A thermodynamically-consistent microplane model for shape memory alloys

In microplane theory, it is assumed that a macroscopic stress tensor is projected to the microplane stresses. It is also assumed that 1D constitutive laws are defined for associated stress and strain components on all microplanes passing through a material point. The macroscopic strain tensor is obtained by strain integration on microplanes of all orientations at a point by using a homogenization process. Traditionally, microplane formulation has been based on the Volumetric–Deviatoric–Tangential split and macroscopic strain tensor was derived using the principle of complementary virtual work. It has been shown that this formulation could violate the second law of thermodynamics in some loading conditions. The present paper focuses on modeling of shape memory alloys using microplane formulation in a thermodynamically-consistent framework. To this end, a free energy potential is defined at the microplane level. Integrating this potential over all orientations provides the macroscopic free energy. Based on this free energy, a new formulation based on Volumetric–Deviatoric split is proposed. This formulation in a thermodynamic-consistent framework captures the behavior of shape memory alloys. Using experimental results for various loading conditions, the validity of the model has been verified.     

Variational principles and generalized variational principles for nonlinear elasticity with finite displacement

In a previous paper (1979)^[1], the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strains from known stresses. This point was not clearly mentioned in the previous paper (1979)^[1]. In this paper, the high order Lagrange multiplier method (1983)^[2] is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V/.V. Novozhilov's results (1958)^[3] for nonlinear elasticity are used.     

A nonhomogeneous nonlocal elasticity model

Nonlocal elasticity with nonhomogeneous elastic moduli and internal length is addressed within a thermodynamic framework suitable to cope with continuum nonlocality. The Clausius–Duhem inequality, enriched by the energy residual, is used to derive the state equations and all other thermodynamic restrictions upon the constitutive equations. A phenomenological nonhomogeneous nonlocal (strain difference-dependent) elasticity model is proposed, in which the stress is the sum of two contributions, local and nonlocal, respectively governed by the standard elastic moduli tensor and the (symmetric positive-definite) nonlocal stiffness tensor. The inhomogeneities of the elastic moduli and of the internal length are conjectured to be each the cause of additional attenuation effects upon the long distance particle interactions. The increased attenuation effects are accounted for by means of the standard attenuation function, but with the standard spatial distance replaced by a suitably larger equivalent distance, and with the spatially variable internal length replaced by the largest value within the domain. Formulae for the computation of the equivalent distance are heuristically suggested and illustrated with numerical examples. The solution uniqueness of the continuum boundary-value problem is proven and the related total potential energy principle given and employed for possible nonlocal-FEM discretizations. A bar in tension is considered for a few numerical applications, showing perfect numerical stability, provided the free energy potential is positive definite.     

Energy theorems and the J-integral in dipolar gradient elasticity

Within the framework of Mindlin’s dipolar gradient elasticity, general energy theorems are proved in this work. These are the theorem of minimum potential energy, the theorem of minimum complementary potential energy, a variational principle analogous to that of the Hellinger–Reissner principle in classical theory, two theorems analogous to those of Castigliano and Engesser in classical theory, a uniqueness theorem of the Kirchhoff–Neumann type, and a reciprocal theorem. These results can be of importance to computational methods for analyzing practical problems. In addition, the J-integral of fracture mechanics is derived within the same framework. The new form of the J-integral is identified with the energy release rate at the tip of a growing crack and its path-independence is proved.The theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory considered here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. The latter case is also treated in the present study.     

Variational analysis of transverse cracking and local delamination in [θm/90n]s laminates

A displacement-based variational model is developed to study the effects of transverse cracking and local delaminations in symmetric composite laminates. In the model, the crack shape is assumed to be a function of crack density and delamination length. Using a variational approach with the principle of minimum potential energy, governing equations are derived. The effective Young’s modulus E_x and energy release rate G are theoretically examined as a result of local delaminations.     

A unified treatment of strain gradient plasticity

A theoretical framework is presented that has potential to cover a large range of strain gradient plasticity effects in isotropic materials. Both incremental plasticity and viscoplasticity models are presented. Many of the alternative models that have been presented in the literature are included as special cases. Based on the expression for plastic dissipation, it is in accordance with Gurtin (J. Mech. Phys. Solids 48 (2000) 989; Int. J. Plast. 19 (2003) 47) argued that the plastic flow direction is governed by a microstress q_ij and not the deviatoric Cauchy stress σ_ij′ that has been assumed by many others. The structure of the governing equations is of second order in the displacements and the plastic strains which makes it comparatively easy to implement in a finite element programme. In addition, a framework for the formulation of consistent boundary conditions is presented. It is shown that there is a close connection between surface energy of an interface and boundary conditions in terms of plastic strains and moment stresses. This should make it possible to study boundary layer effects at the interface between grains or phases. Consistent boundary conditions for an expanding elastic-plastic boundary are as well formulated. As examples, biaxial tension of a thin film on a thick substrate, torsion of a thin wire and a spherical void under remote hydrostatic tension are investigated.     

On the three-dimensional vibrations of elastic prisms with skew cross-section

Three-dimensional (3-D) free vibration of an elastic prism with skew cross-section is investigated using an elasticity-based variational Ritz procedure. Specifically, the associated energy functional minimized in the Ritz procedure is formulated using a simple coordinate mapping to transform the solid skew elastic prism into a unit cube computational domain. The displacements of the prism in each direction are approximately expressed in the form of variable separation. As an enhancement to conventional use of algebraic polynomials trial series in related solid body vibration studies in the associated literature, the assumed skew prism displacement, u, v and w in the computational ξ–η–ζ skew coordinate directions, respectively, are approximated by a set of generalized coefficients multiplied by a finite triplicate Chebyshev polynomial series and boundary functions in ξ–η–ζ to ensure the satisfaction of the geometric boundary conditions of the prism. Upon invoking the stationary condition of the Lagrangian energy functional for the skew elastic prism with respect to the assumed generalized coefficients, the usual characteristic frequency equations of natural vibrations of the skew elastic prism are derived. Upper bound convergence of the first eight non-dimensional frequencies accurate to four significant figures is achieved by using up to 10–15 terms of the assumed skew prism displacement functions. First known 3-D vibration characteristics of skew elastic prisms are examined showing the effects of varying prism length ratios (ranging from skew solids to skew slender beams), as well as, varying cross-sectional side ratios and skewness, which collectively can serve as benchmark studies against which vibration modes predicted by classical Euler and shear deformable skew beam theories as well as alternative methodologies used in elastic prism vibrations of mechanical and structural components.     

Cracked infinite cylinder with two rigid inclusions under axisymmetric tension

This paper considers the problem of an axisymmetric infinite cylinder with a ring shaped crack at z = 0 and two ring-shaped rigid inclusions with negligible thickness at z = ±L. The cylinder is under the action of uniformly distributed axial tension applied at infinity and its lateral surface is free of traction. It is assumed that the material of the cylinder is linearly elastic and isotropic. Crack surfaces are free and the constant displacements are continuous along the rigid inclusions while the stresses have jumps. Formulation of the mixed boundary value problem under consideration is reduced to three singular integral equations in terms of the derivative of the crack surface displacement and the stress jumps on the rigid inclusions. These equations, together with the single-valuedness condition for the displacements around the crack and the equilibrium equations along the inclusions, are converted to a system of linear algebraic equations, which is solved numerically. Stress intensity factors are calculated and presented in graphical form.     

Some issues in the application of cohesive zone models for metal–ceramic interfaces

Cohesive zone models (CZMs) are being increasingly used to simulate discrete fracture processes in a number of homogeneous and inhomogeneous material systems. The models are typically expressed as a function of normal and tangential tractions in terms of separation distances. The forms of the functions and parameters vary from model to model. In this work, two different forms of CZMs (exponential and bilinear) are used to evaluate the response of interfaces in titanium matrix composites reinforced by silicon carbide (SCS-6) fibers. The computational results are then compared to thin slice push-out experimental data. It is observed that the bilinear CZM reproduces the macroscopic mechanical response and the failure process while the exponential form fails to do so. From the numerical simulations, the parameters that describe the bilinear CZM are determined. The sensitivity of the various cohesive zone parameters in predicting the overall interfacial mechanical response (as observed in the thin-slice push out test) is carefully examined. Many researchers have suggested that two independent parameters (the cohesive energy, and either of the cohesive strength or the separation displacement) are sufficient to model cohesive zones implying that the form (shape) of the traction–separation equations is unimportant. However, it is shown in this work that in addition to the two independent parameters, the form of the traction–separation equations for CZMs plays a very critical role in determining the macroscopic mechanical response of the composite system.     

A numerical study of the longitudinal thermoconvective rolls in a mixed convection flow in a horizontal channel with a free surface

This paper presents a numerical study of three-dimensional laminar mixed convection within a liquid flowing on a horizontal channel heated uniformly from below. The upper surface is free and assumed to be flat. The coupled Navier–Stokes and energy equations are solved numerically by the finite volume method taking into account the thermocapillary effects (Marangoni effect). When the strength of the buoyancy, thermocapillary effects and forced convective currents are comparable (Ri ⋍ O(1) and Bd = Ra/Ma ⋍ O(1)), the results show that the development of instabilities in the form of steady longitudinal convective rolls is similar to those encountered in the Poiseuille–Rayleigh–Bénard flow. The number and spatial distribution of these rolls along the channel depend on the flow conditions. The objective of this work is to study the influence of parameters, such as the Reynolds, Rayleigh and Biot numbers, on the flow patterns and heat transfer characteristics. The effects of variations in the surface tension with temperature gradients (Marangoni effect) are also considered.     

On a pure complementary energy principle and a force-based finite element formulation for non-linear elastic cables

A complementary-dual force-based finite element formulation is proposed for the geometrically exact quasi-static analysis of one-dimensional hyperelastic perfectly flexible cables lying in the two-dimensional space. This formulation employs as approximate functions the exact statically admissible force fields, i.e., those that satisfy the equilibrium differential equations in strong form, as well as the equilibrium boundary conditions. The formulation relies on a principle of total complementary energy only expressed in terms of force fields, being therefore called a pure principle. Under the assumption of stress-unilateral behavior, this principle can be regarded as being dual to the principle of minimum total potential energy, corresponding therefore to a maximum principle. Some numerical applications, including cables suspended from two and three points at the same level or at different levels, with both Hookean and Neo-Hookean material behaviors, are presented. As it will be shown, in contrast to the standard two-node displacement-based formulation derived from the principle of minimum total potential energy, the proposed dual force-based formulation is capable of providing the exact solution of a given problem only using a single finite element per cable. Both the proposed principle of pure complementary energy and its corresponding force-based finite element formulation can be easily extended to the case of cables lying in the three-dimensional space.     

On consistent micromechanical estimation of macroscopic elastic energy,coherence energy and phase transformation strains for SMA materials

An apparatus of micromechanics is used to isolate the key ingredients entering macroscopic Gibbs free energy function of a shape memory alloy (SMA) material. A new self-equilibrated eigenstrains influence moduli (SEIM) method is developed for consistent estimation of effective (macroscopic) thermostatic properties of solid materials, which in microscale can be regarded as amalgams of n-phase linear thermoelastic component materials with eigenstrains. The SEIM satisfy the self-consistency conditions, following from elastic reciprocity (Betti) theorem. The method allowed expressing macroscopic coherency energy and elastic complementary energy terms present in the general form of macroscopic Gibbs free energy of SMA materials in the form of semilinear and semiquadratic functions of the phase composition. Consistent SEIM estimates of elastic complementary energy, coherency energy and phase transformation strains corresponding to classical Reuss and Voigt conjectures are explicitly specified. The Voigt explicit relations served as inspiration for working out an original engineering practice-oriented semiexperimental SEIM estimates. They are especially conveniently applicable for an isotropic aggregate (composite) composed of a mixture of n isotropic phases. Using experimental data for NiTi alloy and adopting conjecture that it can be treated as an isotropic aggregate of two isotropic phases, it is shown that the NiTi coherency energy and macroscopic phase strain are practically not influenced by the difference in values of austenite and martensite elastic constants. It is shown that existence of nonzero fluctuating part of phase microeigenstrains field is responsible for building up of so-called stored energy of coherency, which is accumulated in pure martensitic phase after full completion of phase transition. Experimental data for NiTi alloy show that the stored coherency energy cannot be neglected as it considerably influences the characteristic phase transition temperatures of SMA material.     

A linearized theory of elastic dielectrics which couples electrically to flexural vibrations

Assuming that the free energy depends on the deformation gradient and the spatial electric field, we derive the expressions for the Cauchy stress tensor and the spatial electric displacement from an observer invariant quadratic form of the free energy via the strict definitions of these quantities. Specific forms of the Piola-Kirchhoff stress tensor and the material electric displacement are then deduced and linearized in a particular sense. As an application of the resulting theory, we formulate the problem of an electrically driven disc within the context of the classical bending theory of thin plates. The material of the disc is assumed to have at most the symmetry of a hexagonal system of classC _6v.The resulting coupled differential equations for the axial mechanical displacement of the middle surface and the material electric potential indicate that the problem is not empty. This result is of particular interest in view of the fact that it is generally held that the classical theory of piezoelectricity does not permit such couplings to occur.     

A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle

A link is shown to exist between the so-called residual-based strain gradient plasticity theory and the analogous theories based on the (extended) virtual work principle (VWP). To this aim, the former theory is reformulated and cast in a residual-free form, whereby the insulation condition and the (nonlocal) Clausius–Duhem inequality, on which the theory is grounded, are substituted with equivalent residual-free ingredients, namely the energy balance condition and the residual-free form of the Clausius–Duhem inequality. The equivalence of the residual-free formulation to the original one is shown, also in their ability to cope with energetic size effects and interfacial energy ones. It emerges that the residual-free form of the Clausius–Duhem inequality coicides with the way the second thermodynamics principle is enforced within the VWP-based theories, and also that the energy balance condition amounts to the extended VWP enforced in the whole body. This makes the residual-free formulation possess strong similarities with the more general VWP-based theories, such as it constitutes an assessment of the existing link, which can be synthetized by the statement: the insulation condition is equivalent to the extended VWP deprived by the content of the standard principle.     

Analysis of gaseous flow in a micropipe with second order velocity slip and temperature jump boundary conditions

In this paper, second order velocity slip and temperature jump boundary conditions are used to solve the momentum and energy equations along with isoflux thermal boundary condition at the surface of the micropipe. The flow is assumed to be hydrodynamically and thermally fully developed inside the micropipe and viscous dissipation is included in the analysis. The solution yields closed form expressions for the temperature field and Nusselt number (Nu) as a function of various modeling parameters, namely, Knudsen number and Brinkman number (Br). For the given values of Br, the maximum difference of Nu between continuum flow with first order slip model and continuum and, second order slip model is found to be 35.67 and 34.62 %, respectively. Present solution exhibits good agreement with the other theoretical models.     

A local lagrangian analysis of passive particle advection in a gas flow field

A local analysis is performed to study the departure from passive advection by small inertial particles based on a Lagrangian framework. The analysis considers heavy particles immersed in a gaseous flow and is restricted to short times, making it relevant to the PIV technique. A necessary (but not sufficient condition) for passive particle advection of inertial particles is that the quantity Λ_maxτ_p be much smaller than one, where Λ_max is the largest modulus of the eigenvalues corresponding to the velocity gradient tensor. This allows for the inertial and passive time scales to match beyond the initial transient, and consequently for the respective trajectories to remain relatively close. Due to this important role regarding advection behavior, Λ_maxτ_p is offered as a definition of a local Stokes number, St_Λ. Since this quantity is a field quantity, it directly provides indication of when and where passive advection by particles can be expected. A departure equation is obtained in one-dimension, where the influence of initial velocity and gravity are explicitly shown. If the flow is irrotational, the higher dimensional analysis reduces to a set of decoupled one-dimensional equations acting along each respective eigenvector of the velocity gradient tensor. A similar expression is found for the case of a purely temporal flow field.     

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.     

Steady state thermal and mechanical stresses of a poro-piezo-FGM hollow sphere

In this study, an analytical method is developed to obtain mechanical and thermal stress and electrical potential functions, electrical and mechanical displacement in two dimensional steady (r,θ) stat a functionally graded piezo electric porous material hollow sphere (FGPPM). It is assumed that properties of poro, piezoelectric and FGM material is changed through thickness according to power law functions, Heat conduction equation is obtained for obtaining temperature distribution and Navier equations analytically using Legendre polynomials and Euler differential equations system for investigating displacements changes and stress and potential functions for different power indices law and is drawn on a graph. These results are confirmed with the obtained in formations in the paper.