مساله‌ی متقارن کروی در نظریه‌ی‌ گرادیان دوم کرنش

نوع مقاله : مقاله پژوهشی

نویسنده

گروه عمران، دانشکده فنی و مهندسی، دانشگاه الزهرا، تهران، ایران

چکیده

هدف، حل مساله‌ی متقارن کروی یک/دو فازی با استفاده از الاستیسیته‌ی گرادیان دوم کرنش و مقایسه با الاستیسیته‌ی کلاسیک است. نظر به قابلیت گرادیان دوم کرنش در پیش‌بینی اثرات سطح، اندازه و ماهیت گسسته‌ی مواد، مساله‌ی متقارن کروی مدلسازی و تحت اثرات مذکور مطالعه می‌شود. با تبیین گرادیان دوم کرنش در دستگاه مختصات کروی، معادلات تعادل، میدان تنش و کرنش، روابط آنها و نیروهای سطحی استخراج می‌شود. بعد از آسایش سطحی، بارگذاری خارجی متقارن روی سازه اعمال و پاسخ مکانیکی بررسی می‌شود. خواص مکانیکی پوسته‌ تک‌فازی از بلور الماس و پوسته‌ی دوفازی از سیلیکون بلوری با پوشش کربنی مطالعه می‌شود. مقادیر ثابت‌های مواد توسط محاسبات کوانتومی، نظریه‌ی دینامیک شبکه‌ و مدلسازی محیط پیوسته محاسبه می‌شود. محاسبات نشان می‌دهد که پیش‌بینی نظریه‌ی تقویت‌یافته از خواص نانوسازه تفاوت چشمگیری با پیش‌بینی کلاسیک دارد. مثلا، در پوسته‌ی کربنی با شعاع داخلی و خارجی به ترتیب 2 و 10 برابر پارامتر شبکه، تحت بارگذاری فشاری خارجی یکه شده‌ی 0001/0، تنش شعاعی کلاسیک یکه، ثابت و حدود 0001/0- است، درحالیکه در گرادیان کرنش، تنش شعاعی یکه از مقادیر 001/0- و 0002/0- در مرزهای داخلی و خارجی، به ترتیب، به 0003/0 در میانه متغیر است. با افزایش شعاع داخلی، اختلاف دو نظریه در میانه‌ی محیط کاهش می‌یابد اما در نزدیکی سطوح اختلاف قابل ملاحظه است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Spherical Lame-Type Problem in Second Strain Gradient Theory

نویسنده [English]

  • Farzaneh Ojaghnezhad
Engineering Department, Alzahra University, Tehran, Iran
چکیده [English]

Second strain gradient theory is employed to examine the spherical single/double-phase Lame-type problem. Due to the capability of strain gradient theory to capture the effects of the surface, size, and discrete nature of materials, the pertinent relaxed configuration is sought. The theory is written in the spherical coordinate system and the equilibrium equations, stress/strain components, constitutive relations, and tractions are derived. The relaxed configuration is obtained for both the diamond carbon and carbon-coated crystalline silicon shell. Afterwards, the external symmetric loading is applied to the relaxed configuration to analyze the mechanical response. The elastic material parameters are calculated via the quantum computations, lattice dynamics, and material continuum description. The analysis shows that the mechanical response in the augmented theory is significantly different from that in the classical elasticity. For example, in the single-phase problem with an inner and outer radius equal to two and ten lattice parameter, respectively, under a normalized external pressure of about 0.0001, the classic elasticity predicts an approximately constant normalized radial stress of about -0.0001 in the nanoshell. However, in the framework of strain gradient theory, the normalized radial stress is varying from about -0.001 and -0.0002 in the vicinity of the inner and outer boundaries, respectively, to about 0.0003 in the middle of the hollow nanoshell. With increasing the inner radius, the difference between the two results in the middle points decreases.

کلیدواژه‌ها [English]

  • second strain gradient theory
  • Size effect
  • surface effect
  • spherical Lame-type problem
  • spherical symmetry
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