Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models

Internal Polar Continuum Theories for Solid and Fluent Continua

Classical continuum theories are useful in the study of a variety of problems of engineering and applied sciences. However, the emergence of new materials has provided the need for refined theories that account for certain features that are not accounted for in the classical continuum theories. Polar decomposition of the deformation gradient tensor into pure stretch and pure rotation tensors shows that the rotation tensor will in general vary from point to point. Similarly, polar decomposition of the velocity gradient tensor shows that the rate of rotation tensor will vary from point to point. It can also be shown that the strain and strain rate tensors used in classical theories of continuum mechanics do not depend on the rotation tensor or its gradients and therefore neglect the effect of changing rotations and rates of rotations between neighboring material points in Lagrangian description, and between neighboring locations in Eulerian description. Varying rotations and rates of rotations between neighboring material points will, if resisted by the continua, result in internal moments which are conjugate to these rotations and rates of rotations. These internal moments along with the conjugate rotations and rates of rotations will result in energy storage and dissipation, in addition to the energy storage and dissipation resulting from stress and its conjugate strain and strain rate. Based on this observation, it is necessary to modify the existing conservation and balance laws to include internal moments, which results in a more complete thermodynamic framework for solid and fluent continua. In this work, new conservation and balance laws are derived for solid and fluent continua that include internal moments which result from varying rotations and rotation rates. Also, constitutive theories are derived for the stress tensor, moment tensor, and heat vector, resulting in a complete mathematical model internal polar thermoelastic solids and internal polar thermoviscous fluids. This derivation does not rely on the introduction of external micro-rotations or stress couples as is done in the so called micro-polar or couple-stress theories. The theories presented here are therefore referred to as “internal polar continuum theories”, as they are derived using only internal measures of deformation and do not require introduction of external degrees of freedom. We also present a framework for obtaining approximate solutions to the mathematical models resulting from the new continuum theories. Numeric results are presented to show the affect of the internal polar theories presented here

Mechanics and Applications of Pressure Adaptive Honeycomb

A pressure-adaptive wing structure is presented that relies on the pressure-altitude relation to adapt the wing shape to the various flight conditions (e.g. landing and cruise). This structure employs conventional, certified aerospace materials and can be implemented in aircraft ranging from LSA to business jets and high-subsonic transports. The adaptive structure consists of a honeycomb which has cells that extent a significant length perpendicular to the plane of the hexagons. In each cell resides a pouch (bladder) that can be pressurized. Pressurization yields a change in the stiffness of the structure. In combination with a restoring force, this structure shows strains in excess of 50% without any plastic deformation in the honeycomb cell walls. Mass specific energy densities for this pressure adaptive honeycomb is on the par with shape memory alloy, which has the highest mass specific energy density of the adaptive materials. A mathematical model based on the equivalent properties of the pressurized honeycomb is developed and verified against experimental tests. The applicability of this new adaptive structure is proven in the wind tunnel for a pressure adaptive flap on a generic wing section resulting in an increase in maximum lift coefficient of 0.3

Collected Papers in Structural Mechanics Honoring Dr. James H. Starnes, Jr.

This special publication contains a collection of structural mechanics papers honoring Dr. James H. Starnes, Jr. presented at the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference held in Austin, Texas, April 18-21, 2005. Contributors to this publication represent a small number of those influenced by Dr. Starnes' technical leadership, his technical prowess and diversity, and his technical breath and depth in engineering mechanics. These papers cover some of the research areas Dr. Starnes investigated, which included buckling, postbuckling, and collapse of structures; composite structural mechanics, residual strength and damage tolerance of metallic and composite structures; and aircraft structural design, certification and verification. He actively pursued technical understanding and clarity, championed technical excellence, and modeled humility and perseverance