Laminates with thermosetting matrices (resins) reinforced with stitched glass/carbon fabrics have been widely used in aerospace, space, automotive, civil, mechanical, marine, and naval constructions for decades. In the design standards for such constructions, relatively low stress levels leading to linearly elastic/viscoelastic reversible processes are allowed. Effects of viscoelastic properties of laminates, temperature changes and material aging are taken into account by relatively high safety factors specified in relevant standards. Development of rheological modelling of thermosetting polymers and fibre-reinforced thermoset matrix composites will allow new algorithms for the design of laminate shell constructions to be formulated and implemented to the design codes.
This book develops, unique worldwide, an advanced analytical rheological modelling of thermosetting polymers (thermosets) and unidirectional monotropic (transversely isotropic) fibre-reinforced thermoset matrix (UFRT) composites. In engi-neering practice, a UFRT composite is a layer of laminates reinforced with stitched fabrics. New (unaged) polymers and composites under normal conditions (temperature 20ºC, humidity 50%), fully relaxed from curing and post-curing stresses are considered. The theory includes quasi-static short-term / medium-term / long-term reversible rheological processes. According to existing knowledge, thermosets are modelled as isotropic materials exhibiting linearly viscoelastic shear strains and linearly elastic bulk strains, and the fibres are linearly elastic isotropic / monotropic materials.
Experimental studies around the world over the past decades show that a Mittag-Leffler fractional exponential function is an adequate generic function reproducing the viscoelastic characteristics of thermosets. An integral form of this function, applied by the authors, enabled formulation of a new methodology for rheological modelling of both thermosets and UFRT composites. Coupled and uncoupled, standard and inverse constitutive equations of linear elasticity and rheology, mutually analytically transformable, are formulated for thermosets and UFRT composites. New rheological models (coded as H-R/H) for thermosets and UFRT composites are described by the smallest possible numbers of the viscoelastic constants. An isotropic thermoset is described by two independent elastic constants and three independent viscoelastic constants. A homogenized monotropic UFRT composite is described by five inde-pendent elastic constants and four viscoelastic constants, whereby two viscoelastic constants are common to the thermoset matrix and the composite.
The monograph develops selected problems in mechanics of thermosets and UFRT composites, in the authors’ approach, related to the authors’ methodology of elastic and rheological modelling. Chapter 1 presents a critical review of knowledge in rheology of thermosetting polymers and UFRT composites, including publications by other researchers and those by the authors. Against this background, the scope of the monograph is presented. Chapter 2 describes the thermoset and composite mate-rials applied for the experimental and numerical studies, i.e. identification, validation, and cognitive studies.
Chapter 3 develops elastic and rheological modelling of thermosets, preceded by a summary of the material and mechanical assumptions. The considerations were conducted using matrix calculus, taking into account the full stress and strain states. Coupled standard constitutive equations of thermoset elasticity were transformed to an uncoupled form, which were then generalized to the form describing viscoelastic shear strains and elastic bulk strains. Constitutive equations of elasticity and viscoelasticity were formulated in eight possible forms, i.e. coupled and uncoupled, standard and inverse, elastic and rheological equations. The use of the generic function in the form of a Mittag-Leffler fractional exponential function was positively validated experimentally.
For elastic and rheological modelling of UFRT composites with heterogeneous microstructure, a quasi-exact homogenization theory, based on the elasticity theory of isotropic and monotropic media, is needed. Such theory, based on analytical solutions of the selected tasks of the theory of linear elasticity, is formulated in Chapter 4 for monotropic fibres and positively validated experimentally on the enhanced reliabil-ity UFRT composites.
Chapter 5 presents an additional elastic problem, namely the inverse problem in UFRT composite homogenization theory. The analytical solutions describing the elastic constants of monotropic fibres have been shown to be well-conditioned.
Chapter 6 is the most important chapter in this monograph. It contains formula-tion and results of analytical modelling of UFRT composites, in the elastic and rheological terms, using the matrix calculus. Section 1 summarizes assumptions made in the rheological modelling. Standard constitutive equations of elasticity of the homogenized UFRT composite were decoupled using a new concept of quasi-shear and quasi-bulk strains (unique worldwide). These equations allowed standard constitutive equations of viscoelasticity of the composite to be formulated by generalizing the viscoelasticity constitutive equations of the polymer matrix (unique worldwide). The homogenized composite was described by a single generic function consistent with the generic function of the polymer matrix, and by five viscoelasticity constants. Constitutive equations of elasticity and viscoelasticity of the homogenized UFRT composite were formulated in eight possible forms, i.e. coupled and uncoupled, standard and inverse, elastic and rheological equations. These equations are mutually analytically transformable.
Four complementary problems, i.e. numerical integration of selected improper integrals, an experimental unidirectional tension creep test on a thermoset, identifica-tion of the viscoelastic constants of a thermoset, and determination of the effective viscoelastic constants of the homogenized UFRT composite, are formulated and solved analytically in Chapter 7. A Mittag-Leffler fractional exponential function results in some improper integrals, which can be calculated using a high rank Gauss-Legendre quadrature; an original algorithm for the numerical integration is presented in the Section 1 of this chapter. The viscoelastic constants of a thermoset are calculated analytically in an iterative loop, using an experimental long-term unidirectional tension creep test. The viscoelastic constants of the homogenized UFRT composite are calculated analytically using the H-R/H shear/quasi-shear storage compliances and EVCP shear/quasi-shear storage compliances, where EVCP denotes the elastic-viscoelastic correspondence principle (a solution unique worldwide).
The computational algorithms were coded in the Pascal language and described in Chapter 8. Computational paths include: test tasks, experimental identification of viscoelasticity constants of a thermoset, numerical identification of viscoelasticity constants of a homogenized UFRT composite, validation issues, simulation is-sues, UFRP composite homogenization, and inverse problem in the homogenization theory of a UFRT composite. The following test tasks are formulated and presented: testing the Gauss-Legendre quadratures, analysis of a creep function corresponding to a fractional exponential generic function, and analysis of the complex shear compliance of a thermoset.
The experimental and numerical studies are presented in Chapter 9. The follow-ing issues were formulated and analyzed: identification of the elastic and viscoelastic constants of the representative thermosets, variation of the effective elastic constants of the representative UFRT composites vs. a fibre volume fraction, and calculation of the effective viscoelastic constants of the representative UFRT composites. It was shown that one of the viscoelasticity constants of UFRT composites is negligibly small, hence UFRT composites are ultimately described by four viscoelasticity constants. The H-R/H rheological models of thermosets and UFRT composites were validated positively.
Extensive rheological simulations are presented in Chapter 10. Rheological constitutive equations of thermosets and homogenized UFRT composites were reformulated into a practical form that allows rapid simulation of the following basic rheological processes: creep, multiple creep-reverse creep, relaxation, and multiple relaxation-reverse relaxation. Numerical simulations were conducted on representative materials, namely the epoxy resin, a glass/epoxy UFRT composite, a carbon/epoxy UFRT composite. Some original measures of the rheological processes were intro-duced. Conclusions of the simulation studies were formulated.
The final conclusions were summarized in Chapter 11.
The book opens up new research opportunities in the linear rheology of thermosetting polymers and composites with thermoset matrix reinforced with stitched fabrics, e.g.: