局部变形与相变分析
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摘要
在实验中已经发现了与传统局部变形不一样的局部变形,例如吕德斯带,以及镍钛合金管的拉伸或压缩中出现的局部变形,在这些局部变形是在均匀变形场中出现的局部变形。在这些局部变形的边界上,应力、变形梯度不连续。这是与传统局部变形不一样的。现有的描述局部变形的方法,例如引入局部缺陷而导致的局部变形,不能够模拟在变形边界上变形梯度和应力的不连续性。
本文建立一种方法,就是把局部变形区域和未局部变形区域分别看成塑性和弹性两相,采用弹塑性材料的相变理论进行分析。该相变理论考虑了边界上的应力与变形梯度的不连续性并且强加了力的连续性,同时也强加了麦克斯韦尔条件。在这种情况下可以预测出局部变形带的倾斜方向、折曲角以及局部变形区域内的应力场和应变场。
本文模拟了单向压缩下的具有应变软化特性的弹塑性材料的局部变形。作为算例,本文应用相变理论,由细晶粒炭钢在单向拉伸/压缩下的峰值—下降—平台型相变曲线得到了其在单向拉伸/压缩下的力学性质曲线,通过相变分析,可以得到局部变形时的麦克斯韦尔应力、折曲带的倾角、折曲角以及折曲带内的应力与应变。计算结果与有关实验中测量值吻合较好。
本文还分析了具有应变软化特性的弹塑性材料在纯剪切作用下的局部变形。这时候对于平面剪切情况下在理论上仍然可以出现一个局部变形带,这个局部变形带的方向与剪切方向相同。文中已给出平面剪切作用下的镍钛合金板的数值算例。
对于薄壁圆筒在扭转(纯剪切)过程当中观察不到局部变形带这一实验现象,本文做了很好的解释。即薄壁圆筒在扭转作用下局部变形带的倾斜角度为九十度时,不可能观察到局部变形带。
The local deformation was found in experiments is not same to traditional local deformation,such as Lüders-type deformation and local deformation of NiTi alloy tube under uniaxial tension or compression. All of local deformation generate from uniform deformation field. Stress and deformation gradient across interfaces between local deformation field and undeformation field are discontinuity. It is different from traditional local deformation. The present method of characterizing local deformation is that the geometric imperfection was brought in local deformation. But this method can not simulate discontinuity of Stress and deformation gradient across interfaces between local deformation field and undeformation field.
In this paper, we establish a method that we regard local deformation field and undeformation field as elastic phase and plastic phase respectively, and analyze it by theory of phase transition in elastoplastic materials. Discontinuity of Stress and deformation gradient across interfaces between local deformation field and undeformation field are considered and continuity of traction and displacement across interface and the Maxwell relation is imposed. In this case, the inclination angle of the localized deformed band, the kink angle of the strip, the stresses field and strains in the localized deformed field can be predicted.
In present paper, the local deformation in elastoplastic materials with strain-softening behaviour under uniaxial compression is simulated. As an illustration, a mechanical property curve of fine grained steel under uniaxial tension/compression is obtained from its peak-drop-plateau-type curve under uniaxial tension/compression by theory of phase transition. By analysis of phase transformation, the Maxwell stress, the inclination angle of the Lüders band, kink angle of the strip, and both stress and strain in the Lüders band are predicted analytically while local deformation. The predicted results are in reasonable agreement with the experimental measurements.
In this paper, we also analyze local deformation in elastoplastic materials with strain-softening behavior under pure shear. Theoretical, the localized deformed band under plane shear still can appear. The direction of localized deformed band is identical with the direction of shear. As an illustration, a NiTi alloy plate under plane shear is numerically analyzed.
We explain why no locally deformed spiral band was observed in experiments on thin-walled tubes under torsion (pure shear). That is to say, when the inclination angle of the localized deformed band is predicted to be , the localized deformed band cannot be observed in experiments. 90
本文建立一种方法,就是把局部变形区域和未局部变形区域分别看成塑性和弹性两相,采用弹塑性材料的相变理论进行分析。该相变理论考虑了边界上的应力与变形梯度的不连续性并且强加了力的连续性,同时也强加了麦克斯韦尔条件。在这种情况下可以预测出局部变形带的倾斜方向、折曲角以及局部变形区域内的应力场和应变场。
本文模拟了单向压缩下的具有应变软化特性的弹塑性材料的局部变形。作为算例,本文应用相变理论,由细晶粒炭钢在单向拉伸/压缩下的峰值—下降—平台型相变曲线得到了其在单向拉伸/压缩下的力学性质曲线,通过相变分析,可以得到局部变形时的麦克斯韦尔应力、折曲带的倾角、折曲角以及折曲带内的应力与应变。计算结果与有关实验中测量值吻合较好。
本文还分析了具有应变软化特性的弹塑性材料在纯剪切作用下的局部变形。这时候对于平面剪切情况下在理论上仍然可以出现一个局部变形带,这个局部变形带的方向与剪切方向相同。文中已给出平面剪切作用下的镍钛合金板的数值算例。
对于薄壁圆筒在扭转(纯剪切)过程当中观察不到局部变形带这一实验现象,本文做了很好的解释。即薄壁圆筒在扭转作用下局部变形带的倾斜角度为九十度时,不可能观察到局部变形带。
The local deformation was found in experiments is not same to traditional local deformation,such as Lüders-type deformation and local deformation of NiTi alloy tube under uniaxial tension or compression. All of local deformation generate from uniform deformation field. Stress and deformation gradient across interfaces between local deformation field and undeformation field are discontinuity. It is different from traditional local deformation. The present method of characterizing local deformation is that the geometric imperfection was brought in local deformation. But this method can not simulate discontinuity of Stress and deformation gradient across interfaces between local deformation field and undeformation field.
In this paper, we establish a method that we regard local deformation field and undeformation field as elastic phase and plastic phase respectively, and analyze it by theory of phase transition in elastoplastic materials. Discontinuity of Stress and deformation gradient across interfaces between local deformation field and undeformation field are considered and continuity of traction and displacement across interface and the Maxwell relation is imposed. In this case, the inclination angle of the localized deformed band, the kink angle of the strip, the stresses field and strains in the localized deformed field can be predicted.
In present paper, the local deformation in elastoplastic materials with strain-softening behaviour under uniaxial compression is simulated. As an illustration, a mechanical property curve of fine grained steel under uniaxial tension/compression is obtained from its peak-drop-plateau-type curve under uniaxial tension/compression by theory of phase transition. By analysis of phase transformation, the Maxwell stress, the inclination angle of the Lüders band, kink angle of the strip, and both stress and strain in the Lüders band are predicted analytically while local deformation. The predicted results are in reasonable agreement with the experimental measurements.
In this paper, we also analyze local deformation in elastoplastic materials with strain-softening behavior under pure shear. Theoretical, the localized deformed band under plane shear still can appear. The direction of localized deformed band is identical with the direction of shear. As an illustration, a NiTi alloy plate under plane shear is numerically analyzed.
We explain why no locally deformed spiral band was observed in experiments on thin-walled tubes under torsion (pure shear). That is to say, when the inclination angle of the localized deformed band is predicted to be , the localized deformed band cannot be observed in experiments. 90
引文
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[2] Qing-Ping Sun and Zhi-Qi Liu, "Phase transformation in superelastic NiTi polycrystalline micro-tubes under tension and torsion––from localization to homogeneous deformation,"International Journal of Solids and Structures, vol. 39, pp. 3797-3809, 2002.
[3]徐祖耀,相变原理.北京:科学出版社, 1988.
[4] R. Abeyaratne and J. K. Knowles, "Impact-induced phase transitions in thermoelastic solids," Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 355, pp. 843-867, 1997.
[5] J. M. Ball, "Mathematical models of martensitic microstructure," Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing, vol. 378, pp. 61-69, 2004.
[6] P. K. Purohit and K. Bhattacharya, "Dynamics of strings made of phase-transforming materials," Journal of the Mechanics and Physics of Solids, vol. 51, pp. 393-424, 2003.
[7] K. Bhattacharya and R. D. James, "The material is the machine," Science, vol. 307, pp. 53-54, 2005.
[8] R. Abeyaratne, S. J. Kim, and J. K. Knowles, "A One-Dimensional Continuum Model for Shape-Memory Alloys," International Journal of Solids and Structures, vol. 31, pp. 2229-2249, 1994.
[9] R. Abeyaratne and J. K. Knowles, "A Continuum Model of a Thermoelastic Solid Capable of Undergoing Phase-Transitions," Journal of the Mechanics and Physics of Solids, vol. 41, pp. 541-571, 1993.
[10] K. Bhattacharya, "Comparison of the Geometrically Nonlinear and Linear Theories of Martensitic-Transformation," Continuum Mechanics and Thermodynamics, vol. 5, pp. 205-242, 1993.
[11] J. K. E. S. Knowles, E., "On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics," J. Elasticity, vol. 8, pp. 329-379, 1978.
[12] J. L. Ericksen, "Equilibrium of Bars," Journal of Elasticity, vol. 5, pp. 191-201, 1975.
[13] R. Abeyaratne, "An Admissibility Condition for Equilibrium Shocks in Finite Elasticity," Journal of Elasticity, vol. 13, pp. 175-184, 1983.
[14] R. C. Abeyaratne, "Discontinuous Deformation Gradients in the Finite Twisting of an Incompressible Elastic Tube," Journal of Elasticity, vol. 11, pp. 43-80, 1981.
[15] R. Abeyaratne, "Discontinuous Deformation Gradients Away from the Tip of a Crack in Anti-Plane Shear," Journal of Elasticity, vol. 11, pp. 373-393, 1981.
[16] A. R, Bhattacharya.K, and K. J.K, "Strain-energy functions with multiple local minima: modeling phase transformation using finite thermo-elasticity," in Nonlinear Elasticity: theory and Applications, F. Y.B and O. R.W., Eds. Cambridge: Cabdridge University Press, 2001, pp. 431-490.
[17] G. Y. Qiu and T. J. Pence, "Loss of ellipticity in plane deformation of a simple directionally reinforced incompressible nonlinearly elastic solid," Journal of Elasticity, vol. 49, pp. 31-63, 1997.
[18] Q. Jiang and J. K. Knowles, "A Class of Compressible Elastic-Materials Capable of Sustaining Finite Antiplane Shear," Journal of Elasticity, vol. 25, pp. 193-201, 1991.
[19] P. Rosakis and J. K. Knowles, "Unstable kinetic relations and the dynamics of solid-solid phase transitions," Journal of the Mechanics and Physics of Solids, vol. 45, pp. 2055-2081, 1997.
[20] G. Y. Qiu and T. J. Pence, "Remarks on the behavior of simple directionally reinforced incompressible nonlinearly elastic solids," Journal of Elasticity, vol. 49, pp. 1-30, 1997.
[21] Y. B. Fu and A. B. Freidin, "Characterization and stability of two-phase piecewise-homogeneous deformations," Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, vol. 460, pp. 3065-3094, 2004.
[22] Y. B. Fu and Y. T. Zhang, "Continuum-mechanical modelling of kink-band formation in fibre-reinforced composites," International Journal of Solids and Structures, vol. 43, pp. 3306-3323, 2006.
[23] Y. T. Zhang, Y. X. Xie, and S.G.Ren, "Modeling of kinking in elastoplastic continua as a stress-induced phase transformation," International Journal of Plasticity, submitted.
[24] R. Abeyaratne and J. K. Knowles, "Non-Elliptic Elastic-Materials and the Modeling of Elastic-Plastic Behavior for Finite Deformation," Journal of the Mechanics and Physics of Solids, vol. 35, pp. 343-365, 1987.
[25] J. M. Ball, "Mathematical models of martensitic microstructure," Materials Science and Engineering A, vol. 378, pp. 61-69, 2004.
[26] R. Abeyaratne and J. K. Knowles, "On the stability of thermoelastic materials," Journal of Elasticity, vol. 53, pp. 199-213, 1998.
[27] R. Abeyaratne and J. K. Knowles, "Equilibrium Shocks in Plane Deformations of Incompressible Elastic-Materials," Journal of Elasticity, vol. 22, pp. 63-80, 1989.
[28] M.E.Gurtin, An Introduction to Continuum Mechanics. New York: Academic Press, 1982.
[30] R.W.Ogden, "Elements of the theory of finite elasticity," in Nonlinear Elasticity: Theory and Applications, Y.B.Fu&R.W.Ogden, Ed. Cambridge: Cambridge University Press, 2002, pp. 1-57.
[30]黄克智,黄永刚,固体本构关系.北京:清华大学出版社, 1999.
[31] M.E.Gurtin, Configurational forces as basic concepts of continuum physics. New York: Springer, 2000.
[32]李国琛,耶纳,塑性大应变微结构力学,第三版ed.北京:科学出版社, 2003.
[33] A.C.Eringen, Mechanics of Continua, Sencond ed: Robert E. Krieger Publishing Company, 1980.
[34]黄克智,非线性连续介质力学.北京:清华大学出版社,北京大学出版社, 1989.
[35]郭仲衡,非线性弹性理论.北京:科学出版社, 1980.
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