Thermo-mechano-chemical modeling and computation of thermosetting polymers used in post-installed fastening systems in concrete structures

Legal documents require a long service life, especially in structural engineering, no less than 50 years is usual. As a result, experimental procedures are costly and often not feasible. Nonetheless, a few testing techniques and models exist aiming at a performance prediction for that time horizon given the coupled phenomena of post-curing, creep, and fatigue at periodic thermal or mechanical loading. Further complications arise from the high internal moisture of, e.g., concrete or other building materials which are in contact with the thermosets and that give rise to degradation phenomena, such as hydrolytic aging [1‐6] playing a significant role in the design-life of the structure. Different phenomena in many length scales [7‐11] necessitate incorporation of multiscale approaches. Chemical reaction leading to hardening (hydration) needs to be considered by including dissipative effects [12] also caused by the internal friction during deformation [13, 14]. Mechanical response change related to the loading rate [15, 16] or modeling damage in concrete [17‐19] has to be modeled and solved together with the hydration in order to reach accurate prediction of the design-life. Differing bonding mechanisms between structural parts like concrete and steel is an active research area. In the case of a mechanical interlocking, this so-called apparent adhesion [20] based on friction contact provides only a limited bond with respect to a chemical or dispersive adhesion. Amending adhesion is possible by gluing concrete and steel [21‐23] that increases also the time to failure [24]. Such a chemical bond is achieved by a special type of polymer, specifically in this work, we concentrate on this adhesive material at the interface [25] between concrete and steel anchor.

Effected by their high stiffness properties, thermosetting polymers are proposed to be used at the interface. They are already available in the market; mostly an epoxy or vinyl ester is used. By mixing two fluid-like components, one is a multifunctional comonomer and the other one is the hardener, the chemical reaction called curing is ignited forming cross-links (chemical bonds) between polymers leading to an increased stiffness. The curing process never stops until it reaches the fully cured state. This process is of paramount importance for correctly characterizing thermosets. Until the so-called gel point, the process is fast and causes a significant shrinkage leading to residual stresses [26], after the gel point we consider the material as a solid—to be precise a fluid with an infinite viscosity—leading to increased stiffness with further cross-linking. It is possible to track the curing process experimentally [27] by introducing a degree of curing, experimental determination of mechanical response [28‐30] is an important parameter to calculate the long term response [31]. Mechanical characterization [32] and thermomechanical considerations [33] especially below the gel point [34, 35], or considering aging [36, 37] alongside to the curing have been studied. The effect of the degree of cure on the mechanical response is inevitable. Various models have been used by exploiting numerical multiscale approaches [38‐40] or also simple linear models. Multiphysics approaches with adequate numerical implementations are developed as well [41], not only for thermosets using small displacement assumption [42, 43], but also for large displacements [44‐46], has been applied in several applications [47‐49], where the curing is established in a controlled environment (i.e. heat treatment of composite elements in the autoclave) such that the material characteristics yield the chemically best configuration. In other words, the polymer attains a fully cured state and the curing process stops. However, this is rarely possible in case of thermosetting polymers that find application in the construction industry and especially reinforced concrete structures. Owing to the nature of constructing a structure in an uncontrolled environment under definitely not optimal conditions, the thermoset polymer will typically fail to reach a fully cured state after the installation such that the curing phenomenon continues during the life-time of the structure. Considering the composition altering due to the curing and its effect on the design-life of the system is of paramount importance. In order to model the whole system response accurately, in this work, we start with a thermodynamically consistent modeling of mechanochemistry in thermosetting polymers. We present a computational implementation with the finite element method by using the open-source packages developed under the FEniCS project [50, 51].

The underlying thermosetting polymer is a resin being hardened due to the chemical reaction, called curing, leading to cross-links between existing polymer chains. Cross-linking affects (often decreases) the volume per mass, an increase in the overall stiffness, and at the same time altering the temperature due to an exothermal reaction. After mixing an agent to the resin, curing starts by using energy from the environment—mostly heat or radiation energy (UV light) is supplied to level off the temperature. Once the curing starts, from the viscous resin state until the so-called gel point, the curing process is fast. At the gel point the viscosity increases drastically so that we may call the material a solid and the curing process slows down; its rate asymptotically reaches zero at the fully cured state. Degree of cure is a difficult variable to measure. We start with a definition of this variable following [52] by using theory of mixtures in a simplified manner. In a unit volume, V, masses of resin, curing agent, and cured solid are introduced \(m_\text {R}\), \(m_\text {A}\), and \(m_\text {S}\), respectively. Their values vary in time; but the sum is constant, \(m = m_\text {R}+m_\text {A}+m_\text {S}\), throughout the curing process. We introduce mass fractions:leading toNow we introduce the degree of cure, \(\omega =\omega (t)\), as “measured” in time, giving the mass fraction of the solidunder the assumption that the initial mass of solid is zero, \(Y_\text {S}(t=0)=0\), such that we haveConsider that the ratio of masses, \(Y_\text {R}(t=0)/Y_\text {A}(t=0)\), remains the same during the reaction. In other words, same molar mass is used for resin and agent. Now by using \(z = Y_\text {R}(t=0)\), we obtainHence, the whole formulation for calculating the masses of constituents has been subsumed to the degree of cure (conversion degree), \(\omega \). For obtaining its value, an evolution equation is developed dependent on the chemical reaction type. Epoxy resin reaction is based on autocatalysis mechanism [53], effected by hydroxy groups formed during catalysis. Hence, the following evolution equation [54] is found to be usefulwhere amplitudes, \(A_\times \), activation energies, \(E_\times \), and powers, m, n, are constant parameters to be determined. Universal gas constant, R, is known. Temperature, T, is the key variable altering kinetic rates, \(k_\times \), and thus curing rate, , significantly. We refer to [55, Table 1] for parameter values of different materials. Obviously, is used for determining the current degree of cure in each material point. This model fails to incorporate vitrification such that different mechanisms below and above glass transition temperature, \(T_g\), are not modeled accurately. Especially curing at low temperatures (below 100 \(^\circ \)C) leads to incomplete conversion that is of paramount interest in fastening systems, where the hardening occurs in environmental conditions. A possible approach as in [56] characterizes glassy and rubbery states occurring simultaneously. The idea was introduced in [57] based on phenomenological observation combined with the free volume reduction during cure leading to decrease in mobility. Mobility plays a dominant role in the rubbery state as molecules collide and form a network via cross-linking. Beyond a cross-linking density, material vitrifies and in this glassy state the chemical rate is more dominant. By following [58], we use only primary and secondary amines via autocatalysis and impurity catalysis (denoted by c in index), collective kinetic rates, \(K_1\), \(K_{1c}\), involve diffusion controlled mechanism, \(K_\text {diff}\), as well as chemically steered mechanism, \(K_{1,\text {chem}}\), \(K_{1c,\text {chem}}\), as suggested in [59] as follows:The diffusion rate is modeled by the Williams‐Landel‐Ferry (WLF) equation based on [60, 61] in the rubbery regime and greater than chemical rate in many orders in magnitude. In the glassy state, diffusion rate is nearly zero regarding chemical rate. This interplay delivers a realistic prediction of conversion degree, \(\omega \), at low temperatures, where a so-called partial freezing inhibits to attain \(\omega =1\). This model incorporates glass transition temperature, which is often determined by a fit function [62] based on calorimetric measurements. For consistency, we follow [58] and use the relation from [63]with \(\Delta C= \Delta c_\infty /\Delta c_0\) as the ratio of the heat capacity change at the glass transition temperature of the fully cured network with \(\omega =1\), \(T_{g,\infty }\) per the heat capacity change at the glass transition temperature of the monomer with \(\omega =0\), \(T_{g,0}\). The necessary coefficients for the evolution equations (6), (7) are determined by inverse analysis to data obtained by differential scanning calorimetry, we compile them from the literature as presented in Table 1. By knowing this mass fraction and the current mass density of the mixture, we can extract masses of each constituent by using Eq. (5) with z being the mass fraction of resin initially.

Under the simplification that deformation is small such that no distinction is necessary between configurations, we use as the partial time derivative and \(_{,i}\) for the space derivative with respect to \(X_i\), where space coordinates, \(\varvec{X}\), denote material particles composing a material system. Hence, the mass balance is satisfied and the mass density, \(\rho \), is a given function in space—for homogeneous materials we set \(\rho =\text {const}\) in space, the presented formalism holds true for heterogeneous materials as well. We need to solve balances of momentum and internal energy,respectively, where \(\varvec{u}\) in m is the displacement to be calculated; stress, \(\varvec{\sigma }\) in Pa\(\hat{=}\)N/m\(^2\), heat flux, \(\varvec{q}\) in W/m\(^2\), and specific internal energy (energy per mass), u in J/kg, need to be defined; \(\varvec{g}\) in N/kg is the gravitational specific force, and r in W/kg is the given specific supply term. For the sake of simplicity, we use a linear strain measure:However, we emphasize that the formulation is also suitable for nonlinear measures. Internal energy needs to be defined by exploiting thermodynamics. There are several methodologies; we refer to [64] for such examples based on thermodynamics of irreversible processes and non-equilibrium thermodynamics. Briefly, we introduce the so-called specific Helmholtz free energy:with the specific entropy, \(\eta \), to be defined indicating the reversible part of heat flux and temperature, T, to be calculated. Free energy is modeled by assuming that it depends on temperature, T, strain, \(\varvec{\varepsilon }\), and degree of cure, \(\omega \),We stress that we choose to exclude a dependency on the rate of these variables. Now by inserting the free energy into the balance of internal energy, dividing by T, and after rewriting, we obtainBased on the nonequilibrium thermodynamics, we decomposeinto equilibrium and nonequilibrium parts and then use the well-known relations for equilibrium partsFor simplicity we assume that rate of temperature and rate of strain fail to alter mechanical response, leading to \(\eta ^\text {neq}=0\), \(\varvec{\sigma }^\text {neq}=0\). Equation (15)\(_1\) is often introduced as the definition of the entropy and Eq. (15)\(_2\) is Cattaneo’s theorem in its general form for small displacements, we refer to [65] for its derivation from a Lagrange function. We stress that these relations show that (equilibrium) entropy and (equilibrium) stress depend on the same arguments as the free energy, \(\eta =\eta \big ( T, \varvec{\varepsilon }, \omega \big )\) and \(\varvec{\sigma }=\varvec{\sigma }\big ( T, \varvec{\varepsilon }, \omega \big )\), which is often called the principle of equipresence [66, §293.\(\eta \)]. By using these relations, we arrive at the balance of entropy:with the right-hand side called the entropy production that is zero for reversible, and it is positive for irreversible processes. By using this assertion—the second law of thermodynamics—we obtain Fourier’s law and the restrictionassuring that the entropy production is (zero or) positive as long as \(\kappa \ge 0\). The former is well known in thermodynamics. The latter is equivalent to the postulated Karush‐Kuhn‐Tucker relation originally introduced in plasticity, for a general discussion we refer to [67]. We emphasize that the chosen evolution equation for \(\omega \) in Eq. (6) or (7) is positive such that has to be negative. Heat is inserted into the system in a standard calorimetric measurement, \(\updelta Q = c \, \mathrm{d} T\), with specific capacity, c. Now by assuming the relation \(T \, \mathrm{d} \eta = \updelta Q\), as proposed in [68] and used in all thermodynamical formulations, we obtainAll these relations will be fulfilled by defining the free energy adequately. After inserting Eq. (17)\(_1\) into Eq. (13)\(_2\) and using \(\eta =\eta \big ( T, \varvec{\varepsilon }, q \big )\) as well as Eq. (15), we acquireThis governing equation for the temperature is a general formulation under the assumption that the free energy depends “only” on temperature, strain, and degree of cure. By defining the free energy adequately, the governing equation will be closed and then calculated numerically.

We aim at calculating the deformation by satisfyingand the temperature by fulfillingfor a hardening thermosetting polymer, whose state is characterized by the degree of cure, \(\omega \), modeled by the evolution equation (6) or (7). In order to close these governing equations, we need to model the system by defining a specific free energy, . The whole formulation is reduced to define a correct model for the free energy. In the following, we give a possible model and present its relation to the existing models in the literature.

We emphasize that the free energy may depend on temperature, strain, and degree of cure. The real dependence is possibly determined by experiments; there are no thermodynamical restriction about its form. But we know, based on the numerical stability and also partly on our understanding of energy as well as intuition, the energy needs to be positive definite. In full accordance with Eqs. (18), (17)\(_2\), we propose to use the following free energy expression,where all material parameters, c, \(\varvec{C}\), \(\varvec{\alpha }\), \(\varvec{\beta }\), \(H_\text {{ref}}\), may depend on the same arguments as energy, namely temperature, strain, and degree of cure. This model is the simplest case for a thermoelastic material; the form is quadratic providing stability. The reference temperature, \(T_\text {{ref}}\), is often set as the room temperature under two assumptions: the material possesses no thermal stresses and the coefficient of thermal expansion (CTE), \(\varvec{\alpha }\), is measured regarding this temperature. Curing-related shrinkage is given by the material parameter, \(\varvec{\beta }\). Heat release because of the exothermic reaction is given by \(H_\text {{ref}}\), the total heat (per mass) generated by a fully cured specimen. If the material response is such that c, \(\varvec{C}\), \(\varvec{\alpha }\), \(\varvec{\beta }\) are constant in \(\varvec{\varepsilon }\) and T, in other words we neglect hyperelasticity as well as temperature-related changes in material coefficients, then we acquire the (generalized) Hooke’s law with a linear shrinkage and well-known Duhamel–Neumann extensionAnalogously, the specific entropy and its derivative with respect to the strain readIn the general case, we expect to have material parameters depending on the cross-linking. A rather obvious observation is made for stiffness, the material hardens as the cross-linking density increases. Accurate modeling of the relationship between material parameters and degree of cure needs to be achieved by experiments [69] by means of detecting in the aforementioned general formulation. A possible way of introducing the degree of cure is simply to introduce two phases, solid (hardened) and fluid (non-hardened), creating the stiffness in the following simple rule of mixture: \(\varvec{C} = (1-\omega ) \varvec{C}^\text {f}+ \omega \varvec{C}^\text {s}\), \(\varvec{\alpha }=(1-\omega )\varvec{\alpha }^\text {f}+ \omega \varvec{\alpha }^\text {s}\), where \(\varvec{C}^\text {s}\), \(\varvec{\alpha }^\text {s}\) and \(\varvec{C}^\text {f}\), \(\varvec{\alpha }^\text {f}\) mean the maximum stiffness, maximum CTE reached after full cross-linking; and, initial stiffness, initial CTE of the non-hardened fluid alike material, respectively. Even the specific heat capacity, c, may depend on the degree of cure, \(\omega \), we direct to [70] for such a study. The simple relation between material parameters and degree of cure—motivated by the mixture theory—has a formal benefit of constructing thermodynamically sound material properties [71, Sect. 4]. Since the parameter for shrinkage, \(\varvec{\beta }\), is inherently related to the curing process, the explicit (linear) dependence is established in the free energy definition instead of introducing phase depending coefficients. In this manner, we havesuch that we reach from Eq. (21)leading toUntil this stage, we have used the following assumptions:Especially for modeling fastening systems in concrete, all assumptions are easily justified. Furthermore, for the sake of a better comprehension, we introduce more simplifications as follows: Various models in the literature are covered in the methodology presented herein. We continue with the general governing equations to demonstrate their predictive capability by computing academic examples.

We consider a general case and set the objective to compute the primitive variables, \(\omega \), \(\varvec{u}\), T, in space, \(\varvec{x}\), and time, t. By starting with the initial conditions of a partly cured body given by \(\omega _\text {{ref}}\) as follows:within the continuum body \({\mathcal {B}}\), we search for the transient solution of \(\{\varvec{u}, T\}\), by solving Eqs. (20), (21) and determine \(\omega \) by updatingwith the evolution equation for . We use the quadratic free energy as in Eq. (22), leading to linear material equations; yet emphasize that the implementation is designed in such a way that more advanced models are possible to be employed as well. Therefore, we leave the free energy as in the following. All continuous functions are approximated by using their discrete representations in space and time. In order to discretize in time, we use constant time steps and compute in a time serieswith \((\cdot )^0\), \((\cdot )^{00}\) denoting the calculated values of one, two time steps before, respectively. Time derivative is approximated by the finite difference method (Euler backwards scheme)The governing equations becomeWe remark the linearized strain measure in order to find out the value one time step before,For space discretization, we use the finite element method with a triangulation of the continuum space, \({\mathcal {B}}\), and exchanging continuous primitive functions with their counterparts in a Hilbertian Sobolev space [72] by using nth polynomial degree form functions for each discrete element,For the sake of notational easiness, we simply skip to distinguish between continuous functions and their discrete representations. We use the standard procedure to construct the integral forms by building residuals from governing equations and multiplying (weighting) them by corresponding test functions, \(\updelta \varvec{u}\), \(\updelta T\), where they are expressed in the same space as the primitive variables, called the Galerkin procedure,We emphasize that stress already possesses space derivative of primitive variables, because of the additional divergence on it, we need to choose at least \(n=2\) in the space discretization. This continuity condition is weakened by integrating by parts for all second derivative terms,where we have inserted \(\hat{t}_i = n_j \sigma _{ji} = n_j \partial \mathscr {f}/\partial \varepsilon _{ij}\) and \(\hat{q} = n_i q_i = -n_i \kappa T_{,i}\) on boundaries. For the displacement with a given traction, \(\hat{\varvec{t}}\) in Pa, two sets of boundaries appear: Dirichlet boundaries, \(\partial {\mathcal {B}}_D\), and Neumann boundaries, \(\partial {\mathcal {B}}_N\). We restrict the formulation such that these two sets fail to intersect, \(\partial {\mathcal {B}}_D\cap \partial {\mathcal {B}}_N=\{\}\), so the whole boundary is the union of them, \(\partial {\mathcal {B}}=\partial {\mathcal {B}}_D\cup \partial {\mathcal {B}}_N\). At the Dirichlet boundary, the displacement itself is given, hence, no test functions are necessary, \(\updelta \varvec{u}=0 \ \forall \varvec{x}\in \partial {\mathcal {B}}_D\). For the rest, traction is given, \(\hat{\varvec{t}} =\text {given} \ \forall \varvec{x} \in \partial {\mathcal {B}}_N\). Analogously, the temperature may be modeled by given temperature or heat flux; however, we use a more natural approach by using a mixed boundary condition called Robin boundaries, herein applied to the whole boundary,with the convection parameter, h, and the ambient (environment) temperature we use as \(T_\text {a}=T_\text {{ref}}\). Two weak forms are in different units, the former is in the unit of energy and the latter is in the unit of power. After multiplying by the constant time step \(\Delta t\),we can sum them upand use for computing thermomechanics of a curing thermoset.

In order to examine the strength and accuracy in predicting the behavior of realistic systems, we implement the weak form in Eq. (44) by using open source packages developed within the FEniCS project. The formulation is implemented in Python, wrapped into C++, compiled, and solved by mpirun in parallel in a Linux computer (Ubuntu 18.04). This scalable implementation is suitable for larger systems as well. Herein we demonstrate simple examples to comprehend the multiphysics in thermo-mechano-chemical systems. Especially two important features need to be emphasized leading to the implementation presented herein.

First, the weak form is nonlinear such that we use the conventional Newton–Raphson linearization approach for solving a linear problem and update the solution by starting from the last converged solution. Even if the system is governed by stiff differential equations, using small time steps leads to a convergence in each iteration. The linearization is handled by exploiting symbolic differentiation that allows to implement also highly nonlinear forms—for example a hyperelastic material model can be easily implemented with the approach developed herein, we refer to [64] as well as to the supply code of [65] for examples of such an implementation.

Second, the specific free energy in Eq. (22) has been implemented also with the aid of symbolic differentiation. Technically, the whole material formulation is reduced to the free energy definition such that the implementation is very general and can be easily adapted for other systems simply by changing the free energy. We stress that the material modeling is limited by the assumed free energy formulation; however, implementation is novel in the sense that any more advanced formulations are possible to utilize. We make the code publicly available in [73] for encouraging its use under GNU Public licensing [74] for an efficient and transparent scientific exchange.

A general framework has been developed for thermomechanics of thermosetting polymers under curing governed by an evolution equation. The numerical implementation of such a thermo-mechano-chemical process has been established by using open-source packages. Simple geometries are utilized for promoting a better understanding of underlying multiphysics as well as capability of the computational implementation. The following assumptions have been undertaken:Thermodynamically sound modeling of thermo-mechano-chemical processes allows us to accurately model the coupling effects between variables like temperature, deformation, and degree of cure. An important contribution is that we have subsumed all modeling by using the free energy such that any possible restrictions of the simulation is recovered by revisiting the free energy definition. The computational implementation has been utilized by using an automatized generation of relations with the aid of symbolic differentiation of the assumed free energy. Therefore, any more advanced formulation is easily implemented into the framework developed herein. For demonstrating the strength of the numerical implementation as well as comprehending the material model’s capabilities, three different simulations have been utilized: We stress that the code is publicly available in [73] for encouraging its use under GNU Public licensing [74].