CMSG低阶应变梯度塑性理论的发展和应用
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摘要
在经典塑性理论中不包含任何尺度参数,因此无法解释在微米和亚微米量级实验中发现的尺度效应。为了解决工程实践中遇到的日益增多的微尺度设计制造问题,同时为韧性材料的解理断裂提供一种合理的解释,人们建立了应变梯度塑性理论。本文在Huang等2004年提出的CMSG低阶应变梯度塑性理论的基础上,进行了一系列的理论探索和计算研究工作。
1.推导了CMSG理论在混合硬化下的本构方程,考虑了背应力的影响。利用一阶非线性偏微方程的特征线方法,研究了混合硬化下CMSG理论在剪切和自重拉伸两个问题的解。求解得到了CMSG理论在这两个问题中的定解域,并指出了定解域会随着载荷的增大而逐渐变小,直到最后完全消失。如果想得到定解域以外区域上的解,就必须给出附加的非经典边界条件。这也是现在所有低阶应变梯度塑性理论所共有的问题。
2.利用CMSG理论研究了微纳米压痕问题。结果表明:对于包含尺度效应的微压痕问题,压头角度的减小可以显著提高微压痕硬度,Nix-Gao模型能够很好地描述压头角度变化对微压痕硬度的影响。摩擦也会提高微压痕硬度,但提高的程度随压头角度的增加而逐渐减弱,对于Berkovich压头摩擦的影响已经可以忽略。正三棱锥压头和圆锥压头之间的等底面积等价原则已经不再适用,我们在有限元计算结果的基础上提出了两者之间新的等价原则——等角原则。通过引入最大几何必需位错密度的概念,我们改进了CMSG理论的有限元计算模型,计算结果与球形压头微米和纳米压痕实验数据都十分符合。最大几何必需位错密度并不是一个材料常数,其取值与材料属性和压头的几何形状都有关系。
3.利用CMSG理论研究了平面应变I型稳态扩展裂纹问题。计算表明:虽然与MSG理论下的静止裂纹相比稍有下降,在裂尖附近应变梯度明显区域内的等效应力和分离应力都可达到相当高的水平,仍然明显高于HRR场。CMSG理论主导区内应力奇异性很高,达到甚至超过弹性场的平方根奇异性,且与塑性硬化指数无关。CMSG理论在扩展裂纹中的主导区比MSG理论在静止裂纹中要更小一些,且对远场应力强度因子不敏感。
Recent experiments at the micron scale have repeatedly shown that metallic materials display significant size effect. Classical plasticity theories do not possess internal material lengths and therefore cannot explain the observed size dependence of material behavior. Strain gradient theories have been established to deal with design and manufacturing issues at the level of microns and submicrons, and to explain the cleavage fracture in ductile materials. Based on the Conventional theory of Mechanism-based Strain Gradient plasticity (CMSG) developed by Huang et al. 2004, a series of problems have been investigated and some objectives have been achieved.
1. The constitutive relations of CMSG theory for the case that combined isotropic and kinematic hardening rule are introduced, with consideration of the back stress. By the method of characteristics for nonlinear partial differential equations, for the problems of an infinite layer in shear and a uniaxial tension bar subject to a constant body force, we have obtained the“domain of determinacy”for CMSG theory. It is established that, as the applied stress increases, the“domain of determinacy”shrinks and eventually vanishes. Outside the“domain of determinacy”, the solution may not be unique. Additional, non-classical boundary conditions are required to prescribe for the well-posedness of CMSG theory. This peoblem exists not only in CMSG theory but also all the lower-order strain gradient plasticity theories.
2. Some problems in micro- and nanoindentation are studied using finite element method based on CMSG theory.
First of all, the effect of indenter angles on the microindentation hardness is studied. It is shown that the Nix-Gao relation between the microindentation hardness and indentation depth holds for all the six indenter angles. The effect of friction is negligible for relatively flat indenters (e.g., Berkovich indenter), but may be significant for sharp indenters (e.g., cubic indenter).
Secondly, the equivalence rules between triangular pyramid indenter and conical indenter are investigated. It is shown that the base area equivalence rule is no longer applicable in microindentation. We establish a new rule, called the angle equivalence rule, to deal with the angles between the two types of indenters. Numerical results confirmed the angle equivalence rule for indenters with six different angles.
Lastly, the nanoindentation under spherical indenter is studied. We extend the finite element method for CMSG theory by taking into account the maximum allowable density of geometrically necessary dislocations (GND). The numerical results agree very well with both micron- and nanoindentation hardness data for iridium. But the maximum allowable density of GND is not a material constant. Its value depends on not only the material property but also the the indenter tip geometry.
3. Mode I crack under steady-state growth and plane strain is analyzed employing CMSG theory.
The results show that during growth of the crack, although lower than the static crack obtained by MSG theory, the normal separation stress will achieve considerably high value within a sensitive zone of strain gradient near the crack tip. The crack tip stress singularity within the dominance zone of CMSG theory is significantly higher, not only larger than classical plasticity (HRR field), but also equals to or exceeds the square-root singularity of elastic field. Moreover, that singularity is independent of the plastic hardening exponent. The dominance zone size of near-tip field in CMSG plasticity is smaller then that of MSG theory for static crack and insensitive to the level of the remote applied stress intensity.
1.推导了CMSG理论在混合硬化下的本构方程,考虑了背应力的影响。利用一阶非线性偏微方程的特征线方法,研究了混合硬化下CMSG理论在剪切和自重拉伸两个问题的解。求解得到了CMSG理论在这两个问题中的定解域,并指出了定解域会随着载荷的增大而逐渐变小,直到最后完全消失。如果想得到定解域以外区域上的解,就必须给出附加的非经典边界条件。这也是现在所有低阶应变梯度塑性理论所共有的问题。
2.利用CMSG理论研究了微纳米压痕问题。结果表明:对于包含尺度效应的微压痕问题,压头角度的减小可以显著提高微压痕硬度,Nix-Gao模型能够很好地描述压头角度变化对微压痕硬度的影响。摩擦也会提高微压痕硬度,但提高的程度随压头角度的增加而逐渐减弱,对于Berkovich压头摩擦的影响已经可以忽略。正三棱锥压头和圆锥压头之间的等底面积等价原则已经不再适用,我们在有限元计算结果的基础上提出了两者之间新的等价原则——等角原则。通过引入最大几何必需位错密度的概念,我们改进了CMSG理论的有限元计算模型,计算结果与球形压头微米和纳米压痕实验数据都十分符合。最大几何必需位错密度并不是一个材料常数,其取值与材料属性和压头的几何形状都有关系。
3.利用CMSG理论研究了平面应变I型稳态扩展裂纹问题。计算表明:虽然与MSG理论下的静止裂纹相比稍有下降,在裂尖附近应变梯度明显区域内的等效应力和分离应力都可达到相当高的水平,仍然明显高于HRR场。CMSG理论主导区内应力奇异性很高,达到甚至超过弹性场的平方根奇异性,且与塑性硬化指数无关。CMSG理论在扩展裂纹中的主导区比MSG理论在静止裂纹中要更小一些,且对远场应力强度因子不敏感。
Recent experiments at the micron scale have repeatedly shown that metallic materials display significant size effect. Classical plasticity theories do not possess internal material lengths and therefore cannot explain the observed size dependence of material behavior. Strain gradient theories have been established to deal with design and manufacturing issues at the level of microns and submicrons, and to explain the cleavage fracture in ductile materials. Based on the Conventional theory of Mechanism-based Strain Gradient plasticity (CMSG) developed by Huang et al. 2004, a series of problems have been investigated and some objectives have been achieved.
1. The constitutive relations of CMSG theory for the case that combined isotropic and kinematic hardening rule are introduced, with consideration of the back stress. By the method of characteristics for nonlinear partial differential equations, for the problems of an infinite layer in shear and a uniaxial tension bar subject to a constant body force, we have obtained the“domain of determinacy”for CMSG theory. It is established that, as the applied stress increases, the“domain of determinacy”shrinks and eventually vanishes. Outside the“domain of determinacy”, the solution may not be unique. Additional, non-classical boundary conditions are required to prescribe for the well-posedness of CMSG theory. This peoblem exists not only in CMSG theory but also all the lower-order strain gradient plasticity theories.
2. Some problems in micro- and nanoindentation are studied using finite element method based on CMSG theory.
First of all, the effect of indenter angles on the microindentation hardness is studied. It is shown that the Nix-Gao relation between the microindentation hardness and indentation depth holds for all the six indenter angles. The effect of friction is negligible for relatively flat indenters (e.g., Berkovich indenter), but may be significant for sharp indenters (e.g., cubic indenter).
Secondly, the equivalence rules between triangular pyramid indenter and conical indenter are investigated. It is shown that the base area equivalence rule is no longer applicable in microindentation. We establish a new rule, called the angle equivalence rule, to deal with the angles between the two types of indenters. Numerical results confirmed the angle equivalence rule for indenters with six different angles.
Lastly, the nanoindentation under spherical indenter is studied. We extend the finite element method for CMSG theory by taking into account the maximum allowable density of geometrically necessary dislocations (GND). The numerical results agree very well with both micron- and nanoindentation hardness data for iridium. But the maximum allowable density of GND is not a material constant. Its value depends on not only the material property but also the the indenter tip geometry.
3. Mode I crack under steady-state growth and plane strain is analyzed employing CMSG theory.
The results show that during growth of the crack, although lower than the static crack obtained by MSG theory, the normal separation stress will achieve considerably high value within a sensitive zone of strain gradient near the crack tip. The crack tip stress singularity within the dominance zone of CMSG theory is significantly higher, not only larger than classical plasticity (HRR field), but also equals to or exceeds the square-root singularity of elastic field. Moreover, that singularity is independent of the plastic hardening exponent. The dominance zone size of near-tip field in CMSG plasticity is smaller then that of MSG theory for static crack and insensitive to the level of the remote applied stress intensity.
引文
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