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Perforation of Composite Sandwich Panels

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Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of UK Essays.

Published: Wed, 21 Feb 2018

CHAPTER 1

INTRODUCTION

1.1 Introduction

The use of sandwich structures has been increasing in recent years because of their lightweight and high stiffness. Commonly, the naval industry and transportation uses the E-glass fibers while the aerospace industry uses composite structures such as carbon fiber. The use of sandwich panels with composite facesheet in the naval industry is particularly appealing because they have better corrosion and environmental resistance and reduced magnetic signatures when compared to double-hull construction steel ships. On the other hand, composite sandwich panels are easily susceptible to damage by a strange object impact. The damage may be visible, penetration or perforation, or invisible, internal delamination and debonding. Both types of damages will result in stiffness and strength reduction. It is then important to study the impact behavior of composite sandwich panels.

Failure in composite structures can be caused by low, high and extremely high or localized impact. An impact caused by a foreign body initiates two waves from impact point in a panel: a through-thickness wave and a transverse shear wave. Whether or not these waves play an important role in the impact response of the panel depends on the actual contact duration between the projectile and panel and the time it takes the transverse shear wave to reach the panel boundary. Figures 1.1 (a)-(c) show three-impact scenarios: low-velocity, high-velocity and ballistic impact. In low-velocity impact, the contact force duration is long compared to the time it takes the transverse shear wave travel to reach the plate boundary. Many waves reflect back and forth across the side dimension of the panel. In high-velocity impact, the contact force duration is much shorter than the transverse shear wave travel time through the panel. Usually high-velocity impact is the same with perforation and localized damage of the panel. Ballistic impact deals only with through-thickness wave propagation. During ballistic impact, there is complete perforation of the panel with little or no panel deformation. The contact force duration is approximately the wave travel time through the panel thickness. Ballistic impact usually involves the study of penetration mechanics.

  1. Low-velocity
  2. High-velocity
  3. Ballistic Impact

The projectile to panel mass ratio will control whether wave propagation effect dominates the panel impact response and then suggested that a mass ratio be use as a parameter to determine impact response. It was shown that small mass impacts produce more damage than high-mass impacts having same kinetic energy. While small-mass impacts were defined by wave-controlled response, large mass impacts were defined by boundary-controlled response.

Common examples of low-velocity impact are of bird strikes, collision with floating object, and dropped tools, may cause damage. Underwater blast or debris from a faraway explosion and air was considered as a high-velocity impact situation. Examples of ballistic impact would be a bullet or fragments from a nearby explosion hitting the panel.

Another important factor governing the impact on composite structures is the ballistic limit. The ballistic limit is defined as the highest velocity of the projectile to cause perforation. When the residual velocity (exit) of the projectile is zero, then the initial velocity of the projectile that causes perforation is the ballistic limit of the sandwich panel. The ballistic limit may be calculate analytically or determined experimentally. In the experimental method, sandwich panels are shoot with projectiles over narrow range of velocities to either just cause penetration or to just perforate the panel. There exists a striking velocity at which 50% of the panels are completely perforate above this value and remaining 50% are partly penetrate below this value. This striking velocity is expresse as V50, which is the ballistic limit of the panel. In the analytical approach, the ballistic limit is determined by the conservation of energy principle. The approach is complex because it includes a variety of factors like core thickness, facesheet thickness, shape of the projectile, core crushing stress, and so on.

1.2 Problem Statement

This topic was an expansion of the Wan Awis research. He has done only an experimental work. For impact application, we need to predict skin and core material thickness. Since impact phenomena depend on numerous parameters such as material properties or projectile geometry, a numerical model, validated experimentally, is necessary to allow the study of the influence of several parameters without making costly experimental tests. This will definitely enhance the development of our military technology and achievements in the future because of the ability of this software to cut production cost and time consuming of the experimental work. The numerical figures have been compared to modal test results aiming mainly to validate the studies. Simulation based on finite element analysis (FEA) must not exceed ± 15% error or this simulation could be claimed not acceptable.

1.3 Objective

  1. To simulate the damage of composite sandwich structures subjected to high-velocity impact using finite element analysis.
  2. To determine the energy absorption capability of the components on the behavior of the sandwich panel under impact load using ANSYS AUTODYN 13.0
  3. To validate a numerical model with actual experiment.

1.4 Scope of Works

  1. To characterize a mechanical behavior of carbon fiber panel by using tensile and determine the fiber volume force and density.
  2. Design and validate the numerical model.
  3. Conduct a ballistic impact test simulation.
  4. Using the experiments data to calculate the energy absorption on the impact.

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

A great deal of research has been conducted in the area of impact of composite structures. In this chapter, previous work done on the impact response of laminated composite plates and composite sandwich panels will be reviewed.

2.2 Impact of Composite Laminates

A detail study of impact of composite laminates in the three impact regimes ballistic impact, low-velocity and high-velocity is presented in this section.

2.2.1 Low-velocity Impact

Abrate, 1998 give a specific review on different analytical models of impact on composite laminates. He classified impact models into four groups: impact on infinite plate model, energy balance models, spring-mass models, and complete models. In the energy balance model, the initial kinetic energy of the projectile is used to calculate the deformation of the composite laminate. The velocity of the projectile reaches zero at the maximum deflection of the composite laminate. At this point, all of the kinetic energy of the projectile is converted to strain energy needed to deform the composite laminate. Energy balance model assumes that the structure behaves in quasi-static manner. The time history of force and deflection are obtained using the spring-mass model representing the composite laminate. The model shown in Figure 2.1 consists of nonlinear contact stiffness (K), one spring representing the linear stiffness of the structure (Kbs), another spring for the nonlinear membrane stiffness (Km), effective mass of the structure (M2) as well as the mass of the projectile (M1). Equations of motion are written from a free body diagram. The infinite plate model is used when the deformation wavefront has not reached the boundary but if the wave reaches the plate boundary then this model is not an appropriate one to use. In the complete model, the dynamics of the structure and projectile are taken into explanation. Appropriate plate theory has to be selected and used. In many cases the classical plate theory can be used but when transverse shear deformations become significant, higher-order theories must be used.

One of the earliest studies on the impact of composite laminates was by Goldsmith et al, 1995, who conducted high-velocity and quasi-static impact tests on carbon-fiber laminates by using a cylindro-conical projectile. Three different specimen of varying thickness were considered. Energy balance principle was used to predict the dynamic penetration energy, static penetration energy, and also the ballistic limit of the composite laminate. The fiber failure accounted for most of the energy absorbed. The predicted theoretical energy was in good agreement with measured energy for thin laminates but not for the thick laminates. This was approved to the fact that transverse shear deformation played an important responsibility in thick laminates subjected to low-velocity impact. The effect of transverse shear deformation was not dominant due to its quick occurrence in the high-velocity impact of laminates. Therefore, the predicted energy in the dynamic case was always close to but less than the measured energy for the thin and thick laminates. The predicted ballistic limit was less than measured values due to the nonlinear factors.

Cantwell, 2007 studied the influence of target geometry in the low-velocity impact of composite laminate. The tests were performed on GFRP plates with hemispherical indenter on either circular or square supports. He used energy-balance model to predict the plate deflection and the delamination area of the laminated structure. His study stated that there is little or no influence of target geometry on the failure modes. It also suggested that delamination was dependent on interlaminar shear stress and increasing the plate diameter required more energy for damage initiation.

Hou et al., 2000 predicted impact damage in composite laminates using LSDYNA 3D. The numerical results were compared to experimental results on low-velocity impact on composite laminate with an initial velocity of 7.08 m/s The Chang-Chang failure criteria was modified taking the shear stress into consideration and the model was implemented in DYNA 3D.

2.2.2 High-velocity Impact

In 1988, Cantwell performed high-velocity impact tests of CFRP laminates with 6 mm diameter, 1g steel ball as the projectile. The influence of fiber stacking sequence and target geometry was study. The experiments reveal that varying the target geometry had no significance on initial damage caused. While the damage initiated in the distal facesheet in thin laminates, however, in thick laminates it initiated from incident facesheet.

Zhao et al., 2007 investigated the failure modes in composite laminates subjected to high-velocity impact. Three different laminates were subject to impact by hemispherical projectile in the range of 10-300 m/s. An energy balance was considered and equations for residual velocity for the laminates were given in terms of the mass of the projectile and striking velocity. The thickness and stacking sequence were finding to play an important role in the energy absorption.

Cheng et al., 2007 developed an analytical model based on the spring-mass model for high-velocity impact of a blunt ended and a sharp-ended projectile on thick composite laminates. They considered the effect of moving boundary due to the propagation of shear wave. The analysis was modeled using series of quasi-static events. At the end of each quasi-static step, the failed layers were remove based on punch shear damage and fiber damage criteria, and the wave front was moved outwards. While the first spring stiffness constant was measure based on the penetration depth of the projectile, the second spring stiffness constant was measured based on the bottom node of the plate.

2.2.3 Ballistic Impact

Silva et al., 2005 performed numerical simulations of ballistic impact on thin Kevlar 29 composite laminates using a fragment-simulating projectile. The laminate material model was simulating using AUTODYN and the projectile was modeled using Johnson-Cook strength model. Finite element mesh for both laminate and projectile was generating using True Grid. Accurate predictions of ballistic limit (V50) and the failure modes were made. Ballistic limit is the minimum velocity of impact at which a given projectile just perforates a given target. On occasion, the term is also used to identify the maximum impact velocity at which the projectile can penetrate into the target with perforation. It is often defined statistically as the impact velocity for which the projectile has a 50% probability of perforating the target; it is then denoted by V50.

Guild et al., 2007 conducted numerical simulations of ballistic impact on composite laminates and compared them with experimental results. The laminates were made of E-glass/vinyl ester resin with varying thickness and ball bearings of varying mass were use as projectiles. The damage modes included fiber failure, matrix failure, penetration, and delamination. Hashin failure criteria was use to determine the damage mode. Delamination was modeled using an interface between the two plies. As the force increased between two nodes above the specified value, the nodes were untied and the delamination increased. The ballistic limit from experiments was in good agreement with numerical results

Naik et al., 2008 studied the ballistic impact behavior of thick composites. E-glass/epoxy laminates of varying thickness were subject to high-velocity impact. The effects of projectile diameter, projectile mass and laminate thickness on the ballistic limit were studied. Wave theory and an energy balance were use to predict the ballistic limit of the laminate. The contact duration of the projectile with the laminate was maximum when the initial velocity was equal to ballistic limit and decreased when the initial velocity increased beyond the ballistic limit.

Deka et al., 2008 conducted ballisitic impact on E-glass/polypropylene composite laminates with cylinder-shaped projectiles. The experimental results were validating with numerical analysis using LS-DYNA. Although the laminate was modeling in Hypermesh, LS-DYNA was used to analyze failure mechanisms. The analytical model was base on energy conservation and failure in the numerical analysis was predicted based on Hashin’s failure criteria.

2.3 Impact of Composite Sandwich Panels

In this section, a detail study of impact of composite sandwich panels in the three impact regimes low-velocity, high-velocity and ballistic impact is presented.

2.3.1 Low-velocity Impact

Mines et al., 1998 investigated quasi-static loading and low-velocity impact behavior on two different composite sandwich panels. While the first panel was made up of E-glass/vinyl ester skin and Coremat® core, the second panel was made of Eglass/epoxy skin and aluminium honeycomb core. The first panel with Coremat® core had failed in the sequence of core shear, debonding, and distal facesheet damage and incident facesheet failure. The second panel failed by core shear, debonding, incident facesheet failure and then distal facesheet failure later. In the low-velocity impact tests, the failure pattern remained the same in both the panels as of the quasi-static tests. The core properties and impact velocity govern the energy absorption capability of the sandwich panel.

Wen et al., 1998 investigated the penetration and perforation of composite laminates and sandwich panels under quasi-static, drop-weight and ballistic impact tests by flat-faced, hemispherical-ended and conical-nosed indenters/projectiles. They categorized the impact on laminates and sandwich panels into low-velocity impact and wave-dominated (high-velocity/ballistic impact) response. It was also stated in the research that sandwich panels subjected to low-velocity impact have similar load-displacement characteristics as of quasi-static loading case. The perforation energy required by flat faced projectile was more than hemispherical-ended and conical shaped projectiles in high-velocity impact.

Schubel et al., 2005 investigated quasi-static and low-velocity impact behavior of sandwich panels with woven carbon/epoxy facesheets and PVC foam. The low-velocity impact model behaved similar to quasi-static loading case when loads and strain levels were same. The static indentation response was compared to the numerical results obtained using ABAQUS and were in good agreement. A membrane solution, assuming membrane in the core affected region and plate on elastic foundation in the rest of sandwich panel was in poor agreement with the numerical results.

Hoo Fatt et al., 2001, developed static and dynamic models of sandwich panels subjected to low-velocity impact. They investigated the behavior of sandwich panels having carbon/epoxy skins and a Nomex honeycomb core with a hemispherical indenter under various support conditions such as simply supported, fully clamped, and rigidly supported. Spring-mass models were considered to determine the load-displacement curve. They also investigated the damage initiation of sandwich panels under low-velocity impact loading. The initial mode of damage depended upon the panel support conditions, projectile nose shape, geometry of the specimens, and material properties of the facesheet and core. Various failure patterns were studied and solutions based on them were derived separately. The analytical solution for the ballistic limit was also found and results for thick laminates were in better agreement than thin laminates.

Suvorov et al., 2005 performed numerical analysis on sandwich panels with foam core and studied the effect of interlayer in between the top facesheet and foam core. The foam core was modeled as crushable foam in ABAQUS. While the polyurethane (PUR) interlayer reduced the deformations in both the core and the composite facesheets, the elastomeric foam (EF) interlayer offered a better protection for the foam core alone.

Besant et al., 2001 performed numerical analysis on sandwich panels with aluminium honeycomb core. The metal honeycomb core was modeled as elastic perfectly plastic material. A quadratic yield criterion was proposed for the core material, which included both normal and transverse shear stresses. The importance of core plasticity in finite element analysis was explained.

2.3.2 High-velocity Impact

A great deal of work has been done in the area of low-velocity impact of laminates and sandwich panels and high-velocity impact of laminates but limited work has been presented in the domain of high-velocity and ballistic impact of sandwich panels. The following describes some recent studies on the high-velocity impact of composite sandwich panels

Velmurugan et al., 2006 studied the projectile impact on composite sandwich panels in the range of 30-100 m/s. The sandwich models in this study were not the typical sandwich panels in the conventional sense. They had a core height comparable to the facesheet thickness and acted as a bonding agent between the facesheets. Energy-balance model was used to determine the ballistic limit of three different sandwich panels. They assumed the sandwich panel as a single plate since the foam layer was thin and comparable to facesheet thickness. Also uniform failure mechanism along the through thickness direction was assumed in their model.

Skvortsov et al., 2003 developed an analytical model using energy-balance principle to determine the ballistic limit of composite sandwich panels subjected to high velocity impact. Two different sandwich panels were subjected to high velocity impact using three different projectiles. These tests were conducted on simply supported and rigidly supported boundary conditions, and the initial velocity was varied in the range of 70-95 m/s. The predicted panel energy was close to the experimental values and the error was due to the strain-rate effects, plastic behavior, and hardening phenomena, which are not consider in the analysis.

2.3.3 Ballistic Impact

Kepler et al., 2007 conducted ballistic impact on sandwich panels consisting of GFRP plates and Divinycell H80 core, with three different projectiles. Lumped spring mass model was use to calculate force histories and panel response. Concentric rings connected by shear springs represented the sandwich panel. In this model, core shear deformation was assumed as the single significant contributor to the sandwich panel stiffness. The facesheet orthotropic was neglected in the panel response. Four different force histories: constant force, triangular force, sine series, and combination of sine and triangular force were used to calculate the energy loss in the panel. Of these, triangular and combined force gave results in better agreement with experimental results.

2.4 Aluminium Honeycomb

For design and construction of lightweight transportation systems such as satellites, aircraft, high-speed trains and fast ferries, structural weight saving is one of the major considerations. To meet this requirement, sandwich construction is frequently use instead of increasing material thickness. This type of construction consists of thin two facing layers separated by a core material. Potential materials for sandwich facings are aluminium alloys, high tensile steels, titanium, and composites depending on the specific mission requirement. Several types of core shapes and core material are been applied to the construction of sandwich structures. Among them, the honeycomb core that consists of very thin foils in the form of hexagonal cells perpendicular to the facings is the most popular.

A sandwich construction provides excellent structural efficiency, i.e., with high ratio of strength to weight. Other advantages offered by sandwich construction are elimination of welding, superior insulating qualities and design versatility. Even if the concept of sandwich construction is not very new, it has primarily been adopt for non-strength part of structures in the last decade. This is because there are a variety of problem areas to be overcome when the sandwich construction is applied to design of dynamically loaded structures. Other investigators have previously carried out noteworthy theoretical and experimental studies on linear elastic and nonlinear behavior of aluminium sandwich panels.

Kelsey et al., 1985 derived simple theoretical expressions of the shear modulus of honeycomb sandwich cores. Witherell, 1977 performed an extensive theoretical study for structural design of an air cushion vehicle hull structure using aluminium honeycomb sandwich panels. Okuto et al., 1991 showed the validity of the so-called equivalent plate thickness method in which a honeycomb sandwich panel subjected to inplane loads is approximately replaced by a single skin panel with equivalent plate thickness. Kobayashi et al., 1994, studied Elasto plastic bending behavior of sandwich panels. An experimental study was undertaken by Yeh et al., 1991 to investigate the buckling strength characteristics of aluminium honeycomb sandwich panels in axial compression. Kunimo et al., 1989 both, have studied the characteristics of the energy absorption capacity of bare honeycomb cores under lateral crushing loads theoretically and experimentally.

2.5 Ballistic Limit

The ballistic limit may also be defined as the maximum velocity at which a particular projectile is expected to consistently fail to penetrate armor of given thickness and physical properties at a specified angle of obliquity. Because of the expense of firing tests and the impossibility of controlling striking velocity precisely, plus the existence of a zone of mixed results in which a projectile may completely penetrate or only partially penetrate under apparently identical conditions, statistical approaches are necessary, based upon limited firings. Certain approaches lead to approximation of the V50 Point, that is, the velocity at which complete penetration and incomplete penetration are equally likely to occur. Other methods attempt to approximate the V0 Point, that is, the maximum velocity at which no complete penetration will occur

2.6 Energy Absorption Mechanism of Composite Materials

The research was done by Naik and Shrirao at 2004. Impact loads can be categorized into three categories which is low-velocity impact, high-velocity impact and hyper-velocity impact. This classification is made because of change in projectile’s velocity will result in different mechanisms in terms of energy transfer between projectile and target, energy dissipation and damage propagation mechanism.

Basically, ballistic impact is considered as low-mass high velocity impact. In this impact event, a low-mass projectile is launched by source into target at high velocity. It is unlike low-velocity impact that involved high-mass impactor impacting a target at low velocity. In view of the fact that ballistic impact is high velocity event, the effect is localized and near to impact location.

According to Naik et al. (2006), seven possible energy absorbing mechanisms occur at the target during ballistic impact. Those mechanisms are cone formation at the back face of the target, deformation of secondary yarns, tension in primary yarns/fibres, delamination, matrix cracking, shear plugging and friction between the projectile and the target. Then, the researchers formulated all these energies into equation whereby the total energy absorbed by the target is summation of kinetic energy of moving cone EKE, shear plugging ESP, deformation of secondary yarns ED, tensile failure of primary yarns ETF, delamination EDL, matrix cracking EMC and friction energy EF.

ETOTALi = EKEi + ESPi + EDi + ETFi + EDLi + EMCi + EFi

Mines et al. (1999) identified three modes of energy absorption when analysed the ballistic perforation of composites with different shape of projectile. These energy absorptions are local perforation, delamination and friction between the missile and the target. However, the contribution of friction between the missile and the target in energy absorption is low compared to the other two. In terms of local perforation, three through-thickness regimes can be identified, namely: I – shear failure, II – tensile failure and III – tensile failure and delamination. Out of these three regimes, the through-thickness perforation failure is dominated by shear failure. Similar observation has been made by other researcher for thick graphite epoxy laminates whereby the perforation failure is dominated by shear failure. The third main energy absorption mechanism is delamination. Delamination can propagate under Mode I (tensile) and Mode II (shear) loading and each mode can dominate each other depending on structural configuration of the composite as well as material properties. Therefore, it can be predicted that the total perforation energy is a summation of energy absorption due to local perforation, delamination and friction between the missile and the target.

Epred = Ef + Esh + Edl

where Ef = friction between the missile and the target; Esh = local perforation; Edl = delamination

Apart from that, Morye et al. (2000) has studied energy absorption mechanism in thermoplastic fibre reinforced composites through experimental and analytical prediction. They considered three mechanisms that involved in absorbing energy by composite materials upon ballistic impact. The three energy absorption mechanisms are tensile failure of primary yarns, elastic deformation of secondary yarns and the third mechanism is kinetic energy of cone formed at back face of composite materials. They concluded that kinetic energy of the moving cone had a dominant effect as energy absorption mechanism for composite materials. However, they neglected a delamination as one of the factor contributed to the failure of composite materials during ballistic impact.

2.7 Kinetic Energy Equation

Kinetic energy (KE) attack is a penetration of the residual energy of a projectile. A projectile can give a certain amount of energy to attack and damage a vehicle if the projectile sufficient residual energy when it arrive at the target. This residual is very important to overmatch the capability and strength of the target material to resist penetration, and then it will penetrate. Kinetic energy shot can be presented with the simple law of physic.

K.E = ½ Mprojectile Vprojectile2

Increasing the mass (Mprojectile) of the shot increases its energy, but the real payoff comes from increasing its velocity (Vprojectile). If the diameter of the shot fills the whole gun barrel, the projectile becomes heavier and difficult to accelerate to required velocity with the length of the barrel. Additionally, a large diameter solid shot will provide more energy to penetrate the armour plate compared to a projectile which has the same mass but a smaller diameter. Consequently, the larger shot is not only less effective at the target but it is difficult to give it the necessary velocity.

According to Chang et al., 1990, depth of penetration at the target will depend not only on residual energy, but also on shape and size of the projectile. The curve shape at the projectile head is more important, as it must not only able to pierce the armour but the shoulders of the shot must also support the remainder so that it does not break up on its way through the armour. If for given mass the diameter of the shot is reduced and is length increased, then for the same residual energy the shot will penetrate further, as it is working on a smaller cross section area of armour. The ratio of length-to-diameter is called slenderness ratio. Any projectile with ratio in excess of 7:1 cannot be spin stabilized it is not until they reach a ratio approximately 20:1 that they can call long rod.

So, based on those discussions above, we can conclude that energy absorption can be performed by this relation

Eabsor = Ein – Eout

= [½ Mprojectile Vin2] – [½ Mprojectile Vout2]

= ½ Mprojectile (Vin2 – Vout2)

So, Eabsorbed = ½ Mprojectile (Vin2 – Vout2)

2.8 Tsai-Hill Failure Criterion

Hill, 1950 proposed a yield criterion for orthotropic materials:

G+Hσ12+F+Hσ22+F+Gσ32-2Hσ1σ2-2Gσ1σ3-2Fσ2σ3+2Lτ232+2Mτ132+2Nτ122=1

This orthotropic yield criterion will be used as an orthotropic strength or failure criterion in the spirit of both criteria being limits of linear elastic behavior. Thus, Hill’s yield stresses F, G, H, L, M and N will be regarded as failure strengths. Hill’s criterion is an extension of von Mises’ yield criterion. The von Mises criterion, in turn, can be related to the amount of energy that is used to distort the isotropic body rather than to change its volume. However, distortion cannot be separated from dilatation in orthotropic materials, so Equation 2.8 is not related to distortional energy. Unfortunately, some authors still mistakenly call the criterion of Tsai-Hill a distortional energy failure criterion.

The failure strength parameters F, G, H, L, M and N were related to the usual failure strength X, Y, and S for a lamina by Tsai. If only τ12 acts on the body, then, because its maximum value is S,

2N=1S2

Similarly, if only σ1 acts on the body, then

G+H=1X2

And if only σ2 acts, then

F+H=1Y2

If the strength in the 3-direction is denoted by Z and only σ3 acts, then

F+G=1Z2

Then, upon combination of Equations (2.10), (2.11) and (2.12), the following relations between F, G, H and X, Y, Z result:

2F=1Y2+1Z2-1X2

2G=1X2+1Z2-1Y2

2H=1X2+1Y2-1Z2

For plane stress in the 1-2 plane of a unidirectional lamina with fibers in the 1-direction, σ3 = τ13 = τ23 = 0. However, from the cross sectional of such a lamina in Figure 2.3, Y = Z from the obvious geometrical symmetry of the material construction. Thus, Equation (2.8) leads to

σ12X2-σ1σ2X2+σ22Y2+τ122S2=1

as the governing


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