274
P. Franciosi and M. Spagnuolo
then need to be calculated are two pairs selected among those of the form:
t
V
pq jn =
π
θ=0
2π
ϕ=0
ψ V (θ, ϕ)
cos
2l
θ sin
2m
θ cos
2r
ϕ sin
2s
ϕ
sin θ dθ dϕ.
References
Alibert, J. J., Seppecher, P., & dell’Isola, F. (2003). Truss modular beams with deformation energy
depending on higher displacement gradients. Mathematics and Mechanics of Solids, 8(1), 51–73.
Andreaus, U., Spagnuolo, M., Lekszycki, T., & Eugster, S. R. (2018). A Ritz approach for the
static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams.
Continuum Mechanics and Thermodynamics, 30, 1103–1123.
Barboura, S., & Franciosi, P. (2016). Analytical inclusion Green operators in transversally isotropic
media. In Proceeding of the 24th ICTAM conference, Montreal, Canada (vol. 3, pp. 2350–2351).
Barchiesi, E., & Placidi, L. (2017). A review on models for the 3D statics and 2D dynamics of
pantographic fabrics. In Wave dynamics and composite mechanics for microstructured materials
and meta-materials (pp. 239–258). Singapore: Springer.
Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., & Müller, W. H. (2019). Outof-plane buckling of pantographic fabrics in displacement-controlled shear tests: Experimental
results and model validation. Continuum Mechanics and Thermodynamics, 31(1), 33–45.
Bender, J., Jarvoll, P., Nydén, M., & Engström, S. (2008). Structure and dynamics of a sponge phase
in the methyl δ; aminolevulinate/monoolein/water/propylene glycol system. Journal of Colloid
and Interface Science, 317, 577–584.
Benveniste, Y. (1987). A new approach to the application of Mori–Tanaka’s theory in composite
materials. Mechanics of Materials, 6, 147–157.
Berveiller, M., Fassi-Fehri, O., & Hihi, A. (1987). The problem of two plastic and heterogeneous
inclusions in an anisotropic medium. International Journal of Engineering Sciences, 25(6), 691–
709.
Bornert, M., Stolz, C., & Zaoui, A. (1996). Morphologically representative pattern-based bounding
in elasticity. Journal of the Mechanics and Physics of Solids, 44(3), 307–331.
Boucher, S. (1974). On the effective moduli of isotropic two-phase elastic composites. Journal of
Composite Materials, 8(1), 82–89.
Brenner, R., Castelnau, O., & Gilormini, P. (2001). A modified affine theory for the overall properties
of nonlinear composites. Comptes Rendus de l’Académie des Sciences, Paris, Mechanics series,
329(9), 649–654.
Burridge, R. (1967). The singularities on the plane lids of the wave surface of elastic media with
cubic symmetry. The Quarterly Journal of Mechanics and Applied Mathematics, 20(1), 41–56.
Buroni, C., & Saez, A. (2010). Three-dimensional Green functions and its derivatives for materials
with general anisotropic magneto-electro-elastic coupling. Proceedings of the Royal Society A,
466, 515–537.
Buryachenko, V. A. (2001). Multiparticle effective field and related methods in micromechanics of
composite materials. Applied Mechanics Review, 54, 1–47.
Buryachenko, V. A., & Brun, M. (2012). Thermo elastic effective properties and stress concentrator factors of composites reinforced by heterogeneities of non canonical shape. Mechanics of
Materials, 53, 91–110.
Boutin, C., Giorgio, I., & Placidi, L. (2017). Linear pantographic sheets: Asymptotic micro-macro
models identification. Mathematics and Mechanics of Complex Systems, 5(2), 127–162.
P. Franciosi and M. Spagnuolo
then need to be calculated are two pairs selected among those of the form:
t
V
pq jn =
π
θ=0
2π
ϕ=0
ψ V (θ, ϕ)
cos
2l
θ sin
2m
θ cos
2r
ϕ sin
2s
ϕ
sin θ dθ dϕ.
References
Alibert, J. J., Seppecher, P., & dell’Isola, F. (2003). Truss modular beams with deformation energy
depending on higher displacement gradients. Mathematics and Mechanics of Solids, 8(1), 51–73.
Andreaus, U., Spagnuolo, M., Lekszycki, T., & Eugster, S. R. (2018). A Ritz approach for the
static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams.
Continuum Mechanics and Thermodynamics, 30, 1103–1123.
Barboura, S., & Franciosi, P. (2016). Analytical inclusion Green operators in transversally isotropic
media. In Proceeding of the 24th ICTAM conference, Montreal, Canada (vol. 3, pp. 2350–2351).
Barchiesi, E., & Placidi, L. (2017). A review on models for the 3D statics and 2D dynamics of
pantographic fabrics. In Wave dynamics and composite mechanics for microstructured materials
and meta-materials (pp. 239–258). Singapore: Springer.
Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., & Müller, W. H. (2019). Outof-plane buckling of pantographic fabrics in displacement-controlled shear tests: Experimental
results and model validation. Continuum Mechanics and Thermodynamics, 31(1), 33–45.
Bender, J., Jarvoll, P., Nydén, M., & Engström, S. (2008). Structure and dynamics of a sponge phase
in the methyl δ; aminolevulinate/monoolein/water/propylene glycol system. Journal of Colloid
and Interface Science, 317, 577–584.
Benveniste, Y. (1987). A new approach to the application of Mori–Tanaka’s theory in composite
materials. Mechanics of Materials, 6, 147–157.
Berveiller, M., Fassi-Fehri, O., & Hihi, A. (1987). The problem of two plastic and heterogeneous
inclusions in an anisotropic medium. International Journal of Engineering Sciences, 25(6), 691–
709.
Bornert, M., Stolz, C., & Zaoui, A. (1996). Morphologically representative pattern-based bounding
in elasticity. Journal of the Mechanics and Physics of Solids, 44(3), 307–331.
Boucher, S. (1974). On the effective moduli of isotropic two-phase elastic composites. Journal of
Composite Materials, 8(1), 82–89.
Brenner, R., Castelnau, O., & Gilormini, P. (2001). A modified affine theory for the overall properties
of nonlinear composites. Comptes Rendus de l’Académie des Sciences, Paris, Mechanics series,
329(9), 649–654.
Burridge, R. (1967). The singularities on the plane lids of the wave surface of elastic media with
cubic symmetry. The Quarterly Journal of Mechanics and Applied Mathematics, 20(1), 41–56.
Buroni, C., & Saez, A. (2010). Three-dimensional Green functions and its derivatives for materials
with general anisotropic magneto-electro-elastic coupling. Proceedings of the Royal Society A,
466, 515–537.
Buryachenko, V. A. (2001). Multiparticle effective field and related methods in micromechanics of
composite materials. Applied Mechanics Review, 54, 1–47.
Buryachenko, V. A., & Brun, M. (2012). Thermo elastic effective properties and stress concentrator factors of composites reinforced by heterogeneities of non canonical shape. Mechanics of
Materials, 53, 91–110.
Boutin, C., Giorgio, I., & Placidi, L. (2017). Linear pantographic sheets: Asymptotic micro-macro
models identification. Mathematics and Mechanics of Complex Systems, 5(2), 127–162.