372
R. dell’Erba
between the prediction of the discrete model and the predictions of standard FEM
simulations. In particular, some aspects in the geometry of the deformed configurations in the continuous case are found for the discrete system as well. Moreover, a
fracture algorithm was introduced, and some results, including periodic crack formation, were provided. Also in this case, a comparison with the continuous case shows a
good agreement between the fracture front geometry (discrete system) and the most
stressed area (continuous system).
We have also used an ASTM shape whose results will be compared in a real tensile
test experiment, but many questions are still open. We have collected some success
showing plausible deformation in different conditions, but when we consider a beam
under loading, we fail and the need to connect constitutive equations with the parameters of our tool emerges powerfully. However, the tool has demonstrated enough
flexibility to give chances that, once connected with the constitutive parameters, it
can describe the richness of behavior of different materials, including potentially
complex biological tissues.
References
Abali, B. E., Müller, W. H., & Dell’Isola, F. (2017). Theory and computation of higher gradient
elasticity theories based on action principles. Arch. Appl. Mech. 87(9), 1495–1510.
Alibert, J.-J., & Della Corte, A. (2015). Second-gradient continua as homogenized limit of
pantographic microstructured plates: a rigorous proof. Z. Für Angew. Math. Phys., 66(5),
2855–2870.
Alibert, J.-J., Della Corte, A., Giorgio, I., & Battista, A. (2017). Extensional Elastica in large
deformation as $$\Gamma $$-limit of a discrete 1D mechanical system. Z. Für Angew. Math.
Phys., 68(2), 42.
Alibert, J.-J., Seppecher, P., & Dell’isola, F. (2003). Truss modular beams with deformation energy
depending on higher displacement gradients. Math. Mech. Solids, 8(1), 51–73.
Altenbach, J., Altenbach, H., & Eremeyev, V. A. (2010). On generalized Cosserat-type theories of
plates and shells: a short review and bibliography. Arch. Appl. Mech., 80(1), 73–92.
Altenbach, H., Bîrsan, M., & Eremeyev, V. A. (2013). Cosserat-type rods. In Generalized Continua
from the Theory to Engineering Applications (pp. 179–248). Heidelberg: Springer.
Altenbach, H., & Eremeyev, V. A. (2009). On the linear theory of micropolar plates. ZAMM-Journal
Appl. Math. Mech. Für Angew. Math. Mech., 89(4), 242–256.
Altenbach, H., & Eremeyev, V. A. (2013). Generalized Continua From the Theory to Engineering
Applications, CISM Courses and Lectures (Vol. 541). Udine: Springer.
Altenbach, H., Eremeyev, V. A., Lebedev, L. P., & Rendón, L. A. (2010). Acceleration waves and
ellipticity in thermoelastic micropolar media. Arch. Appl. Mech., 80(3), 217–227.
Andreaus, U., Colloca, M., & Iacoviello, D. (2012). An optimal control procedure for bone
adaptation under mechanical stimulus. Control Eng. Pract., 20(6), 575–583.
Andreaus, U., Colloca, M., & Toscano, A. (2008). Mechanical behaviour of physiological and
prosthesized human femurs during stair climbing: A comparative analysis via 3D numerical
simulation. Minerva Ortop. E Traumatol., 59(4), 213–220.
Andreaus, U., Giorgio, I., & Lekszycki, T. (2013). A 2-D Continuum Model of a Mixture of Bone
Tissue and Bio-resorbable Material for Simulating Mass Density Redistribution Under Load
Slowly Variable in Time [Online]. Available at: http://www.scopus.com/inward/record.url?eid=
2-s2.0-84882523299&partnerID=40&md5=e34b49c6b02b11f8e584e3cb57b5db43.
R. dell’Erba
between the prediction of the discrete model and the predictions of standard FEM
simulations. In particular, some aspects in the geometry of the deformed configurations in the continuous case are found for the discrete system as well. Moreover, a
fracture algorithm was introduced, and some results, including periodic crack formation, were provided. Also in this case, a comparison with the continuous case shows a
good agreement between the fracture front geometry (discrete system) and the most
stressed area (continuous system).
We have also used an ASTM shape whose results will be compared in a real tensile
test experiment, but many questions are still open. We have collected some success
showing plausible deformation in different conditions, but when we consider a beam
under loading, we fail and the need to connect constitutive equations with the parameters of our tool emerges powerfully. However, the tool has demonstrated enough
flexibility to give chances that, once connected with the constitutive parameters, it
can describe the richness of behavior of different materials, including potentially
complex biological tissues.
References
Abali, B. E., Müller, W. H., & Dell’Isola, F. (2017). Theory and computation of higher gradient
elasticity theories based on action principles. Arch. Appl. Mech. 87(9), 1495–1510.
Alibert, J.-J., & Della Corte, A. (2015). Second-gradient continua as homogenized limit of
pantographic microstructured plates: a rigorous proof. Z. Für Angew. Math. Phys., 66(5),
2855–2870.
Alibert, J.-J., Della Corte, A., Giorgio, I., & Battista, A. (2017). Extensional Elastica in large
deformation as $$\Gamma $$-limit of a discrete 1D mechanical system. Z. Für Angew. Math.
Phys., 68(2), 42.
Alibert, J.-J., Seppecher, P., & Dell’isola, F. (2003). Truss modular beams with deformation energy
depending on higher displacement gradients. Math. Mech. Solids, 8(1), 51–73.
Altenbach, J., Altenbach, H., & Eremeyev, V. A. (2010). On generalized Cosserat-type theories of
plates and shells: a short review and bibliography. Arch. Appl. Mech., 80(1), 73–92.
Altenbach, H., Bîrsan, M., & Eremeyev, V. A. (2013). Cosserat-type rods. In Generalized Continua
from the Theory to Engineering Applications (pp. 179–248). Heidelberg: Springer.
Altenbach, H., & Eremeyev, V. A. (2009). On the linear theory of micropolar plates. ZAMM-Journal
Appl. Math. Mech. Für Angew. Math. Mech., 89(4), 242–256.
Altenbach, H., & Eremeyev, V. A. (2013). Generalized Continua From the Theory to Engineering
Applications, CISM Courses and Lectures (Vol. 541). Udine: Springer.
Altenbach, H., Eremeyev, V. A., Lebedev, L. P., & Rendón, L. A. (2010). Acceleration waves and
ellipticity in thermoelastic micropolar media. Arch. Appl. Mech., 80(3), 217–227.
Andreaus, U., Colloca, M., & Iacoviello, D. (2012). An optimal control procedure for bone
adaptation under mechanical stimulus. Control Eng. Pract., 20(6), 575–583.
Andreaus, U., Colloca, M., & Toscano, A. (2008). Mechanical behaviour of physiological and
prosthesized human femurs during stair climbing: A comparative analysis via 3D numerical
simulation. Minerva Ortop. E Traumatol., 59(4), 213–220.
Andreaus, U., Giorgio, I., & Lekszycki, T. (2013). A 2-D Continuum Model of a Mixture of Bone
Tissue and Bio-resorbable Material for Simulating Mass Density Redistribution Under Load
Slowly Variable in Time [Online]. Available at: http://www.scopus.com/inward/record.url?eid=
2-s2.0-84882523299&partnerID=40&md5=e34b49c6b02b11f8e584e3cb57b5db43.