Balance Laws Of Continuum Mechanics Research Articles

On dynamic crack propagation in a lattice Boltzmann method for elastodynamics in 2D

AbstractIn recent years, the development of lattice Boltzmann methods (LBMs) for solids has gained attention. Fracture mechanics as a viable application for these methods has been presented before, albeit for mode III cracks in the context of an LBM for antiplane shear deformation. The performance of the LBM itself is promising, while the usage of a regular lattice simplifies the modelling of fractures significantly. Recent advancements in LBMs for solids, especially the description of Dirichlet‐ and Neumann‐type boundary conditions, now make it possible to extend the LBM simulation of crack propagation to the plane strain case with modes I and II crack opening, including growth with non‐uniform speed in arbitrary directions. For this, the configurational force acting on a crack tip is utilised. The definition of the moments of the LBM, which are based on the balance laws of continuum mechanics, render the evaluation of macroscopic fields in the configuration straightforward. In this work, the general in‐plane case of dynamic crack propagation is shown and necessary considerations for the implementation are discussed. Lastly, numerical examples showcase the capabilities of the proposed method to model dynamic fractures and establish a proof‐of‐concept.

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

Thermodynamic consistency of beam theories in the context of classical and non-classical continuum mechanics and a thermodynamically consistent new formulation

In order to enhance currently used beam theories in $$\mathbb {R}^2$$ and $$\mathbb {R}^3$$ to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism in the currently used beam theories. This is the main motivation for the work presented in this paper. The currently used beam theories are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler’s equation(s) extracted from the first variation of this energy functional set to zero yields the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain, the two approaches yield same mathematical models. In this paper we examine whether the currently used beam mathematical models with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler–Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher-order thermodynamically consistent beam theory that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented in this paper ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams.

A Causality Setting for Elasticity Theory

We give a causality approach to nonlinear elasticity theory within a pure mechanics setting. Based on the notion of a fixed frame in and Euclidean invariance considerations, we show how a broadly general and novel evolutionary hypothesis concerning ‘cause’ and ‘effect’ must reduce to the classical statement of the balance of energy, and we obtain all of the classical balance laws of continuum mechanics. The concept of mass and its balance is derived within this theory. The mass density naturally emerges from the theory without preconception as an inertial scalar field for the body which is associated with the speed of its material particles and a measure of their kinetic behavior. Aside from the causality hypothesis and its invariance, the fundamental notions of body, motion, force and internal power are primitive.

Incompressible fluid problems on embedded surfaces: Modeling and variational formulations

Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in $\Bbb{R}^n$. For this case, we present new well-posedness results for the simplified surface fluid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for this surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods.

A general framework for the modeling of porous ferrogels at finite strains

Porous ferrogels are a new class of magneto-active composite materials that deform and alter their material characteristics under the influence of magnetic fields. In the future such materials could find a wide range of application in biomedicine and microfluidics. In this work we present a theoretical and computational framework for the macroscopic, continuum-based modeling of porous ferrogels at finite strains. Departing from the balance laws of continuum mechanics, we derive a thermodynamically consistent model based on Biot’s consolidation theory. Regarding constitutive modeling we limit our attention to isotropic materials. Furthermore we discuss details of the numerical implementation of the coupled three field problem within a nonlinear finite element algorithm. The modeling capabilities and algorithmic performance are demonstrated by means of two representative initial boundary value problems.

Thermal Creep of a Rarefied Gas on the Basis of Non-linear Korteweg-Theory

The study of thermal transpiration, more commonly called thermal creep, is accomplished by use of Korteweg’s theory of capillarity. Incorporation of this theory into the balance laws of continuum mechanics allows resolution of boundary value problems via solutions to systems of ordinary differential equations. The problem was originally considered by Maxwell in his classic 1879 paper Maxwell (Phil Trans Roy Soc (London) 170:231–256, 1879). In that paper Maxwell derived what is now called the Burnett higher order contribution to the Cauchy stress, but was not able to solve his newly derived system of partial differential equations. In this paper the authors note that a more appropriate higher order contribution to the Cauchy stress follows from Korteweg’s 1901 theory Korteweg (Arch Neerl Sci Exactes Nat Ser II 6:1–24, 1901). The appropriateness of Korteweg’s theory is based on the exact summation of the Chapman–Enskog expansion given by Gorban and Karlin. The resulting balance laws are solved exactly, qualitatively, and numerically and the results are qualitatively similar to the numerical and exact results given by Aoki et al., Loyalka et al., and Struchtrup et al.

A homogenization method for thermomechanical continua using extensive physical quantities

This article proposes a continuum thermomechanical homogenization method inspired by the Irving–Kirkwood procedure relating the atomistic equations of motion to the balance laws of continuum mechanics. This method yields expressions for the macroscopic stress and heat flux in terms of microscopic kinematic and kinetic quantities. The resulting equation for macroscopic stress affords a rational comparison with the widely used Hill–Mandel stress-deformation condition, while the one for heat flux reduces, under certain assumptions, to a Hill–Mandel-like condition involving heat flux and the gradient of temperature.

On the interrelation of the basic laws of continuum mechanics

AbstractIt has been shown by the present authors in a recent paper [1] that if some conservation and balance laws of continuum mechanics are represented in a 4 × 4 form, balance of linear physical momentum (i. e., stress) and balance of mass become closely linked. This seemingly novel result was reached in a completely ad‐hoc fashion by treating time on the same level as the spatial coordinates, and not as parameter, as it is usually done.In order to place the above ad‐hoc result on a firmer foundation and since it is in the theory of relativity that space and time are considered on the same footing, an attempt is made to derive several tensors of continuum mechanics in a systematic manner as 4 × 4 invariant objects.

Dynamics of impact-induced phase transition fronts

Most experiments in martensitic phase transformations are performed under quasi-static loading of a specimen. The only well-documented experimental investigation concerning the impact-induced austenite-martensite phase transformations was given by Escobar and Clifton ([1], [2]). As Escobar and Clifton noted, measured velocity profiles provide a difference between the particle velocity and the transverse component of the projectile velocity. This velocity difference, in the absence of any evidence of plastic deformation, is indicative of a stress induced phase transformation that propagates into the crystals from the impact face. We develop a thermomechanical approach to the modeling of phase-transition front propagation based on the balance laws of continuum mechanics in the reference configuration [3] and the thermodynamics of discrete systems [4]. It is shown that the developed model captures the experimentally observed particle velocity difference.

Statically and kinematically exact nonlinear theory of rods and its numerical verification

The entirely general nonlinear theory of rods is developed. Two vector equilibrium equations, valid on the rod axis are derived exactly, by direct integration of the mechanical balance laws of continuum mechanics over the rod cross section. Then the energetically conjugate exact 6-field kinematics of the rod axis is implied in an integral identity, which expresses the principle of the virtual work. The kinematics consists of the displacement vector of the axis, describing the average translation, and the proper orthogonal tensor, representing the average rotation of the cross section. The integral identity also suggests the required general form of the constitutive relations. They are already approximate from the physical side, are then the best place to make geometric approximations and abbreviations to a certain constrained theory. The constitutive relations, the only nonexact segment of the theory, are treated here as an interchangeable element, supplementing the exactly formulated principle of virtual work.The solution of the nonlinear rod problem becomes nonstandard here, for the configuration space is the nonlinear manifold containing the rotation group SO(3). Applying the finite element method, the proper parameterization, linearization, interpolation, accumulation and discretization procedures in SO(3) group are discussed. A library of 2-, 3-, 4-node 1-D isoparametric elements is developed, and the computer program RAM is worked out.The presented numerical tests contain several problems of highly nonlinear plane and spatial deformations of rods with constitutive relations taken, at the moment, from the classical isotropic elastic beams. In all the tests the developed elements, in comparison with the solutions known in the literature, show a good performance.

On the Microstructural Origin of Certain Inelastic Models

A unifying program modeling the evolution of microstructures and their effects on the macroscopic response of materials is outlined. The new physical assumption introduced is the distinction among “normal” states associated with the parent lattice and “excited” states associated with the occurring microstructures. Both states are described by the usual balance laws of continuum mechanics properly modified to account for their mechanical interaction (mass and momentum exchange). Emphasis is put on the spatial evolution of the microstructures and its nonconvex character which is stabilized by considering gradient effects. The program can lead to predictive models for the localization of microstructures and its relation to the localization of macroscopic deformation. Moreover, it provides a rigorous microstructural justification of macroscopic theories of plasticity and viscoplasticity, including isotropic and kinematic hardening, as well as various current inelastic models. Finally, it suggests a proper frame for considering large deformation and rotation effects.