夹芯结构局部压入及弯曲塑性响应的理论研究
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摘要
轻质夹芯结构广泛用于各种工程结构,如航空航天,汽车,船舶等。典型的夹芯结构一般由三层粘结而成,中间芯层采用较厚的轻质材料,两层表皮采用厚度较薄、强度较高的材料。由于夹芯结构几何、材料参数较多,结构变形复杂,研究方法多限于实验及数值模拟,简洁有效的理论模型至今很少。
从工程用途角度来说,夹芯结构可分为三大类,一是作为承载结构,主要利用夹芯结构的比刚度、比承载力较高的性能,质量较轻却能提供较大的刚度和承载力。二是作为吸能防护结构,利用夹芯结构比吸能较高的性能,质量较轻却能吸收较多的塑性耗散能,从而保护易受伤害的对象。三是承载吸能二者兼有,在利用夹芯结构的比承载能力的同时兼顾考虑发生意外碰撞损伤时的缓冲吸能性能。
从材料响应角度来说,当夹芯结构作为承载结构使用时,主要考虑的是材料处于弹性响应阶段,不考虑材料的塑性破坏。理论分析时,主要采用经典弹性理论及其特定条件下的简化方法。另一方面,当夹芯结构作为吸能防护结构时,主要考虑材料发生塑性变形时能量耗散的性能。理论分析时,主要采用能量平衡及有关原理。当同时考虑承载和吸能时,需要进行综合设计。
夹芯结构在使用时的边界条件多种多样,但归纳起来主要有两大类：刚性面支承和边缘支承。其中,刚性面支承是指整个夹芯结构平放在刚性面上,一侧表皮与刚性面完全接触。一般用于防护背面的受保护对象免受外载冲击,结构变形主要表现为局部压入等压缩行为。而边缘支承是指夹芯结构主要在端部或周边受到约束,其它部位为自由状态,在受到横向承载时主要发生弯曲变形。另外,由于夹芯结构一般表皮厚度较薄、芯层强度较低,在受到外载尤其是集中载荷作用时,易于发生局部压入变形。因此,整体弯曲常常伴随有局部压入变形,一方面降低了结构的整体抗弯能力,另一方面增加了理论分析的难度。
本文主要研究夹芯结构的吸能性能,即考虑结构发生塑性变形时的能量耗散特性,而忽略材料的弹性响应。主要内容包括：
1)刚性面支承夹芯梁局部压入响应
分别考虑了两种典型形状的压头即平头及圆柱对夹芯梁的局部压入作用。假定表皮变形区具有一个线性速度场,根据虚速度原理和最小功原理,最后得到表皮变形场分布特点和压头载荷随压入位移的变化关系,并与有限元计算结果进行了比较分析。得到的理论解简洁有效自洽,当压头尺寸趋向于零(相当于线载荷)时两种压头作用下的理论解可以退化为同一结果。进一步根据已得到的理论解,分析了结构表皮及芯层耗散塑性应变能的特点。
2)刚性面支承夹芯圆板局部压入响应
分别考虑了平头及球头两种形状压头的作用。首先沿用前面夹芯梁局部压入响应的研究方法,根据虚速度原理及最小功原理进行理论分析。结果发现,平头作用下的理论结果与有限元值偏差较大,而球头作用下的理论模型存在不自洽的问题。另外,点载荷作用下的压头载荷只与表皮的厚度及流动应力有关,而与芯层强度无关。
由于上述理论模型存在难以解决的问题,改为采用最小势能原理进行分析。假定表皮变形区具有一个二次曲线形式的位移场,根据最小功原理,得到变形区范围和压头载荷随压入位移的变化关系,并与有限元计算结果进行了比较分析。该模型也存在一个问题,虽然当压头尺寸趋向于零时可以退化为同一结果,但无法通过该结果求解得到点载荷作用时的结构响应。最后根据平头及球头作用下的理论解,分析了两种情况下结构不同部位能量耗散的特点。
3)两端固支夹芯梁弯曲行为
根据本文已经得到的夹芯梁局部压入响应,结合弯矩轴力共同作用时的夹芯梁屈服准则,同时通过对夹芯梁整体弯曲时的变形及受力分析,研究了存在局部压入变形时固支夹芯梁的复杂弯曲行为。为简化理论模型,忽略整体弯曲变形对局部压入响应的影响,只考虑了后者对前者的作用。经过理论分析,得到局部压入深度和压头载荷随压头总位移之间的变化关系。另外,通过与有限元计算结果的比较,分析了夹芯结构与实体结构弯曲变形的差别。
Light-weight sandwich constructions have been widely used in various engeering constructures, such as aerospace, automotive and watercraft industries etc. The typical sandwich panels used in structural applications consist of two thin and stiff face skins separated and attached by a thick and relatively flexible core. Sandwich constructions have more geometry and material parameters, and deformation is very complicated. Study methods are mostly limited to tests and numerial simulation, and concisely, effectively analytical models have not been reported up to the present.
From the standpoint of engineering application, sandwich constructions can be divided into three types. Firstly, they are used to carry loads. High specific rigidity and specific load-carrying capacity make sandwich constructions with light weight afford more rigidity and load-carrying capacity. Secondly, they are used as energy absorption and safeguard structrues. High specific energy absorption make sandwich constructions with light weight absorb more plastic dissipation energy and protect objects apt to damage. Finally, they are used as load-carrying and energy-absorbing constructures at the same time. On the basis of utilizing high specific load-carrying capacity of sandwich constructions, buffering and energy-absorbing property is considered on the occasion of accident collision and damage.
From the standpoint of material response, elastic response other than plastic destruction is mostly taken into account when sandwich constructions are used as load-carrying constructures. During theoretical analyzing, classical elastic theory and its simplified method under given condition are mostly applied. On the other hand, when sandwich constructions are used as energy-absorbing and safeguard structures, energy dissipation from plastic deformation is mostly considered. During theoretical analyzing, energy equilibrium principle, etc. is mostly employed. Considering load-carrying and energy-absorbing propery needs compositive design.
Boundary conditions of sandwich constructions in application are in varied forms, and in summary there are two types:rigid-face bearing and edge supporting. Thereinto, rigid-face bearing implies the whole sandwich construction is placed on a rigid face in horizontal direction, and one side of skin is fully contacted with rigid face. Generally, it is used to prevent the sheltered objects behind the back from impact of external loads, and constructural deformation mostly represents compressive behave such as local indentation. On the other side, edge supporting means that sandwich constructions are mostly restricted at the ends or circumference and other parts are free, which would undergo bending deformation when subjected to transverse loading. With thin face sheets and low strength core layer, sandwich constructions are prone to local indentation under external loads especially concentrated ones. So, global bending is generally in accompany with local indentation, which decreases global bending resistance and increases difficulty of analyzing.
The paper focused on property of energy absorption of sandwich constructions, i.e. characteristic of energy dissipation of plastic deformation, and elastic response of materials were neglected. Main content includes:
1) Local indentation response of sandwich beams on a rigid foundation Local indentation effect was considered under indenters of two typical shapes i.e. flat and cylindrical ones, respectively. By assuming that face-sheet deformation zone has a linear velocity field, varying relations of distribution of skin deformation field and indenter loads with indenter displacement were obtained on the basis of virtual velocity and minimal work principles, which were compared and analyzed with results from finite element simulation. Under the two type of indenters, anaytical results, which are concise, effective and harmonious, could reduce to an identical one when the size of indenters tend to zero (Equivalent to a line-shape load). Characteristic of dissipating plastic strain energy of face sheets and core layer were analyzed based on theoretical solutions obtained.
2) Local indentation response of sandwich panels on a rigid foundation Likewise, two type of indenters, i.e. flat and spherical ones were taken into account. By following research method of local indentation response of sandwich beams, the analysis was conducted based on virtual velocity and minimum work principle. Results revealed that theoretical solutions under flat indenter has more deviation from numerical ones and analytical model is not self-consistent. Indenter load under point loading has some connection with thickness and flow stress of face sheets, nothing to do with strength of core layer. Because analytical model above has problems difficult to solve, minimal potential energy principle was applied to analyze, instead. By assuming that skin deformation region has a quadratic field, relation of the range of deformation zone and indenter load with indenter displacement were obtained based on minimal work principle, which were compared and analyzed with results from finite element simulation. In this model there is a problem:although results under flat and spherical indenter can be reduced to an identical one when the size of indenters tend to zero, which can not be solved to obtain the local indentation response under a point load. Finally, energy dissipation characteristic of various parts of the constructions of the two cases were analyzed according to the theoretical solutions under flat and spherical indenters, respectively.
3) Bending behavior of sandwich beams clamped at two ends By utilizing the laws of local indentation of sandwich beams obtained above and the yield criterion for the sandwich cross-sections carrying axial force and bending moment, and analyzing deformation and internal forces on the neutral axis of sandwich beams under global bending, complicated bending havior of clamped sandwich beams including local indentation was investigated. To simplify the analytical model, effect of global bending on local indentation was neglected, and only influence of the latter on the former was considered. Finally, relations of local indentation and indenter loads with indenter displacment were obtained. By comparing with results from finite element simulation, the difference of deformation of sandwich constructions from that of solid constructures was analyzed.
从工程用途角度来说,夹芯结构可分为三大类,一是作为承载结构,主要利用夹芯结构的比刚度、比承载力较高的性能,质量较轻却能提供较大的刚度和承载力。二是作为吸能防护结构,利用夹芯结构比吸能较高的性能,质量较轻却能吸收较多的塑性耗散能,从而保护易受伤害的对象。三是承载吸能二者兼有,在利用夹芯结构的比承载能力的同时兼顾考虑发生意外碰撞损伤时的缓冲吸能性能。
从材料响应角度来说,当夹芯结构作为承载结构使用时,主要考虑的是材料处于弹性响应阶段,不考虑材料的塑性破坏。理论分析时,主要采用经典弹性理论及其特定条件下的简化方法。另一方面,当夹芯结构作为吸能防护结构时,主要考虑材料发生塑性变形时能量耗散的性能。理论分析时,主要采用能量平衡及有关原理。当同时考虑承载和吸能时,需要进行综合设计。
夹芯结构在使用时的边界条件多种多样,但归纳起来主要有两大类：刚性面支承和边缘支承。其中,刚性面支承是指整个夹芯结构平放在刚性面上,一侧表皮与刚性面完全接触。一般用于防护背面的受保护对象免受外载冲击,结构变形主要表现为局部压入等压缩行为。而边缘支承是指夹芯结构主要在端部或周边受到约束,其它部位为自由状态,在受到横向承载时主要发生弯曲变形。另外,由于夹芯结构一般表皮厚度较薄、芯层强度较低,在受到外载尤其是集中载荷作用时,易于发生局部压入变形。因此,整体弯曲常常伴随有局部压入变形,一方面降低了结构的整体抗弯能力,另一方面增加了理论分析的难度。
本文主要研究夹芯结构的吸能性能,即考虑结构发生塑性变形时的能量耗散特性,而忽略材料的弹性响应。主要内容包括：
1)刚性面支承夹芯梁局部压入响应
分别考虑了两种典型形状的压头即平头及圆柱对夹芯梁的局部压入作用。假定表皮变形区具有一个线性速度场,根据虚速度原理和最小功原理,最后得到表皮变形场分布特点和压头载荷随压入位移的变化关系,并与有限元计算结果进行了比较分析。得到的理论解简洁有效自洽,当压头尺寸趋向于零(相当于线载荷)时两种压头作用下的理论解可以退化为同一结果。进一步根据已得到的理论解,分析了结构表皮及芯层耗散塑性应变能的特点。
2)刚性面支承夹芯圆板局部压入响应
分别考虑了平头及球头两种形状压头的作用。首先沿用前面夹芯梁局部压入响应的研究方法,根据虚速度原理及最小功原理进行理论分析。结果发现,平头作用下的理论结果与有限元值偏差较大,而球头作用下的理论模型存在不自洽的问题。另外,点载荷作用下的压头载荷只与表皮的厚度及流动应力有关,而与芯层强度无关。
由于上述理论模型存在难以解决的问题,改为采用最小势能原理进行分析。假定表皮变形区具有一个二次曲线形式的位移场,根据最小功原理,得到变形区范围和压头载荷随压入位移的变化关系,并与有限元计算结果进行了比较分析。该模型也存在一个问题,虽然当压头尺寸趋向于零时可以退化为同一结果,但无法通过该结果求解得到点载荷作用时的结构响应。最后根据平头及球头作用下的理论解,分析了两种情况下结构不同部位能量耗散的特点。
3)两端固支夹芯梁弯曲行为
根据本文已经得到的夹芯梁局部压入响应,结合弯矩轴力共同作用时的夹芯梁屈服准则,同时通过对夹芯梁整体弯曲时的变形及受力分析,研究了存在局部压入变形时固支夹芯梁的复杂弯曲行为。为简化理论模型,忽略整体弯曲变形对局部压入响应的影响,只考虑了后者对前者的作用。经过理论分析,得到局部压入深度和压头载荷随压头总位移之间的变化关系。另外,通过与有限元计算结果的比较,分析了夹芯结构与实体结构弯曲变形的差别。
Light-weight sandwich constructions have been widely used in various engeering constructures, such as aerospace, automotive and watercraft industries etc. The typical sandwich panels used in structural applications consist of two thin and stiff face skins separated and attached by a thick and relatively flexible core. Sandwich constructions have more geometry and material parameters, and deformation is very complicated. Study methods are mostly limited to tests and numerial simulation, and concisely, effectively analytical models have not been reported up to the present.
From the standpoint of engineering application, sandwich constructions can be divided into three types. Firstly, they are used to carry loads. High specific rigidity and specific load-carrying capacity make sandwich constructions with light weight afford more rigidity and load-carrying capacity. Secondly, they are used as energy absorption and safeguard structrues. High specific energy absorption make sandwich constructions with light weight absorb more plastic dissipation energy and protect objects apt to damage. Finally, they are used as load-carrying and energy-absorbing constructures at the same time. On the basis of utilizing high specific load-carrying capacity of sandwich constructions, buffering and energy-absorbing property is considered on the occasion of accident collision and damage.
From the standpoint of material response, elastic response other than plastic destruction is mostly taken into account when sandwich constructions are used as load-carrying constructures. During theoretical analyzing, classical elastic theory and its simplified method under given condition are mostly applied. On the other hand, when sandwich constructions are used as energy-absorbing and safeguard structures, energy dissipation from plastic deformation is mostly considered. During theoretical analyzing, energy equilibrium principle, etc. is mostly employed. Considering load-carrying and energy-absorbing propery needs compositive design.
Boundary conditions of sandwich constructions in application are in varied forms, and in summary there are two types:rigid-face bearing and edge supporting. Thereinto, rigid-face bearing implies the whole sandwich construction is placed on a rigid face in horizontal direction, and one side of skin is fully contacted with rigid face. Generally, it is used to prevent the sheltered objects behind the back from impact of external loads, and constructural deformation mostly represents compressive behave such as local indentation. On the other side, edge supporting means that sandwich constructions are mostly restricted at the ends or circumference and other parts are free, which would undergo bending deformation when subjected to transverse loading. With thin face sheets and low strength core layer, sandwich constructions are prone to local indentation under external loads especially concentrated ones. So, global bending is generally in accompany with local indentation, which decreases global bending resistance and increases difficulty of analyzing.
The paper focused on property of energy absorption of sandwich constructions, i.e. characteristic of energy dissipation of plastic deformation, and elastic response of materials were neglected. Main content includes:
1) Local indentation response of sandwich beams on a rigid foundation Local indentation effect was considered under indenters of two typical shapes i.e. flat and cylindrical ones, respectively. By assuming that face-sheet deformation zone has a linear velocity field, varying relations of distribution of skin deformation field and indenter loads with indenter displacement were obtained on the basis of virtual velocity and minimal work principles, which were compared and analyzed with results from finite element simulation. Under the two type of indenters, anaytical results, which are concise, effective and harmonious, could reduce to an identical one when the size of indenters tend to zero (Equivalent to a line-shape load). Characteristic of dissipating plastic strain energy of face sheets and core layer were analyzed based on theoretical solutions obtained.
2) Local indentation response of sandwich panels on a rigid foundation Likewise, two type of indenters, i.e. flat and spherical ones were taken into account. By following research method of local indentation response of sandwich beams, the analysis was conducted based on virtual velocity and minimum work principle. Results revealed that theoretical solutions under flat indenter has more deviation from numerical ones and analytical model is not self-consistent. Indenter load under point loading has some connection with thickness and flow stress of face sheets, nothing to do with strength of core layer. Because analytical model above has problems difficult to solve, minimal potential energy principle was applied to analyze, instead. By assuming that skin deformation region has a quadratic field, relation of the range of deformation zone and indenter load with indenter displacement were obtained based on minimal work principle, which were compared and analyzed with results from finite element simulation. In this model there is a problem:although results under flat and spherical indenter can be reduced to an identical one when the size of indenters tend to zero, which can not be solved to obtain the local indentation response under a point load. Finally, energy dissipation characteristic of various parts of the constructions of the two cases were analyzed according to the theoretical solutions under flat and spherical indenters, respectively.
3) Bending behavior of sandwich beams clamped at two ends By utilizing the laws of local indentation of sandwich beams obtained above and the yield criterion for the sandwich cross-sections carrying axial force and bending moment, and analyzing deformation and internal forces on the neutral axis of sandwich beams under global bending, complicated bending havior of clamped sandwich beams including local indentation was investigated. To simplify the analytical model, effect of global bending on local indentation was neglected, and only influence of the latter on the former was considered. Finally, relations of local indentation and indenter loads with indenter displacment were obtained. By comparing with results from finite element simulation, the difference of deformation of sandwich constructions from that of solid constructures was analyzed.
引文
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[45]Frostig Y.1993. High-order behavior of sandwich beams with flexible core and transverse diaphragms. Journal of Engineering Mechanics.119(5): 955-972.
[46]Frostig Y.1993. On stress concentration in the bending of sandwich beams with transversely flexible core. Composite Structures.24(2):161-169.
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[52]Frostig Y and Thomsen OT.2005. Localized effects in the nonlinear behavior of sandwich panels with a transversely flexible core. Journal of Sandwich Structures & Materials.7(1):53-75.
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[55]Petras A and Sutcliffe MPF.1999. Indentation resistance of sandwich beams. Composite Structures.46(4):413-424.
[56]Sokolinsky VS, Shen HB, Vaikhanski L, and Nutt SR.2003. Experimental and analytical study of nonlinear bending response of sandwich beams. Composite Structures.60(2):219-229.
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[61]Steeves CA and Fleck NA.2004. Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part 1: analytical models and minimum weight design. International Journal of Mechanical Sciences.46(4):561-583.
[62]Steeves CA and Fleck NA.2004. Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part II: experimental investigation and numerical modelling. International Journal of Mechanical Sciences.46(4):585-608.
[63]Tagarielli VL, Fleck NA, and Deshpande VS.2004. Collapse of clamped and simply supported composite sandwich beams in three-point bending. Composites Part B-Engineering.35(6-8):523-534.
[64]Sadighi M, Pouriayevali H, and Saadati M.2007. A study of indentation energy in three points bending of sandwich beams with composite laminated faces and foam core. Proceedings of World Academy of Science, Engineering and Technology, Vol 26, Parts 1 and 2, December 2007.26(691-697.
[65]Qin QH and Wang TJ.2009. An analytical solution for the large deflections of a slender sandwich beam with a metallic foam core under transverse loading by a flat punch. Composite Structures.88(4):509-518.
[66]Martin JB. Plasticity:fundamentals and general results. Cambridge, Mass: MIT Press,1975.
[67]Abrate S.1997. Localized impact on sandwich structures with laminated facings. Applied Mechanics Reviews.50(2):69-82.
[68]Sadighi M, Pouriayevali H, and Saadati M.2007. Response of fully backed sandwich beams to low velocity transverse impact. Proceedings of World Academy of Science, Engineering and Technology, Vol 26, Parts 1 and 2, December 2007.26(698-703.
[69]Qin QH and Wang TJ.2011. Low-velocity heavy-mass impact response of slender metal foam core sandwich beam. Composite Structures.93(6): 1526-1537.
[70]Fleck NA and Deshpande VS.2004. The resistance of clamped sandwich beams to shock loading. Journal of Applied Mechanics-Transactions of the Asme.71(3):386-401.
[71]Qiu X, Deshpande VS, and Fleck NA.2004. Dynamic response of a clamped circular sandwich plate subject to shock loading. Journal of Applied Mechanics-Transactions of the Asme.71(5):637-645.
[72]Qiu X, Deshpande VS, and Fleck NA.2005. Impulsive loading of clamped monolithic and sandwich beams over a central patch. Journal of the Mechanics and Physics of Solids.53(5):1015-1046.
[73]Tilbrook MT, Deshpande VS, and Fleck NA.2009. Underwater blast loading of sandwich beams:Regimes of behaviour. International Journal of Solids and Structures.46(17):3209-3221.
[74]Qin QH and Wang TJ.2009. A theoretical analysis of the dynamic response of metallic sandwich beam under impulsive loading. European Journal of Mechanics a-Solids.28(5):1014-1025.
[75]Qin QH, Wang TJ, and Zhao SZ.2009. Large deflections of metallic sandwich and monolithic beams under locally impulsive loading. International Journal of Mechanical Sciences.51(11-12):752-773.
[76]Zhu F, Wang ZH, Lu GX, and Nurick G.2010. Some theoretical considerations on the dynamic response of sandwich structures under impulsive loading. International Journal of Impact Engineering.37(6): 625-637.
[77]Frostig Y and Baruch M.1993. High-order buckling analysis of sandwich beams with transversely flexible core. Journal of Engineering Mechanics-Asce. 119(3):476-495.
[78]Frostig Y and Thomsen OT.2004. High-order free vibration of sandwich panels with a flexible core. International Journal of Solids and Structures. 41(5-6):1697-1724.
[79]Frostig Y and Thomsen OT.2009. On the free vibration of sandwich panels with a transversely flexible and temperature-dependent core material-Part Ⅰ: Mathematical formulation. Composites Science and Technology.69(6): 856-862.
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[81]Sokolinsky VS, von Bremen HF, Lesko JJ, and Nutt SR.2004. Higher-order free vibrations of sandwich beams with a locally damaged core. International Journal of Solids and Structures.41(22-23):6529-6547.
[82]Brocca M, Bazant ZP, and Daniel IM.2001. Microplane model for stiff foams and finite element analysis of sandwich failure by core indentation. International Journal of Solids and Structures.38(44-45):8111-8132.
[83]Herup EJ and Palazotto AN.1997. Low-velocity impact damage initiation in graphite/epoxy/Nomex honeycomb-sandwich plates. Composites Science and Technology.57(12):1581-1598.
[84]Mohan K, Yip TH, Sridhar I, and Seow HP.2007. Effect of face sheet material on the indentation response of metallic foams. Journal of Materials Science. 42(11):3714-3723.
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[87]Ferri R and Sankar BV.1997. A comparative study on the impact resistance of composite laminates and sandwich panels. Journal of Thermoplastic Composite Materials.10(4):304-315.
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[89]Wen HM, Reddy TY, Reid SR, and Soden PD. Indentation, penetration and perforation of composite laminates and sandwich panels under quasi-static and projectile loading. Impact Response and Dynamic Failure of Composites and Laminate Materials, Pts 1 and 2. vol.141-1, Kim JK and Yu TX, Eds., ed, 1998, pp.501-552.
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[91]Simone AE and Gibson LJ.1998. Aluminum foams produced by liquid-state processes. Acta Materialia.46(9):3109-3123.
[92]Banhart J.2001. Manufacture, characterisation and application of cellular metals and metal foams. Progress in Materials Science.46(6):559-U3.
[93]Santosa S and Wierzbicki T.1998. On the modeling of crush behavior of a closed-cell aluminum foam structure. Journal of the Mechanics and Physics of Solids.46(4):645-669.
[94]Xie Z, Zheng Z, and Yu J.2012. Localized indentation of sandwich beam with metallic foam core. Journal of Sandwich Structures and Materials.14(2): 197-210.
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