Two Strategies Towards Geometrically Non-Linear Isotropic Gradient Damage
Two Strategies Towards Geometrically Non-Linear Isotropic Gradient Damage
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解决几何非线性各向同性梯度损伤的两种策略
DOI：
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发表时间：
2002
期刊：
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通讯作者：
P. Steinmann
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作者：
T. Liebe;P. Steinmann
本文对几何非线性各向同性梯度损伤的两种不同处理方法进行了比较。在局域理论中的驱动力，即本地存储的能量，被修改为一个准非局域量，以考虑到微观缺陷的相互作用。这个准非局部量进入损伤条件以及历史变量的更新。通过确定特定的本构关系的损伤通量，两种不同的方法被概述：一方面的平均计划，另一方面是一个激励策略。首先，我们考虑所谓的能量梯度公式，其中的非本地存储的能量（NSE）被引入作为一个独立的变量和通量共轭的梯度的NSE计算渲染一个全球平均方程。第二个模型，损伤梯度公式，将损伤场作为一个独立的变量和通量共轭的损伤场的梯度计算。通过引入最大耗散的概念，导出了整体Kuhn-Tucker条件。最后，作为一个模型问题，在张力下的LD杆研究这两种方法。引言标准的连续损伤公式模型材料退化过程经典的局部框架内。作为弹性退化的简单测量，接受物理解释为应力175第13卷，Nos. 3-4，2002各向同性梯度损伤承载面积减少，考虑各向同性损伤就足够了，参见Simo和Ju（1987）。它可以追溯到Kachanov（1958）的早期概念，其特征在于标量损伤变量。当应用局部连续介质公式时，忽略了微观结构的相互作用，这在计算上导致在网格细化后的病态网格依赖的数值解。这使得在消失的局部区域中显示的结果在物理上无意义。局部化表示由于不连续破坏而在窄带内累积的损伤，见Rice（1976）和Rizzi等人（1994）。事实上，在连续介质水平上，局部化表示从均匀应变场到非均匀应变场的过渡，根据Hadamards不稳定性分类。应变软化有利于局部化机制，参见Delaplace等人（1996），这是微观尺度上不均匀性的结果。Kongshavn和Poursartip（1999）对应变软化方法进行了实验研究。因此，通常形成材料最终破裂的前兆的局部区域显示出有限的宽度。因此，总的来说，标准的连续描述，特别是相应的数值解方案未能获得这样的有限宽度，而它是在实验中观察到的。因此，在连续损伤力学的一个最新发展是引入了一个特征的内部长度尺度。在这个尺度内，材料仍然均匀地起作用，并解释了导致非标准（非局部）连续统理论的微观结构相互作用。因此，引入负责软化的那些量的更高梯度提供了非常有效的策略。有关损伤力学的更多趋势、需求和成就，请参见Krajcinovic（2000）。从物理学的角度来看，在塑性力学中引入更高的梯度首先是由Aifantis（1984; 1992）通过考虑单晶中的位错而激发的，最近，从另一个角度来看，由Steinmann（1996）和Menzel和Steinmann（2000）。在这方面，Zbib和Aifantis（1992）提出了在屈服条件中考虑高阶梯度的方法。例如Schreyer和Chen（1986）对高梯度连续体进行了一维研究。Pijaudier-Cabot和Baiant（1987）以及Baiant和Pijaudier-Cabot（1988）提出了连续介质损伤的非局部积分公式。Lasry和Belytschko（1988）创造了本地化限制器的概念。Mllhlhaus和Aifantis（1991）提出了梯度塑性的变分框架; de Borst和Mühlhaus（1992）发展了相应的唯象塑性梯度理论。Vardoulakis和Aifantis（1991年）研究了将较高梯度配方应用于颗粒材料。受Fleck等人（1994）在细铜线上进行的实验的启发，Fleck和哈钦森（1993）提出了一种考虑连续旋转梯度的替代理论。Benallal等人（1993）研究了初始边值的适定性。梯度相关损伤模型和塑性模型，其中梯度相关性基本上通过内部变量的拉普拉斯算子结合在载荷面中，除了她早期的论文外，de Borst等人（1996）、Benallal和Tvergaard（1995）以及最近的Comi（2001）都对这些模型进行了处理。Fleck和哈钦森（1997）提出了一种特殊的应变梯度塑性公式。最后，Nedjar（2001）提出了一个包含损伤梯度的弹塑性-损伤耦合模型。
The paper presents a comparison of two different approaches towards geometrically non-linear isotropic gradient damage. The driving force in local theory, namely the local stored energy, is modified to a quasi-nonlocal quantity in order to take micro defect interactions into account. This quasi-nonlocal quantity enters the damage condition as well as the update of the history variable. By the determination of particular constitutive relations for the damage flux, two different approaches are outlined: on the one hand an averaging scheme and on the other hand a thermodynamically motivated strategy. Firstly, we consider the so-called Energy Gradient Formulation, where the nonlocal stored energy (NSE) is introduced as an independent variable and fluxes conjugated to the gradients of the NSE are computed rendering a global averaging equation. The second model, the Damage Gradient Formulation, treats the damage field as an independent variable and fluxes conjugated to the gradient of the damage field are computed. By following the concept of maximum dissipation, global Kuhn-Tucker conditions are derived. Finally as a model problem, a ld-bar under tension is studied for both approaches. INTRODUCTION Standard continuum damage formulations model material deterioration processes classically within a local framework. As a simple measure of elastic degradation, accepting the physical interpretation as stress175 Vol. 13, Nos. 3-4, 2002 Isotropic Gradient Damage bearing area reduction, it is sufficient to consider isotropic damage, see Simo and Ju (1987). It dates back to the early concept of Kachanov (1958) characterized by a scalar damage variable. When applying local continuum formulations, the microstructural interactions are neglected, which computationally results in pathological mesh dependent numerical solutions upon mesh refinement. This renders physically meaningless results which are displayed in a vanishing localized zone. Localization represents the accumulation of damage within narrow bands due to discontinuous failure, see Rice (1976) and Rizzi et al. (1994). In fact, at the continuum level localization denotes a transition from a homogeneous strain field to a non-homogeneous strain field in terms of Hadamards instability classification. The localization mechanism is favored by strain softening, see Delaplace et al. (1996), which is a result of inhomogeneities on the micro scale. Experimental investigations of a strain softening approach were carried out by Kongshavn and Poursartip (1999). Thereby localized zones, which often form a precursor to the final rupture of the material, display a finite width. Thus, in summary, standard continuum descriptions and in particular the corresponding numerical solution schemes fail to obtain such a finite width, whereas it is observed in experiments. Therefore, a recent development in continuum damage mechanics is the introduction of a characteristic internal length scale. Within this scale the material still acts homogeneously and accounts for microstructural interactions resulting in a non-standard (nonlocal) continuum theory. The incorporation of higher gradients of those quantities which are responsible for softening thereby offers a very effective strategy. For more trends, needs and accomplishments in damage mechanics, see Krajcinovic (2000). From a physical point of view, the incorporation of higher gradients was first motivated in plasticity by considering dislocations in single crystals by Aifantis (1984; 1992) and recently, from an alternative point of view, by Steinmann (1996) and Menzel and Steinmann (2000). In this context, the consideration of gradients of higher order in the yield condition has been proposed by Zbib and Aifantis (1992). One dimensional investigations on a higher gradient continuum were performed, e.g. by Schreyer and Chen (1986). Nonlocal integral formulations of continuum damage were proposed by Pijaudier-Cabot and Baiant (1987) and Baiant and Pijaudier-Cabot (1988). Lasry and Belytschko (1988) coined the notion of localization limiters. A variational framework for gradient plasticity was proposed by Mllhlhaus and Aifantis (1991); a corresponding gradient theory of phenomenological plasticity was developed by de Borst and Mühlhaus (1992). The application of a higher gradient formulation to granular materials was examined by Vardoulakis and Aifantis (1991). Motivated by experiments performed by Fleck et al. (1994) on thin copper wires, Fleck and Hutchinson (1993) proposed an alternative theory which takes the gradient of the continuum rotation into account. The well-posedness of the initial boundary value was studied by Benallal et al. (1993). Gradient dependent damage as well as plasticity models, whereby the gradient dependence is essentially incorporated in the loading surface by the Laplacian of an internal variable, were treated by de Borst et al. (1996), Benallal and Tvergaard (1995) and, most recently, by Comi (2001) in addition to her earlier papers. A particular strain gradient plasticity formulation is advocated by Fleck and Hutchinson (1997). Finally a coupled elastoplastic-damage model including the gradient of damage is proposed by Nedjar (2001).