剪切带演化模型及其在模拟柱壳内外爆剪切失稳中的应用
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摘要
柱壳在内部或外部炸药爆轰驱动下的断裂行为有很强的应用背景,对该问题已经有几十年的研究历史。大量试验研究结果表明：绝热剪切带的形成和演化在爆轰加载柱壳高应变率膨胀断裂和厚壁圆筒向内塌陷破坏的过程中起着关键作用,其是裂纹的先导,断裂面沿着剪切带路径传播发展。目前,对该问题的研究主要以实验和理论探讨为主,数值模拟分析相对较少。本文以柱壳在内部炸药爆轰驱动下的高应变率膨胀断裂和厚壁圆筒在外部炸药驱动下的向内塌陷问题为需求背景,探索适合于模拟多条剪切带自组织行为的材料模型,针对金属材料强烈的粘塑性效应,探索适合于粘塑性本构计算的数值算法。文章主要内容和创新性成果如下：
1)通过在宏观本构中引入概率因子来描述材料缺陷导致的材料屈服应力分布不均匀性,假定该概率因子在空间服从正态分布。在分析绝热剪切带形成过程各阶段特点的基础上,给出了剪切带形成的多阶段本构模型。金属材料中广泛存在着各种缺陷,如位错、晶界、沉淀项、微孔洞、微裂纹等,某些缺陷导致材料局部屈服应力提高,某些缺陷导致材料局部屈服应力降低,屈服强度在材料内呈不均匀分布,但这种不均匀性在现有的宏观本构中并不体现。剪切带的形成是扰动导致变形失稳的结果,扰动源在其形成过程中起关键作用,因此要模拟多条剪切带的形成发展过程,必须在本构中考虑屈服强度的分布不均匀性。本文通过在宏观本构中引入一个概率因子建立概率型本构关系来描述材料内屈服强度的不均匀分布,假定概率因子在空间服从正态分布。绝热剪切带的形成需要经历三个阶段：稳定塑性流动阶段、成核阶段和类流体阶段。在高应变率变形时,材料与环境的热传导可以忽略,因此使用一个绝热的J-C本构描述材料在稳定塑性流动阶段的力学行为；在成核阶段,孔洞软化效果增强,孔洞的发展与材料内的存储能相关,温度软化效果与材料的热能相关,而热能和存储能都由塑性功转化而来,因此温度软化效果和孔洞软化效果可表述为塑性应变的函数；使用粘性流体本构描述材料在类流体阶段抵抗剪切变形的能力。建立起完整的剪切带分阶段模型还需要两个准则,即稳定塑性流动阶段向成核阶段转化的失稳准则和成核阶段向类流体阶段转化的应力塌陷准则。考虑到材料失稳和应力塌陷的应变率相关性,本文采用应变、应变率双变量准则。
2)首次分析了本构计算的稳定性问题,指出“显式精确算法”并不是恒稳定的,给出了其计算稳定的一个必要条件。分别提出了针对粘塑性本构和压力相关本构的迭代算法,理论与计算实例表明,其稳定精确且快速收敛。分析了绝大多数冲击动力学商业软件中普遍采用的“半径回归法”的计算流程,指出只有满足条件(a)屈服准则与压力无关,(b)弹性变形相对于塑性变形很小而可以忽略时,才能使用“半径回归法”。由于计算后继屈服面时忽略了弹性变形,因此“半径回归法”只是一种近似算法,为了避免“半径回归法”中的近似,李永池等提出了“显式精确算法”。本文通过理论分析指出,“显式精确算法”不是恒稳定的,并给出了其计算稳定的一个必要条件。以广义Drucker公设为出发点,本文提出针对粘塑性本构的迭代算法,其同时避免了“半径回归法”中的近似和“显式精确算法”的不稳定性。以上算法都要求屈服准则与压力无关,而混凝土、陶瓷等材料和金属材料在高压下的本构都表现出明显的压力相关性,基于此本文提出一种针对压力相关本构的迭代算法并将其成功应用于内部炸药爆轰驱动混凝土破坏的计算实例中。
3)使用三维数值模拟分析了内部炸药起爆方式对柱壳破坏过程的影响,发现不同起爆方式下柱壳内表所经受的压力、动量、能量历史差别巨大,即使在同一起爆方式下,柱壳内表轴向不同位置的压力载荷历程也存在较大差异。因此,在理论分析和数值模拟中,必须考虑起爆方式和轴向位置的影响。将提出的多阶段概率型本构关系和针对粘塑性本构的迭代算法应用于模拟45钢柱壳在内部JOB9003炸药爆轰驱动下膨胀断裂过程,并与试验结果进行对比,分别分析了柱壳的破坏模式、破片速度和破片尺寸。(a)破坏模式：爆轰初期变形沿环向均匀分布,变形发展到一定程度后,大量剪切带首先在柱壳内表附近萌生并沿最大剪应力的方向向外传播。当剪切带传播到柱壳外表后,剪切带两侧一方面在剪应力的作用下继续滑移,另一方面在拉应力的作用开始分离形成宏观裂纹。当裂纹从柱壳外表延伸到柱壳内表时,柱壳发生贯穿性断裂,爆轰产物泄漏。实验结果表明,从柱壳外表出现裂纹到产物泄漏,柱壳还经历了很长的变形过程。(b)破片速度：当冲击波第一次到达柱壳外表时,柱壳外表的速度立刻产生一个突跃。然后速度出现几次振荡,随后在爆轰产物的驱动下,柱壳逐渐加速并到达最高极限值。试验和数值模拟的结果均表明,破片的速度历史在初期出现明显振荡。这些振荡说明尽管柱壳的壁厚很薄,但是柱面波在柱壳内外表之间的来回传播并不能被忽略,Taylor所假定的环向零应力面并不存在。(c)破片尺寸：分别测量了破片的厚度和宽度,由于回收的破片数量少,所以测量结果的不确定度大。为了提高破片宽度试验结果的可靠性,高速分幅转镜相机拍摄到的同步照片也被用来分析破片的宽度。GK模型尽管是在一维情况下推导得到,但其仍然较为准确地预测了试验中破片的宽度,这说明GK模型中假定的合理性,即剪切带间距由动量扩散所控制。数值模拟结果显示,当剪切带到达柱壳外表后,破片的厚度不再减少,这意味着此时所有的塑性变形都集中在剪切带内,剪切带外的材料被卸载。
4)将提出的多阶段概率型本构关系和针对粘塑性本构的迭代算法应用于模拟厚壁圆筒向内塌陷过程,数值模拟成功再现了厚壁圆筒中出现的剪切带自组织现象。模拟得到的厚壁圆筒破坏早期与晚期,宏观与局部的剪切带形貌均与试验结果吻合得很好。利用数值模拟结果,分析了厚壁圆筒的塌陷速度、剪切带间距和剪切带的传播速度。最后,结合厚壁圆筒的模拟结果,讨论了成核阶段在剪切带形成过程中的重要性。
Adiabatic shear bands play very important roles in the expanding fracture of the explosively driven cylinder and in the collapse of the thick-walled cylinder. Shear bands are the precursors of the fractures. The cracks always propagate along the developed shear bands. In this paper, to simulate the self-organization of multiple shear bands in the expanding cylinder and thick-walled cylinder, a multi-stage probabilistic constitutive model is proposed. The emergences of shear bands are always accompanied by high strain-rate deformation, and most of the metals exhibit a distinct strain rate effect, so a stable and precise algorithm for viscoplastic constitutive relationship is needed to model the behavior of shear bands. Based on the generalized Drucker postulate, an iterative algorithm is proposed.
The main contents of the dissertation are as follows:
1) Various defects exist inherently within polycrystalline ductile metals, such as micro-cracks, micro-holes, grain boundaries, precipitates and dislocations, etc. These defects can change the local yield stress. Some types of defects, such as microcracks and micro-holes, may reduce the local yield stress. Some types of defects, such as precipitates and dislocations, may improve the local yield stress. The yield stress distribution in a material is non-uniform, but the inhomogeneity of the yield stress is hardly considered in a macroscopic constitutive relationship. The perturbation sources play vital roles in emergence of a shear band, because shear bands are the result of deformation instability. So to model the initiation of multiple shear bands, the inhomogeneity of the yield stress should be considered in the constitutive relationship. A probability factor is introduced into the macroscopic constitutive relationship to describe the inhomogeneity. The probability factor is assumed to obey a Gaussian distribution in the material. The formation of an adiabatic shear band experiences three stages:stable plastic flow, nucleation and a fluid-like stage. The thermal conduction between the material and its environment can be neglected while material suffers high strain-rate deformation, so a modified adiabatic J-C constitutive relationship can be used to describe the mechanical behavior of material in stable plastic flow stage. Due to the nucleation and growth of micro-voids, the damage softening is becoming increasingly important in nucleation stage. The damage softening relies on the stored energy of cold work, and the temperature softening relies on the heat energy, yet the stored energy of cold work and heat energy are both converted from the plastic work. So the effect of damage and temperature softening can be described by a function of plastic strain. A viscous fluid constitutive relationship is used to describe the behavior of material in fluid-like stage. Two criteria are needed in the multi-stage model:one is the instability criterion, which is the transition point from the stable plastic flow stage to nucleation stage; the other is the stress collapse criterion, which is the transition point from the nucleation stage to fluid-like stage. Generally speaking, higher strain rates prompt material to lose instability and fail at lower levels of plastic strain, so strain-rate dependent criteria are adopted.
2) The radial return algorithm, which is widely adopted in most of impact dynamic software, is proved to be correct only while (a) the yield criterion is independent on the hydrostatic pressure and (b) the elastic deformation is greatly less than the plastic deformation. The elastic deformation is ignored in the radial return algorithm, so it is just an approximation method. To avoid the errors of the approximation, a new algorithm was proposed by Li et al.[Y.C.Li, et al., Thermo-viscoplastic constitutive relation of damaged materials with application, Explosion and Shock Waves,2004,24(4):289-298.], which was called as explicit precise algorithm. It is found that this algorithm is not absolutely stable. A necessary but not sufficient condition for the numerical stability is deduced. Based on the generalized Drucker postulate, an iterative algorithm was presented. Both of the theoretical and numerical results showed that the iterative algorithm is precise and unconditionally stable. The above algorithms are only applicable for yield criterion which is independent on the hydrostatic pressure, but the hydrostatic pressure has a strong influence on the yield criterion in concrete, soil and metals with high pressure. An iterative algorithm for pressure dependent yield criterion is proposed and is successfully applied in the numerical example of concrete.
3) The explosively induced fracture of metallic cylinder under different detonating methods was simulated. The results show that different detonating methods lead to significant differences on the pressure, impulse and acting history of the cylindrical shell. Even in the same detonation method, the load history in different axial locations are quite different. Therefore, in the theoretical analysis and numerical simulation, the influences of detonation method and axial position must be considered. The proposed multi-stage probabilistic constitutive relationship and iterative algorithm are applied to simulate the expanding fracture process of45steel cylindrical shell. Based on the comparison of the simulation results and experimental observation, the failure mode of cylindrical shell, the velocity and fragmentation fragments size were analyzed separately,(a) Failure mode:In the fracture process, the shear bands initiate near the inner surface of the cylinder and propagate outwardly along the maximum shear stress direction. After the shear bands arrive at the outer surface of the cylinder, the slippage continue to develop inside the shear bands, the cracks initiate at the outer surface and propagate inward along the developed shear bands,(b) When the shock wave arrives at the outer surface of the cylinder for the first time, the velocity exhibits a high increase instantaneously. Then the velocity fluctuates for several times. These fluctuations mean that the cylindrical wave propagating between the inner and outer surface can't be neglected although the thickness of the cylinder is very thin. The simulation shows that the zero hoop stress circle assumed by Taylor isn't exist. Under the following loading of the explosive products, the cylinder is accelerated gradually and reaches the limit,(c) The thickness and width of the recovered fragments are measured separately. The amount of the recovered fragments is so little that the uncertainty of the measured data is very large. To improve the reliability of the experimental results, the photos taken by high speed camera are also used to analyze the width of the fragments. The model developed by Grady and Kipp predicts the width of the fragments accurately, this means that the assumption of the G-K model is right and can be used to estimate the fragment size in the present problem. The simulation results show that after the shear bands arrive at the outer surface of the cylinder, the thickness of the fragments doesn't decrease any more. This means that the plastic deformation are concentrated inside the shear band, the material outside the shear band is unloaded.
4) The proposed multi-stage probabilistic constitutive relationship and iterative algorithm are applied to simulate the fracture process of thick-walled cylinder. The simulation successfully reproduces the fracture process of thick-walled cylinder. The obtained patterns of multiple shear bands in different stages agree well with the experimental results. The collapse velocity, shear band spacing and propagating velocity of shear band are analyzed separately. By combining with the simulation results, the importance of perturbation growth stage is emphasized at last.
1)通过在宏观本构中引入概率因子来描述材料缺陷导致的材料屈服应力分布不均匀性,假定该概率因子在空间服从正态分布。在分析绝热剪切带形成过程各阶段特点的基础上,给出了剪切带形成的多阶段本构模型。金属材料中广泛存在着各种缺陷,如位错、晶界、沉淀项、微孔洞、微裂纹等,某些缺陷导致材料局部屈服应力提高,某些缺陷导致材料局部屈服应力降低,屈服强度在材料内呈不均匀分布,但这种不均匀性在现有的宏观本构中并不体现。剪切带的形成是扰动导致变形失稳的结果,扰动源在其形成过程中起关键作用,因此要模拟多条剪切带的形成发展过程,必须在本构中考虑屈服强度的分布不均匀性。本文通过在宏观本构中引入一个概率因子建立概率型本构关系来描述材料内屈服强度的不均匀分布,假定概率因子在空间服从正态分布。绝热剪切带的形成需要经历三个阶段：稳定塑性流动阶段、成核阶段和类流体阶段。在高应变率变形时,材料与环境的热传导可以忽略,因此使用一个绝热的J-C本构描述材料在稳定塑性流动阶段的力学行为；在成核阶段,孔洞软化效果增强,孔洞的发展与材料内的存储能相关,温度软化效果与材料的热能相关,而热能和存储能都由塑性功转化而来,因此温度软化效果和孔洞软化效果可表述为塑性应变的函数；使用粘性流体本构描述材料在类流体阶段抵抗剪切变形的能力。建立起完整的剪切带分阶段模型还需要两个准则,即稳定塑性流动阶段向成核阶段转化的失稳准则和成核阶段向类流体阶段转化的应力塌陷准则。考虑到材料失稳和应力塌陷的应变率相关性,本文采用应变、应变率双变量准则。
2)首次分析了本构计算的稳定性问题,指出“显式精确算法”并不是恒稳定的,给出了其计算稳定的一个必要条件。分别提出了针对粘塑性本构和压力相关本构的迭代算法,理论与计算实例表明,其稳定精确且快速收敛。分析了绝大多数冲击动力学商业软件中普遍采用的“半径回归法”的计算流程,指出只有满足条件(a)屈服准则与压力无关,(b)弹性变形相对于塑性变形很小而可以忽略时,才能使用“半径回归法”。由于计算后继屈服面时忽略了弹性变形,因此“半径回归法”只是一种近似算法,为了避免“半径回归法”中的近似,李永池等提出了“显式精确算法”。本文通过理论分析指出,“显式精确算法”不是恒稳定的,并给出了其计算稳定的一个必要条件。以广义Drucker公设为出发点,本文提出针对粘塑性本构的迭代算法,其同时避免了“半径回归法”中的近似和“显式精确算法”的不稳定性。以上算法都要求屈服准则与压力无关,而混凝土、陶瓷等材料和金属材料在高压下的本构都表现出明显的压力相关性,基于此本文提出一种针对压力相关本构的迭代算法并将其成功应用于内部炸药爆轰驱动混凝土破坏的计算实例中。
3)使用三维数值模拟分析了内部炸药起爆方式对柱壳破坏过程的影响,发现不同起爆方式下柱壳内表所经受的压力、动量、能量历史差别巨大,即使在同一起爆方式下,柱壳内表轴向不同位置的压力载荷历程也存在较大差异。因此,在理论分析和数值模拟中,必须考虑起爆方式和轴向位置的影响。将提出的多阶段概率型本构关系和针对粘塑性本构的迭代算法应用于模拟45钢柱壳在内部JOB9003炸药爆轰驱动下膨胀断裂过程,并与试验结果进行对比,分别分析了柱壳的破坏模式、破片速度和破片尺寸。(a)破坏模式：爆轰初期变形沿环向均匀分布,变形发展到一定程度后,大量剪切带首先在柱壳内表附近萌生并沿最大剪应力的方向向外传播。当剪切带传播到柱壳外表后,剪切带两侧一方面在剪应力的作用下继续滑移,另一方面在拉应力的作用开始分离形成宏观裂纹。当裂纹从柱壳外表延伸到柱壳内表时,柱壳发生贯穿性断裂,爆轰产物泄漏。实验结果表明,从柱壳外表出现裂纹到产物泄漏,柱壳还经历了很长的变形过程。(b)破片速度：当冲击波第一次到达柱壳外表时,柱壳外表的速度立刻产生一个突跃。然后速度出现几次振荡,随后在爆轰产物的驱动下,柱壳逐渐加速并到达最高极限值。试验和数值模拟的结果均表明,破片的速度历史在初期出现明显振荡。这些振荡说明尽管柱壳的壁厚很薄,但是柱面波在柱壳内外表之间的来回传播并不能被忽略,Taylor所假定的环向零应力面并不存在。(c)破片尺寸：分别测量了破片的厚度和宽度,由于回收的破片数量少,所以测量结果的不确定度大。为了提高破片宽度试验结果的可靠性,高速分幅转镜相机拍摄到的同步照片也被用来分析破片的宽度。GK模型尽管是在一维情况下推导得到,但其仍然较为准确地预测了试验中破片的宽度,这说明GK模型中假定的合理性,即剪切带间距由动量扩散所控制。数值模拟结果显示,当剪切带到达柱壳外表后,破片的厚度不再减少,这意味着此时所有的塑性变形都集中在剪切带内,剪切带外的材料被卸载。
4)将提出的多阶段概率型本构关系和针对粘塑性本构的迭代算法应用于模拟厚壁圆筒向内塌陷过程,数值模拟成功再现了厚壁圆筒中出现的剪切带自组织现象。模拟得到的厚壁圆筒破坏早期与晚期,宏观与局部的剪切带形貌均与试验结果吻合得很好。利用数值模拟结果,分析了厚壁圆筒的塌陷速度、剪切带间距和剪切带的传播速度。最后,结合厚壁圆筒的模拟结果,讨论了成核阶段在剪切带形成过程中的重要性。
Adiabatic shear bands play very important roles in the expanding fracture of the explosively driven cylinder and in the collapse of the thick-walled cylinder. Shear bands are the precursors of the fractures. The cracks always propagate along the developed shear bands. In this paper, to simulate the self-organization of multiple shear bands in the expanding cylinder and thick-walled cylinder, a multi-stage probabilistic constitutive model is proposed. The emergences of shear bands are always accompanied by high strain-rate deformation, and most of the metals exhibit a distinct strain rate effect, so a stable and precise algorithm for viscoplastic constitutive relationship is needed to model the behavior of shear bands. Based on the generalized Drucker postulate, an iterative algorithm is proposed.
The main contents of the dissertation are as follows:
1) Various defects exist inherently within polycrystalline ductile metals, such as micro-cracks, micro-holes, grain boundaries, precipitates and dislocations, etc. These defects can change the local yield stress. Some types of defects, such as microcracks and micro-holes, may reduce the local yield stress. Some types of defects, such as precipitates and dislocations, may improve the local yield stress. The yield stress distribution in a material is non-uniform, but the inhomogeneity of the yield stress is hardly considered in a macroscopic constitutive relationship. The perturbation sources play vital roles in emergence of a shear band, because shear bands are the result of deformation instability. So to model the initiation of multiple shear bands, the inhomogeneity of the yield stress should be considered in the constitutive relationship. A probability factor is introduced into the macroscopic constitutive relationship to describe the inhomogeneity. The probability factor is assumed to obey a Gaussian distribution in the material. The formation of an adiabatic shear band experiences three stages:stable plastic flow, nucleation and a fluid-like stage. The thermal conduction between the material and its environment can be neglected while material suffers high strain-rate deformation, so a modified adiabatic J-C constitutive relationship can be used to describe the mechanical behavior of material in stable plastic flow stage. Due to the nucleation and growth of micro-voids, the damage softening is becoming increasingly important in nucleation stage. The damage softening relies on the stored energy of cold work, and the temperature softening relies on the heat energy, yet the stored energy of cold work and heat energy are both converted from the plastic work. So the effect of damage and temperature softening can be described by a function of plastic strain. A viscous fluid constitutive relationship is used to describe the behavior of material in fluid-like stage. Two criteria are needed in the multi-stage model:one is the instability criterion, which is the transition point from the stable plastic flow stage to nucleation stage; the other is the stress collapse criterion, which is the transition point from the nucleation stage to fluid-like stage. Generally speaking, higher strain rates prompt material to lose instability and fail at lower levels of plastic strain, so strain-rate dependent criteria are adopted.
2) The radial return algorithm, which is widely adopted in most of impact dynamic software, is proved to be correct only while (a) the yield criterion is independent on the hydrostatic pressure and (b) the elastic deformation is greatly less than the plastic deformation. The elastic deformation is ignored in the radial return algorithm, so it is just an approximation method. To avoid the errors of the approximation, a new algorithm was proposed by Li et al.[Y.C.Li, et al., Thermo-viscoplastic constitutive relation of damaged materials with application, Explosion and Shock Waves,2004,24(4):289-298.], which was called as explicit precise algorithm. It is found that this algorithm is not absolutely stable. A necessary but not sufficient condition for the numerical stability is deduced. Based on the generalized Drucker postulate, an iterative algorithm was presented. Both of the theoretical and numerical results showed that the iterative algorithm is precise and unconditionally stable. The above algorithms are only applicable for yield criterion which is independent on the hydrostatic pressure, but the hydrostatic pressure has a strong influence on the yield criterion in concrete, soil and metals with high pressure. An iterative algorithm for pressure dependent yield criterion is proposed and is successfully applied in the numerical example of concrete.
3) The explosively induced fracture of metallic cylinder under different detonating methods was simulated. The results show that different detonating methods lead to significant differences on the pressure, impulse and acting history of the cylindrical shell. Even in the same detonation method, the load history in different axial locations are quite different. Therefore, in the theoretical analysis and numerical simulation, the influences of detonation method and axial position must be considered. The proposed multi-stage probabilistic constitutive relationship and iterative algorithm are applied to simulate the expanding fracture process of45steel cylindrical shell. Based on the comparison of the simulation results and experimental observation, the failure mode of cylindrical shell, the velocity and fragmentation fragments size were analyzed separately,(a) Failure mode:In the fracture process, the shear bands initiate near the inner surface of the cylinder and propagate outwardly along the maximum shear stress direction. After the shear bands arrive at the outer surface of the cylinder, the slippage continue to develop inside the shear bands, the cracks initiate at the outer surface and propagate inward along the developed shear bands,(b) When the shock wave arrives at the outer surface of the cylinder for the first time, the velocity exhibits a high increase instantaneously. Then the velocity fluctuates for several times. These fluctuations mean that the cylindrical wave propagating between the inner and outer surface can't be neglected although the thickness of the cylinder is very thin. The simulation shows that the zero hoop stress circle assumed by Taylor isn't exist. Under the following loading of the explosive products, the cylinder is accelerated gradually and reaches the limit,(c) The thickness and width of the recovered fragments are measured separately. The amount of the recovered fragments is so little that the uncertainty of the measured data is very large. To improve the reliability of the experimental results, the photos taken by high speed camera are also used to analyze the width of the fragments. The model developed by Grady and Kipp predicts the width of the fragments accurately, this means that the assumption of the G-K model is right and can be used to estimate the fragment size in the present problem. The simulation results show that after the shear bands arrive at the outer surface of the cylinder, the thickness of the fragments doesn't decrease any more. This means that the plastic deformation are concentrated inside the shear band, the material outside the shear band is unloaded.
4) The proposed multi-stage probabilistic constitutive relationship and iterative algorithm are applied to simulate the fracture process of thick-walled cylinder. The simulation successfully reproduces the fracture process of thick-walled cylinder. The obtained patterns of multiple shear bands in different stages agree well with the experimental results. The collapse velocity, shear band spacing and propagating velocity of shear band are analyzed separately. By combining with the simulation results, the importance of perturbation growth stage is emphasized at last.
引文
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