ارايهي پاسخ نيمه‌تحليلي براي ضربه‌ي كمسرعت ورق مستطيلي ساخته شده از مواد هدف‌مند

A Semi-Analytical Solution for Low-Velocity Impact Analysis of Functionally Graded Rectangular Plates

تحليل‌هاي ارايه‌شده در زمينه‌ي ضربه‌ي كم سرعت، عموماً به سازه هاي ساخته شده از مواد همسان‌گرد يا كامپوزيت (به‌ويژه، با همسان‌گردي عرضي) تعلق داشته‌ است. از سوي ديگر، در اين مدل‌ها تنها سختي سطح در معرض ضربه در نظر گرفته شده و اثر سختي لايه هاي زيرين در پاسخ هاي ضربه ناديده گرفته شده است. در برخي ديگر از مقالات موجود، از خطي سازي قانون تماس استفاده شده است. در پژوهش حاضر تلاش شده است تا ضمن گسترش تحليل‌هاي ياد‌شده به ورق هاي داراي ساختار هدف‌مند و نيز رفع نارسايي‌هاي ياد‌شده، پاسخي نيمه‌تحليلي براي آن ارايه شود. به همين منظور، اثر پارامترهاي مختلف ضربه زن و ورق با ساختار مواد هدف‌مند بر پاسخ هاي ضربه‌ي كم سرعت، در محدوده‌ي كشسان مورد بررسي قرار گرفته و در تحليل مسيله از قانون غيرخطي تماس هرتز استفاده شده است. از آنجا كه نتايج كنوني با استفاده از پاسخ تحليلي حاصل شده، خطاهاي مدل‌سازي، خطي سازي و تقريب هاي عددي متداول در آن ها آشكار نمي ‌‌شود. نتايج پژوهش كنوني در مقايسه با نتايج ديگر محققين براي حالت‌هاي ويژه، مورد تاييد قرار گرفته است.

Available low-velocity impact analyses have mainly been presented for isotropic or laminated composite (especially those with transversely isotropic material properties) structures. In the mentioned models, only the stiffness of the impacted region is considered and the influence of the stiffness of the underneath layers on the impact responses is discarded. It is evident that the stiffness of the substrate layers may considerably affect the contact stiffness of the contact region. In some other papers, the nonlinear Hertz-type contact law has been replaced by a linear one. In the present paper, in addition to overcoming the mentioned shortcomings, it is intended to extend the previous research to functionally graded plates, whose material properties vary in the transverse direction according to a power law. The apparent contact stiffness is determined through calculation of the volume mean of the material properties and by employing Turner’s procedure. Among other superiorities, the solution is presented in a semi-analytical form. The governing equations of the plate are derived based on the classical plate theory, and the analytical solution is of a Navier-type. The proposed solution is adopted, according to the boundary condition of simply supported edges. Therefore, the plate is modeled as a continuous rather than discrete system. The resulted nonlinear time-dependent equations are solved using the Runge-Kutta numerical time integration method. In the present research, effects of various parameters of the indenter (radius, mass, velocity) and the functionally graded plate (power law exponent or the so-called volume fraction index and thickness of the plate) on the low-velocity responses (indentation, contact force, and lateral deflection) are investigated in the elastic region, employing non-linear Hertz’s contact law. Since the present results are extracted based on an analytical method, modeling, linearization, and common numerical errors are prevented. Results of the present research are verified by results reported by other researchers for special cases.