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Apparent thermal conductivity (k) measurements were made on monolithic silica aerogels and divided low-density xerogels. The objective of the tests on monoliths was to determine the variation of k with the solid content. Additional compressive tests were also made to follow the evolution of bulk modulus with apparent density, in order to further optimize the solid content of xerogel particles. Divided xerogels were characterized under the following conditions: at ambient conditions (T, P, and relative humidity of water vapor in air), under partial vacuum (from atmospheric pressure to 4 · 10 À7 bar) and under different relative humidities. The effect of the divided xerogel granularity on k has been studied by using different size range and distributions. The results of our studies indicate an optimum particle density at 0.15 g cm À3 , and a decrease in k for samples containing fine particles.

Within the framework of the homogenization of heterogeneous media, a non local model is proposed. A field of non-local filtered stiffness tensor is introduced by filtering the solution to the homogenization problem. The filtered stiffness tensor, depending on the filter to heterogeneity size ratio, provides a continuous transition from the actual micro-scale heterogeneous stiffness field to the macro-scale homogenized stiffness tensor. For any intermediate filter size, the homogenization of the filtered stiffness yields exactly the homogenized stiffness, therefore it is called macroscopically consistent. The non-local stiffness tensor is intrinsically non symmetric, but its spatial fluctuations are smoothed, allowing for a less refined discretization in numerical methods. As a by-product, a two step heterogeneous multiscale method is proposed to reduce memory and computational time requirements of existing direct schemes while controlling the accuracy of the result. The first step is the estimation of the filtered stiffness at sampling points by means of an oversampling strategy to reduce boundary effects. The second step is the numerical homogenization of the obtained sampled filtered stiffness.

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Energy absorption during crushing is evaluated using a thermodynamic based continuum damage model inspired from the Matzenmiller-Lubliner-Taylors model. It was found that for crash-worthiness applications, it is necessary to couple the progressive ruin of the material to a representation of the matter openings and debris generation. Element kill technique (erosion) and/or cohesive elements are efficient but not predictive. A technique switching finite elements into discrete particles at rupture is used to create debris and accumulated mater during the crushing of the structure. Switching criteria are evaluated using the contribution of the different ruin modes in the damage evolution, energy absorption, and reaction force generation.