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Mathematical Foundations Of Elasticity

A review of phase-field models, fundamentals and their applications to composite laminates

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Differential Forms With Applications To The Physical Sciences

The objective of the current work is the development of a numerical framework for the simulation of damage in composite structures using explicit time integration. The progressive damage is described using a Continuum Damage Mechanics (CDM) based material model, CODAM2, in which the damage initiation and progression are modelled using Hashin's failure criteria and crack-band theory, respectively. The structural modelling uses higher-order theories based on the Carrera Unified Formulation (CUF). The current work considers 2D-CUF models where Lagrange polynomials are used to represent the displacement field through the thickness of each ply, resulting in a layer-wise element model. Numerical assessments are performed on coupon-level specimens, and the results are shown to be in good agreement with reference numerical predictions and experimental data, thus verifying the current implementation for progressive tensile damage. The capability of the proposed framework in increasing the polynomial expansion order through the ply thickness, and its influence on the global behaviour of the structure in the damaged state, is demonstrated. The advantages of using higher-order structural models in achieving significant improvements in computational efficiency are highlighted.

Progressive damage analysis of composite structures using higher-order layer-wise elements

A peridynamic model for plasticity: Micro-inertia based flow rule, entropy equivalence and localization residuals

A phase-field damage model for orthotropic materials is proposed and used to simulate delamination of orthotropic laminated composites. Using the deviatoric and hydrostatic tensile components of the stress tensor for elastic orthotropic materials, a degraded elastic free energy that can accommodate damage is derived. The governing equations follow from the principle of virtual power and the resulting damage model, by its construction, conforms with the physical relevant condition of no matter interpenetration along the crack faces. The model also dispenses with the traction separation law, an extraneous hypothesis conventionally brought in to model the interlaminar zones. The model is assessed through numerical simulations on delaminations in mode I, mode II, and another such problem with multiple initial notches. The present method is able to reproduce nearly all the features of the experimental load displacement curves, allowing only for small deviations in the softening regime. Numerical results also show forth a superior performance of the proposed method over existing approaches based on a cohesive law.

A Phase-Field Damage Model for Orthotropic Materials and Delamination in Composites

For visco-plasticity in polycrystalline solids under high strain rates, we introduce a dynamic flow rule (also called the micro-force balance) that has a second order time derivative term in the form of micro-inertia. It is revealed that this term, whose physical origin is traced to dynamically evolving dislocations, has a profound effect on the macro-continuum plastic response. Based on energy equivalence between the micro-part of the kinetic energy and that associated with the fictive dislocation mass in the continuous dislocation distribution (CDD) theory, an explicit expression for the micro-inertial length scale is derived. The micro-force balance together with the classical momentum balance equations thus describes the viscoplastic response of the isotropic polycrystalline material. Using rational thermodynamics, we arrive at constitutive equations relating the thermodynamic forces (stresses) and fluxes. A consistent derivation of temperature evolution is also provided, thus replacing the empirical route. The micro-force balance, supplemented with the constitutive relations for the stresses, yields a locally hyperbolic flow rule owing to the micro-inertia term. The implication of micro-inertia on the continuum response is explicitly demonstrated by reproducing experimentally observed stress–strain responses under high strain-rate loadings and varying temperatures. An interesting finding is the identification of micro-inertia as the source of oscillations in the stress–strain response under high strain rates.

A dynamic flow rule for viscoplasticity in polycrystalline solids under high strain rates

Emergence of pseudo-ductility in laminated ceramic composites

We discuss the Hu-Washizu (HW) variational principle from a geometric standpoint. The mainstay of the present approach is to treat quantities defined on the co-tangent bundles of reference and deformed configurations as primal. Such a treatment invites compatibility equations so that the base space (configurations of the solid body) could be realised as a subset of an Euclidean space. Cartan's method of moving frames and the associated structure equations establish this compatibility. Moreover, they permit us to write the metric and connection using 1-forms. With the mathematical machinery provided by differentiable manifolds, we rewrite the deformation gradient and Cauchy-Green deformation tensor in terms of frame and co-frame fields. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. We also show that for a hyperelastic solid, an equation similar to the Doyle-Erciksen formula may be written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite the HW functional in terms of frames and differential forms. Finally, we show that the compatibility of deformation, constitutive rules and equations of equilibrium are obtainable as Euler-Lagrange equations of the HW functional when varied with respect to traction 1-forms, deformation 1-forms and the deformation. This new perspective that involves the notion of kinematic closure precisely explicates the necessary geometrical restrictions on the variational principle, without which the deformed body may not be realized as a subset of the Euclidean space. It also provides a pointer to how these restrictions could be adjusted within a non-Euclidean setting.

A geometric approach to Hu-Washizu variational principle in nonlinear elasticity

http://manuscript.elsevier.com/S0045782522002924/pdf/S0045782522002924.pdf

Bensingh Dhas

Papers

A Phase-Field Damage Model for Orthotropic Materials and Delamination in Composites

A dynamic flow rule for viscoplasticity in polycrystalline solids under high strain rates

Emergence of pseudo-ductility in laminated ceramic composites

A geometric approach to Hu-Washizu variational principle in nonlinear elasticity

A novel four-field mixed FE approximation for Kirchhoff rods using Cartan’s moving frames