A phase-field approach to pneumatic fracture with anisotropic crack resistance

Phase-field models of fracture allow the prediction of crack propagation and crack patterns. In this contribution, externally driven fracture processes in linear and finite elasticity are investigated. Different approaches to consider pneumatic pressure and materials with non-isotropic crack resistance are studied, combined, and examined in detail. The versatility of the proposed models is proven by a series of numerical simulations in two and three dimensions.

Large Deformation Dynamic Analysis of a Cable System by a New Hamiltonian Finite Element Method

This paper develops a new Hamiltonian nodal position finite element method for dynamic analysis of spatial flexible cable systems with large deformation. The dynamic governing equation is derived from finite elasticity theory. Logarithmic strain is applied to construct large deformation Hamiltonian canonical equations. An efficient second-order symplectic difference algorithm is built to solve the canonical equations numerically. A large strain conical pendulum system is analyzed numerically by the proposed method, and the numerical results are compared with those retracted from the existing Hamiltonian methods and Livermore Software Technology Corporation: dynamics (LS-DYNA). The proposed method is further verified by two tethered dynamic experiments involving large displacement motion and large deformation. The comparisons and verifications demonstrate that the proposed method is of symplectic conservation, has high accuracy and has stability for calculating flexible cable system dynamics with large deformation.

Influence of the growth gradient on surface wrinkling and pattern transition in growing tubular tissues

Growth-induced pattern formations in curved film-substrate structures have attracted extensive attention recently. In most existing literature, the growth tensor is assumed to be homogeneous or piecewise homogeneous. In this paper, we aim at clarifying the influence of a growth gradient on pattern formation and pattern evolution in bilayered tubular tissues under plane-strain deformation. In the framework of finite elasticity, a bifurcation condition is derived for a general material model and a generic growth function. Then we suppose that both layers are composed of neo-Hookean materials. In particular, the growth function is assumed to decay linearly either from the inner surface or from the outer surface. It is found that a gradient in the growth has a weak effect on the critical state, compared with the homogeneous growth type where both layers share the same growth factor. Furthermore, a finite-element model is built to validate the theoretical model and to investigate the post-buckling behaviours. It is found that the associated pattern transition is not controlled by the growth gradient but by the ratio of the shear modulus between two layers. Different morphologies can occur when the modulus ratio is varied. The current analysis could provide useful insight into the influence of a growth gradient on surface instabilities and suggests that a homogeneous growth field may provide a good approximation on interpreting complicated morphological formations in multiple systems.

Sharp conditions for the linearization of finite elasticity

We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.

A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies

It is well known that the linearized theory of elasticity admits the logically inconsistent solution of singular strains when applied to certain naive models of fracture while the theory is a first-order approximation to finite elasticity in the asymptotic limit of infinitesimal displacement gradient. Meanwhile, the strain-limiting models, a special subclass of nonlinear implicit constitutive relations, predict uniformly bounded strain in the whole material body including at the strain-concentrator such as a crack tip or reentrant corner. Such a nonlinear approximation cannot be possible within the standard linearization procedure of either Cauchy or Green elasticity. In this work, we examine a finite-element discretization for several boundary value problems to study the state of stress–strain in the solid body of which response is described by a nonlinear strain-limiting theory of elasticity. The problems of notches, oriented cracks, and an interface crack in anti-plane shear are analyzed. The numerical results indicate that the linearized strain remains below a value that can be fixed a priori, therefore, ensuring the validity of the nonlinear model. In addition, we find high stress values in the neighborhood of the crack tip in every example, thereby suggesting that the crack tip acts as a singular energy sink for a stationary crack. We also calculate the stress intensity factor (SIF) in this study. The computed value of SIF in the nonlinear strain-limiting model is corresponding to that of the classical linear model, and thereby providing a tenet for a possible local criterion for fracture. The framework of strain-limiting theories, within which the linearized strain bears a nonlinear relationship with the stress, can provide a rational basis for developing physically meaningful models to study a crack evolution in elastic solids.

Reference Configurations Versus Optimal Rotations: A Derivation of Linear Elasticity from Finite Elasticity for all Traction Forces

We rigorously derive linear elasticity as a low energy limit of pure traction nonlinear elasticity. Unlike previous results, we do not impose any restrictive assumptions on the forces, and obtain a full $$\Gamma $$ Γ -convergence result. The analysis relies on identifying the correct reference configuration to linearize about, and studying its relation to the rotations preferred by the forces (optimal rotations). The $$\Gamma $$ Γ -limit is the standard linear elasticity model, plus a term that penalizes for fluctuations of the reference configurations from the optimal rotations. However, on minimizers this additional term is zero and the limit energy reduces to standard linear elasticity.