Variational-Lagrangian irreversible thermodynamics of initially-stressed solids with thermomolecular diffusion and chemical reactions

A general principle of virtual dissipation in irreversible thermodynamics is applied to a solid under initial stress with small non-isothermal incremental deformations and coupled thermomolecular diffusion and chemical reactions. Dynamical field and Lagrangian equations are obtained directly by variational procedures. In addition, the treatment embodies new fundamental concepts and methods in the thermodynamics of open systems and thermochemistry. The new concept of ‘thermobaric potential’ is briefly outlined. The theory is also applicable to porous solids with ‘diffusionlike’ behaviour of pore-fluid mixtures. General validity of viscoelastic correspondence for chemical or other relaxation processes with internal coordinates is indicated in acoustic propagation and seismic problems.     

A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids

A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids is presented. The coupled thermo-mechanical boundary-value problem under consideration consists of the equilibrium problem for a deformable, inelastic and dissipative solid with the heat conduction problem appended in addition. The variational formulation allows for general dissipative solids, including finite elastic and plastic deformations, non-Newtonian viscosity, rate sensitivity, arbitrary flow and hardening rules, as well as heat conduction. We show that a joint potential function exists such that both the conservation of energy and the balance of linear momentum equations follow as Euler-Lagrange equations. The identification of the joint potential requires a careful distinction between equilibrium and external temperatures, which are equal at equilibrium. The variational framework predicts the fraction of dissipated energy that is converted to heat. A comparison of this prediction and experimental data suggests that α-titanium and Al2024-T conform to the variational framework.     

Variational Lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion

The nonisothermal finite strain dynamics of a porous solid containing a viscous fluid is developed on the basis of a new thermodynamics of open systems and irreversible processes. The same theory is applicable to the mechanics of a nonporous solid with thermomolecular diffusion of a substance in solution. New fundamental concepts of “thermobaric” and “convective” potentials are presented in the context of porous solids. Field equations and Lagrangian equations with generalized coordinates are derived directly from a variational principle of “virtual dissipation”. Inclusion of nonlinear viscoelasticity and plastic behavior is indicated. Partial saturation of pore fluid is discussed. The theory is applicable to the mechanics of a non porous solid with thermolecular diffusion of several molecular species in solution, and under certain conditions to the analogous case of a porous solid containg a fluid mixture. It is shown how the Lagrangian equations provide the foundation of finite element methods.     

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.     

Description of finite deformation of solids in a reference configuration

Variational principles of equilibrium processes of deformation and heat conduction are formulated. The variational relations are written for the initial configuration of solids and can be used without restrictions on the strain. A system of equations for a coupled boundary-value problem for isotropic and anisotropic solids is presented, and initial and boundary conditions are formulated. Results of solving problems of finite deformation of initially cylindrical solids are given.     

Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn-Hilliard equations

Two-phase flows driven by the interfacial dynamics are studied by tracking implicitly interfaces in the framework of the Cahn-Hilliard theory. The fluid dynamics is described by the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a nonlinear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled, in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. The rates of convergence in space and time are evaluated leading to an optimal order in error bounds in space and a second order in time with a backward differentiation formula at the second order. Numerical tests devoted to two-phase flows are provided on ellipsoidal droplet retraction, on the capillary rising of a liquid in a tube, and on the wetting drop over a horizontal solid wall.     

Finite element formulation for transport equations in a mixed co-ordinate system: An application to determine temperature effects on the single-well chemical tracer test

Heterogeneous equation systems in a pair of coupled co-ordinate systems are solved by a finite element method. The specific physical application studied is the effect of temperature on single-well chemical tracer (SWCT) tests to measure residual oil saturation (volume fraction of immobile oil phase) remaining after waterflooding of an oil reservoir. Since temperature effects are caused by injecting cooler surface fluid down a well into a warm reservoir, the vertical temperature profile in the wellbore as well as the temperature distribution in the porous oil-bearing layer must be considered. The entire system is modelled to account for the different transport mechanisms. However, it is expedient to divide the connected geometrical region into two model domains. The equations for each submodel are expressed in an appropriate set of co-ordinates. The variational formulation of each model is then discussed. A significant temperature effect on the estimation of residual oil saturation occurs when the radial temperature and concentration wave propagation speeds in the porous formation are about the same. In this case the temperature gradient is located across the chemical tracer bank, causing the chemical reaction rate to vary radially. The temperature effects are demonstrated for two actual field tests in complex reservoirs.     

Generalized variational principles for boundary value problem of electromagnetic field in electrodynamics

An expression of the generalized principle of virtual work for the boundary value problem of the linear and anisotropic electromagnetic field is given. Using Chien＇s method, a pair of generalized variational principles （GVPs） are established, which directly leads to all four Maxwell＇s equations, two intensity-potential equations, two constitutive equations, and eight boundary conditions. A family of constrained variational principles is derived sequentially. As additional verifications, two degenerated forms are obtained, equivalent to two known variational principles. Two modified GVPs are given to provide the hybrid finite element models for the present problem.     

A point assembly method for stress analysis for two-dimensional solids

A novel point assembly method (PAM) is presented for stress analysis for two-dimensional solids. In the present method, the boundaries of the problem domain are represented by a set of discrete points, and the domain itself is represented by properly scattered points. The displacement in the influence triangular areas of a point is interpolated by the displacements at the point and pairs of surrounding points using shape functions. The shape functions used in this work are obtained in the same way as those of a triangular element in the conventional finite element method (FEM). A variational (weak) form of the equilibrium equation is used to produce a set of system equations. These equations are assembled for all the points in the domain, and solved for the displacement field. Stresses and strains at a point are then computed using the displacements obtained for the point and pairs of the surrounding points. A PAM program with an automatic point-searching algorithm has been developed in fortran. Patch tests and convergence studies have been carried out to verify the convergence of the present method and program. Examples are also presented to demonstrate the efficiency and accuracy of the present method compared with analytical solutions as well as the conventional FEM solutions.     

Inhomogeneous deformation of elastomer gels in equilibrium under saturated and unsaturated conditions

A variational method is employed to obtain governing equations and boundary conditions describing finite strain equilibrium configurations of elastomeric gels. Three situations are considered: a liquid saturated gel, an unsaturated gel, and a gel in equilibrium with a vapor of its own liquid. Surface tractions can lead to equilibrium transitions between these cases. The liquid saturated gel is regarded as immersed in a liquid bath. If this bath becomes depleted, then the gel is unsaturated. The degree of unsaturation - a measure of the amount of liquid that would restore a state of saturation - affects the subsequent mechanical behavior. If the unsaturated system is further allowed to condense or evaporate its liquid component at the gel surface, then a new state of equilibrium is achieved. The transition between the unsaturated case and the case of being in equilibrium with the vapor phase corresponds to the chemical potential variable of the gel changing its value from one that is determined by a volume constraint to the value of the chemical potential in the vapor phase. A finite element method is created on the basis of the variational method and demonstrated in the context of eversion, a deformation that imposes very large finite strains. Liquid migration within the gel is not modeled as our focus is on equilibrium states that occur after all such non-equilibrium processes come to rest.     

Simple equations in terms of displacements for finite axisymmetric deflections of shells of revolution

A set of strain-displacement relations for finite axisymmetric deflections of shells of revolution is obtained in as simple form as is consistent with the basic assumptions of the first approximation shell theory. Corresponding field equations and boundary conditions are identified as the Euler-Lagrange equations of an associated variational problem. The equivalence to Reissner theory is established. Simplified versions of strain-displacement relations are analysed. The theoretical considerations are supported by various numerical examples.     

Studies of nonlinear deformation and stability of noncircular cylindrical shells in transverse bending

The variational finite element method in displacements is used to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with a noncircular contour of the cross-section. Quadrangle finite elements of shells of natural curvature are used. In the approximations of element displacements, the displacements of elements as solids are explicitly separated. The variational Lagrange principle is used to obtain a nonlinear system of algebraic equations for the unknown nodal finite elements. The system is solved by the method of successive loadings and by the Newton-Kantorovich linearization method. The linear system is solved by the Crout method. The critical loads are determined in the process of solving the nonlinear problem by using the Sylvester stability criterion. An algorithm and a computer program are developed to study the problem numerically. The nonlinear deformation and stability of shells with oval and elliptic cross-sections are investigated in a broad range of variation of the elongation and ellipticity parameters. The shell critical loads and buckling modes are determined. The influence of the deformation nonlinearity, elongation, and ellipticity of the shell on the critical loads is examined.     

Variational principles of fluid full-filled elastic solids

IntroductionManypracticalproblemsinengineeringinvolveanalysisoffluidfijll-filledelasticsolids.Energyexplorationand"utilizationaretwoexamples.ThefieldequationsofBlot'sstaticalanddynamicaltheoryoffluidfijll-filledelasticsolidswereestablishedinRefs.[1,Zj.BecausetheitisdifficulttogetexactanswersInumericalmethodsareadopted,especiallythet'initCelementmethod.Theelementmethodbasedonvariationalprinciplesisappliedextensively.GhaboussiandWilsonderivedvariationalprinciplesonthebasisofBlot'sequationsan…     

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.     

Simplified Gurtin-type generalized variational principles for fully dynamic magneto-electro-elasticity with geometrical nonlinearity

The fundamental equations, governing all the variables of the initial boundary value problem in fully dynamic magneto-electro-elasticity with geometrical nonlinearity, are expressed in covariant differential form. The generalized principle of virtual work is given in terms of convolutions for the present problem. Two simplified Gurtin-type generalized variational principles, directly leading to all the fundamental equations, are deduced by using He’s semi-inverse method instead of Laplace transforms. By enforcing some fundamental equations as constraint conditions, one of various constrained variational principles is given as an example. By simply dropping out selected field functions, several reduced variational principles are obtained as special forms for piezoelectricity, elastodynamics, and electromagnetics, respectively. This paper aims at providing a more complete theoretical foundation for the finite element applications for the discussed problem.     

Variational principles for hydrodynamic impact problems

We first establish the rigorous field equations of the two continuous stages before and after entering water. Then correspondently, we obtain the specific variational principles, bounded theorems, and boundary integral equations of the second stage problems. The existence of solutions are proved and the scheme of solving the solutions are provided. Finally, as a numerical example, the ship's wave resistence problem is used to demonstrate the specific application of the second stage problems and its accuracy. Then we provide a rigorous and sound theoretical basis of variational finite element method and boundary element method for calculating the accurately fundamental equations.     

电磁弹性固体三维问题的广义变分原理

提出了以电磁弹性固体所有变量应力、应变、位移、电位移、电场强度、电势、磁感应强度、磁场强度和磁势为自变量的电磁弹性固体三维问题最一般形式的广义变分原理。它们涵盖了电磁弹性固体问题所有的基本方程和边界条件。在此基础上还可以进一步给出部分变量为自变量的其它形式广义变分原理。     

A quadrilateral membrane hybrid stress element with drilling degrees of freedom

A new kind of quadrilateral assumed stress hybrid membrane element with drilling degrees of freedom and a traction-free inclined side has been developed based on an extended Hellinger-Reissner principle which is established by expanding the essential terms of the assumed stress field as polynomials in the natural coordinates of the element. The homogeneous equilibrium equations are imposed in a variational sense through the internal displacements which are also expanded in the natural coordinates, while the tractionfree conditions along the inclined side are satisfied exactly. The use of such special element in the finite element solution is shown to be highly accurate when only a very coarse element mesh is used for plates with V-shaped rounded notches or inclined sides.     

电磁弹性动力学初边值问题12类变量广义变分原理

基于Yao建立的电磁弹性固体广义变分原理,运用关于非传统Hamilton型广义变分原理的方法,建立了电磁弹性动力学初边值问题的12类变量广义变分原理,可反映该问题的全部特征,其独立变分变量为该问题的全部变量,即位移、速度、动量、应变、应力、电位移、磁感应强度、电场强度、磁场强度、电标量势、磁标量势和磁矢量势.本文建立的...