Bounds for the response of viscoelastic composites under antiplane loadings in the time domain
Bounds for the response of viscoelastic composites under antiplane loadings in the time domain
复制标题
时域反平面载荷下粘弹性复合材料响应的界限
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
G. Milton
中科院分区：
文献类型：
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作者：
O. Mattei;G. Milton
为了推导具有粘弹性相的双组分复合材料的应变和应力响应的界限，我们重新审视所谓的解析方法（Bergman 1978），该方法允许人们近似复数有效张量，即组分剪切模量比的函数，作为由正半定剩余矩阵加权的极点之和。这项研究的新颖之处在于应用了这种方法，该方法以前应用于涉及频域循环载荷的问题（Milton 1980；Bergman 1980），以导出反平面粘弹性情况的时域界限。极点的位置和留数矩阵是问题的变分参数：目的是确定这些参数，以便在任何给定时刻获得最小（或最大）响应。对于每个固定极配置，有关复合材料的所有信息都被转换为对残基的约束，即所谓的求和规则。约束的线性性，以及固定时间的响应在残差中呈线性的观察结果，使人们能够使用线性规划理论将问题简化为涉及相对较少的非零残差的问题。最后，通过对极点位置进行数值优化来获得响应的界限。在研究的示例中，结果是非常准确的估计：如果可以获得有关复合材料的足够信息，则在整个时间范围内的界限可能非常严格，从而允许人们预测复合材料的瞬态行为。此外，包含体积分数（以及可能的横向各向同性）的边界在某些特定时间可能非常严格。
To derive bounds on the strain and stress response of a two-component composite material with viscoelastic phases, we revisit the so-called analytic method (Bergman 1978), which allows one to approximate the complex effective tensor, function of the ratio of the component shear moduli, as the sum of poles weighted by positive semidefinite residue matrices. The novelty of this investigation lies in the application of such a method, previously applied (Milton 1980; Bergman 1980) to problems involving cyclic loadings in the frequency domain, to derive bounds in the time domain for the antiplane viscoelasticity case. The position of the poles and the residues matrices are the variational parameters of the problem: the aim is to determine such parameters in order to have the minimum (or maximum) response at any given moment in time. All the information about the composite is translated for each fixed pole configuration into constraints on the residues, the so-called sum rules. The linearity of the constraints, along with the observation that the response at a fixed time is linear in the residues, enables one to use the theory of linear programming to reduce the problem to one involving relatively few non-zero residues. Finally, bounds on the response are obtained by numerically optimizing over the pole positions. In the examples studied, the results turn out to be very accurate estimates: if sufficient information about the composite is available, the bounds can be quite tight over the entire range of time, allowing one to predict the transient behavior of the composite. Furthermore, the bounds incorporating the volume fractions (and possibly transverse isotropy) can be extremely tight at certain specific times.