A linearly conforming point interpolation method (LC‐PIM) for three‐dimensional elasticity problems

Linearly conforming point interpolation method (LC‐PIM) is formulated for three‐dimensional elasticity problems. In this method, shape functions are generated using point interpolation method by adopting polynomial basis functions and local supporting nodes are selected based on the background cells. The shape functions so constructed have the Kronecker delta functions property and it allows straightforward imposition of point essential boundary conditions. Galerkin weak form is used for creating discretized system equations, and a nodal integration scheme with strain‐smoothing operation is used to perform the numerical integration. The present LC‐PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method using linear elements, the LC‐PIM can achieve better efficiency, and higher accuracy especially for stresses. Copyright © 2007 John Wiley & Sons, Ltd.     

Micromechanics of Tension‐Compression Asymmetry of Polycrystalline Shape‐Memory‐Alloys

The asymmetry associated with martensitic transformations observed in tension/compression experiments of shape‐memory‐alloys (SMAs) is investigated on the basis of a recently suggested micromechanical model. The approach is based on crystallographic theory and utilizes a framework of energy minimization in a finite deformation context. Polycrystalline NiTi under tension demonstrates smaller phase‐transformation start‐strain, differe phase‐transformation stress‐levels and flatter phase‐transformation stress‐strain slopes than that under compression in our numerical simulation. The phase‐transformation start‐stress is followed to have a linear relationship with respect to the temperature within a certain range. These results agree well with experimental results reported in the literature.     

A nonlinear plate finite element formulation for shape memory alloy applications

The aim of the present work is to develop a new finite element model for the finite strain analysis of plate structures constituted of shape memory alloy (SMA) material. A three‐dimensional constitutive model for shape memory alloys able to reproduce the special thermomechanical behavior of SMA characterized by pseudoelasticity and shape memory effects is adopted. The finite strain constitutive model is thermodynamically consistent and is completely formulated in the reference configuration. A two‐dimensional plate theory is proposed based on a tensor element shape function formulation. The displacement field is expressed in terms of increasing powers of the transverse coordinate. The equilibrium statement is formulated on the basis of the virtual displacement principle in a total Lagrangian format. The proposed displacement formulation is particularly suitable for the simple derivation of high‐order finite elements. Numerical applications are performed to assess the efficiency and locking performance of the proposed plate finite element. Some additional numerical examples are carried out to study the accuracy and robustness of the proposed computational technique and its capability of describing the structural response of SMA devices. Copyright © 2011 John Wiley & Sons, Ltd.     

A new SMA shell element based on the corotational formulation

Aim of this paper is to develop a new shape memory alloy (SMA) facet-shell finite element accounting for material and geometric nonlinearities. A corotational formulation is exploited, able to filter out large rigid-body motions from the element transformation. Accordingly, a geometrically linear core-element is employed, along with a SMA constitutive model formulated in the small strain framework. In particular, in accordance with the formulation of the classical thin shell theory, a plane-stress SMA model accounting for the pseudo-elastic as well as the shape memory effect is adopted. The time integration of the evolutive equation is performed developing a step-by-step backward-Euler numerical procedure. A highly efficient implementation of the corotational machinery is used, endowed with a fully consistent tangent stiffness. Applications are carried out for assessing the performances of the developed computational procedure and to investigate on some interesting engineering examples. The numerical results show the effectiveness of the proposed shell element, whose simplicity makes it attractive for the design of new advanced SMA-based devices undergoing significant configuration changes during their operation.     

Analysis of SMA composite laminates using a multiscale modelling technique

The present work deals with the analysis of smart laminates, obtained as stacking sequence of fibre‐reinforced composite laminae and composite shape memory alloy (SMA) layers. The behaviour of composite SMA (CSMA) laminate is studied developing a full micro–macro approach. In fact, a non‐linear 4‐node mixed interpolation of tensorial components (MITC4) laminate finite element, based on the first‐order shear deformation theory, is developed. The SMA layer constitutive relationship is determined solving a non‐linear homogenization problem at each non‐linear iteration of each time step for each integration Gauss point. Some numerical applications are developed in order to investigate the influence of the CSMA on the buckling behaviour of plates and on the transversal displacement control of plates subjected to different loading conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

On the robustness and efficiency of integration algorithms for a 3D finite strain phenomenological SMA constitutive model

Most devices based on shape memory alloys experience large rotations and moderate or finite strains. This motivates the development of finite‐strain constitutive models together with the appropriate computational counterparts. To this end, in the present paper a three‐dimensional finite‐strain phenomenological constitutive model is investigated and a robust and efficient integration algorithm is proposed. Properly defining the variables, extensively used regularization schemes are avoided and a nucleation–completion criterion is defined. Moreover, introducing a logarithmic mapping, a new form of time‐discrete equations is proposed. The solution algorithm as well as a suitable initial guess for the resultant nonlinear equations are also deeply discussed. Extensive numerical tests are performed to show robustness as well as efficiency of the proposed integration algorithm. Implementation of the integration algorithm within a user‐defined subroutine UMAT in the commercial nonlinear finite element software ABAQUS/Standard makes also possible the solution of a variety of boundary value problems. The obtained results show the efficiency and robustness of the proposed approach and confirm the improved efficiency (in terms of solution CPU time) when a nucleation–completion criterion is used instead of regularization schemes, as well as when a logarithmic mapping is used for the time‐discrete evolution equation instead of an exponential mapping. Copyright © 2010 John Wiley & Sons, Ltd.     

A Combined Analytic,Numeric, and Experimental Investigation Performed on NiTi/NiTiCu Bi‐Layer Composites under Tensile Loading

Adjusting mechanical behavior and controlling deformation parameters are significant tasks in designing shape memory components. In this paper, an analytical model describes the deformation behavior of NiTi/NiTiCu bi‐layer composites under tensile loading. Different deformation stages are considered based on single mechanical behavior at each stage. Closed‐form equations are derived for stress–strain variations of bi‐layer composites under uniaxial loading–unloading. Bi‐layer composites made via the diffusion bonding method from single layers of NiTi alloy with a composition of Ti‐50.7 at.% Ni, as an austenitic layer, and Ti‐45 at% Ni‐5 at% Cu, as a martensitic layer, are produced by the vacuum arc remelting technique. The tensile behavior of single‐ and bi‐layers is investigated by using loading–unloading experiments to find the nominal stress–strain curves. Numerical simulations are also done by employing Lagoudas constitutive model to simulate stress–strain diagrams. The solutions of the analytical method presented are validated by using the numerical simulations as well as the experimental results. With regard to the results obtained from the analytical modeling, the numerical simulations, and the experiments, it is evident that the bi‐layer composites with different thickness ratios provide adjustable mechanical behavior that can be considered in different application designs, for example, actuators equipped with shape memory components.

     

A finite element method for light activated shape‐memory polymers

Shape‐memory polymers (SMPs) belong to a class of smart materials that have shown promise for a wide range of applications. They are characterized by their ability to maintain a temporary deformed shape and return to an original parent permanent shape. In this paper, we consider the coupled photomechanical behavior of light activated shape‐memory polymers (LASMPs), focusing on the numerical aspects for finite element simulations at the engineering scale. The photomechanical continuum framework is summarized, and some specific constitutive equations for LASMPs are described. Numerical implementation of the multiphysics governing partial differential equations takes the form of a user defined element subroutine within the commercial software package ABAQUS . We verify our two‐dimensional and three‐dimensional finite element procedure for multiple analytically tractable cases. To show the robustness of the numerical implementation, simulations are performed under various geometries and complex photomechanical loading. Copyright © 2016 John Wiley & Sons, Ltd.     

A statistical mechanics approach describing martensitic phase transformation

A novel theoretical approach modeling martensitic phase transformation is demonstrated in the present study. The generally formulated model is based on the block-spin-approach and on renormalization in statistical mechanics and is applied to a representative volume element which is assumed to be in a local thermodynamic equilibrium. Using fundamental physical properties of a shape memory alloy (SMA) single crystal as input data the model predicts the order parameter “total strain”, the martensitic phase fraction and the stress assisted transformation accompanied by pseudoelasticity without the requirement of evolution equations for internal variables and assumptions on the mathematical structure of the classical free energy. In order to demonstrate the novel approach the first computations are carried out for a simple one-dimensional case. Results for total strain and martensitic phase fraction are in good qualitative agreement with well known experimental data according to their macroscopic strain rearrangement when phase transformation occurs. Further a material softening effect during phase transformation in SMAs is predicted by the statistical physics approach.     

FEM Simulation der Prozessschritte Pulverpressen/Sintern

FEM‐Simulation of the sequence Powder Pressing/Sintering The optimisation of powder forming processes is currently based on trial‐and‐error methods. Numerical simulation is a promising tool for reducing production times and costs. Particularly, the Finite Element Method (FEM) is a powerful tool for the simulation of metal forming processes. Within the scope of this work the simulation of powder die compaction for magnesium alloys was formulated by means of an elliptical yield criterion since this theory takes into account the compressibility of porous bodies by plastic deformation. The sintering process was modelled by using constitutive equations which are derived from a time‐dependent creep‐potential. Both models were implemented in a general purpose FEM‐code. In order to demonstrate the capability and numerical stability of the integration algorithms some powder metallurgical processes were simulated. The numerical simulation of the PM sequence die compaction/sintering allows the prediction of the workpiece's quality and the identification of the optimal process parameters. The implemented models enable the computation of the relative density distribution and the prediction of the shape distortion of the green during sintering. Furthermore, The possibility of the sintering process with non‐isothermal conditions and its influence on the shape distortion is demonstrated in this paper.     

Numerical integration on the sphere and its effect on the material symmetry of constitutive equations—A comparative study

In the present paper, we consider a class of constitutive models based on numerical integration on the unit sphere. The directional behaviour of the quadrature schemes and its effect on the symmetry properties of these constitutive models are studied by subjecting the set of integration points on the sphere to arbitrary rigid rotations. We investigate a number of recently proposed integration schemes in application to a full network model of rubber elasticity and to an exponential model for soft tissues. In order to assess and compare these schemes, statistical methods are presented and applied. The analysis discloses a number of integration schemes that offer a good compromise between the numerical error and the number of integration points. However, as a general result it turns out that numerical integration is prone to introduce strong anisotropy into originally isotropic constitutive equations, in particular, for highly non‐linear integrand functions. The consequences for application of the investigated class of constitutive models in finite element calculations are highlighted in a benchmark‐like numerical example. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerically consistent regularization of force‐based frame elements

Recent advances in the literature regularize the strain‐softening response of force‐based frame elements by either modifying the constitutive parameters or scaling selected integration weights. Although the former case maintains numerical accuracy for strain‐hardening behavior, the regularization requires a tight coupling of the element constitutive properties and the numerical integration method. In the latter case, objectivity is maintained for strain‐softening problems; however, there is a lack of convergence for strain‐hardening response. To resolve the dichotomy between strain‐hardening and strain‐softening solutions, a numerically consistent regularization technique is developed for force‐based frame elements using interpolatory quadrature with two integration points of prescribed characteristic lengths at the element ends. Owing to manipulation of the integration weights at the element ends, the solution of a Vandermonde system of equations ensures numerical accuracy in the linear‐elastic range of response. Comparison of closed‐form solutions and published experimental results of reinforced concrete columns demonstrates the effect of the regularization approach on simulating the response of structural members. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐Temperature Shape Memory Effect in Ti‐Ta Thin Films Sputter Deposited at Room Temperature

Ti‐Ta based alloys are potential high‐temperature shape memory materials with operation temperatures above 100 °C. In this study, the room temperature fabrication of Ti‐Ta thin films showing a reversible martensitic transformation and a high temperature shape memory effect above 200 °C is reported. In contrast to other shape memory thin films, no further heat treatment is necessary to obtain the functional properties. A disordered α″ martensite (orthorhombic) phase is formed in the as‐deposited co‐sputtered Ti₇₀Ta₃₀, Ti₆₈Ta₃₂ and Ti₆₇Ta₃₃ films, independent of the substrate. A Ti₇₀Ta₃₀ free‐standing film shows a reversible martensitic transformation, as confirmed by temperature–dependent XRD measurements during thermal cycling between 125 °C to 275 °C. Furthermore, a one‐way shape memory effect is qualitatively confirmed in this film. The observed properties of the Ti‐Ta thin films make them promising for applications on polymer substrates and especially in microsystem technologies.     

Numerical analysis of a three‐dimensional super‐elastic constitutive model

This note deals with the efficient approximation of a non‐linear constitutive relation arising in the study of the three‐dimensional mechanical behaviour of shape memory alloys at constant temperature. In particular, a variable time‐step discretization is investigated. For such an algorithm we prove sharp error estimates of optimal order and exactness for a class of experimentally relevant situations. We also report numerical results relative to proportional and non‐proportional loading tests which fully confirm the theoretical analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

A simulation study on the thermal buckling behavior of laminated composite shells with embedded shape memory alloy (SMA) wires

In this paper, numerical simulation analyses of the thermal buckling behavior of laminated composite shells with embedded shape memory alloy (SMA) wires were performed to investigate the effect of embedded SMA wires on the characteristics of thermal buckling. In order to simulate the thermomechanical behavior of SMA wires, the constitutive equation of the SMA wires was formulated in the form of an ABAQUS user subroutine. The computational program was verified by showing the response of the pseudoelasticity and shape memory effect (SME) at various temperatures and stress levels. Modeling of the laminated composite shells with embedded SMA wires and thermal buckling analyses were performed with the use of the ABAQUS code linked with the subroutine of the formulated SMA constitutive equations. The thermal buckling analyses of the composite shells with embedded SMA wires show that the critical buckling temperature can be increased and the thermal buckling deformation can be decreased by using the activation force of embedded SMA wire actuators.     

A finite element model of shape memory polymer composite beams for space applications

Structures made of shape memory polymer composite (SMPC), due to their ability to be formed into a desired compact loading shape and then transformed back to their original aperture by means of an applied stimulus, are an ideal solution to deployment problems of large and lightweight space structures. In the literature, there is a wide array of work on constitutive laws and qualitative analyses of SMP materials; dynamic equations and numerical solution methods for SMPC structures have rarely been addressed. In this work, a macroscopic model for the shape fixation and shape recovery processes of SMPC structures and a finite element formulation for relevant numerical solutions are developed. To demonstrate basic concepts, a cantilever SMPC beam is used in the presentation. In the development, a quasi‐static beam model that combines geometric nonlinearity in beam deflection with a temperature‐dependent constitutive law of SMP material is obtained, which is followed by derivation of the dynamic equations of the SMPC beam. Furthermore, several finite element models are devised for numerical solutions, which include both beam and shell elements. Finally, in numerical simulation, the quasi‐static SMPC beam model is used to show the physical behaviors of the SMPC beam in shape fixation and shape recovery. Copyright © 2015 John Wiley & Sons, Ltd.     

An implicit unitless error and step‐size control method in integrating unified viscoplastic/creep ODE‐type constitutive equations

A series of numerical analyses are carried out to investigate the difficulties in numerical integration of unified viscoplastic/creep constitutive equations, which are normally represented as a system of ordinary differential equations (ODEs). The problems of numerically integrating the constitutive equations are identified and analysed. To overcome the stiffness problems, implicit methods are used for the numerical integration and a generic technique is introduced to calculate the Jacobian matrix. A normalization technique is introduced in the paper to convert the integration errors for each time increment to unitless errors. Thus, a single tolerance can be used to control the accuracy and step size in integrating a set of unified viscoplastic/creep constitutive equations. In addition, an implicit step‐size control method is proposed and used in the integrations. This method reduces the possibility of rejection of an integration increment due to poor accuracy. Copyright © 2007 John Wiley & Sons, Ltd.     

Explicit expressions for 3D boundary integrals in potential theory

On employing isoparametric, piecewise linear shape functions over a flat triangular domain, exact expressions are derived for all surface potentials involved in the numerical solution of three‐dimensional singular and hyper‐singular boundary integral equations of potential theory. These formulae, which are valid for an arbitrary source point in space, are represented as analytic expressions over the edges of the integration triangle. They can be used to solve integral equations defined on polygonal boundaries via the collocation method or may be utilized as analytic expressions for the inner integrals in the Galerkin technique. In addition, the constant element approximation can be directly obtained with no extra effort. Sample problems solved by the collocation boundary element method for the Laplace equation are included to validate the proposed formulae. Published in 2008 by John Wiley & Sons, Ltd.     

Numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms

Previous studies by the authors and their co‐workers show that the structure of equations representing shape Memory Alloy (SMA) constitutive behaviour can be very similar to those of rate‐independent plasticity models. For example, the Boyd–Lagoudas polynomial hardening model has a stress‐elastic strain constitutive relation that includes the transformation strain as an internal state variable, a transformation function determining the onset of phase transformation, and an evolution equation for the transformation strain. Such a structure allows techniques used in rate‐independent elastoplastic behaviour to be directly applicable to SMAs. In this paper, a comprehensive study on the numerical implementation of SMA thermomechanical constitutive response using return mapping (elastic predictor‐transformation corrector) algorithms is presented. The closest point projection return mapping algorithm which is an implicit scheme is given special attention together with the convex cutting plane return mapping algorithm, an explicit scheme already presented in an earlier work. The closest point algorithm involves relatively large number of tensorial operations than the cutting plane algorithm besides the evaluation of the gradient of the transformation tensor in the flow rule and the inversion of the algorithmic tangent tensor. A unified thermomechanical constitutive model, which does not take into account reorientation of martensitic variants but unifies several of the existing SMA constitutive models, is used for implementation. Remarks on numerical accuracy of both algorithms are given, and it is concluded that both algorithms are applicable for this class of SMA constitutive models and preference can only be given based on the computational cost. Copyright © 2000 John Wiley & Sons, Ltd.