Constitutive modeling of the mechanics associated with crystallizable shape memory polymers

Shape memory polymers are novel materials that can be easily formed into complex shapes, retaining memory of their original shape even after undergoing large deformations. The temporary shape is stable and return to the original shape is triggered by a suitable mechanism such as heating. In this paper, we develop constitutive equations to model the mechanical behavior of crystallizable shape memory polymers. Crystallizable shape memory polymers are called crystallizable because the temporary shape is fixed by a crystalline phase, while return to the original shape is due to the melting of this crystalline phase. The modeling is done using a framework that was developed recently for studying crystallization in polymers ([28], [25], [27], [31]) and is based on the theory of multiple natural configurations. In this paper we formulate constitutive equations for the original amorphous phase and the semi-crystalline phase that is formed after the onset of crystallization. In addition we model the melting of the crystalline phase to capture the return of the polymer to its original shape. The model has been used to simulate a typical uni-axial cycle of deformation, the results of this simulation compare very well with experimental data. In addition to this we also simulate circular shear of a hollow cylinder and present results for different cases in this geometry. Received: January 5, 2005     

Constitutive modeling of the mechanics associated with crystallizable shape memory polymers

Shape memory polymers are novel materials that can be easily formed into complex shapes, retaining memory of their original shape even after undergoing large deformations. The temporary shape is stable and return to the original shape is triggered by a suitable mechanism such as heating. In this paper, we develop constitutive equations to model the mechanical behavior of crystallizable shape memory polymers. Crystallizable shape memory polymers are called crystallizable because the temporary shape is fixed by a crystalline phase, while return to the original shape is due to the melting of this crystalline phase. The modeling is done using a framework that was developed recently for studying crystallization in polymers ([28], [25], [27], [31]) and is based on the theory of multiple natural configurations. In this paper we formulate constitutive equations for the original amorphous phase and the semi-crystalline phase that is formed after the onset of crystallization. In addition we model the melting of the crystalline phase to capture the return of the polymer to its original shape. The model has been used to simulate a typical uni-axial cycle of deformation, the results of this simulation compare very well with experimental data. In addition to this we also simulate circular shear of a hollow cylinder and present results for different cases in this geometry.     

Stress and deformation states in the annulus made from shape memory material

The contribution deals with the stress and deformation states in an annulus made of shape memory material (SMM). The loading‐unloading process is going on at constant temperature, which is below temperature M_f (martensite finish temperature). The research is focused on the determination of the stress and deformation states in the annulus after loading and unloading process. These results are necessary for treating process of constrained recovery in shape memory annulus. The real loading‐unloading function in the stress‐strain coordinate system of shape memory material (Ni–Ti–Cu) is included in the mathematical model.     

形状记忆合金相变过程三维大变形有限元模拟

形状记忆合金（SMA）一直被作为智能材料开发,并被用于阻尼器、促动器和智能传感器元件．形状记忆合金（SMA）的一项重要特性,是它具有恢复在机械加卸载周期下产生的大变形而不表现出永久变形的能力．该文旨在介绍一种由应力产生的相变且可以描述马氏体和奥氏体之间的超弹性滞回环现象本构方程．形状记忆合金的马氏体系数假设为应力偏张量的函数,因此形状记忆合金在相变过程中锁定体积．本构模型是在大变形有限元的基础上执行的,采用了现时构型Lagrange大变形算法．为了方便地使用Cauchy应力和线性应变本构关系,使用了与旋转无关的Jaumann应力增率计算应力．数值分析结果表明,相变引起的超弹性滞回环可以有效地通过该文提出的本构方程和大变形有限元模拟．     

A variational model for the functional fatigue in polycrystalline shape memory alloys

Shape memory alloys show a very complex material behavior associated with a diffusionless solid/solid phase transformation between austenite and martensite. Due to the resulting (thermo-)mechanical properties – namely the effect of pseudoelasticity and pseudoplasticity – they are very promising materials for the current and future technical developments. However, the martensitic phase transformation comes along with a simultaneous plastic deformation and thus, the effect of functional fatigue. We present a variational material model that simulates this effect based on the principle of the minimum of the dissipation potential. We use a combined Voigt/Reuss bound and a coupled dissipation potential to predict the microstructural developments in the polycrystalline material. We present the governing evolution equations for the internal variables and yield functions. In addition, we show some numerical results to prove our model's ability to predict the shape memory alloys' complex inner processes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global solution to the full one-dimensional Frémond model for shape memory alloys

In this paper, we prove the existence and uniqueness of the solution to the one-dimensional initial-boundary value problem resulting from the Frémond thermomechanical model of structural phase transitions in shape memory materials. In this model, the free energy is assumed to depend on temperature, macroscopic deformation and phase fractions. The resulting equilibrium equations are the balance laws of (linear) momentum and energy, coupled with an evolution variational inequality for the phase fractions. Fourth-order regularizing terms in the quasi-stationary momentum balance equation are not necessary, and, as far as we know for the first time, all the non-linear terms of the energy balance equation are taken into account.     

A constitutive model for shape memory alloy fibres and its applicability to prestressed composites

This contribution is concerned with a constitutive model for shape memory fibres. The 1D-constitutive model accounts for the pseudoplastic and shape memory effect (SME). The macroscopic answer of the material is determined by the evolution from a twinned martensitic lattice into a deformed and detwinned one. On the macroscopic scale these effects are responsible for the upper boundary of the hysteresis which is situated around the origin of the stress-strain-diagram. During the phase transition process inelastic strains arise. When the lattice is fully detwinned, a linear elastic branch at the end of the hysteresis is observed. The initial state of the material is recovered by unloading and heating the material subsequently. The constitutive model is derived from the Helmholtz' free energy and fulfils the 2nd law of thermodynamics. For the present model five internal state variables are employed. Two of them are used to describe the inelastic strain and a backstress. The others represent the martensitic volume fraction and are necessary to describe the SME. The latter variables are depending on the deformation state as well as on temperature. A change on temperature goes along with a reduction of the inelastic strain. The model is incorporated in a fibre matrix discretization to prestress the surrounding structure. The boundary value problem is solved for a truss element applying the finite element method. Examples will demonstrate the applicability in engineering structures. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micromechanical model for single‐crystal shape‐memory‐alloys

This work is dealing with solid to solid phase transformations in shape‐memory‐alloys and the simulation of the corresponding characteristic phenomena, e.g. pseudoelasticity and the shape‐memory‐effect. In particular it focuses on the micromechanical behaviour of the material and the appearance of microstructures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of pseudo-plasticity in shape-memory-alloys

The name shape memory originates from the material's capability to recover its original shape after an apparent plastic deformation. The secret of this property lies in the specific microstructure. During mechanical loading, alloys of this particular kind change their crystallographic structure from randomly orientated martensite to ordered martensite. With austenite as high-temperature inter-state induced by heat supply, a recovery from the ordered to the unordered martensite is possible. This is accompanied by a macroscopic "healing" process. We apply our material model for shape-memory alloys to this special property and present numerical results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extending a finite strain hyperelastic micro-sphere framework towards phase transformations

A finite strain micro-sphere framework for hyperelastic solids elaborated by Carol et al. is extended towards the modelling of phase transformations in order to simulate polycrystalline solids under large deformations such as, e.g., shape memory alloys and shape memory polymers. The implemented phase transformation mechanism is based on statistical physics and is not restricted in terms of the number of solid material phases that can be considered, though we restrict the provided examples to two phases for the sake of conceptual clarity. The specifically chosen non-quadratic format of the Helmholtz free energy functions considered on the micro-plane level includes Bain-type transformation strains for each of the phases considered. Following the Voigt assumption on the micro-scale, identical total micro-stretches act in each of the material phases, where a multiplicative decomposition into elastic and transformation-related contributions is applied. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一个新的形状记忆合金模型

借助于Tanaka用一维形核动力学方程导出的指数形式的相变百分数,建立了一个新的形状记忆合金本构模型.提出了不同相变条件下的可恢复形状记忆应变的表达式;考虑了材料在变形过程中马氏体的重定向作用;克服了Tanaka系列模型不能描述当材料为完全马氏体状态时的力学行为的缺点.本模型较现有的形状记忆合金本构模型均简单,便于应用,实验证明了模型的正确性.     

Effect of moisture content on the microdeformation of compact bone tissue in tension

Conclusions The x-ray diffraction results indicate the following major features for the microdeformation of bone tissue. The total deformation in the elastic region is determined by the microdeformation of the mineral bone tissue component. The large yield of the mineral component indicates its relatively low elasticity modulus. The shape of the deformation curves for both dry and moist bone tissue is a factor of the combined deformation of the mineral and organic components. While the total deformation up to fracture in dry bone tissue is determined largely by microdeformation of the crystalline mineral phase, such behavior is found for moist bone tissue only in the first segment of the curve. Deformation in the second, more curved segment of the deformation curve is a factor largely of deformation of the organic bone-tissue component.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 530–535, May–June, 1983.     

Stability of the steady state for the Falk model system of shape memory alloys

In this article a stability result for the Falk model system is proven. The Falk model system describes the martensitic phase transitions in shape memory alloys. In our setting, the steady state is a nonlocal elliptic problem. We show the dynamical stability for the linearized stable critical point of the corresponding functional. Copyright © 2007 John Wiley & Sons, Ltd.     

Quantification of the evolution of shape storage and recovery in thermally responsive shape memory polymers

This study demonstrates the relevance of strain and stress storage/recovery profiles for the thermomechanical behaviour of thermally responsive Shape Memory Polymers (SMPs). It is shown how these experimentally determined profiles may be used for the development, calibration and validation of continuum-based models describing the shape memory effect in polymeric materials. The presented methodology applies both to small and finite strain deformation problems under the assumption of constant strain and cooling/heating rates. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control of thermomechanical phase transitions in shape memory alloys: Necessary conditions of optimality

The necessary conditions for the optimality of an optimal control problem associated with soiid–solid phase transitions in shape memory alloys are established.     

Composite beams with shape memory alloy fibres

We are concerned with the bending problem of fibrous composite beams in which fibres are made of shape memory alloys. These are alloys that may undergo a stress‐induced martensitic phase transformation. The matrix is treated as an elastic medium, and perfect bonding between matrix and fibres is supposed. In our model, the beam is decomposed into layers and the hysteretic behaviour of the shape memory fibres is taken into account. The boundary value problem is formulated in the form of an evolution variational inequality which, after finite element discretization, can be solved incrementally as a sequence of linear complementarity problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study on the thermo-mechanical behavior of superelastic NiTi wires

The strong coupling of thermal and mechanical properties and the highly inhomogeneous strain distribution in tensiontests motivate for thorough investigations on NiTi shape memory alloys. For these tests a complex experimental set-up has been developed which allows for the simultaneous measurement of stress, strain, and temperature with high spatial and temporal resolution. The experimental results show the influence of strain rate, number of cycles, and deformation level on the progress of stress induced phase transformation in the specimens. A critical evaluation of the experimental results in view of a potential constitutive modeling is given. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

First-Order Phase Transitions and Hysteresis

A simple phase-field model for first-order phase transitions with hysteresis is proposed. It describes both temperature- and stress-induced transitions between austenitic and (oriented) martensitic regimes in a shape memory alloy (SMA). Finally, numerical simulations of local paths of the system are performed in the (ε,σ) and (ε,θ) planes, respectively, when either stress or temperature cyclic processes are considered and phase diffusion is neglected.     

Numerical model for vibration damping resulting from the first-order phase transformations

A numerical model is constructed for modelling macroscale damping effects induced by the first-order martensite phase transformations in a shape memory alloy rod. The model is constructed on the basis of the modified Landau–Ginzburg theory that couples nonlinear mechanical and thermal fields. The free energy function for the model is constructed as a double well function at low temperature, such that the external energy can be absorbed during the phase transformation and converted into thermal form. The Chebyshev spectral methods are employed together with backward differentiation for the numerical analysis of the problem. Computational experiments performed for different vibration energies demonstrate the importance of taking into account damping effects induced by phase transformations.     

Harmonic distrotions under nonlinear periodic deformation of fluid polymers

Conclusions 1. It has been shown for a number of viscoelastic fluid systems that under nonlinear periodic deformation, the contribution of the third harmonic of the stress to the fundamental does not exceed 20% of the amplitude.2. In the case of clay soil and melt of filled polyethylene, the shape of the stress waves is essentially definable by the relative phase angle of the third harmonic of the stress and is practically independent of the deformation amplitude in a growing nonlinear range of deformation.3. In the case of the polyethylene melt, the amplitude dependence of the phase angles of the stress harmonics is in satisfactory agreement with the analysis of model I. With increasing deformation amplitude, the modulus vector of the first harmonic rotates counterclockwise and remains in the first trigonometric quadrant; the modulus vector of the third harmonic passes from the second to the third quadrant, and the modulus vector of the fifth harmonic passes from the second to the fourth quadrant via the third.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 893–898, September–October, 1978.