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Considerations of Failure Analysis in a Multi-Layered Composite Structure under Thermomechanical Loading

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Proceedings 2018, 2, 447; doi:10.3390/ICEM18-05329 www.mdpi.com/journal/proceedings
Proceedings
Considerations of Failure Analysis in a Multi-Layered
Composite Structure under Thermomechanical
Loading
Ngeletshedzo Nyambeni * and Boy Raymond Mabuza
Department of Non Destructive Testing and Physics, Vaal University of Technology, Vanderbijlpark 1900,
South Africa; raymondm@vut.ac.za
* Correspondence: ngeletshedzon@vut.ac.za; Tel.: +27-16-950-9338
Presented at the 18th International Conference on Experimental Mechanics (ICEM18), Brussels, Belgium,
1–5 July 2018.
Published: 12 June 2018
Abstract: The study seeks to investigate a failure of laminated composite structure subjected to a
thermomechanical loading. Failure analysis of composite structures is an important design
requirement. The stacking sequence of the structure investigated is restricted to ten thin layers. The
fiber orientation, stacking sequence and material properties influence the response from the
composite structure. Formulas are presented which are used to estimate the response of
multi-layered composite structure to thermomechanical loads. A failure analysis is performed
based on some known failure criteria. The values of the engineering properties for multi-layered
composite structure and the results of the stress and strain distributions subjected to the forces and
bending moments are presented. The numerical results were computed by using MATLAB script.
Selected results of the numerical analysis have been presented.
Keywords: multi-layered composite; failure analysis; thermomechanical loading
1. Introduction
The composite materials are more important in internal combustion engines, machine
components, thermal control and electronic packaging, automobile, train and aircraft structures [1].
They are also important for mechanical components such as brakes, pressure vessels, draft shafts,
flywheels and tanks. Despite their significant advantages, composite materials suffer from different
types of damage mechanism, such as fiber breakage, matrix crack, fiber matrix debonding, edge
cracking and many more [2]. High performance-to-weight-ratio makes laminated composites
contribute to a greater use in critical engineering structures which may be subjected to aggressive
environments such as extreme temperatures [3,4]. A number of laminated composites consist of
multi-directional layers of laminates to achieve certain mechanical properties and design
requirements [3]. The laminates are formed by stacking two or more laminae with varying fiber
orientations. This enables the structure to respond to complex states of stresses [5]. It is therefore
crucial to have knowledge of the mechanical performance of these laminated composites when
subjected to thermomechanical loading.
Quadratic failure theories such as Hoffman, Tsai-Wu and Tsai-Hill criteria are widely
acknowledged for orthotropic materials [6]. For the purpose of this study, Tsai-Hill, Tsai-Wu and
Hoffman theories are used for failure analysis in multi-layered graphite/epoxy structure exposed to
thermomechanical loading. Interfiber failure or interfiber fracture (IFF) has been formulated
depending on the Tsai-Wu and Chang-Chang failure theories. IFF analysis was considered in this
Proceedings 2018, 2, 447 2 of 6
study to account for delamination or matrix failure and also to predict the realistic behavior in
graphite/epoxy material.
2. Mathematical Modeling
The approach used in this study relies on the classical lamination theory. It is therefore
necessary to outline the elastic constants of a lamina and relate them to the engineering constants.
The stress-strain relation of a composite lamina can be properly written in the matrix form 
defined in terms of Young’s modulus, shear modulus and Poisson’s ratio [7]
=
1−,=
1−, =
1−,= (1)
where , are Young’s Moduli in directions 1 and 2;  is the shear modulus in the 1-2 plane,
, are Poison’s ratios in the 1-2 and 2-1 planes. Thus, using the  the lamina stress-strain
relations can be given in a compact form as [6]
= (2)
where  is given in terms of  stated in Equation (1) as follows
=22 (3a)
=−4 (3b)
=−−2−−−2 (3c)
 =22 (3d)
 =−−2−−2 (3e)
=−2−2 (3f)
=cos and =sin. The laminate’s extensional stiffness, , coupling stiffness,  and bending
stiffness,  are given respectively as follows [8]
= 


1 (4)
where ,=1,2 or 6. The forces per unit length, , and  and moments per unit length,
, and  are considered. Using the forces per unit length and the moments per unit length,
the vector of the mid-plane strains and the vector of mid-plane curvatures can be derived from the
following equation [9].
=


 (5)
with being the thermal load vector ,,
, is the thermal moment vector

,
,
and

is the inverse of

. At this stage the stress-strain relation,
accounting for mechanical and thermal effects can be outlined in the form [8]
=  
  
  −
−
−
 (6)
where
=Δ,
=Δ, 
=

=Δ, Δ gives the temperature change and
,, are the thermal expansion coefficients given by [10]
=−2
2
 − −
0 (7)
Proceedings 2018, 2, 447 3 of 6
and is the temperature distribution. The laminate constitutive equations are obtained by
integrating Equation (6) over the thickness and used to obtain the mid-plane strains and mid-plane
curvatures as follows:


=
































 
 








(8)
where 

is the inverse of
. The thermal stress and moment resultants on the right-hand
side of Equation (8) can be stated in the form

=  
  
  


−(9)
and

=  
  
  


1
2
−
(10)
respectively. The local stresses can be related to the global stresses by the following equation
=−2
2
 − −
 (11)
3. Failure Analysis Criteria
3.1. Tsai-Hill Failure Criterion
Tsai-Hill equation for failure theory is stated in the form [11]


−
>1 (12)
where and are 0⟹=,<0⟹=,0⟹=,<0⟹=.
If =0 and =, then the local stresses in Equation (11) can be written as =
−2, =2 and  =−. Applying the above to the
Tsai-Hill failure theory the equation will possess only the global stress in the direction.
3.2. Tsai-Wu Failure Criterion
This failure criterion is based on total strain energy failure theory where failure is assumed to
occur if the following condition is satisfied in the lamina [12];
2
>1 (13)
where
=
,
=
,

=−
×
,

=
,

=
,
=
,

=
.
3.3. Hoffman Failure Criterion
Hoffman’s theory can be used for an orthotropic lamina with unequal tensile and compressive
strengths where the equation is given by [13,14].
||
1
1
1
1

1 (14)
Proceedings 2018, 2, 447 4 of 6
The equation for the interfiber failure can be derived as follows
=1−


 (15)
The Tsai-Wu failure equation for the case of the interfiber failure/fracture where  =0 is
given as
=1− 1

−−
 (16)
The IFF Chang-Chang equation for the tensile fiber mode where 0 can be reduced to
[15,16] =1
1
 (17)
where =0.7 and 1 while that of the compressive matrix mode where 0 can be written as
=1−
2
(18)
4. Numerical Experiment
An orthotropic graphite/epoxy lamina made up of ten layers is considered. The stacking
sequence of the plies is [90;0;45;45;90;90;45;45;0;90]. The engineering constants and strength
properties for this composite material are given in Table 1 below. The thickness of each ply is
assumed to be 0.277 mm.
Table 1. Table showing the stiffness and strength properties of a graphite/epoxy structure.
Stiffness Properties





181 10.3 0.28 7.17
Strength Properties




 
°

°
1500 1500 40 246 68 0.02×10

22.5×10

The orthotropic graphite/epoxy layer is assumed to be loaded by combined axial and shear
stresses. Using the Tsai-Hill criterion, the strengths of the ply as a function of the fibre orientation is
determined at r = 0.05.
5. Results and Discussion
Figure 1 shows that at 0, =7.098×10Pa and =1.469×10Pa. The two values
compare favourably with the health of the composite material since they are smaller than that of the
ultimate strengths. However, when 0, the tensile and compressive loading are decreasing
exponentially until at angle of 90°.
Proceedings 2018, 2, 447 5 of 6
Figure 1. Comparison of tensile and compressive loading as the function of angle .
The results obtained after calculating for the Tsai-Hill, Tsai-Wu and Hoffmann failure criteria
are shown in Table 2. Failure occurs when the set conditions are equal or greater than 1 for Tsai-Wu,
Hoffman and Tsai-Hill failure criteria. From the results obtained, the structure will not fail according
to Tsai-Wu and Hoffman as the values are less than 1. However, according to Tsai-Hill criterion, the
graphite/epoxy structure will fail under the given strength and elastic properties.
Table 2. Results obtained from the Tsai-Hill, Tsai-Wu and Hoffmann failure theories.
Tsai-Hill Tsai-Wu Hoffmann
1.3037 −1.5433 0.7441
Figure 2a shows that Tsai-Wu at >0 has a maximum value of =9.802×10 Pa at
=1.010 Pa For Chang-Chang at <0, the maximum value of =8.02×10 Pa while
that of  =6.728×10 Pa at >0.
(a) (b)
Figure 2. Chang-Chang and Tsai-Wu interfibre fracture; (a) Comparing failure criteria for IFF when
<0 and >0. (b) IFF when >0.
For the Chang-Chang at <0 and >0, the curves complement each other to form a
curve similar to that of Tsai-Wu. The area outside the envelope indicates that interfibre failure occurs
while the area inside the envelope indicates that interfibre failure does not occur. In Figure 2b, the
value of  =8.128×10 Pa when =0 and =0 when =4×10 Pa.
x (MPa)
Proceedings 2018, 2, 447 6 of 6
6. Conclusions
Failure analysis of multi-layered composite structure has been studied in the present work. The
laminate considered for the present failure analysis is a ten-layered ply. The Tsai-Hill and Hoffmann
failure criteria are in good agreement since the graphite/epoxy structure will not fail under the given
set conditions. The results obtained indicate that the failure criteria used are good and can be used to
predict interfiber failure in multi-layered composite structure.
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... When the observed failure index in all failure theories exceeds one, the ply in the composite would fail. However, the ply is safe if the observed failure index is less than one [11]. The failure load for each ply is calculated by using the equation no 15. ...
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Laminated composite structures have started to play a very significant role in today’s aircraft industry. The application of composite materials has now gone beyond the borders of aircraft design and has entered into such fields as automotive, athletics and recreational equipment, etc. The light weight and high specific strength of composite material helps design vehicles with higher fuel efficiency and longevity. In order to understand the influence of design parameters related to the use of composite materials in these applications, a proper study of the laminated composite structures requires a complete failure analysis, which includes both initiation and propagation of damage. In this work a progressive failure methodology is developed and implemented in the commercial Finite Element software package, Abaqus. Out of the numerous failure criteria available in the literature to study damage initiation and propagation in unidirectional fiber reinforced composites, Puck and Schürmann’s failure criteria have been chosen due to their ability to predict results close to those observed experimentally. Key features of the Puck and Schürmann’s failure criteria for three-dimensional deformations of unidirectional fiber reinforced composites have been summarized. Failure modes in the matrix and the fiber are considered separately. The failure criteria are simplified for plane stress deformations. Whereas the failure plane can be analytically identified for plane stress deformations, a numerical search algorithm is needed for three-dimensional problems. Subsequent to the initiation of matrix failure, elastic moduli are degraded and values of these degradation parameters and fracture plane angles are found by using a Continuum Damage Mechanics (CDM) approach. It is found that the assumption that the material response remains transversely isotropic even after the matrix failure has initiated requires the degradation of the transverse Poisson’s ratio. The Puck and Schürmann’s failure criteria and the material degradation process have been implemented as a User Defined Field (USDFLD) subroutine in Abaqus. The implementation has been verified by analytically computing results for simple loadings and comparing them with predictions from using the USDFLD in Abaqus. Subsequently, both two- and three-dimensional problems of more realistic geometries and loadings have been analyzed and computed results compared with either experimental findings or results available in the literature. Major contributions of the work include identifying the degradation parameter for the transverse Poisson’s ratio in terms of the matrix degradation parameter for the matrix failure in compression, development of the USDFLD based on Puck and Schürmann’s failure criteria, implementing the USDFLD in the commercial finite element software, Abaqus, and verifying that results computing using the USDFLD for various laminates and loadings agree with those from either the analytical solution of the problem or those available in the literature.
Book
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This textbook makes use of the popular computer program MATLAB as the major computer tool to study Mechanics of Composite Materials. It is written specifically for students in Engineering and Materials Science examining step-by-step solutions of composite material mechanics problems using MATLAB. Each of the 12 chapters is well structured and includes a summary of the basic equations, MATLAB functions used in the chapter, solved examples and problems for students to solve. The main emphasis of Mechanics of Composite Materials with MATLAB is on learning the composite material mechanics computations and on understanding the underlying concepts. The solutions to most of the given problems appear in an appendix at the end of the book. © Springer-Verlag Berlin Heidelberg 2005. All rights are reserved.
Article
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Composite laminates are currently being pursued for structures which may be subjected to explosive threats in aggressive environments, characterized by extreme temperatures and seawater. A mechanical model is formulated for laminated plates subjected to thermo-mechanical loading and deforming in cylindrical bending; the layers are assumed to be imperfectly bonded, with sliding interfaces and delaminations. The model derives dynamic equilibrium equations which depend on only three unknown displacement functions for arbitrary numbers of layers/interfaces. Close form solutions are obtained which highlight accuracy and limitations of the approach and the influence of the imperfect interfaces on stress and displacement fields.
Article
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The applicability and limitation of several quadratic strength theories were investigated with respect to 2D-C/SiC and 2.5D-C/SiC composites. A kind of damage-based failure criterion, referred to as D-criterion, is proposed for nonlinear ceramic composites. Meanwhile, the newly developed criterion is preliminarily validated under tension-shear combined loadings. The prediction shows that the failure envelope given by D-criterion is lower than that from Tsai-Hill and Hoffman criteria. This reveals that the damage-based criterion is reasonable for evaluation of damage-dominated failure strength.
Article
Safe installation and operation of lightweight composite hydrogen storage cylinders are of primary concern. Typically, the inner liner of the cylinder is made with a high molecular weight polymer or aluminum that serves as a hydrogen gas permeation barrier. A filament-wound, carbon/epoxy composite laminate placed over the liner provides the desired pressure load bearing capacity. In many current designs, a glass/epoxy layer or other material is placed over the carbon/epoxy laminate to provide impact and damage resistance. These cylinders also have pressure/thermal relief devices that are activated in case of an emergency. The difficulty in accurately analyzing the behavior of a filament wound composite storage cylinder derives form the continually varying orientation of the fibers. Most of the analysis reported in filament wound composite cylinders is based on simplifying assumptions and does not account for complexities like thermo-mechanical behavior and highly orthotropic nature of the material. In the present work, a comprehensive finite element simulation tool for the design of hydrogen storage cylinder system is developed. The structural response of the cylinder is analyzed using laminated shell theory accounting for transverse shear deformation and geometric nonlinearity. A composite failure model is used to predict the maximum burst pressure. Results for various thermo-mechanical loading cases are presented.
Article
This paper presents a three dimensional Progressive Failure Analysis (PFA) of laminated composite structures, essential for studying failure in thick laminates. The current work mainly focuses on implementation of a 3D failure criteria originally proposed by Puck and Schurmann. The progressive failure analysis is performed using both analytical and the Finite Element (FEM) approaches. Commercially available finite element software, Abaqus, is used for the FEM implementation. Two fundamental stages of progressive failure analysis, failure initiation and its propagation have been considered in the current work for the structures where 3D efiects are significant. A numerical search method has been adapted to determine the failure initiation in the matrix using the Puck and Schurmann's failure criteria. For the failure progression, the Continuum Damage Mechanics (CDM) approach is used by degrading the stifiness of the laminate. The damage parameters, known as material degradation factors, are found by solving system of non-linear equations, which satisfy the matrix failure criteria. Finally both the failure initiation in the matrix and the failure progression have been programmed with a user defined subroutine known as USDFLD in ABAQUS. Several example problems are studied to validate both the analytical and the Finite Element solutions. Good agreements between the analytical and the Finite Element implementation are found. © 2015, American Institute of Aeronautics and Astronautics Inc. All rights Reserved.
Article
A theoretical and experimental investigation is conducted for the successive failure modes of graphite-epoxy laminated beams, on the basis of the Tsai-Wu and maximum stress failure theories, giving attention to behavior beyond the first failure. It is assumed that, once a ply fails in a laminate, it can carry no further load and its elastic properties are set to zero. The failure analysis is then repeated with the modified laminae on updated matrices, until the next failure point is reached. Theoretical results are compared with experimental ones, and it is found that theory-based failures occur at substantially lower loads than those of actual fracture.
Article
This paper investigates the progressive failure process in a multidirectional composite laminate subjected to an arbitrary thermo-mechanical load condition. A micromechanics modeling approach is employed to predict each-ply failure strength of the laminate and to identify the corresponding failure mode. The analysis has incorporated both material non-linearity and stiffness reduction. The load shared by each lamina in the laminate is determined based on the classical laminate theory, whereas the internal stresses in the constituent fibers and matrix of the lamina are explicitly related to this load by making use of combining the bridging model and the Benveniste and Dvorak’s formula. The lamina failure is assumed whenever any constituent material attains its ultimate stress state. A generalized maximum normal stress criterion is employed to detect the constituent failure. In this approach, all the simulation equations are given explicitly, no iteration is involved, and a minimum number of input data are required. Comparison of the predicted thermo-mechanical strengths of several laminates with available experimental data is favorable.