Mechanics and chemical thermodynamics of phase transition in temperature-sensitive hydrogels

This paper uses the thermodynamic data of aqueous solutions of uncrosslinked poly(N-isopropylacrylamide) (PNIPAM) to study the phase transition of PNIPAM hydrogels. At a low temperature, uncrosslinked PNIPAM can be dissolved in water and form a homogenous liquid solution. When the temperature is increased, the solution separates into two liquid phases with different concentrations of the polymer. Covalently crosslinked PNIPAM, however, does not dissolve in water, but can imbibe water and form a hydrogel. When the temperature is changed, the hydrogel undergoes a phase transition: the amount of water in the hydrogel in equilibrium changes with temperature discontinuously. While the aqueous solution is a liquid and cannot sustain any nonhydrostatic stress in equilibrium, the hydrogel is a solid and can sustain nonhydrostatic stress in equilibrium. The nonhydrostatic stress can markedly affect various aspects of the phase transition in the hydrogel. We adopt the Flory-Rehner model, and show that the interaction parameter as a function of temperature and concentration obtained from the PNIPAM-water solution can be used to analyze diverse phenomena associated with the phase transition of the PNIPAM hydrogel. We analyze free swelling, uniaxially and biaxially constrained swelling of a hydrogel, swelling of a core-shell structure, and coexistent phases in a rod. The analysis is related to available experimental observations. Also outlined is a general theory of coexistent phases undergoing inhomogeneous deformation.     

Transient analysis of temperature-sensitive neutral hydrogels

A model for transient deformation of neutral hydrogels that takes into account conservation of momentum, energy and mass for the solid polymer and fluid phase is derived, nondimensionalized and analyzed. Slow- and fast-response hydrogels are studied for three cases based on the response of (i) a spherical hydrogel, (ii) a constrained hydrogel slab to a step change in temperature, and (iii) the deformation in a temperature gradient. Model predictions for case (i) are shown to agree well with experiments for swelling and shrinking. For case (ii), solvent can be seen entering at the sides and flowing into the interior and towards the corners, such that the corners undergo a faster deformation than the sides. Immersed in a temperature gradient, case (iii), the hydrogel undergoes a bending motion and reaches a curved equilibrium shape, similar to the bending motion of polyelectrolyte hydrogels subjected to an external electric field. The benefit of the scale analysis conducted here, to predict correctly, prior to numerical computations, important characteristics such as stress, osmotic pressure and deformation times, is also highlighted.     

A relationship between effective work of adhesion and peel force for thin hyperelastic films undergoing large deformation

A method to determine the effective work of adhesion for hyperelastic thin films undergoing large deformations is presented. Starting from energy balance equation a relationship between work of adhesion, the peel force, the peel angle, and the stretch is derived. Based on this relation a procedure to compute the energy of adhesion from peel tests is proposed. To this end the peel force as well as the engineering stress vs. engineering strain diagram for thin film is required. The derived relationship shows that the non-linearity of the stress-stain relation must be taken into account in computing the effective work of adhesion from the peel force. The processing of experimental data within the standard linear elasticity approach would lead to an overestimation of effective work of adhesion. The error would increase with a decrease of the peel angle.     

Inhomogeneous large deformation study of temperature-sensitive hydrogel

In this paper, inhomogeneous deformation of a temperature-sensitive hydrogel has been studied and analyzed under arbitrary geometric and boundary conditions. We present the governing equations and equilibrium conditions of an isothermal process based on the monophase gel field theory of hydrogel via a variational approach. We have adopted and implemented an explicit form of energy for temperature-sensitive hydrogel in a three-dimensional finite element method (FEM) using a user-supply subroutine in ABAQUS. For verification purpose, a few numerical results obtained by the proposed approach are compared with existing experimental data and analytical solutions. They are all in good agreement. We also provide several examples to show the possible applications of the proposed method to explain various complex phenomena, including the bifurcation, buckling of membrane, buckling of thin film on compliant substrate and the opening and closure of flowers.     

A theory of coupled diffusion and large deformation in polymeric gels

A large quantity of small molecules may migrate into a network of long polymers, causing the network to swell, forming an aggregate known as a polymeric gel. This paper formulates a theory of the coupled mass transport and large deformation. The free energy of the gel results from two molecular processes: stretching the network and mixing the network with the small molecules. Both the small molecules and the long polymers are taken to be incompressible, a constraint that we enforce by using a Lagrange multiplier, which coincides with the osmosis pressure or the swelling stress. The gel can undergo large deformation of two modes. The first mode results from the fast process of local rearrangement of molecules, allowing the gel to change shape but not volume. The second mode results from the slow process of long-range migration of the small molecules, allowing the gel to change both shape and volume. We assume that the local rearrangement is instantaneous, and model the long-range migration by assuming that the small molecules diffuse inside the gel. The theory is illustrated with a layer of a gel constrained in its plane and subject to a weight in the normal direction. We also predict the scaling behavior of a gel under a conical indenter.     

Nonlinear formulation for flexible multibody system with large deformation

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of the geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation. The project supported by the National Natural Science Foundation of China (10472066, 10372057). The English text was polished by Yunming Chen.     

The effect of large deformation and material nonlinearity on gel indentation

A gel, an aggregate of polymers with solvents, has dual attributes of solid and liquid as solvent migrates in and out of the polymer network. Indentation has recently been used to characterize the mechanical properties of gels. This paper evaluates the effects of large deformation and material nonlinearity on gel indentation through theoretical modeling and finite element analysis. It is found that large deformation significantly affects the interpretation of the experimental observations and the classical relation between indentation force and depth has limitations for large deformation. The material nonlinearity does not play a very important role on indentation experiment so that the poroelasticity is a good approximation. Based on these observations, this paper proposes an alternative approach to measure the mechanical properties of gels, namely, uniaxial compression experiment.     

Finite deformations of Ogden's materials under impact loading

The present paper is devoted to the modeling of finite deformations of a hyperelastic body described by Ogden's model under contact/impact conditions. Frictional contact problems are solved by means of the bi-potential method. The first order algorithm is applied to integrate the equation of motion. The total Lagrangian formulation is adopted to describe the geometrically non-linear behavior. For the finite element implementation, the explicit expression of the tangent operator is derived including the case of repeated eigenvalues. A numerical example is given to illustrate efficiency and accuracy of the method.     

A chemo-electro-mechanical model for simulation of responsive deformation of glucose-sensitive hydrogels with the effect of enzyme catalysis

A multi-effect-coupling glucose-stimulus (MECglu) model is developed and solved numerically for the swelling behavior of soft smart hydrogels responding to changes in the environmental glucose concentration. The model considers the effect of the glucose oxidation reaction catalyzed by enzymes including glucose oxidase and catalase. It is composed of the Nernst-Planck equation for the mobile species in the solvent, the Poisson equation for the electric potential, and a nonlinear mechanical equation for the large deformations of the hydrogel that arise due to the conversion of chemical energy to mechanical. Based on the theory of the chemo-electro-mechanical-coupled fields, the formulation of the fixed charge groups bound onto the cross-linked polymer network is associated with the change of the ambient solution pH. The MECglu model is validated by comparison between the steady-state computation and experimental equilibrium swelling curves, and good agreement is obtained. A parameter study is then conducted by steady-state simulations to ascertain the impact of various solvent parameters on the responsive swelling behavior of the hydrogel. One key parameter is the glucose concentration, which is varied within the range of practical physiological glucose concentrations from 0 to 16.5 mM (300 mg/ml) to support the design and optimization of an insulin delivery system based on a glucose-sensitive hydrogel with immobilized glucose oxidase and catalase. The influence of oxygen and glucose concentrations in the solvent is then further studied for the distributive profiles of reacting and diffusive species concentrations, the electric potential, the displacement, as well as the swelling ratio of the glucose-sensitive hydrogel.     

Effects of large deformation and material nonlinearity on spherical indentation of hyperelastic soft materials

Up to now, the indentation of hyperelastic soft materials has not been completely understood. In this paper, the spherical indentation on hyperelastic soft solids was systematically investigated through theoretical analysis and finite element method (FEM). The validation and application of the Hertzian load-displacement relation for indentation of hyperelastic soft materials were clarified, the effects of large deformation and material nonlinearity on spherical indentation of hyperelastic soft materials were analyzed and discussed. It was found that the complicated indentation behaviors of hyperelastic soft solids mainly depended on the coupling interactions of large deformation and material nonlinearity. Besides, we proposed two new nonlinear elastic contact models to separate the effects of large deformation and material nonlinearity on spherical indentation responses of hyperelastic soft solids. Our efforts might help to enhance the understanding of hyperelastic indentation problems and provided necessary instructions for the mechanical characterization of hyperelastic soft materials.     

Application of MLPG in Large Deformation Analysis

Two-dimensional large deformation analysis of hyperelastic and elasto-plastic solids based on the Meshless Local Petrov–Galerkin method (MLPG) is presented. A material configuration based the nonlinear MLPG formulation is introduced for the large deformation analysis of both path-dependent and path-independent materials. The supports of the MLS approximation functions cover the same sets of nodes during material deformation, thus the shape function needs to be computed only in the initial stage. The multiplicative hyperelasto-plastic constitutive model is adopted to avoid objective time integration for stress update in large rotation. With this constitutive model, the computational formulations for path-dependent and path-independent materials become identical. Computational efficiency of the nonlinear MLPG method is discussed and optimized in several aspects to make the MLPG an O(N) algorithm. The numerical examples indicate that the MLPG method can solve large deformation problems accurately. Moreover, the MLPG computations enjoy better convergence rate than the FEM under very large particle distortion.The project supported by the National Natural Science Foundation of China (10472051). The English text was polished by Keren Wang.     

Geometrically nonlinear large deformation analysis of triangular CNT-reinforced composite plates

A first known investigation on the geometrically nonlinear large deformation behavior of triangular carbon nanotube (CNT) reinforced functionally graded composite plates under transversely distributed loads is investigated. The analysis is carried out using the element-free IMLS-Ritz method. In this study, the first-order shear deformation theory (FSDT) and von Kármán assumption are employed to account for transverse shear strains, rotary inertia and moderate rotations. A convergence study is conducted by varying the supporting size and number of nodes. The effects of transverse shear deformation, CNT distribution and CNT volume fraction on the nonlinear bending characteristics under different boundary conditions are examined.     

On the modelling of anisotropic elastic and inelastic material behaviour at large deformation

The purpose of this work is the formulation and discussion of an approach to the modelling of anisotropic elastic and inelastic material behaviour at large deformation. This is done in the framework of a thermodynamic, internal-variable-based formulation for such a behaviour. In particular, the formulation pursued here is based on a model for plastic or inelastic deformation as a transformation of local reference configuration for each material element. This represents a slight generalization of its modelling as an elastic material isomorphism pursued in earlier work, allowing one in particular to incorporate the effects of isotropic continuum damage directly into the formulation. As for the remaining deformation- and stress-like internal variables of the formulation, these are modelled in a fashion formally analogous to so-called structure tensors. On this basis, it is shown in particular that, while neither the Mandel nor back stress is generally so, the stress measure thermodynamically conjugate to the plastic “velocity gradient”, containing the difference of these two stress measures, is always symmetric with respect to the Euclidean metric, i.e., even in the case of classical or induced anisotropic elastic or inelastic material behaviour. Further, in the context of the assumption that the intermediate configuration is materially uniform, it is shown that the stress measure thermodynamically conjugate to the plastic velocity gradient is directly related to the Eshelby stress. Finally, the approach is applied to the formulation of metal plasticity with isotropic kinematic hardening.     

Experimental analysis of crack tip fields in rubber materials under large deformation

A three-nested-deformation model is proposed to describe crack-tip fields in rubber-like materials with large deformation.The model is inspired by the distribution of the measured in-plane and out-of-plane deformation.The inplane displacement of crack-tip fields under both Mode I and mixed-mode(Mode I-II) fracture conditions is measured by using the digital Moire’ method.The deformation characteristics and experimental sector division mode are investigated by comparing the measured displacement fields under different fracture modes.The out-of-plane displacement field near the crack tip is measured using the three-dimensional digital speckle correlation method.     

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

A phase field approach with a reaction pathways-based potential to model reconstructive martensitic transformations with a large number of variants

A pathway tree is constructed by recursively duplicating a single reconstructive martensitic transformation path with respect to lattice symmetries and point-group rotations. An energy potential built on this pathway is implemented in a phase-field technique in large strain framework, with the transformational strain as the order parameter. A specific splitting between non-dissipative elastic behavior and the dissipative evolution of the order parameter allows for the modeling of acoustic waves during rapid transformations. A simple toy-model transition from hexa- to square-lattice successfully demonstrates the possibility to model reconstructive martensitic transformations for a large number of variants (more than one hundred). Pure traction applied to our toy-model shows that variants can nucleate into previously created variants, with a hierarchical nucleation of variants spanning over five levels of transformation.     

Large deformation and electrochemistry of polyelectrolyte gels

Immersed in an ionic solution, a network of polyelectrolytes imbibes the solution and swells, resulting in a polyelectrolyte gel. The swelling is reversible, and the amount of swelling is regulated by ionic concentrations, mechanical forces, and electric potentials. This paper develops a field theory to couple large deformation and electrochemistry. A specific material model is described, including the effects of stretching the network, mixing the polymers with the solvent and ions, and polarizing the gel. We show that the notion of osmotic pressure in a gel has no experimental significance in general, but acquires a physical interpretation within the specific material model. The theory is used to analyze several phenomena: a gel swells freely in an ionic solution, a gel swells under a constraint of a substrate, electric double layer at the interface between the gel and the external solution, and swelling of a gel of a small size.     

Mechanics of stretchable electronics on balloon catheter under extreme deformation

Stretchable electronics has been applied to balloon catheters for high-efficacy ablation, with tactile sensing integrated on the surface, to establish full and conformal contact with the endocardial surface for elimination of the heart sink caused by blood flow around their surfaces. The balloon of the catheter folds into uniform ‘clover’ patterns driven by the pressure mismatch inside (∼vacuum) and outside of the balloon (pressure ∼1 atm). The balloon catheter, on which microelectrodes and interconnects are printed, undergoes extreme mechanical deformation during its inflation and deflation. An analytic solution is obtained for balloon catheter inflation and deflation, which gives analytically the distribution of curvatures and the maximum strain in the microelectrodes and interconnects. The analytic solution is validated by the finite element analysis. It also accounts for the effect of inflated radius, and is very useful to the optimal design of balloon catheter.     

Lorentz group SOo(5, 1) for perfect elastoplasticity with large deformation and a consistency numerical scheme

How to effectively deal with non-linearity and accurately fulfill the consistency condition is essential for modeling and computing in plasticity. Utilizing the concepts of two phases and homogeneous coordinates, we obtain a linear representation of a constitutive model of perfect elastoplasticity with large deformation, and, furthermore, a linear irreducible representation, which contains a five-order spin tensor. The underlying vector space is found to be the projective realization of a composite space resulting from a surgery on Minkowski spacetime ⁵⁺¹. The irreducible representation in the vector space admits of the projective proper orthochronous Lorentz group PSO_o(5, 1) in the on (or elastoplastic) phase and the special Euclidean group SE(5) in the off (or elastic) phase. The input path dictates symmetry switching between the two groups. Based on such symmetry a numerical scheme is devised which preserves the consistency condition for every time step. The consistency scheme is shown to be stabler, more accurate, and more efficient than the current numerical schemes developed directly based upon the model itself, because the new scheme preserves the internal symmetry SO_o(5, 1) of the model in the on phase so as to locate the stress point automatically on the yield surface at each time step without iterations at all.     

Unsymmetrical large deformation problem of orthotropic plates

Based upon the theory of ardsotropic plates, the unsymmetrical large deformation equations of orthotropic circular plates were derived. By using Fourier series, the partial differential equations of this problem can be transformed into sets of nonlinear differential equations . And the procedure to solve the problem using the iterative method is given .