends only on the kinematic situation existing at the time of measurement of the stress. For many materials, such as viscoelastic solids and liquids, these assumptions are not valid. Since the paper by Oldroyd [13], considerable advances have been made in formulating the continuum mechanics of such materials. In the present paper an account of this work will be given, particularly in relation to viscoelastic solids. Provided that any characteristic lengths which may exist iu the solid can be ignored, we may take as our starting point in the development of appropriate constitutive equations the assumption that the stress tensor: depends not only on the displacement gradients existing in the material considered at the instant at which the stress is measured, but also on those which existed in it at all previous times, i.e., the stress is a functional of the deformation gradient history up to the instant of measurement. If some reference state is adequately specified, the assumption that stress depends on the history of the deformation gradients and the assumption that stress depends on the history of the velocity or acceleration. gradients are quite equivalent, since either of the latter determine the deformation gradient history. The form of the functional dependence of the stress on the deformation gradient history cannot be arbitrarily chosen. It is subject to two major restrictions. The first of these, which is discussed in 2, arises from the fact that a simultaneous time-dependent rotation of the deformed body and reference system leaves the stress components unaltered. The second, which is discussed in 3 and 5, arises from any symmetry which the material may possess in its reference state. The important particular case when the material is isotropic in its reference state is discussed in 5. In certain circumstances, the history of the deformation gradients may be adequately described by a knowledge of the values of the deformation gradients and of the gradients of velocity, acceleration, second acceleration., etc., at the instant of measurement of the stress. The development of canonical forms for the constitutive equations in this case is discussed generally in 4 and for the particular circumstance when the material is isotropic in its reference state in 5. The constitutive equations which arise from these general studies are inevitably complicated. In 6 attention is drawn to the great simplifications whichmay arise