The computational framework for continuum-kinematics-inspired peridynamics

Peridynamics (PD) is a non-local continuum formulation. The original version of PD was restricted to bond-based interactions. Bond-based PD is geometrically exact and its kinematics are similar to classical continuum mechanics (CCM). However, it cannot capture the Poisson effect correctly. This shortcoming was addressed via state-based PD, but the kinematics are not accurately preserved. Continuum-kinematics-inspired peridynamics (CPD) provides a geometrically exact framework whose underlying kinematics coincide with that of CCM and captures the Poisson effect correctly. In CPD, one distinguishes between one-, two- and three-neighbour interactions. One-neighbour interactions are equivalent to the bond-based interactions of the original PD formalism. However, two- and three-neighbour interactions are fundamentally different from state-based interactions as the basic elements of continuum kinematics are preserved precisely. The objective of this contribution is to elaborate on computational aspects of CPD and present detailed derivations that are essential for its implementation. Key features of the resulting computational CPD are elucidated via a series of numerical examples. These include three-dimensional problems at large deformations. The proposed strategy is robust and the quadratic rate of convergence associated with the Newton–Raphson scheme is observed.

A nonlinear thermomechanical formulation for anisotropic volume and surface continua

A thermomechanical, polar continuum formulation under finite strains is proposed for anisotropic materials using a multiplicative decomposition of the deformation gradient. First, the kinematics and conservation laws for three-dimensional, polar, and nonpolar continua are obtained. Next, these kinematics and conservation laws are connected to their corresponding counterparts for surface continua, based on Kirchhoff–Love assumptions. Then the shell material models are extracted from three-dimensional material models for finite-temperature problems using established connections. The weak forms are obtained for both three-dimensional nonpolar continua and Kirchhoff–Love shells. These formulations are expressed in tensorial form so that they can be used in both curvilinear and Cartesian coordinates. They can be used to model anisotropic crystals and soft biological materials.

Constraining the gauge-fixed Lagrangian in minimal Landau gauge

A continuum formulation of gauge-fixing resolving the Gribov-Singer ambiguity remains a challenge. Finding a Lagrangian formulation of operational resolutions in numerical lattice calculations, like minimal Landau gauge, would be one possibility. Such a formulation will here be constrained by reconstructing the Dyson-Schwinger equation for which the lattice minimal-Landau-gauge ghost propagator is a solution. It is found that this requires an additional term. As a by-product new, high precision lattice results for the ghost-gluon vertex in three and four dimensions are obtained.

A New Viscous Potential Function for Developing the Viscohyperelastic Constitutive Model for Bovine Liver Tissue: Continuum Formulation and Finite Element Implementation

The mechanical behavior of very soft tissues, such as liver, brain, kidney, etc. is assumed to be viscohyperelastic. The conventional approaches like quasi-linear viscoelastic (QLV) are limited to low strain rates and are unable to capture the short-term memory effects. In this paper, a new viscohyperelastic constitutive model is presented in which the strain energy function is decomposed into an elastic and a viscous part. The elastic part of the strain energy is assumed as a Mooney–Rivlin model, while a new viscous potential, as a function of strain and its rate, is proposed. Unconfined uniaxial compression tests are conducted up to 10% compression at two loading velocities of 1.3 and 10.56[Formula: see text](mm/min), to determine the material constants. A numerical simulation is also used to investigate the model-predicted material behavior. It is shown that the model is more sensitive to the hyperelastic parameters than the viscous parameters. This model is also able to predict the stress relaxation and hysteresis; however, an instantaneous relaxation is observed.