DYNAMIC RESPONSE OF FOAM-CORE SANDWICH BEAMS UNDER UNIFORM PRESSURE PULSE LOAD. A Thesis. Presented to

Transcription

1 DYNAMIC RESPONSE OF FOAM-CORE SANDWICH BEAMS UNDER UNIFORM PRESSURE PUSE OAD A Thesis Presented t The Graduate Faculty f The University f Akrn In Partial Fulfillment f the Requirements fr the Degree Master f Science Suzana Stelkic December, 0

2 DYNAMIC RESPONSE OF FOAM-CORE SANDWICH BEAMS UNDER UNIFORM PRESSURE PUSE OAD Suzana Stelkic Thesis Apprved: Accepted: Thesis Advisr Dr. Michelle S. H Fatt Department Chair Dr. Wieslaw K. Binienda Academic Advisr Dr. Craig C. Menzemer Dean f the Cllege Dr. Gerge K. Harits Faculty Reader Dr. Anil K. Patnaik Dean f the Graduate Schl Dr. Gerge R. Newkme Date ii

3 ABSTRACT An analytical slutin was derived fr btaining the large amplitude, damped respnse f a crushable fam cre sandwich beam subjected t pressure pulse lading. The simply-supprted sandwich beam, cmprised f aluminum facesheets and a PVC fam cre, was analyzed fr its respnse t a unifrmly distributed pressure pulse lad. The equatins f mtin fr the sandwich beam were develped cnsidering first-rder shear defrmatin and membrane stretching. The initial respnse f the beam was elastic until the nset f plasticity, when transverse shear stresses in the fam cre eceeded the transverse shear yield strength f the fam. The facesheets remained elastic, while the cre was elastic-plastic, thrughut the entire lad unlad cycle. Analytical results were cmpared t results frm finite element analysis using ABAQUS Eplicit and gd agreement was fund between them. The analytical slutin can be used t design sandwich beam eperiments fr etracting fam damping prperties. iii

4 ACKNOWEDGEMENTS I wuld like t thank my academic advisr, Dr. Craig C. Menzemer fr his invaluable guidance and supprt thrughut my graduate studies. Dr. Menzemer always had the mst sund and lgical advice t keep me fcused n my gals. ikewise, Dr. Michelle S. H Fatt s guidance all thrughut my Master s Thesis has been invaluable. Her eperience and passin fr learning have been a psitive influence n my studies as well as my career. She has shwn me that with persistence there is a slutin t even the mst challenging prblem. I am als grateful f all the supprt I received at The University f Akrn. The Department f Civil Engineering fr prviding resurces and help t keep me n schedule. Heidi Cressman f the Wmen In Engineering Prgram fr her encuragement ver the years. Dr. Anil K. Patnaik fr his invaluable input as a faculty reader n my cmmittee. astly, but very imprtantly, I wuld like t thank my family and all my friends fr their many prayers and faith and fr believing in me all these years. I simply culd nt have accmplished this withut each and every ne f yu. iv

5 TABE OF CONTENTS Page IST OF TABES... viii IST OF FIGURES... i CHAPTERS I. INTRODUCTION... II. ITERATURE REVIEW Plymeric Fams under Uniaial Impact ads Fam Cre Sandwich Beams under Impact ads III. MATERIA PROPERTIES AND TESTING ON PVC H00 FOAM Manufacturer Data Specificatin Testing n PVC H00 Fam Mntnic Cmpressin Tests... 4 v

6 3.. Cyclic Cmpressin Tests Test Results Mntnic Test Results Cyclic Test Results Cnclusins Based n Test Results... 0 IV. ANAYSIS OF SANDWICH BEAM UNDER PRESSURE PUSE OADING Prblem Frmulatin Facesheet Material Prperties Fam-Cre Material Behavir Elastic Respnse Elastic-Plastic Respnse Unlading and Viscus Damping Behavir Sandwich Plate Thery Frced Elastic Respnse Elastic-Plastic Respnse vi

7 V. FINITE EEMENT ANAYSIS Finite Element Mdel Elastic Respnse Elastic-Plastic Respnse VI. CONCUDING REMARKS REFERENCES APPENDICES APPENDIX A: IST OF NOTATIONS APPENDIX B: EASTIC-PASTIC INTEGRATION FUNCTIONS... 6 vii

8 IST OF TABES Table Page 3. Material prperties f the Divinycell H00 fam Strain Rate Effects fr PVC H00 fam Material prperties f the Al 606 facesheets. 4 viii

9 IST OF FIGURES Figure Page 3. Test apparatus set-up fr cnducting uniaial tests with MTS machine ISO 844 test standard methd t define the linear prtin f the curve in the elastic regin Cmpressive stress-strain behavir under mntnic lading at varying strain rates Cmpressive stress-strain curves fr specimens at strain amplitude f 0% and varying strain rates Stress-strain curves in cmpressin at a strain rate f 0.50 s - and varius strain amplitudes Stress-strain curves f ten cnsecutive cmpressive cycles at a strain rate f 0.50 s - at varius strain amplitudes 0 i

10 4. Rectangular sandwich beam with crushable fam cre Unifrmly-distributed pressure pulse lad applied t sandwich beam and resulting shear diagram Elastic-Perfectly Plastic Stress-Strain Curve Shear stress vs. shear strain graph f the behavir f the fam cre Displacement and shear rtatin f a symmetric prismatic beam Distances thrugh thickness f the sandwich beam, measured frm the neutral ais f the beam Tranverse shear vs. Transverse shear strain Plastic znes alng the length f the beam Finite element analysis mdel f fam-cre sandwich beam Bundary cnditins f the fam-cre sandwich beam mdel Meshed finite element analysis mdel f the fam-cre sandwich beam Cmparisn f FEA and analytical deflectin at midspan f the beam... 48

11 5.5 Cmparisn f FEA and analytical transverse shear strain at midplane f the fam-cre near the rller supprts Plastic regins in the fam-cre per finite element analysis Cmparisn f FEA and analytical deflectin at midspan f the beam Cmparisn f FEA and analytical transverse shear strain at midplane f the fam-cre near the rller supprts i

12 CHAPTER I INTRODUCTION ightweight sandwich structures with plymer fam cres have significant energy absrptin prperties which have made them especially favrable in prtecting varius structures frm impact r blast lads. Benefits t using plymer fam cre sandwich structures are numerus, including their high strength t weight ratis, resistance t water damage, lw cst, and ability t be prduced frm recycled materials. Fr instance, fam bards and fam granules can be prduced frm recycled plyvinyl chlride (PVC) and used in cnstructin and packaging applicatins. Plymer fam cre sandwich panels can prtect structures frm impact r blast lads by defrming viscelastically and viscplastically, damping the amplitude f vibratin initiated by the impact frce n the system. Energy is absrbed by the micrinertial resistance develped in the material planes f the fam. As the fam crushes, the cellular walls in the fam eventually cllapse resulting in viscelastic behavir at an almst cnstant flw stress. Strain energy and kinetic energy are absrbed by the aluminum facesheets bnded t the fam cre while the fam cre itself absrbs energy by hysteresis which ccurs as a respnse t stress wave prpagatin and structural vibratin. Plymer fam cres ehibit hysteresis behavir due t cellular wall buckling,

13 fracture, and frictin between the material planes. In rder t effectively apply and utilize the energy absrptin prperties f the fam, it is imprtant t understand the behavir and respnse f the fam cre under impact and blast lading. The purpse f this study is t develp analytical slutins fr designing a simple eperiment in which the plastic and viscus damping f the fam may be evaluated. Analytical slutins will be derived and verified by finite element analysis (FEA) in rder t define initial cnditins fr prperly designing the eperiment t etract the desired material prperties. The arrangement analyzed, bth analytically and with FEA, is the same arrangement which will be used in the eperiment. The arrangement cnsists f a symmetric rectangular sandwich beam with aluminum facesheets and a Divinycell H00 fam cre. Aluminum facesheets are ideal in this arrangement because they are cst-effective, have a high yield strength, and are lighter in weight than carbn steel. The Divinycell PVC H00 plymer fam has ecellent shck-absrptive prperties, is lightweight, and is resistant t envirnmental crrsin. An impact r blast lad applied t a cmpsite beam will transfer kinetic energy t the beam system. The lad induces free vibratin respnse f the system which will cntinue until the energy is dissipated and the system cmes t rest. Energy dissipates thrugh cell wall crushing and viscelasticity f the fam cre. If the deflectin is large enugh t eceed the yield strength f the fam cre, it behaves as an elastic-plastic material. Belw the yield strength, the fam cre will behave viscelastically which is

14 elastic behavir with sme damping t be evaluated in the eperiment recmmended later in this study. While many studies have been cnducted t evaluate the elastic-plastic damping plymer fam cre sandwich structures, mst have fcused n plastic defrmatin f the fam cre and failure mdes f the structure. This study will present analytical slutins fr an eperiment that will be able t characterize the viscelastic behavir that ccurs after plastic defrmatin when the structure rebunds frm the impact r blast lad. The characterizatin f the damping that ccurs in the viscelastic phase f the fam cre is state-f-art and has nt yet been evaluated. Cyclic vibratin lads at high strain rates may be applied t fam cre when the sandwich fam cre is used t prtect a structure frm impact r blast lads. As the structure vibrates, it transfers kinetic energy t the fam that will be eventually damped ut by the fam s unique damping prperties bth in the elastic-plastic phase and the viscelastic phase f the stress-strain curve. Frm a design perspective, the stress-strain respnse f the structure and fam cre system is f particular interest s that the plymer fam cre sandwich shck absrbers can be designed and placed such that the capabilities f a system are maimized. A majrity f previus studies dne n the impact respnse f cmpsite sandwich beams evaluated the plastic defrmatin f the fam cre but d nt g beynd this t characterize the viscelastic behavir that ccurs when the beam rebunds frm the lad. The elastic and plastic respnse f a cmpsite sandwich beam subjected t an impact lad will be evaluated analytically in this thesis. A new analytical technique is 3

15 applied, etended frm slutins develped by Vinsn [] and Chapagain [] fr cmpsite sectins with plastic defrmatins. Analytical results are used t design an eperiment in which the viscus damping f the Divinycell H00 fam cre may be effectively evaluated. Viscus damping behavir f the fam cre is ehibited after initial elastic lading and plastic shear defrmatins have taken place. Analytical results f the elastic and plastic respnse f the cmpsite sandwich beam subject t a unifrmly distributed pressure pulse lad are verified using a finite element analysis (FEA). The FEA mdel uses ABAQUS Eplicit sftware t simulate the respnse when a unifrmly distributed pressure pulse lad is applied t the beam. Verifying and evaluating the elastic and plastic respnse f the sectin by bth analytical results and FEA is critical in rder t prperly design the eperiment. Previus studies dne n fam cre sandwich beams and PVC fams are reviewed in Chapter II. Eperiments used t derive material prperties required fr the analysis are described in Chapter III. Analytical techniques fr evaluating the fam cre sandwich beam under blast lads are presented in Chapter IV. The finite element analysis f the fam cre sandwich beam under blast lad is discussed and cmpared t the analytical results in Chapter V. Finally, cnclusins and recmmendatins f this study are made in Chapter VI. 4

16 CHAPTER II ITERATURE REVIEW There are many studies investigating the shck absrptin prperties f fam cre sandwich beams and sectins. Mst f these studies address the bending and aial respnse f a sandwich beam under a drp type impact lad r a cyclic dynamic lad and evaluate the structure s design capacity. Hwever, few studies address the dynamic damped respnse f a sandwich beam due t its viscelastic fam cre. Understanding this characteristic behavir f fam cre sandwich beams is critical in rder t prperly use these materials fr damping and shck absrptin in structures under impact and blast lads. Previus wrk dne n fam cre sandwich structures will be reviewed in this chapter. Wrk dne n testing plymeric fams is discussed, fllwed by a review f wrk dne n impact tests n plymer fam cre sandwich structures, as well as a review f cyclic lad tests cnducted n these type f sandwich structures.. Plymeric Fams under Uniaial Impact ads While much research has been dne n the shck absrptin f plymeric fams under drp-type impact lads, very few studies have analyzed the viscelastic defrmatin that ccurs as a respnse t the impact lads. Instead these studies have 5

17 addressed the penetratin r ttal cmpactin f the fam cre under the cmpressive drp-type lads. This sectin prvides a brief review f sme f these studies. Belingardi et al. [3] studied the impact respnse f a PVC fam with a density f 80 kg/m 3. A drp dart testing machine was used t deliver the impact via a cylindrical impactr with a flat end. Tw different impact velcities (60 m/s and 363 m/s) were used t eamine strain-rate effects fr the fam cre. Within the range f their testing, the results f the dynamic impact tests did nt reveal any significant strain-rate dependence n the dynamic respnse f the fam cre. The PVC fam was used as the cre layer in a cmpsite sectin. They cncluded that the cmpressive stiffness f the fam cre was inadequate t supprt the facesheets fr an impact, and that adding resin membranes between the facesheets wuld greatly imprve the perfrmance f the cmpsite sectin fr impact. Green et al. [4] cnducted dynamic uniaial stress tests n tw different plyurethane fams. One fam was a water-blwn ester plyurethane while the secnd fam tested was frm a castr-il base. Samples f three different densities f the waterblwn ester fam were tested (56 kg/m 3, 5.3 kg/m 3, and 40.3 kg/m 3 ) while the castril base fam nly was available in a 55.4 kg/m 3 density. The samples were dynamically laded at medium strain rates (0-3 in./in./sec t 0 in./in./sec) with a gas perated machine with a mvable pistn. Higher strain rates were als applied with a split-hpkinsn bar device with impact velcities up t 36.6 m/s. In general, they fund that the strength and stiffness f the plyurethane fam increased with the rate f lading. 6

18 The water-blwn ester based fam was mre rigid, and its tensile fracture stress was independent f lad rate. The castr-il based fam was less rigid and shwed mre dependence n lad rate fr the tensile fracture stress. Bth fams shwed increased stiffness at higher lading rates, with a similar stiffness in cmpressin and tensin. The rigid water-blwn ester fam withstd large uniaial cmpressive lads at lw lading rates and crushed at higher lading rates, especially the samples with the highest density. In cntrast, the semi-rigid castr-il based fam withstd large cmpressive lads fr all the lading rates tested. Nemat-Nasser et al. [5] studied the dynamic respnse f aluminum fam cre and metal facesheets independently in rder t characterize the behavir f the cmpnents f the sandwich structure subject t high-rate inertial lads. The aluminum fam specimens were impacted with a prjectile t impse dynamic cmpressive lads. Defrmatin f the specimen was measured using a high-speed camera, and the frce transmitted was measured by a strain gage. The impact velcity ranged frm 30 m/s t 55 m/s. At the lwer impact velcity, they bserved that the face f the specimens cntacting the impactr defrmed mre significantly at first, but then the far end f the specimens began t defrm at mre as time elapsed. The nn-unifrm strain distributin was attributed t the cmpressive stress pulse reflecting thrugh the sample and causing an increase in ttal stress at the far end f the specimen. In cntrast, at the higher impact velcity shwed significant defrmatin at the impactr cntact surface which eventually distributed t a unifrm stress thrughut the specimen. Split-Hpkinsn bar tests were als cnducted n the aluminum fam cre t btain dynamic stress-strain curves. They 7

19 fund that there were n strain-rate effects n the fam in the range tested. When cmpared t the literature regarding uniaial plymeric fam tests, their bservatins indicate metallic fams defrm mre elastically than plymeric fams and stress distributins tend t be mre unifrm in respnse t impact lads in metallic fams [3-5].. Fam Cre Sandwich Structures under Impact ads Many studies have als addressed the failure f fam cre sandwich structures under impact lads in rder t better understand failure mdes and limitatins f such structures. Several studies have attempted t parameterize the fam cre prperties such as density and stiffness with failure mdes and will be briefly reviewed in this sectin. Belingardi et al. [3] als studied the impact respnse f cmpsite sandwich plates cnstructed f the PVC fam cre with glass-fiber and epy cmpsite facesheets. The 00 m 00 m specimens were subjected t bth quasi-static penetratin tests and dynamic impact tests. The quasi-static tests were cnducted using a servhydraulic machine with the lad applied at a cnstant velcity. The results frm these tests were used t determine the required drp height fr the dynamic impact tests. Fr delivering the impact lad, a 0 kg mass was drpped frm a maimum height f m using a drp dart testing machine. The lad-strke graphs frm the impact tests shwed a peak in the lad at the piercing f the tp facesheet, fllwed by a plateau zne which was attributed t the frictin f the drp dart against the fractured layers as well as cmpressin f the fam cre, and lastly fllwed by anther peak at the piercing f the 8

20 bttm facesheet. Based n preliminary tests cnducted t define the material prperties, they fund that the respnse f the sandwich structure depended largely n the strength f the fam cre. N strain-rate effects were bserved fr the facesheets, fam cre, r sandwich structure fr the strain rates applied in this study. im et al. [6] investigated the impact failure mdes and impact energy absrptin characteristics f sandwich beams cnstructed f E-glass/Epy facesheets and a PVC fam cre. Varying densities f the Divinycell HT grade fam cre were studied (54, 70, 97, and 7 kg/m 3 ). Prir t investigating impact lad respnse f the sandwich beams, im et al. [6] studied the respnse under static lading and cnstructed a static failure mde map fr each fam cre material. This failure mde map predicted which failure mde cntrlled (cre shear, cre cmpressin, r facesheet fracture) fr certain dimensinless parameters based n sandwich beams. Similar failure mde maps were cnstructed fr impact lading. These mde maps predict failure in the cre r face based n impact duratin and face thickness fr each fam cre material. Failure mdes due t impact were predicted using finite element analysis sftware and mdeling the fam cre as having elastic-perfectly plastic stress-strain respnse. Impact tests were then cnducted using a pneumatic cylinder with impact speeds up t 30 m/s, the results f which crrelated well with the finite element analysis. im et al. [6] fund that the impact energy absrptin capability f the sandwich beam was ptimized if designed t fail in the facesheet failure mde. 9

21 Cmpstn et al. [7] tested a PVC fam cre sandwich panel fr impact lads delivered by a swinging pendulum. Impact energies started at 5 J and increased in 5 J increments t 5 J. Each impact energy was applied t a new sample. The fam cre cnsisted f Klegecell R00 (density f 00 kg/m 3 ) and the facesheets were a cmpsite. Absrbed energy was calculated by taking the difference between incident ptential energy and the energy at the rebund peaks. A defrmatin prfile was generated and residual indentatin depths were als calculated. In additin, advanced defrmatin and strain analysis using a real-time strain analysis system were utilized t characterize the pst-impact respnse. The same eperiments were als cnducted n aluminum fam cre sandwich specimens. The energy absrptin and defrmatin were then cmpared t the respnse f the PVC fam cre sandwich panels. They fund that the absrbed energy was linear fr the impact lads tested and similar fr bth cre materials. Damage mdes, hwever, fr the tw cre materials varied significantly. w energy impact in the PVC cre sample resulted in a mre lcalized indentatin, whereas the aluminum cre displayed significant ut-f-plane damage as well as sme cell buckling. The higher energy impact results shwed skin fracture and cre crushing fr the PVC cre and nly minr skin fracture fr the aluminum cre alng with cell buckling and ut-f-plane plastic defrmatin. It was bserved that the PVC cre had very minr ut-f-plane damage and less permanent damage than the samples with aluminum cres. Pst-impact respnses f the tw cre materials als varied in that the PVC cre had a higher, mre lcalized strain at the impact pint. Cmpstn et al. [7] cncluded that based n the mre ductile failure mdes and lwer pst-impact strain respnse f the aluminum fam cre 0

22 samples, it appeared that the aluminum cre samples had better damage tlerance when cmpared t the PVC cre samples.

23 CHAPTER III MATERIA PROPERTIES AND TESTING ON PVC H00 FOAM Tests were cnducted n PVC H00 fam specimens in rder predict the behavir f the fam cre in respnse t crushing and energy absrptin. Uniaial cmpressin mntnic and cyclic tests were cnducted at varying strain rates and strain amplitudes t btain the stress-strain respnse f the PVC H00 specimens. Using an MTS 83 servhydraulic machine, the tests were cnducted n ne-inch cube specimens. The fam cre sandwich beam in this study cnsists f tw istrpic aluminum (Al 606) facesheets and a Divinycell H00 fam cre. N material testing f the Al 606 facesheets was required as standard plate stck was used and the analysis was designed nt t eceed the elastic limit f the facesheets at any pint. 3. Manufacturer Data Specificatin The fam cre material used in this study is Divinycell H00 PVC fam. This fam cnsists f a crss-linked clsed cell structure with cell size f 400 μm. Prperties f the PVC H00 fam are listed in Table 3. and have been prvided by DIAB [8].

24 Table 3. Material prperties f the Divinycell H00 fam [8]. The fam was purchased in apprimately ne-inch thick sheets which were inch inch. 3. Testing f PVC H00 Fam The ne-inch cube fam specimens were glued t aluminum grips t maintain unifrm lading in bth uniaial tensin and cmpressin. The aluminum grips were then placed in the MTS 83 serv-hydraulic machine and cnnected t the actuatr arm using a specially designed C clamp shwn in Figure 3.. The cmpressin tests were cnducted accrding t ISO 844 Rigid Cellular plastic-determinatin f Cmpressin Prperties [9]. 3

25 Figure 3. Test apparatus set-up fr cnducting uniaial tests with MTS machine. 3.. Mntnic Cmpressin Tests Mntnic cmpressin tests were cnducted t characterize the stress-strain respnse f the PVC H00 fam specimens prir t the densificatin regin. Strain-rate effects were investigated by using varying strain rates frm 0-4 s - up t 0 - s - and applying strain amplitudes up t 5%. The cmpressive mdulus was btained by the linear prtin f the stress-strain curve per the methd defined by the ISO 844 standard. 4

26 3.. Cyclic Cmpressin Tests Cyclic cmpressin tests were als cnducted t characterize the stress-strain respnse f the fam upn cnsecutive lading. Strain rates varied frm 50-4 s - t 5 s -. Strain amplitudes applied ranged frm % and increased in % increments up t 0%. 3.3 Test Results 3.3. Mntnic Test Results The ISO 844 test standard prvides a methd t define the linear prtin in the elastic regin n the stress-strain curve. The methd prvided is demnstrated in Figure 3., and was used t calculate Yung s mdulus fr varius strain rates. The nnlinear prtins at the beginning and the end f elastic regin are remved, and the remaining prtin is defined as the linear elastic regin. Yung s Mdulus f the material is then calculated by the slpe f. 5

27 Figure 3. ISO 844 test standard methd t define the linear prtin f the curve in the elastic regin. The stress-strain respnse f the PVC H00 fam at varying strain rates shws that the material is viscelastic and viscplastic, as can be seen in Figure 3.3. The viscelasticity f the material is demnstrated by a slight change in cmpressive mdulus with increasing strain rate. Viscplasticity f the material is demnstrated by a change in the yield stress with increasing strain rate. 6

28 Figure 3.3 Cmpressive stress-strain behavir under mntnic lading at varying strain rates. Strain rate effects n the stress-strain behavir f the fam specimens are summarized in Table 3.. With increasing strain rates, it can be bserved that there is an increase in the cmpressive mdulus, the cmpressive strength, and plateau stress. Fr all the strain rates tested, yielding ccurred at abut 3% strain. Table 3. Strain Rate Effects fr PVC H00 fam. 7

29 3.3. Cyclic Test Results Cyclic cmpressin tests were als cnducted n PVC H00 fam specimens. Varying strain rates and amplitudes were applied t the specimens. The stress-strain respnses f the specimens at strain amplitude f 0% and five different strain rates are shwn in Figure 3.4. Viscplasticity f the material is demnstrated by the permanent (plastic) strain n the stress-strain curves in Figures 3.4, as the material des nt re-trace its initial lading path n cnsecutive cycles. Figure 3.4 Cmpressive stress-strain curves fr specimens at strain amplitude f 0% and varying strain rates. Inspectin f the stress-strain curves at the same strain rate and varying strain amplitudes reveals that the cmpressive engineering stress is reduced fr each cnsecutive cycle at all strain amplitudes tested. This is particularly evident fr a strain 8

30 rate f 0.50 s -, as demnstrated in Figure 3.5. Cyclic behavir f specimens laded in ten cnsecutive cmpressive cycles are shwn in Figure 3.6. It can be seen that each cnsecutive cycle eperiences sme amunt f additinal damage t the cell structure. The damage that ccurs t the internal structure is what prduces the damping behavir f the fam cre in a sandwich beam applicatin. Figure 3.5 Stress-strain curves in cmpressin at a strain rate f 0.50 s - and varius strain amplitudes. 9

31 Figure 3.6 Stress-strain curves f ten cnsecutive cmpressive cycles at a strain rate f 0.50 s - at varius strain amplitudes. 3.4 Cnclusins Based n Test Results Material testing and a review f the material prperties allw several cnclusins t be made abut the impact respnse behavir f PVC H00 fam. Based n the cmpressive mntnic lad tests, it can be cncluded that PVC H00 fam is linear elastic-plastic. Bth the mntnic and cyclic cmpressive stress-strain curves at varius strain rates shw that the PVC H00 fam is viscelastic and viscplastic. The cmpressive cyclic stress-strain curves shw that damage is sustained by the fam n cnsecutive lading cycles which will prvide damping t a dynamic system. This damping is primarily viscelastic. It is indicative f the energy absrptin that can ccur in a fam-cre sandwich beam under pressure pulse lading. 0

32 CHAPTER IV ANAYSIS OF SANDWICH BEAM UNDER PRESSURE PUSE OADING A pressure pulse r blast lad applied t a sandwich beam will transfer kinetic energy t the beam system. The pressure pulse lad is a frm f impulsive lad which is applied unifrmly acrss the surface f ne f the facesheets. It is characterized by a steep rise in amplitude and then an epnential decay in time. An idealizatin f the pressure pulse is a triangular distributin with a steep rise and a linear decrease in amplitude ver a shrt perid f time. When the pressure pulse lad is remved frm the system, the energy transferred t the system causes the sandwich beam t vibrate until the energy has been dissipated by damping primarily in the fam cre. The sandwich beam deflects in the directin f the lad until the velcity decreases t zer and becmes negative, creating a reverse deflectin. This cycle will cntinue in the frm f free vibratin respnse f the beam system until the energy is cmpletely dissipated and the system cmes t rest. 4. Prblem Frmulatin Cnsider a simply-supprted rectangular sandwich beam as shwn in Figure 4.. The sandwich beam is cmprised f tw istrpic facesheets f thickness h and density

33 ρ f, as well as a crushable fam cre f thickness H and density ρ c. The ntatin can be fund in Appendi A. Figure 4. Rectangular sandwich beam with crushable fam cre. et the Cartesian crdinate system (, y, z) be riented as shwn if Figure 4. with the facesheets lcated in the y plane and z-ais alng the thrugh-thickness directin. The sandwich beam is symmetric abut the -ais (the facesheets are bth equidistant frm the centerline f the fam cre). A unifrmly-distributed pressure pulse p is applied acrss the surface f ne facesheet f the rectangular sandwich beam as shwn in Figure 4..

34 Figure 4. Unifrmly-distributed pressure pulse lad applied t sandwich beam and resulting static shear diagram. The pressure pulse lad p can be described by t p0, t ΔT p( t) = ΔT 0, t > ΔT (4.) where p0 is the peak pressure, ΔT is the lad duratin and t is time. 4.. Facesheet Material Prperties The Al 606 facesheets are cnsidered t be less than ¼ inch thick. The facesheets are assumed t be perfectly bnded t the fam cre fr the entire duratin f the respnse f the sandwich beam t a pressure pulse lad. Only elastic prperties fr the facesheets are cnsidered since the facesheets are nt t eceed the elastic limit at any pint in the analysis. Prperties fr the Al 606 facesheets are summarized in Table 4., per ASME Sectin II Part D [0]. 3

35 Table 4. Material prperties f the Al 606 facesheets [0]. 4.. Fam-Cre Material Behavir As demnstrated by the mntnic cmpressin test results discussed in Chapter III, the PVC H00 fam cre is elastic-plastic. Fr simplicity, an idealized elasticperfectly plastic curve is shwn in Figure 4.3. The initial stress-strain curve is linear up until the yield stress where the stress-strain curve becmes flat at a cnstant flw stress. In this study, strain rates will be limited t less than 0% and therefre any changes in the density f fam as it underges lading will be insignificant. Figure 4.3 Elastic-Perfectly Plastic Stress-Strain Curve. 4

36 4..3 Elastic Respnse At the nset f lading, energy is transferred thrugh the elastic deflectin f the aluminum facesheets as well as viscelastic behavir f the fam cre. The facesheets are assumed t remain elastic thrughut the entire respnse until the system cmes t rest. The transverse shear stress τ z in the fam cre will cntinue t increase linearly with increasing deflectin f the sandwich beam until the shear stress reaches the shear yield stress τ. Under the pressure pulse lad p 0, the maimum transverse shear stress ccurs at the supprts f the beam as can be seen in the transverse shear frce diagram in Figure 4.. The material behavir transitins frm elastic t plastic at Pint A f Figure 4.4, and crrespnds t the pint at which τ z equals τ and the transverse shear strain γ z equals γ. τ A B τ A B D D G γ γ C C Figure 4.4 Shear stress vs. shear strain graph f the behavir f the fam cre. 5

37 4..4 Elastic Plastic Respnse If the deflectin f the beam is large enugh t cause the transverse shear stress in the fam cre t eceed the transverse shear yield stress τ, the behavir f the fam cre will be assumed t be perfectly plastic as is shwn by the flat sectin f the shear stressstrain curve between Pint A and Pint B in Figure Unlading and Viscus Damping Behavir As the beam deflects back twards the neutral psitin, r the unlading f the system, the fam cre will behave viscelastically with sme shear viscsity. The shear viscsity will prvide damping t the fam cre sandwich beam system in the unlading prtin f the curve (between Pints B and C n the curve in Figure 4.4). An eperiment fr evaluating the shear viscsity f the fam cre will be suggested in Chapter VI f this study. If the energy applied t the system is nt enugh t cause reverse yielding, the fam cre will prduce a viscelastic hysteresis curve and cycle between Pints C and D n Figure 4.4 until the system cmes t rest. 4. Sandwich Plate Thery Sandwich plate thery develped by Vinsn [] and Chapagain [] will be applied fr analyzing a sandwich beam structure. Cnsider the elastic respnse due t a frced 6

38 vibratin f a fam cre sandwich beam with gemetry as shwn in Figure 4.. Plastic effects will be discussed later in this chapter. The assumed displacement field is given by u = u(, t) + zα (, t) w = w(, t) (4.) (4.3) where u is the in-plane defrmatin, w is the transverse deflectin at the beam midplane, and α is the shear rtatin in the -directin as shwn in Figure 4.5. The in-plane defrmatin u is cnsidered t be the sum f a translatin and α is the rtatin f linear element in the thrugh-thickness directin f the beam. Defrmatins in the y-directin are nt cnsidered fr a beam structure because it is s narrw (classical beam assumptin). z α ~FACESHEET~ ~FOAM CORE~ u ~FACESHEET~ W Figure 4.5 Displacement and shear rtatin f a symmetric prismatic beam. 7

39 The general agrangian tensr epressin fr strain cmpnents f a three dimensinal bdy is the fllwing Ref. []: ε = + jk u j, k + uk, j ui, jui, k (4.4) where the cmmas dente partial differentiatin with respect t the fllwing subscripted symbl, i, j, k =, y, z. Recall that the y directin will nt be cnsidered in the analysis f a sandwich beam. Substituting Equatins (4.) and (4.3) int (4.4) prvides the eplicit frm f the straindisplacement relatinships: ε u0 = α w + z + γ z w = + α (4.5) ε z = 0 Accrding t classical sandwich beam thery, there are additinal epressins t apprimate stress at a pint. The membrane stress resultant (N ), the bending stress resultant (M ), and the transverse shear resultant (Q ) are given by N = N h i= h σ dz i (4.6) 8

40 M = N h i= h σ zdz i (4.7) Q = N h i= h τ zi dz (4.8) where i is the layer number, N is the ttal number f layers, h is the ttal thickness f the sandwich beam and the depths f the layers measured frm the neutral ais are as shwn in Figure 4.6. ~FACESHEET~ Z 3 ~FOAM CORE~ Z Z Z 0 ~FACESHEET~ Figure 4.6 Distances thrugh thickness f the sandwich beam, measured frm the neutral ais f the beam. 9

41 Since bth the facesheets and fam cre are cnsidered t be elastic in this phase, the fllwing equatins define the stresses in the beam: σ = E i i ε (4.9) τ = z G i i γ y (4.0) Per the gemetry defined in Figure 4.: h = h + H (4.) The mtin f the beam is a special case f the Euler-agrangian equatins f mtin f a sandwich plate under unifrm pressure pulse lad which are presented by Chapagain []. These equatins are given in terms f displacement and rtatin fr fully elastic respnse t an applied pressure pulse lad: M w N t w Q = p (4.) I α M t + Q = 0 (4.3) where p is the pressure pulse lad described by Equatin (4.), M is effective mass and I is effective rtary inertia f the membrane, bending and transverse shear resultants are N = A ε (4.4) 30

42 M = D α (4.5) w = A + α Q 55 (4.6) where A is the membrane stiffness, D is the bending stiffness and A 55 is the transverse shear stiffness. The membrane stiffness term A is A = h h be i dz (4.7) Distances thrugh the thickness f the sandwich beam, measuring frm the neutral ais t the inner and uter surfaces f the facesheet are shwn in Figure 4.6. Integrating ver the thickness f the sandwich beam, this epressin reduces t A = ( E h E H )b f + c (4.8) where E f is the elastic mdulus f the facesheet and E c is the tensile mdulus f the fam cre. 3

43 3 The bending stiffness D in Equatin (4.5) is defined by (4.9) Integrating thrugh the thickness f the sandwich beam this epressin becmes (4.0) ikewise, the transverse shear stiffness A 55 fr a sandwich beam can be calculated frm (4.) Integrating ver the thickness f the sandwich beam, Equatin (4.) reduces t (4.) where G f is the shear mdulus f the facesheet and G c is the shear mdulus f the fam cre. ( ) = h h i dz h z Q b A + + = H h H G H h H h h G b A c f = h h i dz be z D b H E h H H b E D c f + =

44 Per Vinsn [], if h << H per the gemetry defined in Figure 4., Equatin (4.) can be apprimated by A 55 = G Hb c (4.3) Fr the symmetric sandwich beam, the equivalent mass and rtary inertia are h M = ρ i dz h (4.4) and I = h h ρ z i dz (4.5) Integrating thrugh the thickness f the beam gives M = ρ h + ρ H f c (4.6) and I ( z z ) + ρ ( z z ) + ( z z ) = ρ f c ρ f (4.7) where ρ c is the density f the fam cre and ρ f is the density f the facesheets. 33

45 Assume the fllwing shape functins which satisfy the simply-supprted bundary cnditin f the beam: π w = W sin (4.8) π α = Γ cs (4.9) Equatins (4.8) and (4.9) are a first-term apprimatin f a cmplete Furier series representatin f w and α. and =: Fr a simply-supprted beam, the fllwing bundary cnditins apply at =0 u =0 w=0 (4.30) M=0 The membrane strain ε can be written as u0 w ε = + 0 (4.3) Assume that membrane strain ε is cnstant s that ε = 0 (4.3) 34

46 35 Nw, let (4.33) Integrating bth sides f Equatin (4.33) ver the length f the beam and slving fr Δ gives (4.34) Applying bundary cnditins fr u in Equatin (4.30) gives (4.35) Upn substituting the shape functin defined in Equatin (4.8) int (4.35) ne gets (4.36) After integrating ver the length f the beam, Equatin (4.36) reduces t (4.37) h w u Δ = + = ε + = Δ d w u h 0 = Δ d w h 0 = Δ d W h 0 cs π π 3 W h = Δ π

47 36 Substituting Equatin (4.4) int (4.) gives (4.38) Differentiating with respect t, this epressin becmes (4.39) Substituting the shape functin defined in Equatin (4.8), the epressin fr ε defined in Equatin (4.33) and the epressin fr Δ defined in Equatin (4.37), ne gets (4.40) 4.. Frced Elastic Respnse Substituting Equatins (4.4) thrugh (4.6), (4.8) and (4.9) int Equatins (4.) and (4.3) gives (4.4) and (4.4) = w A w N ε w A w N = ε W A w N π π sin = = Γ + + Γ + D A W A I π π && = Γ p A W A W A M W π π π π &&

48 where the unifrmly distributed pressure pulse lad is 4 p( t) p = π (4.43) and p(t) is defined in Equatin (4.). Nte that p is the first term f a Furier series representatin f a unifrm pressure lad. 4.. Elastic-Plastic Respnse During elastic-plastic respnse plasticity is nly induced in the fam cre. The facesheets remain linear elastic. T derive equatins f mtin fr elastic-plastic respnse, it is assumed that the bending mment and membrane stress resultant, M and N, are elastic and given by epressins used in Sectin 4... This is because M and N are primarily due t the facesheets, which are linear elastic. Plasticity is intrduced int the beam in the transverse shear resultant Q. Recall that the PVC H00 is assumed t behave perfectly plastic nce the deflectin f the beam is large enugh t cause the transverse shear stress in the fam cre t eceed the transverse shear yield stress τ. The perfectly plastic behavir is shwn by the flat sectin f the shear stress-strain curve between Pint A and Pint B in Figure 4.4. The transverse shear Q can be calculated frm the yielding transverse shear 37

49 stress and the gemetry given in Figure 4. by Q = τ H (4.44) The relatinship between transverse shear resultant Q and transverse shear strains γ z is shwn in Figure 4.7. The slpe f the linear prtin f the curve is A 55 the transverse shear stiffness f the beam. Q Q 0 A 55 γ γ z Figure 4.7 Tranverse shear vs. Transverse shear strain. The plastic regin f the fam cre will first develp at the pint f maimum transverse shear stress and prpagate dwn the length f the beam as lng as the transverse shear stress remains at r abve the yielding transverse shear stress. The transverse shear strain is given by γ z π = γ 3 cs (4.45) 38

50 Fr a unifrm pressure pulse lad n a simply-supprted prismatic beam, the maimum transverse shear stresses ccur at the supprts as shwn by the static shear frce diagram in Figure 4.. In this analysis, the length f the plastic regin f the PVC H00 fam will be referred t as ζ, as shwn in Figure 4.8. ζ -ζ ζ Figure 4.8 Plastic znes alng the length f the beam. The amplitude f the transverse shear strain can be lcally defined as γ π 3 = W + Γ (4.46) At the yielding transverse shear strain, Equatin (4.46) is equal t γ. 39

51 40 Based n the dimensins defined in Figure 4.7 and substituting Equatin (4.46) int (4.45), ne gets (4.47) Slving Equatin (4.47) fr the regin ζ gives (4.48) The transverse shear resultant then becmes (4.49) Substituting Equatin (4.49) int the Euler-agrangian equatins f mtin defined generically in Equatins (4.) and (4.3) gives equatins f mtin in three separate regins. + Γ = W πς π γ cs + Γ = cs W π γ π ς < < < < + < < = Q w A Q Q ς ς ς α ς,, 0, 55

52 4 Fr the plastic regin ς < < 0 : (4.50) (4.5) Fr the elastic regin ς ς < < : (4.5) (4.53) Fr the plastic regin < < ς : (4.54) (4.55) As mentined earlier, the bending and membrane frce resultants, M and, N remain the same as in purely elastic respnse and are given by epressins in Sectin 4... p w N t w M = 0 = + Q M t I α p w A w N t w M = + α = + + α α w A M t I p w N t w M = 0 = Q M t I α

53 One can slve these equatins f mtin, Equatins (4.50) thrugh (4.55), fr the entire beam using the methd described in Appendi B. The elastic-plastic equatins f mtin then becme 4 A 3 M W&& π π π + W + fa55 W + Γ = 4 p (4.56) and I Γ&& + D π π Γ + f A55 W + Γ + Qd = 0 (4.57) where f ς πς = sin π + (4.58) f ς πς = sin π (4.59) Q d 4 πς = Q sin π (4.60) 4

54 CHAPTER V FINITE EEMENT ANAYSIS Finite element analysis using ABAQUS Eplicit was cmpleted n a fam cre sandwich beam subjected t a pressure pulse lad in rder t verify the analytical slutin discussed in Chapter IV. The beam gemetry and lading are described in Figures 4. and 4.. Details and parameters used t develp the finite element analysis (FEA) mdel will be discussed in this chapter. 5. Finite Element Analysis Mdel A symmetric simply-supprted fam-cre sandwich beam with aluminum facesheets was cnsidered. The full span f the beam is =76mm, but due t symmetry abut the centerline f the beam, nly half f the beam was mdeled. Hence a prtin cntaining ne supprt and a length f 38mm was used. The width f the facesheets and fam cre was b=5mm. The thickness f each aluminum facesheet was h=3.75mm, while the thickness f the PVC H00 fam cre was the purchased thickness f the fam sheet, H=5mm. The FEA mdel fr the simply-supprted fam cre sandwich beam and the glbal crdinate system are shwn in Figure

55 Figure 5. Finite element analysis mdel f fam-cre sandwich beam. The facesheet material is aluminum (Al 606). Prperties f the facesheet material are listed in Table 4.. The fam cre is Divinycell PVC H00, the prperties f which are listed in Table 3.. The material prperties fr the aluminum facesheets were specified as elastic and istrpic. Material prperties f the fam cre included the elastic istrpic definitin, as well as a plastic definitin. The plastic definitin was specified as *Crushable Fam, hardening=isotropic, and the plastic Pissn s ratin was specified t be zer. The plasticity curve fr the fam cre was taken frm Mines et al. []. A unifrm pressure pulse lad was applied t the tp facesheet f the fam cre sandwich beam. The pressure pulse lad was specified as a triangular pressure pulse 44

56 lad, with magnitude was p (.758 MPa) and time duratin T (0. ms). The pressure lad time histry is described by Equatin (3.). The simply-supprted bundary cnditin fr the beam supprt was created using frictinless, rigid rllers n the tp and bttm facesheet. A shrt length f beam was set t verhang the rller supprts t prevent slippage at any pint during the vibratin f the beam. Per Chapagain [], the rller supprts were required due t cncentrated lcalized stresses that wuld ccur in the plymeric fam if supprts were directly attached t the fam material. The frictinless rllers were fied t a reference pint. Since nly half f the beam was mdeled, the XSYMM bundary cnditin was used t cnstrain the symmetry plane f the beam. Bundary cnstraints used in the FEA mdel are shwn in Figure 5.. RIGID ROER (TYP.) X-SYMMETRY Figure 5. Bundary cnditins f the fam-cre sandwich beam mdel. 45

57 Interactin relatinships were applied at several pints in the mdel. The interactin between the frictinless rllers and the facesheets was defined as *Interactin, surface-t-surface cntact (Eplicit), with the mechanical cnstraint frmulatin *Penalty cntact methd. The interactin prperties assigned t the frictinless rller t facesheet interface was *Interactin, tangential behavir, frictin frmulatin, frictinless. In the nrmal directin the prperty assigned was *Interactin, nrmal behavir, hard cntact. Bth the tp and bttm facesheet were tied t the fam cre using *Tie cnstraint, surface t surface discretizatin methd. The tp and bttm surface f the fam cre were each tied as slave surfaces t mve similarly with the facesheet (master surface). The facesheets and cre were meshed using eight-nde linear brick elements (C3D8). Full-integratin and default distrtin cntrl was chsen fr the integratin type. A finer mesh was used near the supprt where the regin f plasticity was epected t develp in the fam cre. The mesh is shwn in Figure 5.3. Figure 5.3 Meshed finite element analysis mdel f the fam-cre sandwich beam. 46

58 The mdel was then analyzed in ABAQUS sftware using the Dynamic Eplicit slver. Effects due t nnlinear gemetry were activated. 5. Elastic Respnse In the purely elastic respnse, the fam was mdeled as a linear elastic istrpic material (i.e., the crushable fam prperties were suppressed). The purely elastic respnse f the sandwich beam t an applied pressure pulse lad was btained frm the FEA. The utput f the FEA was graphed and cmpared t the analytical results btained by the analytical methd described in Chapter IV. The deflectin f the centerline f the beam (the ais f symmetry) was pltted against analytical results in Figure 5.4. The deflectins were measured at the midplane f the fam (the neutral ais f the beam). In additin, the transverse shear strains at the midplane f the fam cre near the frictinless rller supprt were pltted against the transverse shear strains calculated by MATHCAD sftware using the analytical methd and are shwn in Figure 5.5. The FEA results agree well with the analytical results. 47

59 FEA Analytical Displacement (m) Time (ms) Figure 5.4 Cmparisn f FEA and analytical deflectin at midspan f the beam Transverse Shear Strain (γ 3 ) FEA Analytical Time (ms) Figure 5.5 Cmparisn f FEA and analytical transverse shear strain at midplane f the fam-cre near the rller supprts. 48

60 5.3 Elastic-Plastic Respnse In the elastic-plastic respnse, the fam plasticity was included in additin t elastic behavir. The PVC H00 fam cre was mdeled as crushable fam with istrpic hardening with a cmpressin yield stress rati f.73. Plasticity develps in the fam cre when the transverse shear strains reach the transverse yielding strain (γ in Figure 4.4). Plasticity in the fam cre will first develp near the frictinless rller supprts where the transverse shear stresses in the beam are at maimum. The transverse shear strains in the fam cre frm the FEA are shwn in Figure 5.6. Plasticity ccurs when the transverse shear strain is greater than 0.095, and thus plastic regins ccur at the lcatins epected by analytical results (i.e. plastic regin shwn in Figure 4.8). The facesheets have been remved in the FEA s that transverse strains in the cre are mre visible. The deflectins at the centerline f the beam taken at the midplane f the fam cre were pltted against analytical results in Figure 5.7. There is relatively gd agreement in the centerline deflectin. Transverse shear strains at the midplane f the fam cre near the frictinless rller supprt were als pltted against analytical results in Figure 5.8. There is als relatively gd agreement between the tw results with the eceptin that the peak strain in the FEA ccurred a little later. The results btained by FEA using ABAQUS agree well with the analytical results calculated using MATHCAD. 49

61 Figure 5.6 Plastic regins in the fam-cre per finite element analysis Displacement (m) FEA Plasticity Begins -0.0 Analytical Time (ms) Figure 5.7 Cmparisn f FEA and analytical deflectin at midspan f the beam. 50

62 Transverse Shear Strain (γ 3 ) Plasticity Begins Analytical FEA Time (ms) Figure 5.8 Cmparisn f FEA and analytical transverse shear strain at midplane f the fam-cre near the rller supprts. 5

63 CHAPTER VI CONCUDING REMARKS An analytical slutin fr btaining the large amplitude, damped respnse f a crushable fam-cre sandwich beam subjected t pressure pulse lading has been presented in this study. The simply-supprted sandwich beam cmprised f aluminum facesheets and a PVC fam cre was analyzed fr its respnse t an applied unifrmly distributed pressure pulse lad. Material testing was first cnducted n PVC H00 fam specimens under uniaial mntnic and cyclic cmpressive lading. Based n the cmpressive mntnic lad tests, it can be cncluded that PVC H00 fam is linear elastic-perfectly plastic. Bth the mntnic and cyclic cmpressive stress-strain curves at varius strain rates shw that the PVC H00 fam is viscelastic and viscplastic. The cmpressive cyclic stress-strain curves shw that damage is sustained by the fam n cnsecutive lading cycles which will prvide additinal damping t a dynamic system. This damping is primarily viscelastic. 5

64 The equatins f mtin fr the sandwich beam were develped cnsidering first rder shear defrmatin and membrane stretching. The facesheets f the sandwich beam were cnsidered t be elastic and istrpic whereas the plymeric fam cre material behavir was elastic-plastic. The initial respnse f the beam was elastic until the nset f plasticity when stresses in the fam cre eceeded the yield strength f the fam. Analytical results were verified by finite element analysis using ABAQUS Eplicit. The deflectins f the beam at the midspan and the transverse shear strains btained by finite element analysis agreed well with the analytical results. The analytical mdel f the fam-cre sandwich beam culd be further develped t incrprate cnsecutive unlading / lading cycles with viscelastic damping. This culd be the fundatin f an eperiment t etract viscelastic damping prperties f the fam. In rder t evaluate the damping prperties f the PVC H00 fam, an eperiment is recmmended which wuld evaluate the respnse f a simply-supprted beam under an impact lad. The sandwich beam wuld be cmprised f elastic facesheets and a PVC H00 fam cre which wuld be bnded t the facesheets. T avid high lcalized stresses in the fam cre, it is recmmended that frictinless rllers be used t supprt the beam. The frictinless rllers shuld be placed symmetrically n bth facesheets in rder t maintain cnsistent supprt regardless f the directin f the beam deflectin. Analytical results using the methd utlined in this study shuld be used t predict the lad which will induce plasticity in the fam cre, but limit the plastic regin t nly a prtin f the beam span. It is recmmended that the impact lad applied t the 53