Geometry and anisotropy: analytical structural model

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Residual stresses and geometrical distortion in a uncon- strained unidirectional composite tube

In the present Section, we used the analytical model presented in Section 2.3.1 to assess the thermal stresses induced in a unconstrained unidirectional composite tube of internal radius Rin and total thickness htot with stacking sequence [90◦], that is the fibers oriented along the circumferential direction, subjected to a uniform temperature differ- ence ∆T . The resultant bending moment is used in the curved beam model, presented in Section 2.3.2, to assess the opening or closure of the tube.
The geometrical and loading parameters are: Rin = 20 mm, htot = 0.96 mm and ∆T = −400◦C. The material considered is Carbon/PEEK and the elastic material pa- rameters are the following (Grogan et al., 2015): EL = 134 GPa, ET = 10.3 GPa, µL = 6 GPa, νLT = 0.32, νT T = 0.4. The coefficients of thermal expansion used are (Grogan et al., 2015): α1 = 0.2(10−6)/◦C, α2 = 28.8(10−6)/◦C. The radial ( ), circumferential ( ) and axial ( ) stresses along the thickness ofσrrthe tube are depictedσθθ in Figure (2.σ4),zz while the shear stress (σθz) is obviously null.

Finite element composites modeling

The mechanism of the combined effect of the curved geometry of the structure and the anisotropy of the composite material on resid-ual stresses, explained in Section 2.2, has important consequences in terms of finite element modeling.
The same simulation performed on the unidirectional composite tube with fibers oriented along the circumferential direction in Section 2.3.3 with the structural model is performed using the commercial fi-nite element software Abaqus. The uniform temperature difference used is ◦C. Two ∆typesT= of−400elements were used: solid shell elements (called Con-tinuum Shell elements in Abaqus) and 3D volumetric elements (called 3D Stress elements in Abaqus). The results of these simulations are depicted in Figure (2.12) and in Figure (2.13) for the Abaqus color map.
As can be noticed from the results depicted in Figure (2.12) and in Figure (2.13), the solid shell element, which is formulated on a volumet-ric support with only displacement degrees of freedom, but with under-lying classical shell kinematics, is not able to assess residual stresses in this scenario. This is because one of the assumptions of the classical shell theory is that the transverse normal strain is null ( with respect to the material basis), therefore the solid shell elementε33= 0is not able to take into account the effect of the curved geometry, which cou-ples the in-plane kinematics with the out-plane kinematics of the tube (Section 2.2). Since the only cause which generates residual stresses in the unidirectional ◦ tube subjected to a uniform temperature difference [90C ]is the combined effect of the geometry and the anisotropy,∆T the=−Abaqus400 finite element simulation performed with the solid shell element gives null residual stresses (see Figure (2.12) and Fig-ure (2.13) left for the circumferential stress). The circumferential stress obtained by the FEM simulation with the 3D volumetric elements, de-picted in Figure (2.12), is the same as the circumferential stress assessed with the structural model shown in Figure (2.4) left. The inability of the shell model to correctly predict the stresses in-duced in a composite tube by free thermal expansion was initially high-lighted by Whitney (1971). Other authors Reddy (1982); Barbero et al. (1990); Dicarlo et al. (2001) proposed solutions that provide an enrich-ment of the out-of-plane kinematics of the shell (modified shell model).

Abaqus composite modelling strategies

In the present Subsection, two composite laminate modelling strate-gies on Abaqus are addressed. The first approach explored is to use one mesh element for each layer of the laminate, which is the classical modeling technique to de-scribe composites in 3D. In order to do that, each layer of the compos-ite laminate needs to be discretized, therefore the material parame-ters, the thickness and the fibers orientations need to be defined.
The second approach is the Abaqus Composite Layup option. This strategy, usually adopted in conjunction with shell elements, can also be applied to 3D volumetric elements as well within the Abaqus soft-ware. It enables us to use only one 3D volumetric element of the mesh in the structure’s thickness, which contains all of the information about the stacking sequence of the laminate. Figure (2.14) shows the Abaqus Composite Layup option window, where the number of composite lay-ers, the material, the relative thickness of each layer and the fibers an-gle can be defined. In particular, the whole composite laminate stacking sequence (shown here for a ◦ composite tube) with its inte-gration points is shown in Figure [(2±.5515).
The two approaches discussed above were tested on a unconstrained ◦ tube subjected to a uniform temperature difference ( [±55 ◦]C)3. The comparison between the two approaches is performed∆T= based−400 on the shear stresses, since in this case are the dominant ones (see Figure (2.9) and Figure (2.10)). The shear stresses results for the simulation with the Composite Layup approach are depicted in Fig-ure (2.16) left. The shear stresses results for the simulation with six 3D stress element in the thickness of the structure (one 3D stress element for each layer) are depicted in Figure (2.16) right. Results in both cases are exactly the same as those shown in Figure (2.10).

Transversely isotropic elastic behavior based on Cartan decomposition

The key point of the proposed model consists in defining the trans-versely isotropic elastic behavior of a unidirectional composite ply in terms of elementary, uncoupled material parameters, which can be physically related to the material parameters of the two underlying constituents, the fibers and the matrix. As it is discussed in the fol-lowing, these uncoupled parameters do not exactly correspond to the classically defined elastic constants , , , and.
The key to the definition of such anEL uncoupledETµLµT behaviorνLT is to derive the elastic fourth order tensor from a quadratic energy function, for-mulated in terms of an appropriate minimal integrity basis (Boehler, 1987). Such an integrity basis, in turn, is defined by finding an irreducible decomposition (Golubitsky et al., 1988) of the second order stressO(2)(and strain) tensor, where is the orthogonal group of the Euclidean plane, modeling the anisotropyO(2).
The idea to use group representation theory (Fulton and Harris, 1991) and make an irreducible decomposition of the second order stress or strain tensor has already been used in the case of the cubic aniso-tropy (Desmorat and Marull, 2011; Desmorat et al., 2017; Mattiello et al., 2018; Bertram and Olschewski, 1996; Biegler and Mehrabadi, 1995). Furthermore, finding an explicit integrity basis can be straightforward using the recent publicationO(2)(Desmorat et al., 2020), while some work was done in Ranaivomiarana (2019) about irreducible decomposition of strain tensor. O(2) This rigorous mathematical formulation turns out to have deep phys-ical significance, as it is discussed in the following. In particular, it en-ables us to clearly separate the material parameters which are influ-enced by the deviatoric response of the matrix and, as such, which should display a viscoelastic long term behavior, from those which should remain elastic throughout the analysis. This approach is radically dif-ferent from the one adopted in many viscoelastic models for trans-versely isotropic materials (Zocher et al., 1997; Kaliske, 2000; Petter-mann and DeSimone, 2018), which introduce different viscous relax-ation functions for each of the classical elastic constants ( , , , and ), with no clear link to the mechanical behavior ofEtheLEunderTµL- lyingµT constituentsνLT. A similar, although not irreducible, decomposition of the stress (and strain) tensor, was proposed by Spencer (1984) and used to model the viscoelastic response of unidirectional plies by Ned-jar (2011), but only partial uncoupling is achieved in that case.
Let us introduce the fibers’ direction vector , defining the normal to the isotropic plane (v2, v3). The transverse isotropyvf is then modelled on the group generated by the rotations of axis , as well as the change of direction of (equivalently the rotation of axisvf and angle ). Such a group is in factv the group of orthogonal transformationv2π of the plane . An irreducibleO(2) decomposition of the stress tensor, also known(v2,v3 )as a CartanO(2) decomposition (Golubitsky et al., 1988), is for instance given as follows: σ = sf Mf + shMh + sfs + sd, (3.1).

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Evaluation of residual stresses in tubes with different stacking sequences

The CETIM manufactured several tubes of thermoplastic matrix composite material (carbon/PEEK) using the LATP manufacturing process with two different mandrel temperatures. In one case tubes were manufactured winding the composite tapes around a cold mandrel (Tmand = 20◦C), in the other case tubes were manufactured using a hot mandrel (Tmand = 180◦C). The main consequence of the two different mandrel temperatures lies on the induced temperature gradients within the structure during the manufacturing process, therefore in the overall temperature history experienced by the composite structure. In particular, higher thermal gradients are expected in the manufacturing process with the cold mandrel.
The manufactured tubes are 0.96 mm thick, with an inner radius of Rin = 20 mm and 6 layers (each layer is 0.16 mm thick) with different stacking sequence: [±85◦]3 and [±55◦]3. After the specimens were manufactured, a cut along the axial direction was performed (see Fig-ure (2.6)) and the opening/closure (see Figure (2.7) right) and twisting (see Figure (2.8) right) displacements were measured in order to indirectly quantify the residual stresses due to manufacturing.
The initial cut width is din = 3.4 mm (see Figure (2.7) left) and the experimental measurements are detailed in Table (2.2). The circumferential relative displacements reported in the tables are the differences between the final cut widths and the initial cut width (de−di, see Figure (2.7)).
A negative experimental result for the circumferential relative displacement (see Table (2.2)) corresponds to the closing of the tube, while a positive one corresponds to the opening of the tube. The experimental twist offset is measured through the axial offset between the free edges of the tube after the axial cut (see Figure (2.8) right).

Table of contents :

Introduction
Motivation and goal
Outline of the manuscript
1 Initial state of the manufactured structure: sources of residual stresses and strains 
1.1 Laser Assisted Tape Placement manufacturing process
1.2 Geometrical distortions and residual stresses induced by the manufacturing process
1.3 Sources of incompatibility
1.3.1 Micro-scale
1.3.2 Meso-scale
1.3.3 Macro-scale
1.4 LATP Residual stresses and strains
1.5 Modeling approach
2 Structural modelling: the effect of geometry and anisotropy 
2.1 Introduction
2.2 Curved geometry and anisotropy
2.3 Geometry and anisotropy: analytical structural model
2.3.1 Residual stresses in the tubes: three dimensional model
2.3.2 Opening and closing of tubes: curved beam model
2.3.3 Residual stresses and geometrical distortion in a unconstrained unidirectional composite tube
2.4 Evaluation of residual stresses in tubes with different stacking sequences
2.4.1 Experimental results
2.4.2 Analytical results
2.4.3 Comparison between experimental and analytical results
2.5 Finite element composites modeling
2.5.1 Abaqus composite modelling strategies
2.6 Conclusions
3 A new mechanism-based temperature dependent viscoelastic model for unidirectional polymer matrix composites based on Cartan decomposition 
3.1 Introduction
3.2 Transversely isotropic elastic behavior based on Cartan decomposition
3.2.1 Remarks
3.3 Analytical homogenisation
3.4 Linear viscoelasticity
3.4.1 Remarks
3.5 Effects of temperature: thermal strains and time-temperature superposition .
3.6 Numerical simulations
3.6.1 Loading histories on a single material point
3.6.2 Structural simulations
3.7 Conclusions and perspectives
4 Thermal model of Laser Assisted Tape Placement 
4.1 Introduction
4.2 Thermal Model
4.2.1 Heat transfer equation
4.2.2 Boundary conditions
4.2.3 Initial condition
4.3 Analytical solution of the 1D heat transfer equation
4.3.1 Steady-state problem solution
4.3.2 Transient problem solution
4.4 Partial results: temperature evolution during ply deposition
4.4.1 Initial temperature distribution induced by different heating phase models
4.4.2 Time temperature evolution in the composite material laminate: a comparison
4.4.3 Thermal history induced by the Laser Assisted Tape Placement
4.5 Conclusions
5 Residual stresses and strains induced in tubes manufactured by LATP 
5.1 Introduction
5.2 Tubes simulations: residual stresses results
5.2.1 LATP manufacturing process modeling
5.2.2 Thermal histories induced by the LATP manufacturing processes with hot and cold mandrel
5.2.3 Numerical simulations on a [90◦]6 cylindrical tube
5.2.4 Numerical simulations on a [±85◦]3 tube
5.2.5 Numerical simulations on a [0◦]6 tube
5.2.6 Numerical simulations on a [±55◦]3 cylindrical tube
5.3 Geometrical distortions of the cylindrical tubes after the axial cut
5.3.1 Comparison of numerical and experimental results
5.4 Conclusions and perspectives
Conclusions and perspectives 
A Proof of modes orthogonality for the one-dimensional thermal model 

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