Thermomechanical behavior of thermoset shape memory polymer programmed by cold-compression: Testing and constitutive modeling

Programming is a key process for thermally activated stress or strain recovery of shape memory polymers (SMPs). Typically, programming requires an initial heating above the glass transition temperature (T_g), subsequent cooling below T_g and removal of the applied load, in order to fix a temporary shape. This work adopted a new approach to program thermoset SMPs directly at temperatures well below T_g, which effectively simplified the shape fixing process. 1-D compression programming below T_g and free shape recovery of a thermoset SMP were experimentally investigated. Functional stability of the shape fixity under various environmental attacks was also experimentally evaluated. A mechanism-based thermoviscoelastic-thermoviscoplastic constitutive model incorporating structural and stress relaxation was then developed to predict the nonlinear shape memory behavior of the SMP trained below T_g. Comparison between the prediction and the experiment showed good agreement. The structure dependence of the thermomechanical behavior of the SMP was further discussed through a parametric study per the validated constitutive model. This study validates that programming by cold-compression is a viable alternative for thermally responsive thermoset SMPs.     

Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers

Shape memory polymers (SMPs) are polymers that can demonstrate programmable shape memory effects. Typically, an SMP is pre-deformed from an initial shape to a deformed shape by applying a mechanical load at the temperature T_H>T_g. It will maintain this deformed shape after subsequently lowering the temperature to T_L<T_g and removing the externally mechanical load. The shape memory effect is activated by increasing the temperature to T_D>T_g, where the initial shape is recovered. In this paper, the finite deformation thermo-mechanical behaviors of amorphous SMPs are experimentally investigated. Based on the experimental observations and an understanding of the underlying physical mechanism of the shape memory behavior, a three-dimensional (3D) constitutive model is developed to describe the finite deformation thermo-mechanical response of SMPs. The model in this paper has been implemented into an ABAQUS user material subroutine (UMAT) for finite element analysis, and numerical simulations of the thermo-mechanical experiments verify the efficiency of the model. This model will serve as a modeling tool for the design of more complicated SMP-based structures and devices.     

On thermodynamic potentials in linear thermoelasticity

The four thermodynamic potentials, the internal energy u=u(ε_ij,s), the Helmholtz free energy f=f(ε_ij,T), the Gibbs energy g=g(σ_ij,T) and the enthalpy h=h(σ_ij,s) are derived, independently of each other, by using the Duhamel–Neumann extension of Hooke's law and an assumed linear dependence of the specific heat on temperature. A systematic procedure is then presented to express all thermodynamic potentials in terms of four possible pairs of independent state variables. This procedure circumvents a tedious transition from one potential to another, based on the formal change of variables, and inversions of the stress–strain and entropy–temperature relations. The general results are applied to uniaxial loading paths under isothermal, adiabatic, constant stress, and constant strain conditions. An interplay of adiabatic and isothermal elastic constants in the expressions for exchanged heat along certain thermodynamic paths is indicated.     

In-plane and out-of-plane constraint effects under pressurized thermal shocks

Transferability of fracture toughness data obtained on small scale specimens to a full-scale cracked structure is one of the key issues in integrity assessment of engineering structures. In order to transfer fracture toughness under different constraints, both in-plane and out-of-plane constraint effect should be considered for the specimens and structures. In this paper both in-plane and out-of-plane constraint effects of a crack in a reference reactor pressure vessel (RPV) subjected to pressurized thermal shocks (PTSs) are analyzed by two-parameter and three-parameter methods. The comparison between elastic and elastic–plastic analysis shows that the constraint effect varies with the material property. T₁₁ (the second term of William’s extension acting parallel to the crack plane) generally displays a reversed relation to the stress intensity factor (SIF) with the transient time, which indicates that the loading (SIF) plays an important role on the in-plane constraint effect. The thickness at the crack tip contributes more than the loading to the out-of-plane constraint, such that T₃₃ (the second term of William’s extension acting along the thickness) displays a similar relation to ε₃₃ (strain along the thickness direction) and a different relation to T₁₁ during the transient. The results demonstrate that both in-plane and out-of-plane constraint effect should be analyzed separately in order to describe precisely the stress distribution ahead of the crack tip.     

On the Ohno–Wang kinematic hardening rules for multiaxial ratcheting modeling of medium carbon steel

This paper evaluates the performance of four Ohno–Wang type constitutive models in predicting ratcheting responses of medium carbon steel S45C for a set of axial/torsional loading paths. Suggestions are also made for further modification. The four models are the Ohno–Wang model, the McDowell model, the Jiang–Sehitoglu model and the AbdelKarim–Ohno model. It is shown that the Ohno–Wang model and the McDowell model overestimate the multiaxial ratcheting. Whereas, the Jiang–Sehitoglu model yields good predictions for most loading conditions used in this study with an appropriate modification of the dynamic recovery term. The AbdelKarim–Ohno model gives acceptable predictions for all considered multiaxial conditions when used with an evolution function for μ_i, but gives poor predictions of uniaxial ratcheting if the parameter μ_i is determined from a multiaxial ratcheting response. A new modified Ohno–Wang hardening rule is proposed for better adaptability under diverse situations by multiplying a factor to the dynamic recovery term, which is dependent on noncoaxiality of the plastic strain rate and back stress. This new model predicts ratcheting strain reasonably well for the test cases.     

Thermal tests of a steam-turbine nozzle-chamber model

This paper describes the measurement of thermal strains and temperatures in a 0.25-scale aluminum-filled epoxy model of a double-flow large-steam-turbine nozzle chamber. A temperature gradient was induced by circulating chilled methyl alcohol through the interior. Strain gages and thermocouples were used to determine surface strains and temperatures at various locations for comparison with a finite-element analysis under development. A uniform cylinder was included on the inlet section of the model for calibration. The maximum measured tensile strain on the interior surface was 0.96 αΔT _n , where ΔT _n was the average temperature difference between the interior surface and the initial temperature. A maximum compressive strain of 0.50 αΔT _n was measured on the outside surface of the nozzle-bowl sidewall.     

Brittle fracture: Variation of fracture toughness with constraint and crack curving under mode I conditions

The effect of constraint on brittle fracture of solids under predominantly elastic deformation and mode I loading conditions is studied. Using different cracked specimen geometry, the variation of constraint is achieved in this work. Fracture tests of polymethyl methacrylate were performed using single edge notch, compact tension and double cantilever beam specimens to cover a bread range of constraint. The test data demonstrate that the apparent fracture toughness of the material varies with the specimen geometry or the constraint level. Theory is developed using the critical stress (strain) as the fracture criterion to show that this variation can be interpreted using the critical stress intensity factorK _Cand a second parameterT orA ₃,whereT andA ₃are the amplitudes of the second and the third term in the Williams series solution, respectively. The implication of this constraint effect to the ASTM fracture toughness value, crack tip opening displacement fracture criterion and energy release rateG _Cis discussed. Using the same critical stress (strain) as the fracture criterion, the theory further predicts crack curving or instability under mode I loading conditions. Experimental data are presented and compared with the theory.     

Effects of the strain-hardening exponent on two-parameter characterizations of surface-cracks under large-scale yielding

The influence of strain hardening exponent on two-parameter J-Q near tip opening stress field characterization with modified boundary layer formulation and the corresponding validity limits are explored in detail. Finite element simulations of surface cracked plates under uniaxial tension are implemented for loads exceeding net-section yield. The results from this study provide numerical methodology for limit analysis and demonstrate the strong material dependencies of fracture parameterization under large scale yielding. Sufficient strain hardening is shown to be necessary to maintain J-Q predicted fields when plastic flow progresses through the remaining ligament. Lower strain hardening amplifies constraint loss due to stress redistribution in the plastic zone and increases the ratio of tip deformation to J. The onset of plastic collapse is marked by shape change and/or rapid relaxation of tip fields compared to those predicted by MBL solutions and thus defining the limits of J-Q dominance. A radially independent Q-parameter cannot be evaluated for the low strain hardening material at larger deformations within a range where both cleavage and ductile fracture mechanisms are present. The geometric deformation limit of near tip stress field characterization is shown to be directly proportional to the level of stress the material is capable of carrying within the plastic zone. Accounting for the strain hardening of a material provides a more adjusted and less conservative limit methodology compared to those generalized by the yield strength alone. Results from this study are of relevance to establishing testing standards for surface cracked tensile geometries.     

Thermo-damage-viscoelastic constitutive model of HTPB composite propellant

The main goal of this work is to establish a thermo-damage-viscoelastic model for Hydroxyl-Terminated Polybutadiene (HTPB) composite propellant based on the thermodynamic theory and elastic–viscoelastic correspondence principles. The model will also consider the influence of temperatures. The parameter α which represent the damage evolution rate and the material constants a, b are defined as exponential functions of temperature T, i.e. α(T), a(T) and b(T). Relaxation tests and uniaxial constant rate tensile tests are used to acquire the model parameters, and C(S) curves of different rates under the same temperature states are considered to be overlapped in this paper, while noncoincidence under the different temperature states. Then, uniaxial constant rate tensile tests and multi-step tensile-relaxation tests are used to verify the accuracy of the model. The results show that, the model is highly accurate in describing the mechanical property of HTPB under various loading conditions, but some drawback in describing the relaxation property inside of the HTPB’s nonlinear viscoelastic segments.     

Basis free relations for the conjugate stresses of the strains based on the right stretch tensor

In this paper, general relations between two different stress tensors T^f and T^g, respectively conjugate to strain measure tensors f(U) and g(U) are found. The strain class f(U) is based on the right stretch tensor U which includes the Seth–Hill strain tensors. The method is based on the definition of energy conjugacy and Hill’s principal axis method. The relations are derived for the cases of distinct as well as coalescent principal stretches. As a special case, conjugate stresses of the Seth–Hill strain measures are then more investigated in their general form. The relations are first obtained in the principal axes of the tensor U. Then they are used to obtain basis free tensorial equations between different conjugate stresses. These basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes, with a possible tensor expansion relation between the stresses with unknown coefficients, the unknown coefficients to be obtained. In this regard, some relations are also obtained for T⁽⁰⁾ which is the stress conjugate to the logarithmic strain tensor lnU.     

The plastic deformation of non-homogeneous polycrystals

To describe the work hardening process of polycrystals processed using various thermomechanical cycles with isochronal annealing from 500 to 900 °C, a dislocation based strain hardening model constructed in the basis of the so-called Kocks–Mecking model is proposed. The time and temperature dependence of flow stress is accounted via grain boundary migration, and the migration is related to annihilation of extrinsic grain boundary dislocations (EGBD’s) by climb via lattice diffusion of vacancies at the triple points. Recovery of yield stress is associated with changes in the total dislocation density term ρ^T. A sequence of deformation and annealing steps generally result in reduction of flow stress via the annihilation of the total dislocation density ρ^T defined as the sum of geometrically necessary dislocations ρ^G and statistically stored dislocations ρ^S. The predicted variation of yield stress with annealing temperature and cold working stages is in agreement with experimental observations. An attempt is made to determine the mathematical expressions which best describe the deformation behaviour of polycrystals in uniaxial deformation.     

Elastomers in large-amplitude oscillatory uniaxial extension

In this work, we subject elastomers to a fixed pre-stretch in uniaxial extension, ε _p , upon which a large-amplitude, ε ₀, oscillatory uniaxial extensional (LAOE) deformation is superposed. We find that if both ε _p and ε ₀ are large enough, the stress responds with a rich set of higher harmonics, both even and odd. We further find the Lissajous-Bowditch plots of our measured stress responses versus uniaxial strain to be without twofold symmetry and, specifically, to be shaped like convex bananas. Our new continuum model for this behavior combines a new nonlinear spring, in parallel with a Newtonian dashpot, and we call this the Voigt model with strain-hardening. We consider this three-parameter (Young’s modulus, viscosity, and strain-hardening coefficient) model to be the simplest relevant one for the observed convex bananas. We fit the parameters to both our LAOE measurements and then to our uniaxial elongation measurements at constant extension rate. We develop analytical expressions for the Fourier components of the stress response, parts both in-phase and out-of-phase with the extensional strain, for the zeroth, first, second, and third harmonics. We find that the part of the second harmonic that is out-of-phase with the strain must be negative for proper banana convexity.     

A finite element investigation of the effect of crack tip constraint on hole growth under mode i and mixed mode loading

In this work, the effect of constraint on hole growth near a notch tip in a ductile material under mode I and mixed mode loading (involving modes I and II) is investigated. To this end, a 2-D plane strain, modified boundary layer formulation is employed in which the mixed mode elastic K–T field is prescribed as remote boundary conditions. A finite element procedure that accounts for finite deformations and rotations is used along with an appropriate version of J₂ flow theory of plasticity with small elastic strains. Several analyses are carried out corresponding to different values of T-stress and remote elastic mode-mixity. The interaction between the notch and hole is studied by examining the distribution of hydrostatic stress and equivalent plastic strain in the ligament between the notch tip and the hole, as well as the growth of the hole. The implications of the above results on ductile fracture initiation due to micro-void coalescence are discussed.     

Coarse-grained simulation of molecular mechanisms of recovery in thermally activated shape-memory polymers

Thermally actuated shape-memory polymers (SMPs) are capable of being programmed into a temporary shape and then recovering their permanent reference shape upon exposure to heat, which facilitates a phase transition that allows dramatic increase in molecular mobility. Experimental, analytical, and computational studies have established empirical relations of the thermomechanical behavior of SMPs that have been instrumental in device design. However, the underlying mechanisms of the recovery behavior and dependence on polymer microstructure remain to be fully understood for copolymer systems. This presents an opportunity for bottom-up studies through molecular modeling; however, the limited time-scales of atomistic simulations prohibit the study of key performance metrics pertaining to recovery. In order to elucidate the effects of phase fraction, recovery temperature, and deformation temperature on shape recovery, here we investigate the shape-memory behavior in a copolymer model with coarse-grained potentials using a two-phase molecular model that reproduces physical crosslinking. Our simulation protocol allows observation of upwards of 90% strain recovery in some cases, at time-scales that are on the order of the timescale of the relevant relaxation mechanism (stress relaxation in the unentangled soft-phase). Partial disintegration of the glassy phase during mechanical deformation is found to contribute to irrecoverable strain. Temperature dependence of the recovery indicates nearly full elastic recovery above the trigger temperature, which is near the glass-transition temperature of the rubbery switching matrix. We find that the trigger temperature is also directly correlated with the deformation temperature, indicating that deformation temperature influences the recovery temperatures required to obtain a given amount of shape recovery, until the plateau regions overlap above the transition region. Increasing the fraction of glassy phase results in higher strain recovery at low to intermediate temperatures, a widening of the transition region, and an eventual crossover at high temperatures. Our results corroborate experimental findings on shape-memory behavior and provide new insight into factors governing deformation recovery that can be leveraged in biomaterials design. The established computational methodology can be extended in straightforward ways to investigate the effects of monomer chemistry, low-molecular-weight solvents, physical and chemical crosslinking, different phase-separation morphologies, and more complicated mechanical deformation toward predictive modeling capabilities for stimuli-responsive polymers.     

A simple model for dislocation behavior,strain and strain rate hardening evolution in deforming aluminum alloys

In this work, modeling of the stress–strain behavior is carried out using a simple dislocation model. This model uses three variables to characterize the dislocation population: The average forest and mobile dislocation densities, ρ_f and ρ_m, and the average dislocation mean free path L. However, it is shown that within reasonable assumptions, only two of these variables are independent. The mathematical form derived from this dislocation-based model was applied to experimental stress–strain data determined at room temperature for pure aluminum, 3003-O, 2008-T4, 6022-T4, 5182-O and 5032-T4 aluminum alloy sheets. The evolution of the state variables was calculated for these materials from a single stress–strain curve. The average dislocation mean free paths at a strain of 0.5 were compared with TEM observations of dislocation cell sizes or inter-dislocation spacing for specimens deformed equal biaxially with the hydraulic bulge test. A very good agreement was obtained between predictions and experiments.     

Visco-plastic constitutive modeling on Ohno-Wang kinematic hardening rule for uniaxial ratcheting behavior of Z2CND18.12N steel

Experimental results of monotonic uniaxial tensile tests at different strain rates and the reversed strain cycling test showed the characteristics of rate-dependence and cyclic hardening of Z2CND18.12N austenitic stainless steel at room temperature, respectively. Based on the Ohno-Wang kinematic hardening rule, a visco-plastic constitutive model incorporated with isotropic hardening was developed to describe the uniaxial ratcheting behavior of Z2CND18.12N steel under various stress-controlled loading conditions. Predicted results of the developed model agreed better with experimental results when the ratcheting strain level became higher, but the developed model overestimated the ratcheting deformation in other cases. A modified model was proposed to improve the prediction accuracy. In the modified model, the parameter m_i of the Ohno-Wang kinematic hardening rule was developed to evolve with the accumulated plastic strain. Simulation results of the modified model proved much better agreement with experiments.     

水蒸汽驱动高聚物形状记忆效应的力学模拟

部分形状记忆高聚物在相对湿度较高的环境中会从其临时形状恢复到永久形状,这种效应被称之为水蒸汽驱动形状记忆效应。由于不需要升高温度就可实现形状恢复,水蒸汽驱动的形状记忆效应在多个领域都有着潜在的应用价值。本文拟建立一个热-力-化学多场耦合的理论模型来模拟非晶态高聚物的水蒸汽驱动形状记忆行为。该理论模型采用自由体积的概念来模拟玻璃态转变,采用Fick定律来模拟水蒸汽在高聚物基体中的扩散行为。相关有限元模拟结果表明,该模型能定性地描述文献中观察到的恢复温度、相对湿度以及溶剂分子扩散速度对形状恢复行为的影响,也能模拟复杂变形条件下水蒸汽驱动的形状记忆效应。