粘弹性材料动态扩展裂纹尖端场
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摘要
裂纹尖端渐近场的研究是断裂力学研究的重要课题之一，裂纹尖端应力、应变和其它物理量的确定为讨论材料参数对裂纹尖端场的影响及材料破坏断裂准则的建立提供了理论的依据。动态裂纹的扩展在材料学、地质学和结构工程领域等有着广泛的应用，因而研究粘弹性材料中裂纹动态扩展问题具有理论意义和广阔的应用前景。
随着科学技术的不断发展，许多新型材料不断涌现，这类材料的一个共同特点是具有明显的粘弹性特征。材料的粘弹性不仅会影响结构的刚度，也会影响到其强度，在研究裂纹尖端渐近场时，应该考虑到材料的粘性效应，这不仅更加符合实际情况，得到更精确的解，而且能解决率无关渐近解中存在的一些问题，因而材料的粘弹性性质对材料断裂性能影响的研究受到越来越大的重视。本文分别采用简单而且实用的粘弹性模型及刚性-粘弹性界面模型，对平面应变不可压缩材料的Ⅰ型和Ⅱ型的动态扩展裂纹尖端的应力、应变和位移场进行了具体的分析和计算；又采用粘弹性模型，对平面应变不可压缩材料的混合型动态扩展裂纹尖端场进行了具体的分析和计算。围绕这一问题，本文的主要工作有以下几个方面：
1．通过对裂尖场的渐近分析，确定了动态扩展裂纹尖端的应力和应变场的指数奇异性阶次，得出应力、应变具有相同的奇异量级，即σ～ε～γ~(-1/(n-1)。
2．通过对粘弹性本构理论的分析，给出了稳态蠕变阶段，粘性和弹性共同占主导作用的本构方程，并结合运动和协调方程，推导出粘弹性材料动态扩展裂尖场的控制方程。
3．根据问题的边界条件，通过对控制方程进行数值求解，得到了裂纹尖端的连续的应力、应变和位移场。
4．分析了动态解的性质，并讨论了裂尖应力、应变和位移场随各参数的变化规律，指出了材料的蠕变指数、蠕变系数和马赫数等物理常数对裂纹尖端渐近场的影响。
5．通过对两个模型的基本方程的推导、分析，得到当材料的蠕变指数趋
哈尔滨工程大学博士学位论文
于无穷时，本文中的幂硬化粘弹性材料动态扩展裂纹尖端场与Freund给出的
理想弹塑性材料动态扩展裂纹尖端场具有相近的形式;当马赫数趋于零且蠕
变指数趋于无穷时，粘弹性材料的渐近解与静态的理想塑性材料的渐近解具
有相近的形式。
6.通过对动态扩展裂纹尖端场的渐近分析，得出在粘弹性材料1型裂纹
前方，环向应变达到最大值，在粘弹性材料H型裂纹，刚性一粘弹性界面I型
和11型裂纹及混合型裂纹前方，剪应变达到最大值，因此，可以考虑从应变
角度出发建立局部的断裂准则。
总之，通过考虑扩展裂纹尖端材料的弹性和粘性效应，本文建立了不可
压缩幂硬化粘弹性材料中动态扩展裂纹尖端场的力学模型。通过理论分析和
相应的数值计算，验证了本模型的合理性和有效性。本文所作的研究，将为
最终解决裂纹尖端渐近场问题提供一种有益的探索，并且对于解决工程实践
中所遇到的相应的问题和建立材料的破坏准则提供理论上的参考。
The studying of crack-tip asymptotical field is one of the important subjects of fracture mechanics. After determining the stress, strain and other physical quantity in crack-tip field, the related theoretical reference may be put forward to discuss material parameter's influence to crack-tip field and found the fracture rule of material breakage. The propagating of dynamic crack has extensive application in materials science, geology and structural engineering field, therefore the study of dynamic propagation of crack in viscoelastic material possess theoretical meaning and broad application foreground.
With the continuous development of science and technology, coming forth much new type material, the common feature of them is possessing obvious viscoelastic character, which not only influences the rigidity of structure, but also the intensity. Then when studying the asymptotic crack-tip field, viscous effects of material should be considered, which not only accords with reality, but also can solve some problem existing in asymptotic solution of rate irrelated material, therefore the research of viscoelastic character's influence on fracture capability of material attracts much more recognition.
Adopting separately a rather simple but practicable viscoelastic model and rigid-viscoelastic Bi-material interface model in the dissertation, it has been asymptotically analyzed and calculated that the stress, strain and displacement of dynamic propagating crack-tip fields of plane strain mode I, II mode crack under the condition of incompressibility; and further adopting viscoelastic model to analyze and calculate the stress, strain and displacement of mixed mode crack under the condition of incompressibility. With regard to this problem, the main research items in this dissertation are as follows:
1. Through the asymptotical analyzing of singular crack-tip field, it is determined that the same exponent of singularity of stress and strain in dynamic propagating crack-tip field, namely,
2. By analyzing to the constitutive theory of viscoelastic material, the constitutive equations of viscosity and elasticity with the same dominant effect is obtained in the stable creep stage, and combining the movement and harmonization equations, the governing equations in dynamic growing crack-tip
field is deduced.
3. Numerical solutions of governing equations are obtained in combination with boundary conditions of each problem, and the fully continuous stress, strain and displacement in crack-tip field is found.
4. Having analyzed the nature of each dynamic solution, the laws of variation of stress, strain and displacement in crack-tip field according to each parameter are discussed and the influence of physical constant, such as creep exponent, creep coefficient and the Mach number on asymptotical crack-tip field is pointed out further.
5. By deducing and analyzing the basic equations of two models, it is obtained that when creeping exponent is close to infinitude, the solution of dynamic crack-tip field of viscoelastic power-hardening material in the dissertation is close formally to the solution of elastic-ideally plastic material given by Freund, in addition, when the Mach number is close to zero and creeping exponent is close to infinitude, the asymptotical solution of viscoelastic material is close formally to the static solution of elastic-ideally plastic material.
6. Through the asymptotical analyzing of dynamic propagation crack-tip field, the result is produced that the hoop strain reaches to maximum in the front of mode I crack of viscoelastic material, and the shearing strain reaches to maximum in the front of mode II and mixed mode crack of viscoelastic material, mode I and mode II interfacial crack of rigid-viscoelastic material, so local fracture rules may be founded from the point of view of strain.
In conclusion, by consideration of elasticity and viscosity effect of material in propagation crack-tip field, it is established that the mechanics model of viscoelasti
随着科学技术的不断发展，许多新型材料不断涌现，这类材料的一个共同特点是具有明显的粘弹性特征。材料的粘弹性不仅会影响结构的刚度，也会影响到其强度，在研究裂纹尖端渐近场时，应该考虑到材料的粘性效应，这不仅更加符合实际情况，得到更精确的解，而且能解决率无关渐近解中存在的一些问题，因而材料的粘弹性性质对材料断裂性能影响的研究受到越来越大的重视。本文分别采用简单而且实用的粘弹性模型及刚性-粘弹性界面模型，对平面应变不可压缩材料的Ⅰ型和Ⅱ型的动态扩展裂纹尖端的应力、应变和位移场进行了具体的分析和计算；又采用粘弹性模型，对平面应变不可压缩材料的混合型动态扩展裂纹尖端场进行了具体的分析和计算。围绕这一问题，本文的主要工作有以下几个方面：
1．通过对裂尖场的渐近分析，确定了动态扩展裂纹尖端的应力和应变场的指数奇异性阶次，得出应力、应变具有相同的奇异量级，即σ～ε～γ~(-1/(n-1)。
2．通过对粘弹性本构理论的分析，给出了稳态蠕变阶段，粘性和弹性共同占主导作用的本构方程，并结合运动和协调方程，推导出粘弹性材料动态扩展裂尖场的控制方程。
3．根据问题的边界条件，通过对控制方程进行数值求解，得到了裂纹尖端的连续的应力、应变和位移场。
4．分析了动态解的性质，并讨论了裂尖应力、应变和位移场随各参数的变化规律，指出了材料的蠕变指数、蠕变系数和马赫数等物理常数对裂纹尖端渐近场的影响。
5．通过对两个模型的基本方程的推导、分析，得到当材料的蠕变指数趋
哈尔滨工程大学博士学位论文
于无穷时，本文中的幂硬化粘弹性材料动态扩展裂纹尖端场与Freund给出的
理想弹塑性材料动态扩展裂纹尖端场具有相近的形式;当马赫数趋于零且蠕
变指数趋于无穷时，粘弹性材料的渐近解与静态的理想塑性材料的渐近解具
有相近的形式。
6.通过对动态扩展裂纹尖端场的渐近分析，得出在粘弹性材料1型裂纹
前方，环向应变达到最大值，在粘弹性材料H型裂纹，刚性一粘弹性界面I型
和11型裂纹及混合型裂纹前方，剪应变达到最大值，因此，可以考虑从应变
角度出发建立局部的断裂准则。
总之，通过考虑扩展裂纹尖端材料的弹性和粘性效应，本文建立了不可
压缩幂硬化粘弹性材料中动态扩展裂纹尖端场的力学模型。通过理论分析和
相应的数值计算，验证了本模型的合理性和有效性。本文所作的研究，将为
最终解决裂纹尖端渐近场问题提供一种有益的探索，并且对于解决工程实践
中所遇到的相应的问题和建立材料的破坏准则提供理论上的参考。
The studying of crack-tip asymptotical field is one of the important subjects of fracture mechanics. After determining the stress, strain and other physical quantity in crack-tip field, the related theoretical reference may be put forward to discuss material parameter's influence to crack-tip field and found the fracture rule of material breakage. The propagating of dynamic crack has extensive application in materials science, geology and structural engineering field, therefore the study of dynamic propagation of crack in viscoelastic material possess theoretical meaning and broad application foreground.
With the continuous development of science and technology, coming forth much new type material, the common feature of them is possessing obvious viscoelastic character, which not only influences the rigidity of structure, but also the intensity. Then when studying the asymptotic crack-tip field, viscous effects of material should be considered, which not only accords with reality, but also can solve some problem existing in asymptotic solution of rate irrelated material, therefore the research of viscoelastic character's influence on fracture capability of material attracts much more recognition.
Adopting separately a rather simple but practicable viscoelastic model and rigid-viscoelastic Bi-material interface model in the dissertation, it has been asymptotically analyzed and calculated that the stress, strain and displacement of dynamic propagating crack-tip fields of plane strain mode I, II mode crack under the condition of incompressibility; and further adopting viscoelastic model to analyze and calculate the stress, strain and displacement of mixed mode crack under the condition of incompressibility. With regard to this problem, the main research items in this dissertation are as follows:
1. Through the asymptotical analyzing of singular crack-tip field, it is determined that the same exponent of singularity of stress and strain in dynamic propagating crack-tip field, namely,
2. By analyzing to the constitutive theory of viscoelastic material, the constitutive equations of viscosity and elasticity with the same dominant effect is obtained in the stable creep stage, and combining the movement and harmonization equations, the governing equations in dynamic growing crack-tip
field is deduced.
3. Numerical solutions of governing equations are obtained in combination with boundary conditions of each problem, and the fully continuous stress, strain and displacement in crack-tip field is found.
4. Having analyzed the nature of each dynamic solution, the laws of variation of stress, strain and displacement in crack-tip field according to each parameter are discussed and the influence of physical constant, such as creep exponent, creep coefficient and the Mach number on asymptotical crack-tip field is pointed out further.
5. By deducing and analyzing the basic equations of two models, it is obtained that when creeping exponent is close to infinitude, the solution of dynamic crack-tip field of viscoelastic power-hardening material in the dissertation is close formally to the solution of elastic-ideally plastic material given by Freund, in addition, when the Mach number is close to zero and creeping exponent is close to infinitude, the asymptotical solution of viscoelastic material is close formally to the static solution of elastic-ideally plastic material.
6. Through the asymptotical analyzing of dynamic propagation crack-tip field, the result is produced that the hoop strain reaches to maximum in the front of mode I crack of viscoelastic material, and the shearing strain reaches to maximum in the front of mode II and mixed mode crack of viscoelastic material, mode I and mode II interfacial crack of rigid-viscoelastic material, so local fracture rules may be founded from the point of view of strain.
In conclusion, by consideration of elasticity and viscosity effect of material in propagation crack-tip field, it is established that the mechanics model of viscoelasti
引文
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