Modeling and Analysis of Swelling and Instability of Polymer Gels and a Consistent Beam Theory
高聚物凝膠膨脹及不穩定性的建模和分析以及一致性梁理論
Student thesis: Doctoral Thesis
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Award date  4 May 2016 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(c93855fdf3374d869e54e93b5da94144).html 

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Abstract
This thesis is on the swelling and instability of polymer gels as well as the derivation of a consistent beam theory.
In the first part, we study a cylindrical gel undergoing three homogeneous deformations including free swelling, uniaxial and biaxial swellings and the inhomogeneous deformation of a spherical coreshell gel. In particular, in the homogeneous deformations, one key controlling parameter related with chemical potential is identified and regular perturbation method is applied to solve the governing nonlinear algebraic equations. Analytical formulas for stretches and stresses are obtained and the valid domain for them is also provided. The inhomogeneous deformation is governed by variablecoefficient ordinary differential equations which can not be solved exactly. However, it is found that to the leading order field equations, analytical (or semianalytical) solutions can be constructed when the above mentioned controlling parameter lies in a large (or small) region. These analytical solutions lead to useful informations hardly derived from numerical simulations. For example, they suggest that the best strategy for preventing debonding is to increase the prestretch of the inner core which significantly reduces the radial stress while has no influence on the whole volume expansion. In the second part, the swelling and instability of a cylindrical coreshell gel is studied. The initial inhomogeneous swelling is investigated in a similar way as that in the first part and the leading order solution is derived. Insights are then deduced based on the analytical solutions. The restrictive swelling of the gel generates compressive hoop stress and, at a critical state, instability takes place and wrinkle patterns appear. To understand the instability, an incremental deformation theory in nonlinear elasticity is used to conduct linear bifurcation analysis which gives the critical load for the onset of wrinkling. Behaviors of various physical quantities are discussed in detail at the critical state. It is found that the critical mode number, though insensitive to the material parameter, is greatly influenced by the ratio of outer and inner radii of the coreshell gel. Also, an interesting finding is that the critical swelling ratio is an increasing function of this geometrical ratio which implies a thinner annulus is more likely to be unstable than a thicker one.
In the third part, we study the swelling and instability of a spherical coreshell gel. Compared with the first part, a different energy density function is employed. It is based on the nonGaussian statistics and a new parameter is introduced to account for the limited extensibility of the polymer chain. Analytical solutions as well as their valid domains are given when this parameter is relatively large and numerical investigations are conducted when this parameter is small. It is also found that the distribution of solvent concentration stay almost invariant throughout the polymer matrix under certain parameter domains, which is a surprising result. When the inhomogeneous swelling reaches certain stage, instability takes place. A linear bifurcation analysis is then conducted to obtain the critical loading and relevant quantities at this critical state are examined in detail. The results share some similarities with that in the second part. However, for the incremental deformation, an extra assumption is taken. In the fourth part, we study the crease formation in a swelling gel layer. Unlike the smooth patterns of wrinkle, crease has a singular surface profile which adds complexity to analysis. We start by adopting a series expansion for the displacement vector about the top surface. Then the recursion relations are established from the field equations together with the top traction conditions. Based on these recursion relations, consistent beam equations are deduced which transform the original nonlinear eigenvalue problem of partial differential equations to nonlinear eigenvalue problem of ordinary differential equations. The beam equations are then solved analytically and together with a set of algebraic boundary conditions, all postbifurcation branches are constructed semianalytically. With the available analytical results, a number of insights on creaseformations are given, including three pathways leading to crease, determination of the bifurcation type, establishment of a lower bound for mode numbers and two scaling laws. Based on the scaling laws, it is shown that several physical quantities are independent of initial layer thickness at the moment of crease formation which interpret the experimental results. A discussion about the mechanism of crease formation is provided which also explains why the initiation of crease can not be deduced by a linear analysis.
In the fifth part, we study the planar deformation of a linearly elastic material. Staring from the linear planestress relations and utilizing the series expansion derivation just as in part four, beam equations with two unknown leading displacements are obtained. The improvement compared with part four is that the remainders of the expansion series are carried over to the final beam equations. Then, based on the beam equations, pointwise error estimates for displacement and stress fields are rigorously established. Boundary layer effects for clamped and welded edges are also discussed. Under further assumptions, the beam theory can also provide asymptoticallyvalid results outside the boundary layer. Based on above obtained results, three benchmark problems are considered, for which the twodimensional exact solutions are available. It is shown that this new beam theory recovers the exact solutions for these problems.
In the first part, we study a cylindrical gel undergoing three homogeneous deformations including free swelling, uniaxial and biaxial swellings and the inhomogeneous deformation of a spherical coreshell gel. In particular, in the homogeneous deformations, one key controlling parameter related with chemical potential is identified and regular perturbation method is applied to solve the governing nonlinear algebraic equations. Analytical formulas for stretches and stresses are obtained and the valid domain for them is also provided. The inhomogeneous deformation is governed by variablecoefficient ordinary differential equations which can not be solved exactly. However, it is found that to the leading order field equations, analytical (or semianalytical) solutions can be constructed when the above mentioned controlling parameter lies in a large (or small) region. These analytical solutions lead to useful informations hardly derived from numerical simulations. For example, they suggest that the best strategy for preventing debonding is to increase the prestretch of the inner core which significantly reduces the radial stress while has no influence on the whole volume expansion. In the second part, the swelling and instability of a cylindrical coreshell gel is studied. The initial inhomogeneous swelling is investigated in a similar way as that in the first part and the leading order solution is derived. Insights are then deduced based on the analytical solutions. The restrictive swelling of the gel generates compressive hoop stress and, at a critical state, instability takes place and wrinkle patterns appear. To understand the instability, an incremental deformation theory in nonlinear elasticity is used to conduct linear bifurcation analysis which gives the critical load for the onset of wrinkling. Behaviors of various physical quantities are discussed in detail at the critical state. It is found that the critical mode number, though insensitive to the material parameter, is greatly influenced by the ratio of outer and inner radii of the coreshell gel. Also, an interesting finding is that the critical swelling ratio is an increasing function of this geometrical ratio which implies a thinner annulus is more likely to be unstable than a thicker one.
In the third part, we study the swelling and instability of a spherical coreshell gel. Compared with the first part, a different energy density function is employed. It is based on the nonGaussian statistics and a new parameter is introduced to account for the limited extensibility of the polymer chain. Analytical solutions as well as their valid domains are given when this parameter is relatively large and numerical investigations are conducted when this parameter is small. It is also found that the distribution of solvent concentration stay almost invariant throughout the polymer matrix under certain parameter domains, which is a surprising result. When the inhomogeneous swelling reaches certain stage, instability takes place. A linear bifurcation analysis is then conducted to obtain the critical loading and relevant quantities at this critical state are examined in detail. The results share some similarities with that in the second part. However, for the incremental deformation, an extra assumption is taken. In the fourth part, we study the crease formation in a swelling gel layer. Unlike the smooth patterns of wrinkle, crease has a singular surface profile which adds complexity to analysis. We start by adopting a series expansion for the displacement vector about the top surface. Then the recursion relations are established from the field equations together with the top traction conditions. Based on these recursion relations, consistent beam equations are deduced which transform the original nonlinear eigenvalue problem of partial differential equations to nonlinear eigenvalue problem of ordinary differential equations. The beam equations are then solved analytically and together with a set of algebraic boundary conditions, all postbifurcation branches are constructed semianalytically. With the available analytical results, a number of insights on creaseformations are given, including three pathways leading to crease, determination of the bifurcation type, establishment of a lower bound for mode numbers and two scaling laws. Based on the scaling laws, it is shown that several physical quantities are independent of initial layer thickness at the moment of crease formation which interpret the experimental results. A discussion about the mechanism of crease formation is provided which also explains why the initiation of crease can not be deduced by a linear analysis.
In the fifth part, we study the planar deformation of a linearly elastic material. Staring from the linear planestress relations and utilizing the series expansion derivation just as in part four, beam equations with two unknown leading displacements are obtained. The improvement compared with part four is that the remainders of the expansion series are carried over to the final beam equations. Then, based on the beam equations, pointwise error estimates for displacement and stress fields are rigorously established. Boundary layer effects for clamped and welded edges are also discussed. Under further assumptions, the beam theory can also provide asymptoticallyvalid results outside the boundary layer. Based on above obtained results, three benchmark problems are considered, for which the twodimensional exact solutions are available. It is shown that this new beam theory recovers the exact solutions for these problems.