拉压不同模量弹性问题的数值研究
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摘要
众所周知,经典弹性理论认为材料的拉伸弹性模量和压缩弹性模量是相等的。但是,实际上许多工程材料都在不同程度上表现出拉、压不同的弹性性质,如陶瓷、玻璃钢、塑料、钢筋混凝土、石墨、粉末冶金材料、聚合材料及复合材料等。它们的抗拉强度与抗压强度不仅相差较大,拉、压弹性模量也是不同的。随着科学技术的日益发展,对材料力学性质的研究提出了更高的要求,研制新型的材料以及挖掘材料自身特性的潜力,已成为新的研究趋向。对这类拉压模量不同材料制成的构件或结构其力学性能的研究,是弹性理论发展的一个新方向,也是工程实践的迫切需要。拉压不同弹性模量的材料,其弹性系数与结构的材料有关,与结构的形状、边界条件及外载荷有关,是诸多因素所致具有非线性现象的力学问题。
近二十年来,由于矩阵结构理论、有限元方法以及大型计算机软硬件的发展,使得计算力学获得了突飞猛进的进展。一方面有限元方法已广泛应用于实际和重要的工程问题中,另一方面关于这些材料的物理性质上尚未完全透彻地研究清楚,以至于拉压不同模量的有限元方法还处在不断发展和完善阶段。而在现代技术中,具有不同模量的材料应用日益广泛,因此要正确地对这类材料进行力学分析,发展行之有效的数值求解方法,显得十分必要。
本文主要研究思路是通过建立和完善拉压不同模量三维有限元格式和算法,对具有拉压不同模量结构的力学性能做进一步深入探讨和研究。
利用等效和完备的思想,对Ambartsumyan有限元计算模型、Jones有限元计算模型、叶志明等有限元计算模型进行改进和探讨,建立和完善了相应的有限元格式、误差估计和无量纲化公式。对叶志明等提出的主应变判定法则,和早期提出并广泛应用的主应力判定法则给出了等价性的证明。通过对主材料矩阵和刚度矩阵的无量纲化,完善了拉压不同模量有限元模型的通用性。
利用改进的拉压不同模量有限元模型,编制了相应的有限元程序,计算分析了拉压不同模量结构的弹性问题,探讨了不同模量弯曲结构的中性轴问题,在不同荷载以及在不同边界条件下拉压模量比的变化对计算结果的影响。
以改进的叶志明等有限元模型为基础,发展了拉压不同模量大位移有限元格式和计算方法,并编制了相应的三维实体单元有限元程序。
利用建立的拉压不同模量小位移和大位移有限元格式和计算方法,对拉压不同模量结构在不同工况下进行了数值计算和分析。进一步探讨了结构极限承载的影响因素,如约束条件、加载条件、拉压模量比等。
It is well known that tensile modulus is assumed the same as compressive modulus in the classical theory of elasticity. However, many engineering materials, especially new materials, that are widely developed and applied, such as powder metallurgical materials, polymeric material, composite materials, etc, have different strength when they are loaded with tension and compression, respectively. Most of them behave distinctive mechanical characteristics, one of which is different Young's moduli. With the development of science and technology, some research turns into a new study trend to develop new materials and to explore potency of material speciality in itself. The theory breaks through the assumption the elastic modulus is only related to the properties of material itself. The elastic modulus is related to the material, shape, boundary condition and external loadings of structures. So it is a nonlinear problem contributed by many factors.
The finite element method (FEM) has been applied to many engineering fields. As the mechanical properties of the materials have not been clarified, the finite element method for the different modulus problem is not widely developed and applied in practical engineering. With the development of contemporary engineering materials, the materials with different Young's moduli in tension and compression will be widely used in engineering. Therefore, it is necessary to develop an effective numerical method in order to correctly analyze the mechanical properties of these materials.
This thesis focuses on developing a new finite element formulation and iterative algorithm for different Young's moduli in tension and compression. Numerical study of mechanical properties is presented for the structures having different moduli in tension and compression.
Ambartsumyan's FE model, Jones' FE model and Ye's FE model are improved and investigated by an equivalent and complete concept. The FEM for different Young's moduli in tension and compression is further perfected with error estimation and dimensionless formulations. Equivalent is proven between the criterion of the principal strain presented by Ye's and the criterion of the principal stress widely used by predecessors. The dimensionless formulations of the principal material matrix and stiffness matrix develop the finite element method for different moduli.
The corresponding FEM program is established with Matlab platform. Numerical study is presented for the elastic problems having different moduli. The neutral axis problem and the ratio influence of tensile modulus to compressive modulus are discussed for bending structures under various conditions, such as different geometric models, different loads. With increasing the difference between tensile modulus and compressive modulus, there exits large errors in the value and distribution of displacement and stress, if classical theory is used to solve the problems having different Young's moduli in tension and compression.
The different tension-compression elastic moduli are introduced into small-displacement and large-displacement finite element formulations and numerical methods are developed using modified. Three-dimensional finite element iterative program for geometric and material nonlinear analysis is established.
Numerical study is presented for load-carrying capacity problem of different modulus structures. The influential factors of load-carrying capacity are further discussed, including constraint conditions, couple moments, ratios of tensile modulus to compressive modulus, etc. Finally, results show that there are big errors of load-carrying capacity in the bending-compression members having different moduli by using uniform modulus model.
近二十年来,由于矩阵结构理论、有限元方法以及大型计算机软硬件的发展,使得计算力学获得了突飞猛进的进展。一方面有限元方法已广泛应用于实际和重要的工程问题中,另一方面关于这些材料的物理性质上尚未完全透彻地研究清楚,以至于拉压不同模量的有限元方法还处在不断发展和完善阶段。而在现代技术中,具有不同模量的材料应用日益广泛,因此要正确地对这类材料进行力学分析,发展行之有效的数值求解方法,显得十分必要。
本文主要研究思路是通过建立和完善拉压不同模量三维有限元格式和算法,对具有拉压不同模量结构的力学性能做进一步深入探讨和研究。
利用等效和完备的思想,对Ambartsumyan有限元计算模型、Jones有限元计算模型、叶志明等有限元计算模型进行改进和探讨,建立和完善了相应的有限元格式、误差估计和无量纲化公式。对叶志明等提出的主应变判定法则,和早期提出并广泛应用的主应力判定法则给出了等价性的证明。通过对主材料矩阵和刚度矩阵的无量纲化,完善了拉压不同模量有限元模型的通用性。
利用改进的拉压不同模量有限元模型,编制了相应的有限元程序,计算分析了拉压不同模量结构的弹性问题,探讨了不同模量弯曲结构的中性轴问题,在不同荷载以及在不同边界条件下拉压模量比的变化对计算结果的影响。
以改进的叶志明等有限元模型为基础,发展了拉压不同模量大位移有限元格式和计算方法,并编制了相应的三维实体单元有限元程序。
利用建立的拉压不同模量小位移和大位移有限元格式和计算方法,对拉压不同模量结构在不同工况下进行了数值计算和分析。进一步探讨了结构极限承载的影响因素,如约束条件、加载条件、拉压模量比等。
It is well known that tensile modulus is assumed the same as compressive modulus in the classical theory of elasticity. However, many engineering materials, especially new materials, that are widely developed and applied, such as powder metallurgical materials, polymeric material, composite materials, etc, have different strength when they are loaded with tension and compression, respectively. Most of them behave distinctive mechanical characteristics, one of which is different Young's moduli. With the development of science and technology, some research turns into a new study trend to develop new materials and to explore potency of material speciality in itself. The theory breaks through the assumption the elastic modulus is only related to the properties of material itself. The elastic modulus is related to the material, shape, boundary condition and external loadings of structures. So it is a nonlinear problem contributed by many factors.
The finite element method (FEM) has been applied to many engineering fields. As the mechanical properties of the materials have not been clarified, the finite element method for the different modulus problem is not widely developed and applied in practical engineering. With the development of contemporary engineering materials, the materials with different Young's moduli in tension and compression will be widely used in engineering. Therefore, it is necessary to develop an effective numerical method in order to correctly analyze the mechanical properties of these materials.
This thesis focuses on developing a new finite element formulation and iterative algorithm for different Young's moduli in tension and compression. Numerical study of mechanical properties is presented for the structures having different moduli in tension and compression.
Ambartsumyan's FE model, Jones' FE model and Ye's FE model are improved and investigated by an equivalent and complete concept. The FEM for different Young's moduli in tension and compression is further perfected with error estimation and dimensionless formulations. Equivalent is proven between the criterion of the principal strain presented by Ye's and the criterion of the principal stress widely used by predecessors. The dimensionless formulations of the principal material matrix and stiffness matrix develop the finite element method for different moduli.
The corresponding FEM program is established with Matlab platform. Numerical study is presented for the elastic problems having different moduli. The neutral axis problem and the ratio influence of tensile modulus to compressive modulus are discussed for bending structures under various conditions, such as different geometric models, different loads. With increasing the difference between tensile modulus and compressive modulus, there exits large errors in the value and distribution of displacement and stress, if classical theory is used to solve the problems having different Young's moduli in tension and compression.
The different tension-compression elastic moduli are introduced into small-displacement and large-displacement finite element formulations and numerical methods are developed using modified. Three-dimensional finite element iterative program for geometric and material nonlinear analysis is established.
Numerical study is presented for load-carrying capacity problem of different modulus structures. The influential factors of load-carrying capacity are further discussed, including constraint conditions, couple moments, ratios of tensile modulus to compressive modulus, etc. Finally, results show that there are big errors of load-carrying capacity in the bending-compression members having different moduli by using uniform modulus model.
引文
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