TY - JOUR

T1 - Stability of 2D discrete mass-spring systems with negative stiffness springs

AU - Esin, Maxim

AU - Pasternak, Elena

AU - Dyskin, Arcady

PY - 2016

Y1 - 2016

N2 - © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimNegative stiffness elements (elements with direction of the force opposite to the direction of the deformation) are unstable as the energy is no longer positive definite. Nevertheless, materials and structures with negative stiffness elements can exist when the element negative energy is compensated by the energy of the rest of the system or an encompassing system that provide stabilisation. In this paper, we study stability of two-dimensional square packing discrete mass-spring systems with some spring stiffnesses being negative. Each mass (particle) in the system has three degrees of freedom. The particles are connected by normal, shear and rotational springs to simulate all possible elastic interactions between the masses. The stability is investigated by considering three simple problems: (i) a system consisting of only one particle, (ii) a “channel” of two particles, (iii) two-by-two and three-by-three systems of particles. The particles are connected to rigid external boundary. We found that two-dimensional square packing systems with fixed boundary particles consisting of positive and negative stiffness springs can be stable when the total number of negative stiffness springs does not exceed the total number of degrees of freedom of the system. This necessary condition is also generalised to three-dimensional cubic packing systems. The presence of negative stiffness springs leads to a decrease in the eigenfrequencies: the smallest eigenfrequency becomes zero when the absolute value of the negative stiffness spring reaches its critical value.

AB - © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimNegative stiffness elements (elements with direction of the force opposite to the direction of the deformation) are unstable as the energy is no longer positive definite. Nevertheless, materials and structures with negative stiffness elements can exist when the element negative energy is compensated by the energy of the rest of the system or an encompassing system that provide stabilisation. In this paper, we study stability of two-dimensional square packing discrete mass-spring systems with some spring stiffnesses being negative. Each mass (particle) in the system has three degrees of freedom. The particles are connected by normal, shear and rotational springs to simulate all possible elastic interactions between the masses. The stability is investigated by considering three simple problems: (i) a system consisting of only one particle, (ii) a “channel” of two particles, (iii) two-by-two and three-by-three systems of particles. The particles are connected to rigid external boundary. We found that two-dimensional square packing systems with fixed boundary particles consisting of positive and negative stiffness springs can be stable when the total number of negative stiffness springs does not exceed the total number of degrees of freedom of the system. This necessary condition is also generalised to three-dimensional cubic packing systems. The presence of negative stiffness springs leads to a decrease in the eigenfrequencies: the smallest eigenfrequency becomes zero when the absolute value of the negative stiffness spring reaches its critical value.

U2 - 10.1002/pssb.201552763

DO - 10.1002/pssb.201552763

M3 - Article

VL - 253

SP - 1395

EP - 1409

JO - Physica Status Solidi. B: Basic Solid State Physics

JF - Physica Status Solidi. B: Basic Solid State Physics

SN - 0370-1972

IS - 7

ER -