response-of-piezoelectric-elasto-plastic-laminate

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this article is about Nonlinear dynamic response of piezoelectric elasto-plastic laminated plates with damage_fu_china
response-of-piezoelectric-elasto-plastic-laminate
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<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/625956ce92dc900e62b90def/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625956ce92dc900e62b90def/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Nonlinear<span class="_ _0"> </span>dynamic<span class="_ _0"> </span>response<span class="_ _0"> </span>of<span class="_ _0"> </span>piezoelectric<span class="_ _0"> </span>elasto-plastic<span class="_ _0"> </span>laminated<span class="_ _0"> </span>plates</div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">with<span class="_ _0"> </span>damage</div><div class="t m0 x1 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0">Y.P.<span class="_ _1"> </span>Tian</div><div class="t m1 x2 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">a,</div><div class="t m1 x3 h5 y5 ff2 fs2 fc0 sc0 ls0 ws0">*</div><div class="t m0 x4 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0">,<span class="_ _1"> </span>Y.M.<span class="_ _1"> </span>Fu</div><div class="t m1 x5 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">a</div><div class="t m0 x6 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0">,<span class="_ _1"> </span>Y.<span class="_ _1"> </span>Wang</div><div class="t m1 x7 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">b</div><div class="t m2 x1 h6 y6 ff1 fs3 fc0 sc0 ls0 ws0">a</div><div class="t m3 x8 h7 y7 ff3 fs4 fc0 sc0 ls0 ws0">College<span class="_ _2"> </span>of<span class="_ _2"> </span>Mechanics<span class="_ _2"> </span>and<span class="_ _2"> </span>Aerospace,<span class="_ _2"> </span>Hunan<span class="_ _2"> </span>University,<span class="_ _2"> </span>Changsha<span class="_ _2"> </span>410082,<span class="_ _2"> </span>PR<span class="_ _2"> </span>China</div><div class="t m2 x1 h6 y8 ff1 fs3 fc0 sc0 ls0 ws0">b</div><div class="t m3 x8 h7 y9 ff3 fs4 fc0 sc0 ls0 ws0">Department<span class="_ _2"> </span>of<span class="_ _2"> </span>Mechanics,<span class="_ _2"> </span>Zhejiang<span class="_ _2"> </span>University,<span class="_ _2"> </span>Hangzhou<span class="_ _2"> </span>310027,<span class="_ _2"> </span>PR<span class="_ _2"> </span>China</div><div class="t m0 x1 h8 ya ff4 fs5 fc0 sc0 ls1 ws0">article<span class="_ _3"> </span>info</div><div class="t m3 x1 h7 yb ff3 fs4 fc0 sc0 ls0 ws0">Article<span class="_ _2"> </span>history:</div><div class="t m3 x1 h9 yc ff1 fs4 fc0 sc0 ls0 ws0">Available<span class="_ _2"> </span>online<span class="_ _2"> </span>7<span class="_ _2"> </span>March<span class="_ _2"> </span>2008</div><div class="t m3 x1 h7 yd ff3 fs4 fc0 sc0 ls0 ws0">Keywords:</div><div class="t m3 x1 h9 ye ff1 fs4 fc0 sc0 ls0 ws0">Piezoelectric<span class="_ _2"> </span>laminated<span class="_ _2"> </span>plates</div><div class="t m3 x1 h9 yf ff1 fs4 fc0 sc0 ls0 ws0">Elasto-plastic</div><div class="t m3 x1 h9 y10 ff1 fs4 fc0 sc0 ls0 ws0">Damage<span class="_ _2"> </span>effects</div><div class="t m3 x1 h9 y11 ff1 fs4 fc0 sc0 ls0 ws0">Nonlinear<span class="_ _2"> </span>dynamic<span class="_ _2"> </span>response</div><div class="t m0 x9 h8 y12 ff4 fs5 fc0 sc0 ls2 ws0">abstract</div><div class="t m4 x9 ha y13 ff1 fs6 fc0 sc0 ls0 ws0">Based<span class="_ _2"> </span>on<span class="_ _4"> </span>the<span class="_ _2"> </span>elasto-plastic<span class="_ _4"> </span>mechanics<span class="_ _2"> </span>and<span class="_ _4"> </span>continuum<span class="_ _2"> </span>dama<span class="_ _5"></span>ge<span class="_ _2"> </span>theory,<span class="_ _4"> </span>a<span class="_ _2"> </span>yield<span class="_ _4"> </span>criterion<span class="_ _2"> </span>that<span class="_ _4"> </span>is<span class="_ _2"> </span>related<span class="_ _4"> </span>to</div><div class="t m4 x9 ha y14 ff1 fs6 fc0 sc0 ls0 ws0">the<span class="_ _6"> </span>spherical<span class="_ _6"> </span>stress<span class="_ _6"> </span>tensor<span class="_ _6"> </span>is<span class="_ _6"> </span>proposed<span class="_ _2"> </span>to<span class="_ _6"> </span>describe<span class="_ _6"> </span>the<span class="_ _6"> </span>mixed<span class="_ _6"> </span>hardening<span class="_ _6"> </span>of<span class="_ _6"> </span>damaged<span class="_ _6"> </span>orthotropic<span class="_ _6"> </span>materials,</div><div class="t m4 x9 ha y15 ff1 fs6 fc0 sc0 ls0 ws0">and<span class="_ _6"> </span>the<span class="_ _2"> </span>dimensionless<span class="_ _2"> </span>form<span class="_ _6"> </span>of<span class="_ _2"> </span>which<span class="_ _6"> </span>is<span class="_ _2"> </span>isomorphic<span class="_ _6"> </span>with<span class="_ _2"> </span>the<span class="_ _6"> </span>Mises<span class="_ _2"> </span>criterion<span class="_ _2"> </span>for<span class="_ _6"> </span>isotropic<span class="_ _2"> </span>materials.<span class="_ _6"> </span>Then,</div><div class="t m4 x9 ha y16 ff1 fs6 fc0 sc0 ls0 ws0">the<span class="_ _2"> </span>incremental<span class="_ _2"> </span>elasto-plastic<span class="_ _4"> </span>damage<span class="_ _2"> </span>constitutive<span class="_ _4"> </span>equations<span class="_ _2"> </span>and<span class="_ _4"> </span>damage<span class="_ _2"> </span>evolution<span class="_ _2"> </span>equations<span class="_ _4"> </span>of<span class="_ _2"> </span>ortho-</div><div class="t m4 x9 ha y17 ff1 fs6 fc0 sc0 ls0 ws0">tropic<span class="_ _2"> </span>materials<span class="_ _2"> </span>are<span class="_ _2"> </span>established.<span class="_ _4"> </span>By<span class="_ _2"> </span>using<span class="_ _2"> </span>the<span class="_ _2"> </span>classical<span class="_ _4"> </span>nonlinear<span class="_ _2"> </span>plate<span class="_ _2"> </span>theory,<span class="_ _2"> </span>the<span class="_ _4"> </span>incremental<span class="_ _2"> </span>nonlinear</div><div class="t m4 x9 ha y18 ff1 fs6 fc0 sc0 ls0 ws0">dynamic<span class="_ _7"> </span>governing<span class="_ _7"> </span>equations<span class="_ _7"> </span>of<span class="_ _7"> </span>the<span class="_ _7"> </span>piezoelectric<span class="_ _8"> </span>elasto-plastic<span class="_ _7"> </span>laminated<span class="_ _7"> </span>plates<span class="_ _7"> </span>with<span class="_ _8"> </span>damage<span class="_ _7"> </span>are</div><div class="t m4 x9 ha y19 ff1 fs6 fc0 sc0 ls0 ws0">obtained,<span class="_ _6"> </span>and<span class="_ _6"> </span>the<span class="_ _6"> </span>equations<span class="_ _6"> </span>are<span class="_ _2"> </span>solved<span class="_ _6"> </span>by<span class="_ _6"> </span>&#64257;nite<span class="_ _6"> </span>difference<span class="_ _6"> </span>method<span class="_ _6"> </span>and<span class="_ _6"> </span>iteration<span class="_ _6"> </span>method.<span class="_ _2"> </span>In<span class="_ _6"> </span>the<span class="_ _6"> </span>numerical</div><div class="t m4 x9 ha y1a ff1 fs6 fc0 sc0 ls0 ws0">examples,<span class="_ _7"> </span>the<span class="_ _1"> </span>effect<span class="_ _7"> </span>of<span class="_ _7"> </span>damage<span class="_ _7"> </span>evolution,<span class="_ _7"> </span>piezoelectric<span class="_ _7"> </span>effect<span class="_ _1"> </span>and<span class="_ _7"> </span>external<span class="_ _7"> </span>loads<span class="_ _7"> </span>on<span class="_ _7"> </span>the<span class="_ _7"> </span>nonlinear</div><div class="t m4 x9 ha y1b ff1 fs6 fc0 sc0 ls0 ws0">dynamic<span class="_ _4"> </span>response<span class="_ _4"> </span>of<span class="_ _2"> </span>elasto-plastic<span class="_ _4"> </span>laminated<span class="_ _4"> </span>plates<span class="_ _4"> </span>are<span class="_ _4"> </span>discussed<span class="_ _4"> </span>detailed.<span class="_ _4"> </span>The<span class="_ _4"> </span>numerical<span class="_ _4"> </span>results<span class="_ _4"> </span>show</div><div class="t m4 x9 ha y1c ff1 fs6 fc0 sc0 ls0 ws0">that<span class="_ _2"> </span>the<span class="_ _6"> </span>mechanical<span class="_ _2"> </span>properties<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _6"> </span>structures<span class="_ _2"> </span>change<span class="_ _2"> </span>remarkably<span class="_ _2"> </span>when<span class="_ _6"> </span>considering<span class="_ _2"> </span>the<span class="_ _2"> </span>damage,<span class="_ _2"> </span>dam-</div><div class="t m4 x9 ha y1d ff1 fs6 fc0 sc0 ls0 ws0">age<span class="_ _2"> </span>evolu<span class="_ _5"></span>tion<span class="_ _2"> </span>and<span class="_ _4"> </span>piezoelectric<span class="_ _4"> </span>effects.</div><div class="t m4 xa ha y1e ff5 fs6 fc0 sc0 ls0 ws0">&#211;<span class="_ _4"> </span><span class="ff1">2008<span class="_ _2"> </span>Elsevier<span class="_ _4"> </span>Ltd.<span class="_ _4"> </span>All<span class="_ _4"> </span>rights<span class="_ _4"> </span>reserved.</span></div><div class="t m4 x1 hb y1f ff6 fs7 fc0 sc0 ls0 ws0">1.<span class="_ _7"> </span>Introduction</div><div class="t m4 xb hc y20 ff1 fs7 fc0 sc0 ls0 ws0">Lighter<span class="_ _9"> </span>weight,<span class="_ _9"> </span>higher<span class="_ _9"> </span>strength<span class="_ _9"> </span>and<span class="_ _9"> </span>higher<span class="_ _9"> </span>energy<span class="_ _9"> </span>dissipation</div><div class="t m4 x1 hc y21 ff1 fs7 fc0 sc0 ls0 ws0">capability<span class="_ _1"> </span>makes<span class="_ _1"> </span>the<span class="_ _1"> </span>laminated<span class="_ _1"> </span>composite<span class="_ _1"> </span>materials<span class="_ _1"> </span>extensively</div><div class="t m4 x1 hc y22 ff1 fs7 fc0 sc0 ls0 ws0">used<span class="_ _a"> </span>in<span class="_ _a"> </span>automotive<span class="_ _a"> </span>and<span class="_ _a"> </span>aerospace<span class="_ _a"> </span>applications.<span class="_ _a"> </span>As<span class="_ _4"> </span>structures<span class="_ _a"> </span>can</div><div class="t m4 x1 hc y23 ff1 fs7 fc0 sc0 ls0 ws0">still<span class="_ _a"> </span>bear<span class="_ _a"> </span>loads<span class="_ _a"> </span>after<span class="_ _a"> </span>exceed<span class="_ _a"> </span>their<span class="_ _a"> </span>yield<span class="_ _4"> </span>limits,<span class="_ _a"> </span>it<span class="_ _a"> </span>is<span class="_ _a"> </span>diseconomy<span class="_ _a"> </span>to</div><div class="t m4 x1 hc y24 ff1 fs7 fc0 sc0 ls0 ws0">design<span class="_ _a"> </span>structures<span class="_ _a"> </span>by<span class="_ _a"> </span>using<span class="_ _a"> </span>the<span class="_ _a"> </span>elastic<span class="_ _a"> </span>theory,<span class="_ _a"> </span>and<span class="_ _a"> </span>the<span class="_ _a"> </span>mechanical</div><div class="t m4 x1 hc y25 ff1 fs7 fc0 sc0 ls0 ws0">properties<span class="_ _9"> </span>of<span class="_ _9"> </span>structures<span class="_ _9"> </span>in<span class="_ _a"> </span>plastic<span class="_ _9"> </span>stage<span class="_ _9"> </span>are<span class="_ _9"> </span>necessary<span class="_ _9"> </span>to<span class="_ _9"> </span>be<span class="_ _9"> </span>ana-</div><div class="t m4 x1 hc y26 ff1 fs7 fc0 sc0 ls0 ws0">lyzed.<span class="_ _7"> </span>Moreover,<span class="_ _7"> </span>due<span class="_ _7"> </span>to<span class="_ _7"> </span>the<span class="_ _7"> </span>material<span class="_ _7"> </span>defects<span class="_ _7"> </span>or<span class="_ _7"> </span>the<span class="_ _7"> </span>technology</div><div class="t m4 x1 hc y27 ff1 fs7 fc0 sc0 ls0 ws0">problems<span class="_ _a"> </span>in<span class="_ _4"> </span>the<span class="_ _a"> </span>machine-shaping,<span class="_ _4"> </span>some<span class="_ _a"> </span>damages<span class="_ _a"> </span>may<span class="_ _4"> </span>be<span class="_ _a"> </span>created</div><div class="t m4 x1 hc y28 ff1 fs7 fc0 sc0 ls0 ws0">inevitably,<span class="_ _a"> </span>inner<span class="_ _9"> </span>or<span class="_ _a"> </span>outside<span class="_ _9"> </span>of<span class="_ _9"> </span>the<span class="_ _a"> </span>structures,<span class="_ _a"> </span>these<span class="_ _9"> </span>damages<span class="_ _9"> </span>will</div><div class="t m4 x1 hc y29 ff1 fs7 fc0 sc0 ls0 ws0">decrease<span class="_ _7"> </span>the<span class="_ _7"> </span>stiffness<span class="_ _7"> </span>and<span class="_ _8"> </span>change<span class="_ _7"> </span>the<span class="_ _7"> </span>mechanical<span class="_ _7"> </span>properties<span class="_ _7"> </span>of</div><div class="t m4 x1 hc y2a ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _4"> </span>structures.</div><div class="t m4 xb hc y2b ff1 fs7 fc0 sc0 ls0 ws0">In<span class="_ _2"> </span>recent<span class="_ _2"> </span>years,<span class="_ _2"> </span>the<span class="_ _2"> </span>study<span class="_ _2"> </span>of<span class="_ _4"> </span>elasto-plastic<span class="_ _2"> </span>constitutive<span class="_ _2"> </span>relations</div><div class="t m4 x1 hc y2c ff1 fs7 fc0 sc0 ls0 ws0">of<span class="_ _2"> </span>materials<span class="_ _4"> </span>has<span class="_ _2"> </span>achieved<span class="_ _4"> </span>great<span class="_ _2"> </span>progress.<span class="_ _4"> </span>Zhu<span class="_ _2"> </span><span class="fc1">[1]<span class="_ _4"> </span></span>developed<span class="_ _2"> </span>an<span class="_ _4"> </span>en-</div><div class="t m4 x1 hc y2d ff1 fs7 fc0 sc0 ls0 ws0">ergy-based<span class="_ _a"> </span>anisotropic<span class="_ _a"> </span>damage<span class="_ _9"> </span>model<span class="_ _a"> </span>at<span class="_ _9"> </span>&#64257;nite<span class="_ _a"> </span>strains<span class="_ _a"> </span>for<span class="_ _9"> </span>ductile</div><div class="t m4 x1 hc y2e ff1 fs7 fc0 sc0 ls0 ws0">fracture.<span class="_ _2"> </span>Since<span class="_ _6"> </span>the<span class="_ _2"> </span>onset<span class="_ _6"> </span>of<span class="_ _2"> </span>damage<span class="_ _2"> </span>and<span class="_ _6"> </span>plastic<span class="_ _2"> </span>yielding<span class="_ _6"> </span>begins<span class="_ _2"> </span>very</div><div class="t m4 x1 hc y2f ff1 fs7 fc0 sc0 ls0 ws0">early<span class="_ _2"> </span>in<span class="_ _2"> </span>the<span class="_ _2"> </span>loading<span class="_ _2"> </span>process,<span class="_ _2"> </span>Taehyo<span class="_ _2"> </span>and<span class="_ _4"> </span>Voyiadjis<span class="_ _2"> </span><span class="fc1">[2]<span class="_ _2"> </span></span>analyzed<span class="_ _2"> </span>the</div><div class="t m4 x1 hc y30 ff1 fs7 fc0 sc0 ls0 ws0">damage<span class="_ _9"> </span>and<span class="_ _1"> </span>elasto-plastic<span class="_ _1"> </span>behavior<span class="_ _1"> </span>of<span class="_ _1"> </span>metal<span class="_ _1"> </span>matrix<span class="_ _9"> </span>composites.</div><div class="t m4 x1 hc y31 ff1 fs7 fc0 sc0 ls0 ws0">Using<span class="_ _b"> </span>the<span class="_ _b"> </span>irreversible<span class="_ _b"> </span>thermodynamics<span class="_ _b"> </span>theory,<span class="_ _b"> </span>Hayakawa<span class="_ _b"> </span><span class="fc1">[3]</span></div><div class="t m4 x1 hc y32 ff1 fs7 fc0 sc0 ls0 ws0">developed<span class="_ _7"> </span>the<span class="_ _7"> </span>constitutive<span class="_ _1"> </span>and<span class="_ _7"> </span>damage<span class="_ _7"> </span>evolution<span class="_ _7"> </span>equations<span class="_ _7"> </span>for</div><div class="t m4 x1 hc y33 ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _a"> </span>elasto-plastic<span class="_ _a"> </span>damage<span class="_ _4"> </span>materials,<span class="_ _a"> </span>and<span class="_ _a"> </span>applied<span class="_ _a"> </span>the<span class="_ _a"> </span>equations<span class="_ _4"> </span>to</div><div class="t m4 x1 hc y34 ff1 fs7 fc0 sc0 ls0 ws0">elucidate<span class="_ _2"> </span>the<span class="_ _6"> </span>damage<span class="_ _2"> </span>process<span class="_ _2"> </span>of<span class="_ _6"> </span>tubular<span class="_ _2"> </span>specimens<span class="_ _6"> </span>of<span class="_ _2"> </span>spheroidized</div><div class="t m4 x1 hc y35 ff1 fs7 fc0 sc0 ls0 ws0">graphite<span class="_ _6"> </span>cast<span class="_ _2"> </span>iron<span class="_ _6"> </span>under<span class="_ _2"> </span>tension<span class="_ _6"> </span>and<span class="_ _2"> </span>torsion.<span class="_ _6"> </span>Voyiadjis<span class="_ _2"> </span>and<span class="_ _6"> </span>Deliktas</div><div class="t m4 x1 hc y36 ff1 fs7 fc1 sc0 ls0 ws0">[4]<span class="_ _9"> </span><span class="fc0">introduced<span class="_ _9"> </span>the<span class="_ _9"> </span>elasto-plastic<span class="_ _9"> </span>damage<span class="_ _9"> </span>model<span class="_ _9"> </span>of<span class="_ _9"> </span>rate-indepen-</span></div><div class="t m4 xc hc y37 ff1 fs7 fc0 sc0 ls0 ws0">dent<span class="_ _9"> </span>and<span class="_ _1"> </span>rate-dependent,<span class="_ _1"> </span>respectively,<span class="_ _1"> </span>and<span class="_ _9"> </span>based<span class="_ _1"> </span>on<span class="_ _1"> </span>the<span class="_ _1"> </span>model,</div><div class="t m4 xc hc y38 ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _9"> </span>stress&#8211;strain<span class="_ _1"> </span>relationships<span class="_ _1"> </span>of<span class="_ _1"> </span>composite<span class="_ _1"> </span>materials<span class="_ _1"> </span>are<span class="_ _1"> </span>calcu-</div><div class="t m4 xc hc y39 ff1 fs7 fc0 sc0 ls0 ws0">lated,<span class="_ _4"> </span>then<span class="_ _4"> </span>compared<span class="_ _4"> </span>them<span class="_ _4"> </span>to<span class="_ _4"> </span>experimental<span class="_ _4"> </span>results.<span class="_ _4"> </span>By<span class="_ _4"> </span>taking<span class="_ _4"> </span>into</div><div class="t m4 xc hc y3a ff1 fs7 fc0 sc0 ls0 ws0">account<span class="_ _2"> </span>the<span class="_ _4"> </span>kinematic<span class="_ _4"> </span>description<span class="_ _2"> </span>of<span class="_ _4"> </span>damage,<span class="_ _4"> </span>the<span class="_ _4"> </span>anisotropic<span class="_ _2"> </span>duc-</div><div class="t m4 xc hc y3b ff1 fs7 fc0 sc0 ls0 ws0">tile<span class="_ _7"> </span>models<span class="_ _7"> </span>were<span class="_ _7"> </span>proposed<span class="_ _7"> </span>by<span class="_ _7"> </span>Br&#252;nig<span class="_ _7"> </span><span class="fc1">[5]</span>,<span class="_ _8"> </span>Br&#252;nig<span class="_ _7"> </span>and<span class="_ _7"> </span>Ricci<span class="_ _7"> </span><span class="fc1">[6]</span>.</div><div class="t m4 xc hc y3c ff1 fs7 fc0 sc0 ls0 ws0">Tham<span class="_ _a"> </span><span class="fc1">[7]<span class="_ _9"> </span></span>presented<span class="_ _a"> </span>an<span class="_ _9"> </span>elasto-plastic<span class="_ _a"> </span>damage<span class="_ _a"> </span>model<span class="_ _9"> </span>on<span class="_ _a"> </span>the<span class="_ _9"> </span>basis</div><div class="t m4 xc hc y3d ff1 fs7 fc0 sc0 ls0 ws0">of<span class="_ _2"> </span>irreversible<span class="_ _4"> </span>thermodynamics<span class="_ _2"> </span>and<span class="_ _4"> </span>internal<span class="_ _2"> </span>state<span class="_ _4"> </span>variable<span class="_ _2"> </span>formal-</div><div class="t m4 xc hc y3e ff1 fs7 fc0 sc0 ls0 ws0">ism<span class="_ _a"> </span>for<span class="_ _9"> </span>the<span class="_ _a"> </span>analysis<span class="_ _9"> </span>of<span class="_ _a"> </span>multi-layered<span class="_ _9"> </span>composites.<span class="_ _a"> </span>On<span class="_ _a"> </span>the<span class="_ _9"> </span>basis<span class="_ _a"> </span>of</div><div class="t m4 xc hc y3f ff1 fs7 fc0 sc0 ls0 ws0">Hill&#8217;s<span class="_ _6"> </span>yield<span class="_ _2"> </span>criterion,<span class="_ _6"> </span>Boutaous<span class="_ _2"> </span><span class="fc1">[8]<span class="_ _6"> </span></span>developed<span class="_ _2"> </span>an<span class="_ _6"> </span>elasto-plastic<span class="_ _2"> </span>dam-</div><div class="t m4 xc hc y40 ff1 fs7 fc0 sc0 ls0 ws0">age<span class="_ _2"> </span>model<span class="_ _2"> </span>and<span class="_ _2"> </span>analyzed<span class="_ _2"> </span>the<span class="_ _2"> </span>deformation<span class="_ _6"> </span>of<span class="_ _2"> </span>carbon<span class="_ _2"> </span>&#64257;bre<span class="_ _2"> </span>laminated</div><div class="t m4 xc hc y41 ff1 fs7 fc0 sc0 ls0 ws0">plates.<span class="_ _4"> </span>Within<span class="_ _a"> </span>a<span class="_ _4"> </span>thermodynamic<span class="_ _4"> </span>framework<span class="_ _4"> </span>and<span class="_ _a"> </span>using<span class="_ _4"> </span>continuum</div><div class="t m4 xc hc y42 ff1 fs7 fc0 sc0 ls0 ws0">damage<span class="_ _8"> </span>mechanics,<span class="_ _8"> </span>Edlund<span class="_ _0"> </span>and<span class="_ _8"> </span>Volgers<span class="_ _0"> </span><span class="fc1">[9]<span class="_ _8"> </span></span>derived<span class="_ _8"> </span>a<span class="_ _0"> </span>material</div><div class="t m4 xc hc y43 ff1 fs7 fc0 sc0 ls0 ws0">model<span class="_ _2"> </span>including<span class="_ _2"> </span>the<span class="_ _2"> </span>failure<span class="_ _2"> </span>behavior<span class="_ _2"> </span>for<span class="_ _2"> </span>a<span class="_ _2"> </span>thin<span class="_ _2"> </span>unidirectional<span class="_ _2"> </span>com-</div><div class="t m4 xc hc y44 ff1 fs7 fc0 sc0 ls0 ws0">posite<span class="_ _2"> </span>ply.<span class="_ _4"> </span>Jason<span class="_ _2"> </span><span class="fc1">[10]<span class="_ _2"> </span></span>thought<span class="_ _4"> </span>the<span class="_ _2"> </span>current<span class="_ _4"> </span>elastic<span class="_ _2"> </span>and<span class="_ _2"> </span>elasto-plastic</div><div class="t m4 xc hc y45 ff1 fs7 fc0 sc0 ls0 ws0">damage<span class="_ _9"> </span>models<span class="_ _1"> </span>failed<span class="_ _9"> </span>to<span class="_ _1"> </span>reproduce<span class="_ _1"> </span>the<span class="_ _9"> </span>unloading<span class="_ _1"> </span>slopes<span class="_ _9"> </span>during</div><div class="t m4 xc hc y46 ff1 fs7 fc0 sc0 ls0 ws0">cyclic<span class="_ _a"> </span>loading,<span class="_ _a"> </span>so<span class="_ _a"> </span>combined<span class="_ _a"> </span>the<span class="_ _9"> </span>plastic<span class="_ _a"> </span>hardening<span class="_ _a"> </span>model<span class="_ _a"> </span>and<span class="_ _a"> </span>iso-</div><div class="t m4 xc hc y47 ff1 fs7 fc0 sc0 ls0 ws0">tropic<span class="_ _a"> </span>damage<span class="_ _9"> </span>model,<span class="_ _9"> </span>a<span class="_ _a"> </span>simple<span class="_ _9"> </span>elasto-plastic<span class="_ _a"> </span>damage<span class="_ _9"> </span>model<span class="_ _a"> </span>was</div><div class="t m4 xc hc y48 ff1 fs7 fc0 sc0 ls0 ws0">established,<span class="_ _9"> </span>and<span class="_ _9"> </span>it<span class="_ _a"> </span>was<span class="_ _9"> </span>veri&#64257;ed<span class="_ _9"> </span>by<span class="_ _9"> </span>three<span class="_ _9"> </span>elementary<span class="_ _9"> </span>tests.<span class="_ _9"> </span>Based</div><div class="t m4 xc hc y49 ff1 fs7 fc0 sc0 ls0 ws0">on<span class="_ _9"> </span>the<span class="_ _1"> </span>modi&#64257;ed<span class="_ _1"> </span>multiplicative<span class="_ _9"> </span>decomposition<span class="_ _1"> </span>of<span class="_ _1"> </span>the<span class="_ _9"> </span>left<span class="_ _1"> </span>stretch</div><div class="t m4 xc hc y4a ff1 fs7 fc0 sc0 ls0 ws0">tensor,<span class="_ _2"> </span>the<span class="_ _4"> </span>&#64257;nite<span class="_ _4"> </span>elastic-plastic<span class="_ _4"> </span>deformations<span class="_ _4"> </span>of<span class="_ _4"> </span>hardening<span class="_ _2"> </span>materi-</div><div class="t m4 xc hc y4b ff1 fs7 fc0 sc0 ls0 ws0">als<span class="_ _a"> </span>were<span class="_ _9"> </span>analyzed<span class="_ _9"> </span>by<span class="_ _a"> </span>Ghavam<span class="_ _9"> </span>and<span class="_ _a"> </span>Naghdabadi<span class="_ _9"> </span><span class="fc1">[11]</span>.<span class="_ _9"> </span>But<span class="_ _a"> </span>most<span class="_ _9"> </span>of</div><div class="t m4 xc hc y4c ff1 fs7 fc0 sc0 ls0 ws0">these<span class="_ _a"> </span>models<span class="_ _4"> </span>were<span class="_ _a"> </span>built<span class="_ _a"> </span>based<span class="_ _a"> </span>on<span class="_ _a"> </span>the<span class="_ _4"> </span>Hill&#8217;s<span class="_ _a"> </span>yield<span class="_ _a"> </span>criterion,<span class="_ _a"> </span>which</div><div class="t m4 xc hc y4d ff1 fs7 fc0 sc0 ls0 ws0">supposed<span class="_ _4"> </span>the<span class="_ _a"> </span>material<span class="_ _4"> </span>yield<span class="_ _4"> </span>is<span class="_ _a"> </span>independent<span class="_ _4"> </span>of<span class="_ _4"> </span>the<span class="_ _a"> </span>spherical<span class="_ _4"> </span>stress</div><div class="t m4 xc hc y4e ff1 fs7 fc0 sc0 ls0 ws0">tensor.<span class="_ _2"> </span>Actually,<span class="_ _2"> </span>under<span class="_ _2"> </span>the<span class="_ _2"> </span>acting<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _2"> </span>spherical<span class="_ _2"> </span>stress<span class="_ _2"> </span>tensor,<span class="_ _2"> </span>the</div><div class="t m4 xc hc y4f ff1 fs7 fc0 sc0 ls0 ws0">structures<span class="_ _2"> </span>will<span class="_ _2"> </span>be<span class="_ _2"> </span>distorted<span class="_ _2"> </span>due<span class="_ _2"> </span>to<span class="_ _2"> </span>the<span class="_ _2"> </span>different<span class="_ _6"> </span>elastic<span class="_ _2"> </span>constants<span class="_ _2"> </span>in</div><div class="t m4 xc hc y50 ff1 fs7 fc0 sc0 ls0 ws0">every<span class="_ _2"> </span>principal<span class="_ _4"> </span>direction.<span class="_ _2"> </span>According<span class="_ _4"> </span>to<span class="_ _4"> </span>the<span class="_ _2"> </span>Mises<span class="_ _4"> </span>distortion<span class="_ _2"> </span>energy</div><div class="t m4 xc hc y51 ff1 fs7 fc0 sc0 ls0 ws0">yield<span class="_ _4"> </span>criteria,<span class="_ _4"> </span>the<span class="_ _4"> </span>materials<span class="_ _4"> </span>will<span class="_ _4"> </span>get<span class="_ _a"> </span>into<span class="_ _2"> </span>plastic<span class="_ _4"> </span>stage<span class="_ _a"> </span>if<span class="_ _2"> </span>the<span class="_ _4"> </span>distor-</div><div class="t m4 xc hc y52 ff1 fs7 fc0 sc0 ls0 ws0">tion<span class="_ _7"> </span>energy<span class="_ _7"> </span>reaches<span class="_ _8"> </span>to<span class="_ _7"> </span>a<span class="_ _8"> </span>certain<span class="_ _7"> </span>value,<span class="_ _8"> </span>and<span class="_ _7"> </span>the<span class="_ _8"> </span>materials<span class="_ _7"> </span>have</div><div class="t m3 x1 h9 y53 ff1 fs4 fc0 sc0 ls0 ws0">0263-8223/$<span class="_ _2"> </span>-<span class="_ _2"> </span>see<span class="_ _2"> </span>front<span class="_ _2"> </span>matter<span class="_ _2"> </span><span class="ff5">&#211;<span class="_ _2"> </span></span>2008<span class="_ _2"> </span>Elsevier<span class="_ _2"> </span>Ltd.<span class="_ _2"> </span>All<span class="_ _2"> </span>rights<span class="_ _2"> </span>reserved.</div><div class="t m3 x1 h9 y54 ff1 fs4 fc0 sc0 ls0 ws0">doi:10.1016/j.compstruct.2008.02.019</div><div class="t m3 xd h9 y55 ff2 fs4 fc0 sc0 ls0 ws0">*<span class="_ _9"> </span><span class="ff1">Corresponding<span class="_ _2"> </span>author.<span class="_ _2"> </span>Tel.:<span class="_ _2"> </span>+86<span class="_ _2"> </span>731<span class="_ _2"> </span>8822421.</span></div><div class="t m3 xb h9 y56 ff3 fs4 fc0 sc0 ls0 ws0">E-mail<span class="_ _2"> </span>address:<span class="_ _2"> </span><span class="ff1 fc1">tianyanp@gmail.com<span class="_ _2"> </span><span class="fc0">(Y.P.<span class="_ _2"> </span>Tian).</span></span></div><div class="t m3 xe h9 y57 ff1 fs4 fc0 sc0 ls0 ws0">Composite<span class="_ _2"> </span>Structures<span class="_ _2"> </span>88<span class="_ _2"> </span>(2009)<span class="_ _2"> </span>169&#8211;178</div><div class="t m0 xf hd y58 ff7 fs7 fc0 sc0 ls0 ws0">Contents<span class="_ _4"> </span>lists<span class="_ _a"> </span>available<span class="_ _4"> </span>at<span class="_ _a"> </span><span class="fc1">ScienceDirect</span></div><div class="t m5 x10 he y59 ff1 fs8 fc0 sc0 ls0 ws0">Composite<span class="_ _0"> </span>Structures</div><div class="t m4 x11 hd y5a ff7 fs7 fc0 sc0 ls3 ws0">journal<span class="_ _4"> </span>homepage:<span class="_ _a"> </span>www.elsevi<span class="ls4">er.com/locate/compstruct</span></div><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m6"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d 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