压力敏感性材料球形孔洞膨胀问题的弹塑性分析
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摘要
压力敏感性材料(包括岩石、土壤、泡沫金属、聚合物材料、橡胶等)是自然界中应用最广泛的材料。由于材料中存在微结构(孔洞、微缺陷、微裂纹等),在外载荷作用下材料变形和破坏机理复杂,因而对压力敏感性材料的变形和破坏机理进行深入的力学研究已成为当前固体力学中的一个重要研究课题。
在压力敏感性材料变形和破坏机理的研究中,球形孔洞膨胀模型因其具有良好的对称性、简便明确并易于通过理论推导给出应力和应变场解,该研究结果具有明确物理意义,从而可以揭示材料的变形本质,因而无论是固体力学、材料科学、固体物理,还是爆炸力学等学科,研究人员都十分重视球形孔洞膨胀弹塑性分析问题的研究。
本文在阐述了有限变形弹塑性理论的基础上,指出对于次弹-塑性理论,解决问题的关键在于屈服函数的选择。讨论了三类压力敏感性材料的双独立参数屈服准则。由于采用椭圆型方程很好地保持了从弹性变形到塑性变形的连续性,论文中采用椭圆型屈服函数,对压力敏感性材料球形孔洞膨胀问题进行了弹塑性分析。本文的主要工作如下：
1、在次-弹塑性有限变形理论的框架下,建立了材料的本构模型。在球坐标系下推导出球形孔洞膨胀有限弹塑性变形问题的本构方程及平衡方程的表达式。并利用对数应变,得出几何方程及协调方程和边界条件。通过数值计算,给出在内压作用下,压力敏感性材料中球形孔洞应力和应变的分布,讨论了压力敏感性系数对应力和应变场的影响。
2、研究了理想弹塑性材料有限变形弹塑性球形孔洞膨胀问题。对于应力场,在Euler坐标系中,求解的问题是“静定问题”,即可以用屈服条件和平衡方程求解球形孔洞的膨胀问题。由变形前后Lagrange坐标向Euler坐标的转换规律,利用对数应变,通过数值计算,给出在内压作用下理想弹塑性压力敏感性材料中球形孔洞应变的分布,讨论了压力敏感性系数对应力和应变场的影响。
3、采用椭圆型压力敏感性材料屈服准则和自相似假设,采用三区模型,研究球形孔洞动态扩展问题。通过对弹性区的推导得出应力的分布和弹塑性交界处连续条件；在塑性区给出求解问题的关于∑r和∑θ非线性微分方程,给出基本物理量(∑θ,∑r,V,ρ／ρ0)数值结果并讨论了材料参数对场量的影响。
4、研究了理想压力敏感性弹塑性材料孔洞动态扩展问题。
将工程中的实际问题抽象出理论模型,研究其变形的普遍规律,用以设计实验和建立数值计算的模型,将理论研究、数值计算和实验紧密结合,互相渗透、互相补充,深刻理解工程的物理本质找出一般性规律并指导实践,是力学学科发展的必然趋势。随着新材料的涌现,科学技术的发展,在军事工程、建筑工程和航天航空工程中提出许多有待于解决的复杂问题,需要从微观、细观以及宏观等不同层次上深入认识材料和结构的力学行为。因此压力敏感性材料中球形孔洞膨胀的深入研究更具有理论意义和广泛的工程应用价值。
The pressure-sensitive dilatant materials, such as rock, soil, foam metal, polymeric material, rubber and so on, are the most widely applied materials in nature. As these materials contain micro-structures (such as micro-voids, defects, inclusions and cracks), the deformation and failure mechanism are complicated. Thus, in-depth study on the deformation and failure mechanism of the pressure-sensitive dilatant material is an important research subject in solid mechanics at present.
During the study on the deformation and failure mechanism of the pressure-sensitive dilatant material, researchers paid great attention to spherical cavity inflation model for its symmetry, simplicity, and convenience to get the stress and strain field solutions by theoretical deduction and reveal the deformation law. So investigation on spherical cavity inflation through elastoplastic analysis was devoted much attention by researchers from no matter solid mechanics, materials science, solid state physics, explosion mechanics etc.
In this paper, based on expatiation of the finite deformation elastic-plastic theory, for the hypoelastoplastic problem, the key to solve it is how to choose the yield functions, so three kinds of two-independent-parameters yield functions were discussed. Then the elliptic-equation yield function was chosen to make elastoplastic analysis on spherical cavity inflation problem in pressure-sensitive dilatant materials, owing to it can maintain the continuity from the elastic deformation to the plastic deformation. In this paper, the main jobs are as follows:
1. The constitutive model of the pressure-sensitive dilatant materials was established based on the frame of the hypoelastoplastic finite deformation theory. The expressions of the constitutive equations and the equilibrium equations were deduced in spherical coordinate system. Then the geometry equations, the compatibility equations and the boundary conditions were obtained by the use of the logarithmic strain. Accordingly, through the numerical calculation, the stress distribution and the strain distribution of the spherical cavity were given in the pressure-sensitive dilatant materials under internal pressure. The influence of the pressure sensitive coefficient on the stress field and the strain field were also discussed.
2. The problems on the spherical cavity inflation with finite elastoplastic deformation in the elastic-perfectly plastic materials were studied. It is the statically indeterminate problem for the stress fields in the Euler coordinate system, which can be solved by using the yield functions and the equilibrium equations. Via the transforming rules from the Lagrange coordinate system to the Euler coordinate system pre-and post-deformation, the logarithmic strain and the numerical calculation, the stress distribution and the strain distribution of the spherical cavity are given in the elastic-perfectly plastic pressure-sensitive dilatant materials under internal pressure. The influence of the pressure sensitive coefficient on the stress field and the strain field are also discussed.
3. The problem on the spherical cavity dynamic expansion was studied, through the use of the elliptic-equation yield function, the self-similarity hypothesis and the three region model. Then the stress distribution and the continuous conditions in the intersection of plastic and elastic areas were deduced in the elastic region; the nonlinear differential equations on∑r and∑θare also given in the plastic region. The numerical calculation results on the basic physical quantities (∑θ,∑r,V,ρ/ρ0) are obtained, at last, the effects of the material parameters on the stress and strain fields are discussed.
4. The problem on the spherical cavity dynamic expansion was studied in the perfectly pressure-sensitive dilatant materials.
Drawing out the theoretical model from the practical problems in engineering to investigate the general deformation law for the experiment design and the numerical calculation model, combining the theoretical research, numerical calculation with the experiment closely and permeating each other to understand the physical essence of the engineering problem deeply and find the general law for guiding practice, all of these things above are inevitable trends in the development of mechanics. With the emergence of the new materials as well as the development of science and technology, there are many complex problems to solve in aviation, navigation, national defense, textile industry, which demand profound comprehension on the mechanical behavior of the materials and structure at microcosmic, microscopic and macroscopic level respectively. Therefore, further study on the spherical cavity inflation by elasto-plastic analysis in the pressure-sensitive dilatant materials has theoretical significance and practical value.
在压力敏感性材料变形和破坏机理的研究中,球形孔洞膨胀模型因其具有良好的对称性、简便明确并易于通过理论推导给出应力和应变场解,该研究结果具有明确物理意义,从而可以揭示材料的变形本质,因而无论是固体力学、材料科学、固体物理,还是爆炸力学等学科,研究人员都十分重视球形孔洞膨胀弹塑性分析问题的研究。
本文在阐述了有限变形弹塑性理论的基础上,指出对于次弹-塑性理论,解决问题的关键在于屈服函数的选择。讨论了三类压力敏感性材料的双独立参数屈服准则。由于采用椭圆型方程很好地保持了从弹性变形到塑性变形的连续性,论文中采用椭圆型屈服函数,对压力敏感性材料球形孔洞膨胀问题进行了弹塑性分析。本文的主要工作如下：
1、在次-弹塑性有限变形理论的框架下,建立了材料的本构模型。在球坐标系下推导出球形孔洞膨胀有限弹塑性变形问题的本构方程及平衡方程的表达式。并利用对数应变,得出几何方程及协调方程和边界条件。通过数值计算,给出在内压作用下,压力敏感性材料中球形孔洞应力和应变的分布,讨论了压力敏感性系数对应力和应变场的影响。
2、研究了理想弹塑性材料有限变形弹塑性球形孔洞膨胀问题。对于应力场,在Euler坐标系中,求解的问题是“静定问题”,即可以用屈服条件和平衡方程求解球形孔洞的膨胀问题。由变形前后Lagrange坐标向Euler坐标的转换规律,利用对数应变,通过数值计算,给出在内压作用下理想弹塑性压力敏感性材料中球形孔洞应变的分布,讨论了压力敏感性系数对应力和应变场的影响。
3、采用椭圆型压力敏感性材料屈服准则和自相似假设,采用三区模型,研究球形孔洞动态扩展问题。通过对弹性区的推导得出应力的分布和弹塑性交界处连续条件；在塑性区给出求解问题的关于∑r和∑θ非线性微分方程,给出基本物理量(∑θ,∑r,V,ρ／ρ0)数值结果并讨论了材料参数对场量的影响。
4、研究了理想压力敏感性弹塑性材料孔洞动态扩展问题。
将工程中的实际问题抽象出理论模型,研究其变形的普遍规律,用以设计实验和建立数值计算的模型,将理论研究、数值计算和实验紧密结合,互相渗透、互相补充,深刻理解工程的物理本质找出一般性规律并指导实践,是力学学科发展的必然趋势。随着新材料的涌现,科学技术的发展,在军事工程、建筑工程和航天航空工程中提出许多有待于解决的复杂问题,需要从微观、细观以及宏观等不同层次上深入认识材料和结构的力学行为。因此压力敏感性材料中球形孔洞膨胀的深入研究更具有理论意义和广泛的工程应用价值。
The pressure-sensitive dilatant materials, such as rock, soil, foam metal, polymeric material, rubber and so on, are the most widely applied materials in nature. As these materials contain micro-structures (such as micro-voids, defects, inclusions and cracks), the deformation and failure mechanism are complicated. Thus, in-depth study on the deformation and failure mechanism of the pressure-sensitive dilatant material is an important research subject in solid mechanics at present.
During the study on the deformation and failure mechanism of the pressure-sensitive dilatant material, researchers paid great attention to spherical cavity inflation model for its symmetry, simplicity, and convenience to get the stress and strain field solutions by theoretical deduction and reveal the deformation law. So investigation on spherical cavity inflation through elastoplastic analysis was devoted much attention by researchers from no matter solid mechanics, materials science, solid state physics, explosion mechanics etc.
In this paper, based on expatiation of the finite deformation elastic-plastic theory, for the hypoelastoplastic problem, the key to solve it is how to choose the yield functions, so three kinds of two-independent-parameters yield functions were discussed. Then the elliptic-equation yield function was chosen to make elastoplastic analysis on spherical cavity inflation problem in pressure-sensitive dilatant materials, owing to it can maintain the continuity from the elastic deformation to the plastic deformation. In this paper, the main jobs are as follows:
1. The constitutive model of the pressure-sensitive dilatant materials was established based on the frame of the hypoelastoplastic finite deformation theory. The expressions of the constitutive equations and the equilibrium equations were deduced in spherical coordinate system. Then the geometry equations, the compatibility equations and the boundary conditions were obtained by the use of the logarithmic strain. Accordingly, through the numerical calculation, the stress distribution and the strain distribution of the spherical cavity were given in the pressure-sensitive dilatant materials under internal pressure. The influence of the pressure sensitive coefficient on the stress field and the strain field were also discussed.
2. The problems on the spherical cavity inflation with finite elastoplastic deformation in the elastic-perfectly plastic materials were studied. It is the statically indeterminate problem for the stress fields in the Euler coordinate system, which can be solved by using the yield functions and the equilibrium equations. Via the transforming rules from the Lagrange coordinate system to the Euler coordinate system pre-and post-deformation, the logarithmic strain and the numerical calculation, the stress distribution and the strain distribution of the spherical cavity are given in the elastic-perfectly plastic pressure-sensitive dilatant materials under internal pressure. The influence of the pressure sensitive coefficient on the stress field and the strain field are also discussed.
3. The problem on the spherical cavity dynamic expansion was studied, through the use of the elliptic-equation yield function, the self-similarity hypothesis and the three region model. Then the stress distribution and the continuous conditions in the intersection of plastic and elastic areas were deduced in the elastic region; the nonlinear differential equations on∑r and∑θare also given in the plastic region. The numerical calculation results on the basic physical quantities (∑θ,∑r,V,ρ/ρ0) are obtained, at last, the effects of the material parameters on the stress and strain fields are discussed.
4. The problem on the spherical cavity dynamic expansion was studied in the perfectly pressure-sensitive dilatant materials.
Drawing out the theoretical model from the practical problems in engineering to investigate the general deformation law for the experiment design and the numerical calculation model, combining the theoretical research, numerical calculation with the experiment closely and permeating each other to understand the physical essence of the engineering problem deeply and find the general law for guiding practice, all of these things above are inevitable trends in the development of mechanics. With the emergence of the new materials as well as the development of science and technology, there are many complex problems to solve in aviation, navigation, national defense, textile industry, which demand profound comprehension on the mechanical behavior of the materials and structure at microcosmic, microscopic and macroscopic level respectively. Therefore, further study on the spherical cavity inflation by elasto-plastic analysis in the pressure-sensitive dilatant materials has theoretical significance and practical value.
引文
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