粘弹性复合材料在反平面载荷作用下时域响应的界
Bounds for the response of viscoelastic composites under antiplane loadings in the time domain
摘要：
为了得到粘弹性相的双组分复合材料的应变和应力响应的界限，我们重新审视了所谓的分析方法（Bergman 1978），该方法允许将复有效张量（组分剪切模量之比的函数）近似为半正定剩余矩阵加权的极点之和。这项调查的新颖性在于应用这种方法，以前应用（米尔顿1980年，伯格曼1980年）的问题，涉及在频域中的循环载荷，在时域中的反平面粘弹性的情况下，得到的界限。极点的位置和剩余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具有最小（或最大）响应。关于复合的所有信息对于每个固定的极点配置被转换为对剩余的约束，即所谓的求和规则。线性的约束，以及观察到的响应在一个固定的时间是线性的残留物，使人们能够使用线性规划的理论，以减少问题，涉及相对较少的非零残留物。最后，通过数值优化的极点位置得到的响应的界限。在研究的例子中，结果是非常准确的估计：如果有足够的关于复合材料的信息，在整个时间范围内，边界可以非常紧，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以非常紧密。
英文摘要：
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.
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