A new perspective on variational methods for stability analysis of columns

Complementary energy approach to contact problems based on consistent augmented Lagrangian formulation

Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function , we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

Slip systems and flow patterns in viscoplastic metallic sheets with dislocations

Slab method of analysis has been used for solving metal forming problems for a long time. However it has been restricted to plane strain and axisymmetric problems due to limitations in its formulations. In this paper a new formulation has been proposed so that it could be applied to three dimensional problems in metal forming. A parametric slab has been considered in this analysis and the force balance on the slab was carried to obtain equilibrium equations in terms of these parameters. The parameters in fact are related to the geometry of the final extruded shape, the die and the material flow regime assumed in the formulation. In this way most of the limitations encountered in previous formulations were surpassed. The effect of reduction of area, frictional conditions and other process parameters on the extrusion pressure was investigated. The theoretical results obtained in this paper were compared with the results of finite element method and a good agreement was observed between them.

Most aluminum cylinder blocks produced using high-pressure or low-pressure die-casting processes require gray cast iron liners (GCI liners) to compensate for their insufficient wear resistance and heat resistance of the Al-Si-Cu alloys. However, the cast-in liners cause excessive residual stress at the cylinder bore region. The resultant residual stress induces distortion of the cylinder liner. These inconveniences hinder development of more efficient engines. Therefore, an accurate thermal stress analysis technique has been sought to predict the residual stress and distortion of the cylinder liner. For accurate thermal stress analysis, we have already developed an elastoplastic-creep constitutive equation for which the inelastic strain developed at high temperatures does not contribute to strain hardening that occurs at low temperatures by duplicating the recovery behavior. Our earlier investigation using this equation has already revealed that incorporation of the recovery in the alloy constitutive equation is effective for improving the prediction accuracy of the thermal stress developed during casting. However, this conclusion was obtained only for a simple shape casting with a uniaxial thermal stress state. Effects of the developed constitutive equation have not been discussed for a casting closer to an actual cylinder block. For this study, a cylindrical aluminum casting with GCI (ISO 300) insert was produced. Then, the circumferential strain of the GCI liner was measured in-situ during casting. Measurements were taken of the residual stresses of the cylindrical aluminum casting and GCI liner, and of the liner deformation at a room temperature. The experimentally obtained results supported a discussion of the predictive accuracies of the elastoplastic-creep constitutive equation and the classical elastoplastic constitutive equation. A comparison revealed that the elastoplastic-creep constitutive equation for the aluminum casting has better predictive accuracy than the classical elastoplastic equation for residual stress, liner deformation, and the circumferential strain of a GCI liner during casting. Investigation of the simulated strain components of the cylindrical aluminum casting during casting indicated incorporation of the recovery in the alloy constitutive equation as a main factor improving the predictive accuracy.

Summary This work addresses the development of some alternative hybrid and mixed variational formulations for the geometrically-exact three-dimensional first-order-shear shell boundary value problem [1,2]. In the framework of the complementaryenergy-based formulations, a Legendre transformation is used to introduce the complementary energy density in the variational statements as a function of the cross-sectional resultants only. The corresponding variational principles are shown to feature stationarity within the framework of the boundary-value-problem. The main features of the principles are highlighted, giving special attention to their relationships from both theoretical and numerical point of view. Variational principles constitute the core of the development of numerical methods in Solid Mechanics. The utility of such principles is two-fold: first, they provide a very convenient method for the derivation of the governing equations and natural boundary conditions of the boundary value problem and, second, they provide the mathematical foundation required to produce consistent numerical approximations. In this second role, the variational methods have been most useful in Computational Solid Mechanics [3]. Many different variational principles can be constructed depending on the equations enforced in the weak form. Perhaps the most remarkable variational principle is the Principle of Stationary Total Potential Energy, which states that, among all kinematically admissible displacement fields, those that satisfy the equilibrium conditions in the domain and at the boundary, lead to a stationary value of the total potential energy functional. In this paper, after the presentation of many mixed variational principles for the nonlinear analysis of shells, we derive a hybrid form of the Two-field Principle of Total Complementary Potential Energy, which is used on a totally consistent derivation of equilibrated co-rotational finite elements [4].

Due to their tunable mechanical properties and broad applicability, three dimensional compressible–twistable mechanical metamaterials (3D-CTMM) have attracted significant attention in recent years. While the recent research on these configurations is limited to linear mechanical behavior analysis, the nonlinear characteristics of these metamaterials are crucial in practical applications such as adaptive vibration isolation, impact protection, and energy harvesting. This study introduces a combined nonlinear stress-strain relationship and strain-energy-based analytical framework, supported by finite-element analysis (FEA) and experimental validation for investigating the nonlinear mechanical load-deformation behavior of a 3D-CTMM, with potential applicability to a broader class of the 3D-CTMM designs. The constitutive model and coupled load-deformation and load-torsion responses of the unit cells are investigated by incorporating the intrinsic material nonlinearity and complementary strain energy theory into the analysis and then expanded to the overall phase of the 3D-CTMM. The accuracy of the mechanical modeling is evaluated through both numerical simulations and experimental characterization. The results demonstrate strong agreement across the multiscale analyses and reveal pronounced nonlinear behavior in the deformation responses particularly in relatively high global strain values of up to 16%. The nonlinear behavior suggests that the conventional assumption of constant Young’s modulus and Poisson’s ratio for the overall phase of the 3D-CTMM may not fully capture the response, as these parameters vary with nonlinearity and depend on the number of unit cells in the configuration. Furthermore, as a tangible application, the capability of the studied 3D-CTMM to absorb impact energy is also presented where a 4×4×4 unit cell assembly fabricated by thermoplastic polyurethane (TPU) exhibits superior potential for absorbing impact energy.

In this paper by the use of “convex analysis” the general nonlinear unilateral static problem of structural mechanics is formulated. By means of the “subdifferentiability” concept the unilateral nonlinear boundary conditions and the unilateral nonlinear constitutive laws are formulated as “subdifferential relations”, permitting the derivation of the variational inequalities of virtual and complementary virtual work. Thus the minimum potential and complementary energy theorems are proved. These minimum theorems are used to formulate the analysis as a nonlinear programming problem. Numerical examples on structures with nonlinear unilateral constitutive laws and nonlinear unilateral boundary conditions illustrate the theory.

Figure 19.1 illustrates the place of rheometry in the process for solving a non-Newtonian fluid mechanics problem for an incompressible fluid. Given that the flow problem and boundary conditions can be specified, the basic and established tools to effect a solution for the problem are the Cauchy momentum equations (conservation of momentum) and the continuity equation (conservation of mass). However, these equations can only be solved when the stresses are related to the velocity field via a constitutive equation. With the exception of Newtonian fluids and some simple non-Newtonian fluids, e.g. power-law fluids, where fluid elasticity is of no importance, the relevant constitutive equation is not known and must be chosen from a range of available possibilities. It is the parameters in the constitutive equation, such as viscosity, relaxation time and retardation time in the Oldroyd B model, which must be measured in rheometry in order to define the flow problem completely so that it can be solved by numerical methods and sometimes by analytical methods, albeit rarely. Having solved the fluid mechanics problem for the stress and velocity field, the solution must be checked because of the assumed constitutive equation. At this stage, flow visualization techniques can be used to observe the velocity and/or stress field for comparison with prediction. If the comparison is good, the constitutive equation is adequate and the problem is solved. Unfortunately, this process has only been successful for relatively simple elastic liquids [1] in squeeze film flow [2], Stokes flow around a sphere [3, 4], isothermal spinning [5], and drop breakup in filament extension [6].

PREFACE. INTRODUCTION. Generalities. Ordinary differential equations. Supplementary conditions associated to ODEs. The Cauchy (initial) problem.The two-point problem. 1: LINEAR ODEs OF FIRST AND SECOND ORDER. 1.1 Linear first order ODEs. 1.1.1 Equations of the form . 1.1.2 The linear homogeneous equation. 1.1.3 The general case. 1.1.4 The method of variation of parameters (Lagrange's method). 1.1.5 Differential polynomials. 1.2 Linear second order ODEs. 1.2.1 Homogeneous equations. 1.2.2 Non-homogeneous equations. Lagrange's method. 1.2.3 ODEs with constant coefficients. 1.2.4 Order reduction. 1.2.5 The Cauchy problem. Analytical methods to obtain the solution. 1.2.6 Two-point problems (Picard). 1.2.7 Sturm-Liouville problems. 1.2.8 Linear ODEs of special form. 1.3. Applications 2: LINEAR ODEs OF HIGHER ORDER (n >2). 2.1 The general study of linear ODEs of order . 2.1.1 Generalities. 2.1.2 Linear homogeneous ODEs. 2.1.3 The general solution of the non-homogeneous ODE. 2.1.4 Order reduction. 2.2 Linear ODEs with constant coefficients. 2.2.1 The general solution of the homogeneous equation. 2.2.2 The non-homogeneous ODE. 2.2.3 Euler type ODEs. 2.3 Fundamental solution. Green function. 2.3.1 The fundamental solution. 2.3.2 The Green function. 2.3.3 The non-homogeneous problem. 2.3.4 The homogeneous two-point problem. Eigenvalues. 2.4 Applications. 3: LINEAR ODSs OF FIRST ORDER. 3.1 The general study of linear first order ODSs. 3.1.1 Generalities. 3.1.2 The general solution of the homogeneous ODS. 3.1.3 The general solution of the non-homogeneous ODS. 3.1.4 Order reduction of homogeneous ODSs. 3.1.5 Boundary value problems for ODSs. 3.2 ODSs with constant coefficients. 3.2.1 The general solution of the homogeneous ODS. 3.2.2 Solutions in matrix form for linear ODSs with constant coefficients. 3.3 Applications. 4: NON-LINEAR ODEs OF FIRST AND SECOND ORDER. 4.1 First order non-linear ODEs. 4.1.1 Forms of first order ODEs and oftheir solutions. 4.1.2 Geometric interpretation. The theorem of existence and uniqueness. 4.1.3 Analytic methods for solving first order non-linear ODEs. 4.1.4. First order ODEs integrable by quadratures. 4.2 Non-linear second order ODEs. 4.2.1 Cauchy problems. 4.2.2 Two-point problems. 4.2.3 Order reduction of second order ODEs. 4.2.4 The Bernoulli-Euler equation. 4.2.5 Elliptic integrals. 4.3 Applications. 5: NON-LINEAR ODSs OF FIRST ORDER. 5.1 Generalities. 5.1.1 The general form of a first order ODS. 5.1.2 The existence and uniqueness theorem for the solution of the Cauchy problem. 5.1.3 The particle dynamics. 5.2 First integrals of an ODS. 5.2.1 Generalities. 5.2.2 The theorem of conservation of the kinetic energy. 5.2.3 The symmetric form of an ODS. Integral combinations. 5.2.4 Jacobi's multiplier. The method of the last multiplier. 5.3 Analytical methods of solving the Cauchy problem for non-linear ODSs. 5.3.1 The method of successive approximations (Picard-Lindeloff). 5.3.2 The method of the Taylor series expansion. 5.3.3 The linear equivalence method (LEM). 5.4 Applications. 6: VARIATIONAL CALCULUS. 6.1 Necessary condition of extremum for functionals of integral type. 6.1.1 Generalities. 6.1.2 Functionals of the form..... 6.1.3 Functionals of the form..... 6.1.4 Functionals of integral type, depending on n functions. 6.2 Conditional extrema. 6.2.1 Isoperimetric problems. 6.2.2 Lagrange's problem. 6.3 Applications. 7: STABILITY. 7.1 Lyapunov Stability. 7.1.1 Generalities. 7.1.2 Lyapunov's theorem of stability. 7.2 The stability of the solutions of dynamical systems. 7.2.1 Autonomous dynamical systems. 7.2.2 Long term behaviour of the solutions. 7.3 Applications. INDEX. REFERENCES.

A general method of estimating constitutive equation of material from experiment of free oscillation is presented. The information on attenuation and frequency in the oscillation changing the environment temperature is utilized to estimate the linear visco-elastic constitutive equation with deformation process. Linear visco-elastic models of 2-element, 3-element or 4-element are adapted with the activation process in the dash-pot elements. Method of examining the profile in the attenuation versus temperature curve is proposed to obtain the elements constants using the Arrhenius plots, as an additional method to the ordinary Arrhenius plot for the frequency versus peak attenuation temperature relation. Deformation characteristic of solid material is generally expressed as elastic, plastic, viscous or their combinational form of constitutive equation. Most of materials exhibit more or less strain rate depen- dency in the stress-strain relationship and the strain rate dependency is expressed as viscous terms in the constitutive equation. Especially as the environmental temperature increases the rate dependency becomes more remarkable so that viscous effect becomes crucial at elevated temperature. This temper- ature dependency is often explained by the activation theory and has also been confirmed for various materials by experiments. Constitutive equation that expresses the activation process includes activation energy terms in the equation and the values of activation energy obtained from experiments lead to know the deformation mechanism of the material. The activation process is generally given by the equation,

New classes of electro-elastic and thermo-electro-elastic bodies that are not Green elastic

Interfacial localisation in simple shear tests on a granular medium modelled as a Cosserat continuum

Applied Mathematics and Mechanics

A new constitutive equation has been developed for describing the nonlinear viscoelastic properties, including yield, of polymer solids. The nonlinear viscoelastic constitutive equation is the logical extension of the standard three‐dimensional linear viscoelastic constitutive equation to include the effects of deformation‐induced changes in the thermodynamic state of the polymer on the rate of viscoelastic relaxation. The rate of viscoelastic relaxation depends upon the fraction of unoccupied lattice sites (i.e., holes) in the Simha‐Somcynsky statistical thermodynamic equation of state, and the hole fraction is a known function of the temperature, pressure, and specific volume. The nonlinear viscoelastic yield and postyield behavior are a direct consequence of the deformation‐induced dilation. All material constants contained in the constitutive equation can be determined from independent elastic, linear viscoelastic, and pressure‐volume‐temperature measurements—there are no adjustable constants. The constitutive equation has been solved for model viscoelastic materials in uniaxial extension and shear deformations where the applied strain rate is constant. It is shown that an experimentally realizable “shear” deformation is only isochoric in the linear viscoelastic limit. Yield and postyield softening are predicted in both uniaxial extension and experimental shear, and the predicted yield stress exhibits the appropriate strain rate and temperature dependence. The effects of using various finite strains measures in the nonlinear viscoelastic constitutive equation are analyzed.