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Continuum Mechanics and Thermodynamics
Abstract
Continuum mechanics and thermodynamics are foundational theories of many fields of science and engineering. This book presents a fresh perspective on these fundamental topics, connecting micro- and nanoscopic theories and emphasizing topics relevant to understanding solid-state thermo-mechanical behavior. Providing clear, in-depth coverage, the book gives a self-contained treatment of topics directly related to nonlinear materials modeling. It starts with vectors and tensors, finite deformation kinematics, the fundamental balance and conservation laws, and classical thermodynamics. It then discusses the principles of constitutive theory and examples of constitutive models, presents a foundational treatment of energy principles and stability theory, and concludes with example closed-form solutions and the essentials of finite elements. Together with its companion book, Modeling Materials, (Cambridge University Press, 2011), this work presents the fundamentals of multiscale materials modeling for graduate students and researchers in physics, materials science, chemistry and engineering.
... 8,47 For example, copper is a cubic crystal with mirror planes and threefold rotation axes (point group m 3m), and such symmetry results in a number of only three independent components. Formally, the material symmetry imposes the following constraints on the elasticity tensor: 45,49 ...
... 49 Of the eight classes, one is for isotropic materials, and each of the other seven corresponds to a crystal system. 45,50 In our opinion, there is still signicant confusion on this topic. For example, the categorization by Wallace 51 and populated by Nye, 47 which incorrectly gives two unique classes for each of the tetragonal and trigonal cases (Fig. S1 in the ESI †), is still widely cited in recent works. ...
... For example, the categorization by Wallace 51 and populated by Nye, 47 which incorrectly gives two unique classes for each of the tetragonal and trigonal cases (Fig. S1 in the ESI †), is still widely cited in recent works. [52][53][54] We refer to Section 6.5 of ref. 45 for a historical note on the development of the categorization. ...
The elasticity tensor is a fundamental material property that describes the elastic response of a material to external force. The availability of full elasticity tensors for inorganic crystalline compounds, however, is limited due to experimental and computational challenges. Here, we report the materials tensor (MatTen) model for rapid and accurate prediction of the full fourth-rank elasticity tensors of crystals. Based on equivariant graph neural networks, MatTen satisfies two essential requirements for elasticity tensors: independence of the frame of reference and preservation of material symmetry. Consequently, it provides a unified treatment of elasticity tensors for all seven crystal systems across diverse chemical spaces, without the need to deal with each separately. MatTen was trained on a dataset of first-principles elasticity tensors garnered by the Materials Project over the past several years (we are releasing the data herein) and has broad applications in predicting the isotropic elastic properties of polycrystalline materials, examining the anisotropic behavior of single crystals, and discovering materials with exceptional mechanical properties. Using MatTen, we have found a hundred crystals with extremely large maximum directional Young's modulus and eleven polymorphs of elemental cubic metals with unconventional spatial orientation of Young's modulus.
... First, the standard balances of macro-forces and moments are expressed in a Lagrangian description [76] as follows ...
... For isothermal processes, the second law of thermodynamics can be simplified as follows [76]: ...
... To this end, we introduce the following function spaces in addition to the spaces (S h d,k and V h d,k ) which were defined in Eqs. (76) and (77): ...
Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton's principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee -continuity of the solution at inter-element boundaries, and enforces the required regularity with the aid of additional variational and interior penalty terms in the weak form. The phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods.
... Having accepted the postulate of local thermodynamic equilibrium, we assumed that at each point, there exists a thermodynamic equilibrium locally at each step of loading. Thus, the fields of state variables would describe the transition made by external perturbations (for further information, look at [36]). The first law of thermodynamics is explained in terms of the rate of kinetic energy _ K ,the rate of internal energy _ U , the external work P ext and the heat supply R as: ...
... The external power leads to the body deformation, which is expressed by multiplication of stress (σ) and strain (ɛ) variables. Lastly, applying the divergence theorem and combining terms, yields the conservation of energy for an infinitesimal continuum particle as [36]: ...
... Thermodynamically, the mechanical degradation of a system is a set of irreversible processes deal with the entropy of the system, which increases with time up to the final failure. The Clausius-Duhem inequality, in the rating form, states that the specific internal entropy production rate (_ s int ) never decreases as follows [36]: ...
Experimental investigation of recent research shows that the accumulated generated entropy during low-cycle fatigue due to irreversible thermodynamic processes could be a constant value as a material property in metals and composite laminates. The present study aims to evaluate the constancy of accumulated generated entropy in static loading in plain weave fabric composites. A unit-cell-based model, considering different defects, is assembled with continuum damage mechanics (CDM) at the tows level. Equivalent elastic properties of a layer, the initiation, and evolution of damage, as well as irreversible deformation in uniaxial tension and in-plane shear, are simulated in a meaningful physical scale up to the final failure. The obtained results will confirm that Accumulated Entropy Generation (AEG) can be an intrinsic material property in static loading and a measure of the final failure, which can be used in mixed loading conditions.
... Having accepted the postulate of local thermodynamic equilibrium, we assumed that at each point, there exists a thermodynamic equilibrium locally at each step of loading. Thus, the fields of state variables would describe the transition made by external perturbations (for further information, look at [36]). The first law of thermodynamics is explained in terms of the rate of kinetic energy _ K ,the rate of internal energy _ U , the external work P ext and the heat supply R as: ...
... The external power leads to the body deformation, which is expressed by multiplication of stress (σ) and strain (ɛ) variables. Lastly, applying the divergence theorem and combining terms, yields the conservation of energy for an infinitesimal continuum particle as [36]: ...
... Thermodynamically, the mechanical degradation of a system is a set of irreversible processes deal with the entropy of the system, which increases with time up to the final failure. The Clausius-Duhem inequality, in the rating form, states that the specific internal entropy production rate (_ s int ) never decreases as follows [36]: ...
Experimental investigation of recent research shows that the accumulated generated entropy during low-cycle fatigue due to irreversible thermodynamic processes could be a constant value as a material property in metals and composite laminates. The present study aims to evaluate the constancy of accumulated generated entropy in static loading in plain weave fabric composites. A unit-cell-based model, considering different defects, is assembled with continuum damage mechanics (CDM) at the tows level. Equivalent elastic properties of a layer, the initiation, and evolution of damage, as well as irreversible deformation in uniaxial tension and in-plane shear, are simulated in a meaningful physical scale up to the final failure. The obtained results will confirm that Accumulated Entropy Generation (AEG) can be an intrinsic material property in static loading and a measure of the final failure, which can be used in mixed loading conditions.
... Although these equations describe the evolution of the conserved quantities, we can not deduce from these equations any restriction about admissible direction of the underlying physical processes, cf. [29, 44, 48, 59, 81, 85, 87]. More precisely, in order to come to reasonable statements about admissible directions of physical processes, we have to introduce an other quantity: The specific entropy s [ ...
... [19, p. 25]. However, regarding a detailed presentation of the classical results and the history of thermodynamics, we refer, e.g., to [23, 49, 58, 59, 65, 81, 85, 87]. We proceed by deriving an explicit expression for the entropy production rate σ. ...
... In these situations, we have to choose among the various constitutive laws for viscoelastic materials instead, cf. [46, 77, 81, 82]. ...
This paper presents a thermodynamically consistent model for multicomponent electrolyte solutions. The first part of this paper derives the general governing equations for nonequilibrium systems within the theory of nonequilibrium thermodynamics. Here, we consider electrolyte solutions as general mixtures of charged constituents. Furthermore, in this part of the paper we combine the general theory of nonequilibrium thermodynamics with the well-known splittings of the entropy and the energy into a pure substance part and a part due to mixing. Thereby, we successfully establish evolution equations for both parts. Furthermore, we derive for both parts explicit expressions of the respective entropy production rates. Hence, we provide an approach that allows to study the entropy of mixing independently of the pure substance entropy and vice versa. This is of great value, in particular for a better understanding of the complex phenomena due to mixing in multicomponent systems. In the second part of this paper, we close the system of general balance equations by applying constitutive laws. This is the crucial step in the modeling procedure. ...
... First, the standard balances of macro-forces and moments are expressed in a Lagrangian description [82] as follows ...
... For isothermal processes, the second law of thermodynamics can be simplified as follows [82]: ...
Modeling crack initiation and propagation in brittle materials is of great importance to be able to predict sudden loss of load-carrying capacity and prevent catastrophic failure under severe dynamic loading conditions. Second-order phase-field fracture models have gained wide adoption given their ability to capture the formation of complex fracture patterns, e.g. via crack merging and branching, and their suitability for implementation within the context of the conventional finite element method. Higher-order phase-field models have also been proposed to increase the regularity of the exact solution and thus increase the spatial convergence rate of its numerical approximation. However, they require special numerical techniques to enforce the necessary continuity of the phase field solution. In this paper, we derive a fourth-order phase-field model of fracture in two independent ways; namely, from Hamilton’s principle and from a higher-order micromechanics-based approach. The latter approach is novel, and provides a physical interpretation of the higher-order terms in the model. In addition, we propose a continuous/discontinuous Galerkin (C/DG) method for use in computing the approximate phase-field solution. This method employs Lagrange polynomial shape functions to guarantee C0-continuity of the solution at inter-element boundaries, and enforces the required C1 regularity with the aid of additional variational and interior penalty terms in the weak form. The phase-field equation is coupled with the momentum balance equation to model dynamic fracture problems in hyper-elastic materials. Two benchmark problems are presented to compare the numerical behavior of the C/DG method with mixed finite element methods.
... We consider the classical problem of seeking the static equilibrium of an elastic body undergoing finite deformations, see for example Odgen [36], Ciarlet and Philippe [37], Gurtin et al. [38], Tadmor et al. [39] and Bonnet et al. [40] for additional details. ...
... Incompressible neo-Hookean (NHK-I), [39] : ...
In this work we introduce a dG framework for nonlinear elasticity based on a Bassi-Rebay (BR2) formulation. The framework encompasses compressible and incompressible hyperelastic materials and is capable of dealing with large deformations. In order to achieve stability, we combine higher-order lifting operators for the BR2 stabilization term with an adaptive stabilization strategy which relies on the BR2 Laplace operator stabilization and a penalty parameter based on the spectrum of the fourth-order elasticity tensor. Dirichlet boundary conditions for the displacement can be imposed by means of Lagrange multipliers and Nitsche method. Efficiency of the solution strategy is achieved by means of state-of-the-art agglomeration based h-multigrid preconditioners and the code implementation supports distributed memory execution on modern parallel architectures. Several benchmark test cases are proposed in order to investigate some relevant computational aspects, namely the performance of the h-multigrid iterative solver varying the stabilization parameters and the influence of Dirichlet boundary conditions on Newton's method globalisation strategy.
... (a) Linearized solid and structural elements such as two-and three-node truss (i.e., lin2DTruss2, lin2DTruss3, lin3DTruss2, and lin3DTruss3), three-and six-node triangular (i.e., lin2DTria3 and lin2DTria6), four-and eight-node quadrilateral (i.e., lin2DQuad4 and lin2DQuad8), four-and ten-node tetrahedron (i.e., lin3DTetra4 and lin3DTetra10), eight-and twenty-node hexahedron (i.e., lin3DHexa8 and lin3D-Hexa20), two-node frame (i.e., lin2DFrame2 and lin3DFrame2), and four-node shell (i.e., lin3DShell4) elements are currently available. (b) Finite kinematics solid and structural elements such as two-node truss (i.e., kin-2DTruss2 and kin3DTruss2), four-node quadrilateral (i.e., kin2DQuad4), eight-node hexahedron (i.e., kin3DHexa8), and two-node frame (i.e., kin2DFrame2 and kin3D-Frame2) elements currently allow large deformation [3,73,74]. (c) The perfectly matched layer (PML) can be specified for emulating semi-infinite halfspaces in 2D and 3D simulations. Currently, four-and eight-node quadrilateral (PML2DQuad4 and PML2DQuad8), and eight-and twenty-node hexahedron (PML3DHexa8 and PML3DHexa20) elements are implemented. ...
In the fields of structural and geotechnical engineering, improving the understanding of soil–structure interaction (SSI) effects is critical for earthquake-resistant design. Engineers and practitioners often resort to finite element (FE) software to advance this objective. Unfortunately, the availability of software equipped with boundary representation for absorbing scattered waves and ensuring consistent input ground motion prescriptions, which is necessary for accurately representing SSI effects, is currently limited. To address such limitations, the authors developed Seismo-VLAB (SVL v1.0-stable) an open-source software designed to perform SSI simulations. The methodology considers the integration of advanced techniques, including the domain decomposition method (DDM), perfectly matched layers (PMLs), and domain reduction method (DRM), in addition to parallel computing capabilities to accelerate the solution of large-scale problems. In this work, the authors provide a detailed description of the implementation for addressing SSI modeling, validate some of the SVL’s features needed for such purpose, and demonstrate that the coupled DRM–PML technique is a necessary condition for accurately solving SSI problems. It is expected that SVL provides a significant contribution to the SSI research community, offering a self-contained and versatile alternative. The software’s practical application in analyzing SSI and directionality effects on 3D structures under seismic loading demonstrates its capability to model real-world earthquake responses in structural engineering.
... where ∇ 0 denotes the gradient operator in the reference configuration, P is the first Piola-Kirchhoff stress tensor described by an underlying constitutive law discussed in more detail below in Section 4.2, and u ∂Ωmve is a prescribed displacement on the boundary ∂Ω mve , see, e.g., Tadmor et al. (2011) for more details on continuum mechanics. Note that because only essential boundary conditions are of interest, no tractions are prescribed to the model. ...
Micromechanical parameters are essential in understanding the behavior of materials with a heterogeneous structure, which helps to predict complex physical processes such as delamination, cracks, and plasticity. However, identifying these parameters is challenging due to micro-macro length scale differences, required high resolution, and ambiguity in boundary conditions, among others. The Integrated Digital Image Correlation (IDIC) method, a state-of-the-art full-field deterministic approach to parameter identification, is widely used but suffers from high sensitivity to boundary data errors and is limited to identification of parameters within well-posed problems. This article employs Bayesian approach to estimate micromechanical shear and bulk moduli of fiber-reinforced composite samples under plane strain assumption, and to improve handling of boundary noise. The main purpose of this article is to quantify the effect of uncertainty in the boundary conditions in the stochastic setting. To this end, the Metropolis–Hastings Algorithm (MHA) is employed to estimate probability distributions of bulk and shear moduli and boundary condition parameters using IDIC, considering a fiber-reinforced composite sample under plane strain assumption. The performance and robustness of the MHA are compared to two versions of deterministic IDIC method, under artificially introduced random and systematic errors in kinematic boundary conditions. Although MHA is shown to be computationally more expensive and in certain cases less accurate than the recently introduced Boundary-Enriched IDIC, it offers significant advantages, in particular being able to optimize a large number of parameters while obtaining statistical characterization as well as insights into individual parameter relationships. The paper furthermore highlights the benefits of the non-normalized approach to parameter identification with MHA (leading, within deterministic IDIC, to an ill-posed formulation), which significantly improves the robustness in handling the boundary noise.
... where ∇ 0 denotes the gradient operator in the reference configuration, P is the first Piola-Kirchhoff stress tensor, and u ∂Ωmve is a prescribed displacement on the boundary ∂Ω mve , see, e.g., Tadmor et al. (2011) for more details on continuum mechanics. The solution is typically discretized with the Finite Element Method (FEM). ...
Materials with heterogeneous structures exhibit complex physical processes such as delamination, cracks, and plasticity, which require micromechanical parameters for understanding. However, identifying these parameters is challenging due to micro-macro length scale differences, ambiguity in boundary conditions, and required high resolution, among others. While the Integrated Digital Image Correlation (IDIC) method is widely used for this purpose, it suffers from sensitivity to boundary data errors and limited identification of parameters within well-posed problems. To address these issues, a Bayesian approach is proposed for micromechanical parameter identification using the Metropolis--Hastings Algorithm (MHA) to estimate probability distributions of bulk and shear moduli and boundary condition parameters. The proposed approach is compared to two versions of deterministic IDIC method under artificially introduced errors. Although MHA is computationally more expensive and in certain cases less accurate than BE-IDIC, it offers significant advantages, including the ability to optimize a large number of parameters, obtain statistical characterization and insights into individual parameter relationships. The paper highlights the benefits of the non-normalized approach to parameter identification with MHA, which significantly improves the robustness in handling boundary noise, compared to deterministic IDIC. The study considers a fiber-reinforced composite sample under plane strain assumption.
... The present manuscript focuses on the multiphysics modeling of protein motility along advecting animal cell membranes, overviewing the state of the art and proposing suitable physical laws to couple receptor relocation on membranes with cellular mechanical deformation. From a conceptual point of view, physical theories and mathematical tools allow us to relate the mechanical principles with the behavior of living matter: thermomechanics of continua [1,2] is the ideal framework to model nature's laws. Due to its intrinsic interdisciplinarity, a multi-physics approach to biological phenomena may have the potential to highlight key and limiting factors, providing innovative pathways for analysis and interpretation. ...
This work aims to overview multiphysics mechanobiological computational models for receptor dynamics along advecting cell membranes. Continuum and statistical models of receptor motility are the two main modeling methodologies identified in reviewing the state of the art. Within the former modeling class, a further subdivision based on different biological purposes and processes of proteins’ motion is recognized; cell adhesion, cell contractility, endocytosis, and receptor relocations on advecting membranes are the most relevant biological processes identified in which receptor motility is pivotal. Numerical and/or experimental methods and approaches are highlighted in the exposure of the reviewed works provided by the literature, pertinent to the topic of the present manuscript. With a main focus on the continuum models of receptor motility, we discuss appropriate multiphyisics laws to model the mass flux of receptor proteins in the reproduction of receptor relocation and recruitment along cell membranes to describe receptor–ligand chemical interactions, and the cell’s structural response. The mass flux of receptor modeling is further supported by a discussion on the methodology utilized to evaluate the protein diffusion coefficient developed over the years.
... 21 The velocity gradient tensor, , can be decomposed in a unique manner into its symmetric part, D, and an anti-symmetric part, Ω, so that = + , where = 1 2 (( ) + ( ) ) and = 1 2 (( ) − ( ) ). D is often referred to as stretching tensor and Ω is the spin tensor associated with rotation. 8,11,66 Their magnitudes are given by ...
von Willebrand Factor is a mechano-sensitive protein circulating in blood that mediates platelet adhesion to subendothelial collagen and platelet aggregation at high shear rates. Its hemostatic function and thrombogenic effect, as well as susceptibility to enzymatic cleavage, are regulated by a conformational change from a collapsed globular state to a stretched state. Therefore, it is essential to account for the conformation of the vWF multimers when modeling vWF-mediated thrombosis or vWF degradation. We introduce a continuum model of vWF unfolding that is developed within the framework of our multi-constituent model of platelet-mediated thrombosis. The model considers two interconvertible vWF species corresponding to the collapsed and stretched conformational states. vWF unfolding takes place via two regimes: tumbling in simple shear and strong unfolding in flows with dominant extensional component. These two regimes were demonstrated in a Couette flow between parallel plates and an extensional flow in a cross-slot geometry. The vWF unfolding model was then verified in several microfluidic systems designed for inducing high-shear vWF-mediated thrombosis and screening for von Willebrand Disease. The model predicted high concentration of stretched vWF in key regions where occlusive thrombosis was observed experimentally. Strong unfolding caused by the extensional flow was limited to the center axis or middle plane of the channels, whereas vWF unfolding near the channel walls relied upon the shear tumbling mechanism. The continuum model of vWF unfolding presented in this work can be employed in numerical simulations of vWF-mediated thrombosis or vWF degradation in complex geometries. However, extending the model to 3-D arbitrary flows and turbulent flows will pose considerable challenges.
... In a non-hydrostatically stressed elastic solid, instead of mechanical equilibrium involving the conjugate pair of scalars p and V, the thermodynamics is written in terms of second-rank tensors for stress and strain (see the appropriate sections of a continuum mechanics textbook, for example, Fung & Tong, 2001;Gurtin, Fried, & Anand, 2010;Lai, Rubin, & Krempl, 2010;Malvern, 1969;Nye, 1985;Tadmor, Miller, & Elliot, 2012). Minerals do not strain much elastically even with quite large stresses so it is reasonable to use the small strain approximation in the thermodynamics in the way quantified by Gurtin et al. (2010, chapter 52), for example. ...
This essay in honour of Mike Brown addresses aspects of chemical equilibrium and equilibration in rocks, with a focus on the role that chemical potentials play. Chemical equilibrium is achieved by diffusive attening of chemical potential gradients. The idea of equilibration volume is developed, and the way equilibration volumes may evolve along a pressure‐temperature path is discussed. The effect of the environment of an equilibration volume is key to understanding the evolution of the equilibration volume with changing conditions. The likely behaviour of equilibration volumes is used to suggest why preservation of equilibrium mineral assemblages and mineral compositions from metamorphism tends to occur. This line of logic then provides the conceptual support to conventional equilibrium thermodynamic approaches to studying rocks, using, for example, thermobarometry and pseudosections.
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... In a non-hydrostatically stressed elastic solid, instead of mechanical equilibrium involving the conjugate pair of scalars p and V, the thermodynamics is written in terms of second-rank tensors for stress and strain (see the appropriate sections of a continuum mechanics textbook, for example, Fung & Tong, 2001;Gurtin, Fried, & Anand, 2010;Lai, Rubin, & Krempl, 2010;Malvern, 1969;Nye, 1985;Tadmor, Miller, & Elliot, 2012). Minerals do not strain much elastically even with quite large stresses so it is reasonable to use the small strain approximation in the thermodynamics in the way quantified by Gurtin et al. (2010, chapter 52), for example. ...
Metamorphic geology has accumulated a huge body of observation on mineral assemblages that reveal strong patterns in occurrence, summarised for example in the idea of metamorphic facies. On the realisation that such patterns needed a simple explanation, there has been considerable a posteriori success from adopting the idea that equilibrium thermodynamics can be used on mineral assemblages to make sense of the patterns in terms of, for example, the pressure and temperature of formation of mineral assemblages. In doing so, a particularly simple implicit assumption is made, that mineral assemblages operate essentially hydrostatically. Structural geologists have studied the same rocks for different ends, but, remarkably, the phenomena they are interested in depend on non-hydrostatic stress. We look at the effect of such behaviour on mineral equilibria. With adoption of some plausible assumptions about how metamorphism in the crust works, the consequence of minerals being non-hydrostatically stressed is commonly second order in equilibrium calculations.
... Furthermore, N i is defined to be zero in any element not touching node i. This property of shape functions, also called compact support, limits the region of interpolation to that of an element domain [53]. Most finite element softwares use the isoparametric formulation, permitting the use of arbitrarily shaped elements like non-rectangular hexahedra. ...
Atomistic simulations play an important role in advancing our understanding of the mechanical properties of materials. Currently, most atomistic simulations are performed using relatively simple geometries under homogeneous loading conditions, and a significant part of the computer time is spent calculating the elastic response of the material, while the focus of the studies lies usually on the mechanisms of plastic deformation and failure. Here we present a simple but versatile approach called FE2AT to use finite element calculations to provide appropriate initial and boundary conditions for atomistic simulations. FE2AT allows to forgo the simulation of large parts of the elastic loading process, even in the case of complex sample geometries and loading conditions. FE2AT is open source and can be used in combination with different atomistic simulation codes and methods. Its application is demonstrated using the bending of a nano-beam and the determination of the displacement field around a crack tip as examples.
... Readers are referred to the literature on continuum mechanics like [29,30] for the continuum form of Clausius inequality and [2] on the derivation of the Fourier law from the second law of thermodynamics. By applying the Clausius inequality to any two reversible cycles (A + C) and (B + C), where A, B, C are three reversible processes in figure 1a, we have This shows that ( 2 1 δq/T) rev depends only on States 1 and 2. Here, subscript rev stands for reversible processes. ...
Recent studies have shown that precipitation hardening effectively enhances the mechanical quality factor of ferroelectric material. In this work, Li‐doped , a material system with elliptical precipitates is investigated. We present a mechanical model to determine energetically stable precipitate shapes by minimizing the total energy, consisting of elastic and interface energy. Furthermore, we investigate the influence of external loads on the precipitate topology. Correct elastic constants as well as lattice misfits for the simulation are provided. The shapes determined from finite element simulations agree well with observed results.
For a given material, controllable deformations are those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials, universal deformations are those deformations that are controllable for any material within the class. In this paper, we characterize the universal deformations in compressible isotropic implicit elasticity defined by solids whose constitutive equations, in terms of the Cauchy stress σ and the left Cauchy-Green strain b, have the implicit form f(σ,b)=0. We prove that universal deformations are homogeneous. However, an important observation is that, unlike Cauchy (and Green) elasticity, not every homogeneous deformation is constitutively admissible for a given implicit-elastic solid. In other words, the set of universal deformations is material-dependent, yet it remains a subset of homogeneous deformations.
Soft robotics technology has the potential for a wide range of applications; however, their implementation is hindered by the lack of accurate mathematical models and control methods. For example, a method to control multiple pneumatic inflatable actuators in turn by a passive mechanism has been proposed in a literature, but no mathematical model was presented to develop control laws for the system. This study proposes a mathematical model for the inflation characteristics of sequential soft actuators, and theoretical control law for passive sequential control of the inflation of multiple actuators by using flow resistors. The model is constructed from the physical requirements of the actuators based on fluid dynamics and thermodynamics. The study shows that the pressure change characteristics can be derived from the differential equations even for complex parallel and series circuits. The model mathematically represents the physical principle by which the actuators' inflation order depends on the relationship between changes in the input pressure and internal pressure with time. Experiments with real circuits of simple and complex layouts are presented. The actuators are inflated in sequence from a single air pressure source, and the mathematical model was able to describe the pressure and flow rate change characteristics at each point in the circuit.
The fundamental physics of continua is governed by the balance laws of four conserved quantities, namely mass, linear momentum, angular momentum, and internal energy.
To complete the mechanical model, we still need to define a constitutive law that describes the stress tensor . For simplicity, and to expose the fundamental behavior of the system clearly, we consider a class of simple, fundamental, linear models. To set up our multidimensional constitutive equations, we start by describing the standard model variants in one space dimension, and then generalize from there.
Many manufacturing processes in the process industry are modeled using the theory of axially moving continua. In this book, we apply this perspective to the additive manufacturing of metal products. This chapter briefly reviews the fundamental kinematics, setting up the stage for deriving the governing equations for our model.
We would like to construct a model that can handle elastic and viscoelastic solids, as well as Newtonian fluids. The standard treatment of Newtonian fluids differs slightly from that of a Newtonian dashpot in the context of viscous solids, so we begin by a brief review of standard fluid mechanics for Newtonian fluids. We then look at the similarities and differences between fluids and solids, and at possibilities to develop a unified constitutive law that can be locally specialized into either type as needed.
This chapter summarizes the basic relation needed to formulate the deformation of solids in the linear and nonlinear range. It is subdivided into the sections kinematics, balance laws, variational formulations and constitutive equations. This part of the book is not meant for studying continuum mechanics, it only summarizes results that provide essential background and notation for understanding the discretization techniques related to the virtual element method in the following chapters.
ChatGPT is a language model trained by OpenAI to follow an instruction in a prompt and to provide a detailed response. We investigate the capabilities of ChatGPT to generate codes which implement the finite element method. The finite element method (FEM) is a popular technique for the numerical solution of partial differential equations (PDEs). More specifically, we analyze the codes generated for two open source platforms: deal.II, a C++ software library, and FEniCS, for which we focus on its Python interface. We consider as benchmark problems the Poisson equation and a linear advection problem. The outcomes suggest that ChatGPT can be employed as an initial building block to write finite element codes, but certain limitations and failures, which require further improvement of the machine learning model and human supervision, are still present.
The scientific community has witnessed, lately, a tremendous progress in the fabrication and synthesis of nanomaterials. As a result, it is essential to develop new and efficient numerical techniques that are capable of modeling the behavior of materials at nanoscale with sufficient accuracy. In this work, a novel approach is presented for the multiscale analysis of brittle failure in nanostructures using the phase-field modeling. The specimen at microscale is discretized using finite elements (FEs), whose integration points lie in the representative volume elements (RVEs) at nanoscale. The displacement computed in upper scale for a microstructure that contains an evolving crack is imposed on the boundaries of the representative volume element in lower scale. On the other hand, the stresses and material properties obtained for the representative volume element in lower scale are transferred to upper scale to compute stiffness matrices and load vectors. The evolution of the phase-field variable indicates the initiation and propagation of cracks at microscale. In order to avoid time-consuming molecular dynamics (MD) simulations at nanoscale in each step of the analysis, the Mooney-Rivlin material model is used to simulate the behavior of Aluminum (AL) nanostructure at this scale. The approach that is utilized to compute the material constants and the formulation for the multiscale technique combined with the phase-field modeling in upper scale are described in detail. It is discussed how the phase-field variable in microstructure is evolved based on the properties of the representative volume element in nanostructure. Many numerical examples are presented to demonstrate the application of the proposed multiscale technique in the solution of engineering problems.
Inhomogeneously swollen elastomers are an emergent class of materials, comprising elastic matrices with inclusion phases in the form of microgel particles or osmolytes. Inclusion phases can undergo osmotically driven swelling and deswelling over orders of magnitude. In the swollen state, the inclusions typically have negligible Young's modulus, and the matrix is strongly deformed. In that regime, the effective mechanical properties of the composite are governed by the matrix. Laying the groundwork for a generic analysis of inhomogeneously swollen elastomers, we develop a model based on incremental mean-field homogenization of a hyperelastic matrix. The framework allows for the computation of the macroscopic effective stiffness for arbitrary hyperelastic matrix materials. For an in-depth quantification of the local effective stiffness, we extend the concept of elastic stiffness maps to incompressible materials. For strain-stiffening materials, stiffness maps in the swollen state highlight pronounced radial stiffening with a non-monotonic change in stiffness in the hoop direction. Stiffening characteristics are sensitive to the form of constitutive models, which may be exploited in the design of hydrated actuators, soft composites and metamaterials. For validation, we apply this framework to a Yeoh material, and compare to recently published data. Model predictions agree well with experimental data on elastomers with highly swollen embedded microgel particles. We identify three distinct regimes related to an increasing degree of particle swelling: first, an initial decrease in composite stiffness is attributed to particle softening upon liquid intake. Second, dilute particle swelling leads to matrix stiffening dominating over particle softening, resulting in an increase in composite stiffness. Third, for swelling degrees beyond the dilute limit, particle interactions dominate further matrix stiffening.
This article includes a short survey of selected averaging and dimension reduction techniques for deterministic fast–slow systems. This survey includes, among others, classical techniques, such as the WKB approximation or the averaging method, as well as modern techniques, such as the GENERIC formalism. The main part of this article combines ideas of some of these techniques and addresses the problem of deriving a reduced system for the slow degrees of freedom (DOF) of a fast–slow Hamiltonian system. In the first part, we derive an asymptotic expansion of the averaged evolution of the fast–slow system up to second order, using weak convergence techniques and two-scale convergence. In the second part, we determine quantities which can be interpreted as temperature and entropy of the system and expand these quantities up to second order, using results from the first part. The results give new insights into the thermodynamic interpretation of the fast–slow system at different scales.
The equation of state of the triclinic compound 1,3,5‐triamino‐2,4,6‐trinitrobenzene (TATB) as well as its second‐order isothermal elastic tensor were computed through classical molecular dynamics simulations under various temperature and pressure conditions. Hydrostatic pressures similar to previous diamond anvil cell experiments were imposed up to 60 GPa and temperatures chosen between 100 and 900 K in conjunction with the most recent version of an all‐atom fully flexible molecule force field. The isothermal elastic constants were computed using the generalized Hooke's law by fitting Cauchy stress vs. linear strain curves for pressures below 50 GPa. Along isobaric loadings, TATB single crystal stiffnesses are found to undergo softening, less pronounced at high pressure, while maintaining its elastic anisotropy. On the other hand, along an isothermal loading, a non‐linear increase is observed in the elastic constants with respect to pressure with a significant decrease in anisotropy. Towards a precise mesoscopic modelling of TATB single crystal mechanical behavior, we provide “ready to plug‐in” analytical formulations of the P, V, T equation of state and pressure‐temperature dependent non‐linear elasticity.
In this work we introduce a DG framework for nonlinear elasticity based on a Bassi-Rebay (BR2) formulation. The framework encompasses compressible and incompressible hyperelastic materials and is capable of dealing with large deformations. In order to achieve stability, we combine higher-order lifting operators for the BR2 stabilisation term with an adaptive stabilisation strategy which relies on the BR2 Laplace operator stabilisation and a penalty parameter based on the spectrum of the fourth-order elasticity tensor. Dirichlet boundary conditions for the displacement can be imposed by means of Lagrange multipliers and Nitsche method. Efficiency of the solution strategy is achieved by means of state-of-the-art agglomeration based h-multigrid preconditioners and the code implementation supports distributed memory execution on modern parallel architectures. Several benchmark test cases are proposed in order to investigate some relevant computational aspects, namely the performance of the h-multigrid iterative solver varying the stabilisation parameters and the influence of Dirichlet boundary conditions on Newton's method globalisation strategy.
>>> Link : https://www.theses.fr/2021IPPAX015 <<<
Future developments of lighter, more compact and powerful motors -- driven by environmental and sustainability considerations in the transportation industry – involve higher stresses, currents and electromagnetic fields. For the components used in electric motors – especially for the ferromagnetic ones – strong couplings between mechanical, thermal and electromagnetic effects arise, which are amplified by the higher loads. They affect the machine’s performance, thus requiring a consistent multiphysics modeling for the motors’ design. Understanding and modeling these couplings has recently become an important subject of research. The work presented here proposes a coupled electromagnetic-thermomechanical continuum theory together with analytical and numerical (finite element) tools for the solutions of boundary value problems arising in electric motors.In the first part of the work, using the direct approach of continuum mechanics, based on a Eulerian (current configuration) approach, a general modeling framework coupling the electromagnetic, thermal and mechanical fields is derived from the basic principles of thermodynamics using the eddy current approximation. Although the proposed theory is general enough to account for a wide range of material behaviors, particular attention is paid to the derivation of the coupled constitutive equations for isotropic materials under small strain but arbitrary magnetization. As a first application, the theory is employed for the analytical modeling of the rotor and stator of idealized electric motor configurations for which we calculate the electric current, magnetic, stress and temperature fields. At the rotor, the different components of the stress tensor and body force vector are compared to their purely mechanical counterparts due to inertia, quantifying the significant influence of electromagnetic phenomena. At the stator, comparison with coarser models found in the electrical engineering literature is provided, quantifying the influence of the proposed model on the elastic stress and strains’ amplitudes.In the second part of the work, a variational formulation of the problem is presented based on a Lagrangian (reference configuration) approach and shown to be equivalent to the direct approach. The numerical implementation of the proposed model – via a user element in a general purpose finite element code and accounting for non-linear material behavior – is validated by comparison of the results from the analytical models of the simplified stator configuration to the numerical results for small values of the magnetic field (range of linear behavior for the magnetic field). Calculations are then performed on more complex stator configurations with a more intense magnetic field, using a non-linear magnetic response that accounts for magnetic saturation (a Langevin-type model), in order to put forward the capacities of the proposed formulation and obtain results for realistic engineering applications.
von Willebrand Factor is a mechano-sensitive protein circulating in blood that mediates platelet adhesion to subendothelial collagen and platelet aggregation at high shear rates. Its hemostatic function and thrombogenic effect, as well as susceptibility to enzymatic cleavage, are regulated by a conformational change from a collapsed globular state to a stretched state. Therefore, it is essential to account for the conformation of the vWF multimers when modeling vWF-mediated thrombosis or vWF degradation. We introduce a continuum model of vWF unfolding that is developed within the framework of our multi-constituent model of platelet-mediated thrombosis. The model considers two interconvertible vWF species corresponding to the collapsed and stretched conformational states. vWF unfolding takes place via two regimes: tumbling in simple shear and strong unfolding in flows with dominant extensional component. These two regimes were demonstrated in a Couette flow between parallel plates and an extensional flow in a cross-slot geometry. The vWF unfolding model was then verified in several microfluidic systems designed for inducing high-shear vWF-mediated thrombosis and screening for von Willebrand Disease. The model predicted high concentration of stretched vWF in key regions where occlusive thrombosis was observed experimentally. Strong unfolding caused by the extensional flow was limited to the center axis or middle plane of the channels, whereas vWF unfolding near the channel walls relied upon the shear tumbling mechanism. The continuum model of vWF unfolding presented in this work can be employed in numerical simulations of vWF-mediated thrombosis or vWF degradation in complex geometries. However, extending the model to 3-D arbitrary flows and turbulent flows will pose considerable challenges.
The response of cells during spreading and motility is dictated by several multi-physics events, which are triggered by extracellular cues and occur at different time-scales. For this sake, it is not completely appropriate to provide a cell with classical notions of the mechanics of materials, as for “rheology” or “mechanical response”. Rather, a cell is an alive system with constituents that show a reproducible response, as for the contractility for single stress fibers or for the mechanical response of a biopolymer actin network, but that reorganize in response to external cues in a non-exactly-predictable and reproducible way. Aware of such complexity, in this note we aim at formulating a multi-physics framework for modeling cells spreading and motility, accounting for the relocation of proteins on advecting lipid membranes.
Graphic Abstract
We study the mechanical response under compression/extension of an assembly composed of 8 helical rods, pin-jointed and arranged in pairs with opposite chirality. In compression we find that, whereas a single rod buckles (a), the rods of the assembly deform as stable helical shapes (b). We investigate the effect of different boundary conditions and elastic properties on the mechanical response, and find that the deformed geometries exhibit a common central region where rods remain circular helices. Our findings highlight the key role of mutual interactions in the ensemble response and shed some light on the reasons why tubular helical assemblies are so common and persistent.
This paper presents a passive control method for multiple degrees of freedom in a soft pneumatic robot through the combination of flow resistor tubes with series inflatable actuators. We designed and developed these 3D printed resistors based on the pressure drop principle of multiple capillary orifices, which allows a passive control of its sequential activation from a single source of pressure. Our design fits in standard tube connectors, making it easy to adopt it on any other type of actuator with pneumatic inlets. We present its characterization of pressure drop and evaluation of the activation sequence for series and parallel circuits of actuators. Moreover, we present an application for the assistance of postural transition from lying to sitting. We embedded it in a wearable garment robot-suit designed for infants with cerebral palsy. Then, we performed the test with a dummy baby for emulating the upper-body motion control. The results show a sequential motion control of the sitting and lying transitions validating the proposed system for flow control and its application on the robot-suit.
The study focuses on the development of technology for volumetric noncontact liquid heating based on the dissipative liquid heating effect and realized in a high-gradient flow in the rotor–stator system of a hydrodynamic apparatus. In this paper, the heating performance of a prototype of a rotor-type dissipative liquid heater is studied, and the dynamics of several parameters, such as the heat power generated, the electrical power consumed and the efficiency factor, are defined. It is shown that the time evolution of the efficiency factor along with the other parameters displays a nonlinear character and depends on the flow pattern of the water–vapor system. It is found that the highest efficiency of the dissipative liquid heating of 91.6% is observed in the froth flow of the two-phase water–vapor system. In addition, potential industrial applications of the dissipative liquid heaters are discussed.
The final publication is available at https://link.springer.com/article/10.1007/s40095-020-00354-0
Energetic material condensed phase constituents come into contact, chemically react, and simultaneously undergo a phase change. Phase change in a given molecular material is often considered separately from the chemical reaction. We use a self‐consistent continuum, thermo‐mechanical model based on specified Gibbs potentials for all relevant component species that are present in a single material. A single stress tensor and a single temperature are assumed for the material.
Physically‐based mass fractions of the components are used to describe composition and phase and chemical change and replace order parameters used in older phase‐field variable theories. Our formulation is entirely analogous to non‐equilibrium theories used for gas‐phase combustion but generalized to include condensed phases. We describe the formulation and illustrate its use with three examples. Example 1. considers the quasi‐static heating of an oxide‐coated aluminum particle, that is slowly heated so that the temperature within the particle can be considered a constant. We compute the stress and displacement within the particle by using the derived multicomponent equation of state. Example 2. shows how to model the melting and vaporization in a rapidly heated, confined slab of initially solid aluminum. The model treats multi‐phase transitions and illustrates how the multicomponent equation of state is used. Example 3. describes the modeling of mechanical thermo‐chemical ignition processes in a nano‐slab of HMX and considers six components: Two solid (beta and delta HMX), one liquid HMX component, and two gas components for vapor HMX and reacted products. We describe some possible next steps for further study and model improvements and how to interact with modern reactive molecular dynamics.
In this contribution, we present a characterization methodology to obtain pseudo experimental deformation data from CG MD simulations of polymers as an inevitable prerequisite to choose and calibrate continuum mechanical constitutive laws. Without restriction of generality, we employ a well established CG model of atactic polystyrene as exemplary model system and simulate its mechanical behavior under various uniaxial tension and compression load cases. To demonstrate the applicability of the obtained data, we exemplarily calibrate a viscoelastic continuum mechanical constitutive law. We conclude our contribution by a thorough discussion of the findings obtained in the numerical pseudo experiments and give an outline of subsequent research activities. Thus, this work contributes to the field of multiscale simulation methods and adds a specific application to the body of knowledge of CG MD simulations.
When investigated as a function of deformation rate and time, model nano‐crystals exhibit a clear threshold in mechanical and relaxation behavior, this observation being valid in both the Hooke (elastic) and plastic regime. At low deformation rate (ϵ˙<ϵ˙c≃5×105 s−1), materials are able to equilibrate with ongoing deformation, whereas ultra‐fast deformations bring the system out of equilibrium on the same nanosecond timescale. The time evolution of stress is investigated and suggests features that are typical of glassy relaxation with a stretched exponential behavior for stress relaxation that is only encountered for large deformation rates. Related relaxation times furthermore reveal a rapid increase with ϵ˙. These results actually link general aspects of plastic deformation, irreversibility and thermal equilibrium.
Acoustic and elastic metamaterials are media with a subwavelength structure that behave as effective materials displaying atypical effective dynamic properties. These material systems are of interest because the design of their sub-wavelength structure allows for direct control of macroscopic wave dispersion. One major design limitation of most metamaterial structures is that the dynamic response cannot be altered once the microstructure is manufactured. However, the ability to modify wave propagation in the metamaterial with an external stimulus is highly desirable for numerous applications and therefore remains a significant challenge in elastic metamaterials research. In this work, a honeycomb structure composed of a doubly periodic array of curved beams, known as a negative stiffness honeycomb (NSH), is analyzed as a tunable elastic metamaterial. The nonlinear static elastic response that results from large deformations of the NSH unit cell leads to a large variation in linear elastic wave dispersion associated with infinitesimal motion superposed on the externally imposed pre-strain. A finite element model is utilized to model the static deformation and subsequent linear wave motion at the pre-strained state. Analysis of the slowness surface and group velocity demonstrates that the NSH exhibits significant tunability and a high degree of anisotropy which can be used to guide wave energy depending on static pre-strain levels. In addition, it is shown that partial band gaps exist where only longitudinal waves propagate. The NSH therefore behaves as a meta-fluid, or pentamode metamaterial, which may be of use for applications of transformation elastodynamics such as cloaking and gradient index lens devices.
With rapidly increasing numbers of studies of new and exotic material uses for perovskites and quasicrystals, these demand newer instrumentation and simulation developments to resolve the revealed complexities. One such set of observational mechanics at the nanoscale is presented here for somewhat simpler material systems. The expectation is that these approaches will assist those materials scientists and physicists needing to verify atomistic potentials appropriate to the nanomechanical understanding of increasingly complex solids. The five following segments from nine University, National and Industrial Laboratories both review and forecast where some of the important approaches will allow a confirming of how in situ mechanics and nanometric visualization might unravel complex phenomena. These address two-dimensional structures, temporal models for the nanoscale, atomistic and multiscale friction fundamentals, nanoparticle surfaces and interfaces and nanomechanical fracture measurements, all coupled to in situ observational techniques. Rapid future advances in the applicability of such materials science solutions appear guaranteed.
Soft robots are able to complete tasks which traditional robots cannot, thus providing new opportunities for robots to navigate confined spaces. These tasks include pipe inspection and endoluminal surgical applications. The research objective of this work is to present a design methodology for soft robots capable of traversing a cannula or pipe using only passive elements, a single pressure source, and while avoiding blockage, that is, avoiding full occlusion of the cannula to still allow fluid flow. The robot consists of three segments, each with actuators and valves, and is driven by hydraulics (water). The actuators were built using the FREE method and optimized for the specific task of traversing a cannula with a diameter of 19mm. An experimental approach was used to create a pressure-volume relationship from the kinematic fiberreinforced elastomer enclosure (FREE) model. The passive valves were built as flow restrictors and modeled with the nonlinear orifice equation. A simulation and grid search was performed over a range of valve coefficients. An objective function was then maximized to produce orifice coefficients which augmented extension of the robot per input pressure cycle. The physical robot was then constructed and tested in a cannula. This resulted in a robot which was serially actuated, never fully occluded the cannula, and was able to successfully locomote a cannula at a rate of 274mm per two second cycle. This experimental result was close to the simulated result of 28mm per cycle.
This review focuses on energy storage materials modeling, with particular emphasis on Li-ion batteries. Theoretical and computational analyses not only provide a better understanding of the intimate behavior of actual batteries under operational and extreme conditions, but they may tailor new materials and shape new architectures in a complementary way to experimental approaches. Modeling can therefore play a very valuable role in the design and lifetime prediction of energy storage materials and devices. Batteries are inherently multi-scale, in space and time. The macro-structural characteristic lengths (the thickness of a single cell, for instance) are order of magnitudes larger than the particles that form the microstructure of the porous electrodes, which in turn are scale-separated from interface layers at which atomistic intercalations occur. Multi-physics modeling concepts, methodologies, and simulations at different scales, as well as scale transition strategies proposed in the recent literature are here revised. Finally, computational challenges toward the next generation of Li-ion batteries are discussed.
The aim of this work is to provide an improved information exchange in hierarchical atomistic-to-continuum settings by applying stochastic approximation methods. For this purpose a typical model belonging to this class is chosen and enhanced. On the macroscale of this particular two-scale model, the balance equations of continuum mechanics are solved using a nonlinear finite element formulation. The microscale, on which a canonical ensemble of statistical mechanics is simulated using molecular dynamics, replaces a classic material formulation. The constitutive behavior is computed on the microscale by computing time averages. However, these time averages are thermal noise-corrupted as the microscale may practically not be tracked for a sufficiently long period of time due to limited computational resources. This noise prevents the model from a classical convergence behavior and creates a setting that shows remarkable resemblance to iteration schemes known from stochastic approximation. This resemblance justifies the use of two averaging strategies known to improve the convergence behavior in stochastic approximation schemes under certain, fairly general, conditions. To demonstrate the effectiveness of the proposed strategies, three numerical examples are studied.
The Parrinello-Rahman algorithm for imposing a general state of stress in periodic molecular dynamics simulations is widely used in the literature and has been implemented in many readily available molecular dynamics codes. However, what is often overlooked is that this algorithm controls the second Piola-Kirchhoff stress as opposed to the true (Cauchy) stress. This can lead to misinterpretation of simulation results because (1) the true stress that is imposed during the simulation depends on the deformation of the periodic cell, (2) the true stress is potentially very different from the imposed second Piola-Kirchhoff stress, and (3) the true stress can vary significantly during the simulation even if the imposed second Piola-Kirchhoff is constant. We propose a simple modification to the algorithm that allows the true Cauchy stress to be controlled directly. We then demonstrate the efficacy of the new algorithm with the example of martensitic phase transformations under applied stress.
Mechanistic computational modeling of environmental systems presupposes the existence of a properly posed set of equations that adequately describes the underlying physical, chemical, and biological processes at time and length scales consistent with the problem being addressed.
Recent computational simulations of ionic conductivity across the electrolyte of commercial batteries by
Salvadori et al. (2015) have shown that the concentration of ions exceeds half the saturation limit near
the electrodes. This observation, which is in agreement with other approaches by Danilov and Notten
(2008) , implies that the widespread assumption of infinite dilution far from saturation is questionable.
The present contribution is therefore devoted to investigate the role of saturation in modeling ionic
transport in the electrolyte of Li-ion batteries. An important result is found, that saturation has no effect
on the diffusivity, whereby the condition of electroneutrality is well approximated in the solution.
However saturation affects the electric potential up to 40% near the electrodes for all charge rates.
A novel approach in modeling the ionic transport in the electrolyte of Li-ion batteries is here presented.
Diffusion and migration processes govern the transport of ions in solution in the absence of convection.
In the porous electrode theory [1] it is common to model these processes via mass balance equations and
electroneutrality. A parabolic set of equations arises, in terms of a non constant electric field which is
afflicted by the paradox of being generated without electrical charges. To remedy this contradiction,
Maxwell's equations have been used here, coupled to Faraday's law of electrochemical charge transfer.
The set of continuity equations for mass and Maxwell's equations lead to a consistent model, with
distinctive energy characteristics. Numerical examples show the robustness of the approach, which is
well suited for multi-scale analyses [2,3].
Irreversible quasi-surface metallurgical phase transformations are the specific response of some metallic materials—such as metals and alloys—subjected to high thermomechanical loads applied very near their surface during the manufacturing processes or after being put into operation. These solid/solid phase transformations can be observed, for example, on the tread of many rails in railroad networks frequented by freight trains. The severe thermal and mechanical loads imposed on the surface of the rails and in the immediate vicinity of the surface by the wheel/rail contact often result in highly localized irreversible metallurgical transformations. A new kinetic model based on a previous study is presented here, which accounts more realistically for the nucleation and growth of these irreversible solid/solid phase transformations resulting from high thermomechanical loads. This metallurgical behavioral model was developed in the framework of continuum thermodynamics with gradients of temperature and internal variables.
The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen in Appl Mech Rev 47, 1994, Stability and nonlinear solid mechanics. Wiley, New York,2000; Mielke in Handbook of differential equations, evolutionary equations. Elsevier, Amsterdam,2005; Bourdin et al. in The variational approach to fracture. Springer, Berlin,2008). In a recent publication (Salvadori and Carini in Int J Solids Struct 48:1362–1369,
2011) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini’s theorem (Ceradini in Rendiconti Istituto Lombardo di Scienze e Lettere A99, 1965, Meccanica 1:77–82,1966) in plasticity. Following the cornerstone work of Rice (1989)on
weight function theories, Leblond et al. (Leblond in Int J Solids Struct 36:79–103, 1999 ; Leblond et al. in
Int J Solids Struct 36:105–142, 1999) proposed asymptotic expansions for stress intensity factors in three
dimensions—see also Lazarus (J Mech Phys Solids 59:121–144, 2011 ). As formerly in 2D, expansions can
be given a Colonnetti’s decomposition interpretation. In view of the expression of the expansions proposed in Leblond (Int J Solids Struct 36:79–103,1999 ), Leblond et al. (Int J Solids Struct 36:105–142,1999 ) however, symmetry of Ceradini’stheorem operators was not evident and the extension of outcomes proposed in Salvadori and Carini (Int J Solids Struct 48:1362–1369, 2011) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived.
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