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1. Non-linear Motion Equations for Rod with Large Deformation | |||
XIAO Jianhua | |||
Mechanics 24 December 2012 | |||
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Abstract:Referring to the central line of rod to establish local dragging coordinator system, the S+R decomposition of deformation tensor is used to obtain the two sets of non-linear motion equations of rod with large deformation. One set is about the central line, another set is out-central line regions. For simple bending and simple torsion, simplifed non-linear motion equations are obtained. The research shows that: 1) for sharp bending, the exact motion equations for elastic line is obtained; 2) for simple torsion, the intrinsic stretching along the rotation direction is indispensable. This is the main distinction between rigid rotation and the local whole rotation in continuum; 3) when the local rotation angle is big enough, the deformation will jump to the another kind of formation form; 4) the S+R decomposition can be viewed as decomposing the general deformation into one torsion-free local stretching S and one deformation with torsion R. | |||
TO cite this article:XIAO Jianhua. Non-linear Motion Equations for Rod with Large Deformation[OL].[24 December 2012] http://en.paper.edu.cn/en_releasepaper/content/4506394 |
2. Mixture Stress Tensor and Non-linear Motion Equations for Large Deformation | |||
XIAO Jianhua | |||
Mechanics 28 August 2012 | |||
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Abstract:The non-linear motion equations of deformation are open problem for rational mechanics. For this research, the deformation is defined as the base vector transformation of commoving dragging coordinator system in geometrical field theory. As a logic conclusion, the strain and stress is defined as mixture tensors with the lower index related with current configuration and the upper index related with initial configuration. In this research, after a simply review about the mixture stress tensor, the corresponding motion equations are introduced. Then, the S+R decomposition is used to simplify the non-linear motion equations. To compare the general motion equations obtained in this research with the well-known non-linear motion equations, the motion equations in curvature coordinator systems are derived from general equations, respectively. Finally, the strategy to solve the general motion equations for large deformation is discussed. | |||
TO cite this article:XIAO Jianhua. Mixture Stress Tensor and Non-linear Motion Equations for Large Deformation[OL].[28 August 2012] http://en.paper.edu.cn/en_releasepaper/content/4487977 |
3. Determining Stress and Effective Elastic Parameters for Spherical Contact Problems II: Dynamic Contact | |||
Xiao Jianhua | |||
Mechanics 19 March 2008 | |||
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Abstract:The time dependent dynamic solutions are obtained in analytical form. The solution shows that: (1) for elastic contact, the time rate of stress increasing is determined by the contact geometry and medium feature (no dependence on load force); (2) for plastic contact, the time rate of stress increasing is determined by the medium feature (no dependence on load force and contact geometry); (3) for cracking contact, reducing the radium of sphere will increase the cracking range size and the cracking tip transportation speed; (4) for a given sphere size, increase the load will increase the cracking range and increase the cracking tip transportation speed. The solution in this paper forms the basic solution for the dynamic contact problem. | |||
TO cite this article:Xiao Jianhua. Determining Stress and Effective Elastic Parameters for Spherical Contact Problems II: Dynamic Contact[OL].[19 March 2008] http://en.paper.edu.cn/en_releasepaper/content/19443 |
4. Determining Stress and Effective Elastic Parameters for Spherical Contact Problems I: Static Contact | |||
Xiao Jianhua | |||
Mechanics 11 March 2008 | |||
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Abstract:For intrinsic incompressible medium, the spherical static contact problems are researched. The exact elastic analytical solutions are obtained. For the elastic-plastic deformation, the plastic stress and plastic deformation are obtained by the medium features and spherical radium parameters. The effective elastic-plastic constants are expressed explicitly. Furthermore, the cracked depth is formulated explicitly with the medium features and spherical radium parameters, also. Therefore, they form a complete solution for the problem under discussion. The results are compared with observed phenomenon. The evolution rate problem is discussed also. They are valuable for industry application. | |||
TO cite this article:Xiao Jianhua. Determining Stress and Effective Elastic Parameters for Spherical Contact Problems I: Static Contact[OL].[11 March 2008] http://en.paper.edu.cn/en_releasepaper/content/19191 |
5. Intrinsic Knots Produced by Large Deformation in 3-Space II: Multiscale | |||
Xiao Jianhua | |||
Mechanics 10 January 2008 | |||
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Abstract:Once micro-scale intrinsic curvatures are produced within the material by large deformation in 3-space, what physical feature can be used to identify their existence? Based on the traditional continuity concept, the physical features still can be measured by the macro displacement field. The effects produced by micro-scale intrinsic variation can be expressed by the variations of macro physical constant, where its variation is interpreted by micro dynamics or micro substructure variation. Along this theoretic line, multi-scale concept is introduced in mechanics. This concept is well supported by experiments. So, theoretically, formulating a suitable method to express the multi-scale feature by the macro deformation field should be valuable. In this research, the multi-scale features of material under large deformation are studied by the macro deformation field. That is to say, the macro deformation field is used to evaluate the possible micro-variations. In mathematics, it is equivalent with the inverse problem. Its physical foundation is that the same macro deformation field can be produced by several possible micro-scale deformations. As the curvature is an intrinsic local feature, the research shows that the multi-scale feature of materials under large deformation can be well explained by the curvature concept. The mechanical multi-scales can be obtained by the curvature parameter and the initial elastic parameters of macro scale of material. Therefore, the multi-scale features of material under large deformation are formulated by the deformation field itself rather than introducing other artificial assumption. | |||
TO cite this article:Xiao Jianhua. Intrinsic Knots Produced by Large Deformation in 3-Space II: Multiscale[OL].[10 January 2008] http://en.paper.edu.cn/en_releasepaper/content/17949 |
6. Intrinsic Knots Produced by Large Deformation in 3-Space I: Curvatures | |||
Xiao Jianhua | |||
Mechanics 17 October 2007 | |||
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Abstract:For a medium suffered large deformation, knots may be produced internally. Such a kind of knotted medium will reduce the elasticity of the medium. In this research, intrinsic knot production conditions are formulated by the deformation tensor in 3-space. Hence, the traditional continuity concept is replaced with path-connected continuity concept. For a spring produced by deformation, the curvatures of spring are formulated by the intrinsic local rotation angular which is determined by the asymmetrical displacement gradient components. The corresponding deformation energy functionals are formulated, also. They form the basic theoretic formulation of knots for finite deformation mechanics in Chen rational mechanics frame. Finally, the deformation energy of knotted medium is discussed with the knot representation polynomials. These results is expected to be valuable for many practical mechanic engineering problems, such as fatigue-crack, damage/risk evaluation, and elastoplastic instability, especially for understanding and formulating the non-linear dynamics of finite deformation. | |||
TO cite this article:Xiao Jianhua. Intrinsic Knots Produced by Large Deformation in 3-Space I: Curvatures[OL].[17 October 2007] http://en.paper.edu.cn/en_releasepaper/content/15736 |
7. Chen Rational Mechanics XII: Calculating Material Constants for Complicated Stress-Strain Experimental Curves | |||
Xiao Jianhua | |||
Mechanics 04 May 2007 | |||
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Abstract:In many experimental cases, the stress is directly measured by the stress-sensor mounted on force action face and the strain is directly measured by the strain-sensor mounted on the material free boundary. The strain measured by the sensor is different from the strain calculated by the length variation in that, at the local braking-down zone, the strain sensor gives out zero strain while the calculated length-variation strain is non-zero. Such a kind of measurement way gives out the intrinsic stress-strain relationship. This topic is studied by Chen Zhida in rational mechanics frame. Chen rational mechanics theory is based on the additive decomposition of deformation gradient. The new theoretic formulation introduces the local relative rotation concept which can be expressed by the local rotation angular parameter. On intrinsic sense, this angular represents the non-linear feature of stress-strain relation. By the new theory, the plastic parameter is expressed by the local rotation angular and the breaking-down stress is expressed by the critical rotation angular. For the experimental stress-strain data from elastic deformation to breaking-down, the calculating method of material constants is studied. By this method, the complicated deformation curve can be well expressed by the initial elastic constants and the local rotation angular. For initial isotropic material, the non-linear feature of stress-strain relation is expressed by two elastic constants and one path-dependent local rotation angular parameter. | |||
TO cite this article:Xiao Jianhua. Chen Rational Mechanics XII: Calculating Material Constants for Complicated Stress-Strain Experimental Curves[OL].[ 4 May 2007] http://en.paper.edu.cn/en_releasepaper/content/12628 |
8. Chen Rational Mechanics XI. Multi-scaling and Non-locality for Solids with Periodic Structure | |||
Xiao Jianhua | |||
Mechanics 04 April 2007 | |||
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Abstract:The non-locality problem is explained by the geometrical equation of finite deformation. Here, the global geometrical feature is determined by the intrinsic local deformation. This excludes the possibility of giving fixed geometry for deformation problem. This, in fact, is the physical underlying to use dragging coordinator system for large deformation. The multi-scaling problem is explained by the intrinsic physical variation. The micro physical energy variation is determined by quantum wave function solution. Combining with the non-locality feature of macro deformation, the mechanical parameters are expressed by micro physical parameters. This research shows how that the multi-scaling and non-locality are determined by the micro physical variation and measurable macro geometrical variation. | |||
TO cite this article:Xiao Jianhua. Chen Rational Mechanics XI. Multi-scaling and Non-locality for Solids with Periodic Structure[OL].[ 4 April 2007] http://en.paper.edu.cn/en_releasepaper/content/11931 |
9. Chen Rational Mechanics X. Path-dependency Description for One-dimension Large Deformation | |||
Xiao Jianhua | |||
Mechanics 19 March 2007 | |||
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Abstract:This research uses the invariant gauge deformation to introduce the local rotation concept. This local rotation can be expressed by the asymmetrical displacement. For one-dimension deformation problems, the research shows that the traditional strain definition cannot express the true stress field for curving deformation even the deformation is infinitesimal locally. Then, using the new theory, the analytical solutions for typical loading cases are obtained. The solution is highly non-linear even for infinitesimal local deformation. Furthermore, the local solution and global solution are not independent. The non-locality for curving deformation is the main difficulty for the traditional mechanics in treating the curving deformation. This problem is overcome by the new theoretic understanding and formulation. | |||
TO cite this article:Xiao Jianhua. Chen Rational Mechanics X. Path-dependency Description for One-dimension Large Deformation[OL].[19 March 2007] http://en.paper.edu.cn/en_releasepaper/content/11499 |
10. Chen Rational Mechanics IX. Dynamic Instability and Fatigue-Cracking Deformation | |||
Xiao Jianhua | |||
Mechanics 12 March 2007 | |||
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Abstract: Mechanical vibration not only produces noise but also trigs dynamic failure of material. Experientially, non-symmetric strain or stress is the main cause of instability of dynamic deformation. Based on geometrical field mechanics, it is found that, when the non-symmetric components of deformation are big enough, the vibration will become instable, which usually will trig the failure of material. The research shows that for a dynamic deformation the stress field is not symmetric even the deformation tensor is symmetric. However, for static deformation, the stress is always symmetric. This explains the non-linear behavior of vibration as the difference between the static deformation design with symmetric stress and the dynamic deformation with non-symmetrical stress. The vibration equation is introduced and discussed by finite geometrical field mechanics theory. Based on above results, the instability criteria for dynamic deformation are given in general form. The results will be applicable for the control of vibration. | |||
TO cite this article:Xiao Jianhua. Chen Rational Mechanics IX. Dynamic Instability and Fatigue-Cracking Deformation[OL].[12 March 2007] http://en.paper.edu.cn/en_releasepaper/content/11364 |
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