Deformation In Continuum Mechanics Term Paper

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ABSTRACT

Continuum mechanics is a combination of physical laws and mathematics that estimate the behavior of matter that is prone to mechanical loading. The basic equations involved in continuum mechanics may be formed in two distinct but fundamentally parallel formulations. In continuum mechanics, deformation is the transformation of certain bodies or objects from a reference shape to a present shape. This paper explores the different types of deformation in continuum mechanics.

Keyword: continuum mechanics, deformation, stretch, rotation, strain, physics laws, mathematics

INTRODUCTION
The examination of the mechanical and kinematic characteristic of materials demonstrated on the continuum postulation is termed as continuum mechanics. Continuum mechanics is divided into two primary concepts. The first concept focuses on the derivation of the basic equations that are application for continuous means. The equations are grounded on physics’ universal laws that include the law on conservation of mass, momentum, the energy principles, and a lot more. The second concept focuses on the development of constitutive equations that describe the behavior of certain flawless materials, the elastic solid, as well as the viscous fluid. These equations offer the central ideas around which investigations in plasticity, elasticity, fluid mechanics, and viscoelasticity proceed.
On the mathematical sense, the basic equations involved in continuum mechanics may be formed in two distinct but fundamentally parallel formulations. The global or integral form originates from a deliberation of the fundamental concepts used to a finite volume of the object. The field or differential approach results to equations taken from the fundamental concepts used to a small component of volume. In practical applications, it is often convenient and beneficial to assume the field equations from their counterparts in the global sense. Based from the outcome of the continuum assumptions, velocity and densities are field quantities that mirror the kinematic and mechanical components of continuum bodies. These continuum bodies are stated as continuous functions of time and space variables.
In continuum mechanics, deformation is the transformation of certain bodies or objects from a reference shape to a present shape. A shape is made up of the positions of all body particles. Deformation can be brought about by body forces including electromagnetic forces or gravity, external loads, moisture content, temperature changes, or even chemical responses. In terms of continuous body, a field of deformation derives from a stress field brought by applied pressures or changes in the field temperature within the body. The link between induced strains and stress is stated through the use of constitutive equations. Deformations that are mended following the eradication of the stress field are elastic deformations. In this aspect, the continuum totally regains the original shape. On the contrary, irreversible deformations still continue even after elimination of stresses.
This paper explores deformation in continuum mechanics. Specifically, this paper covers pertinent topics that involve continuum mechanics and deformation.

WHAT DEFORMATION MEANS

Deformation can best be illustrated by denoting the initial shape of a physical body, for instance an airplane part, through the symbol “Ωo”. Using the illustration used by Brannon,
Figure 1: A 2D Deformation
Figure 1 shows a deformed slice of putty that created a new shape in pseudo 2D manner. The shape’s cross-section as well as the out-of-plane length transformed indiscriminately, however no out-of-plane shearing is possible. According to Brannon (2008), the grid was painted conceptually to aid easy view of the deformation. The X is figure 1 denotes the starting position of a reference in the body. Prohibiting fracture in which elements “break” into two detached particles, each reference in a body may possess only a single starting location. X is also the starting position vector that can be considered as a distinct name of any particle. When expressing particle X, it means that the particle was primarily situated at X position. Each single particle moves to a new position and since no two unique particles are permitted to deform into similar place, the primary association between the mapping operations is associated with the beginning and final position of the vectors. At most times, a mapping function need not necessarily be found. Rather, kinematics is established on the concept that there is presence of a mapping function.
Because every single particle has its own distinct position in space during the start and deformed shapes, it can be asserted that there is a mapping function or a one-to-one function. In a mapping function, no material element must be allowed to invert. Such limitation is termed local admissibility and it is guaranteed particularly when the matrix possesses a positive determinant.
In stimulations of finite element, a method termed as hourglass control is generally applied to guarantee local admissibility. Nevertheless, such local admissibility is not enough to ensure global admissibility because the latter necessitates that the mapping function be invertible and thus disallowed interpenetration of materials.
Figure 2: A locally admissible deformation
Figure 2 demonstrates a locally admissible deformation but not globally admissible. In computations, contact algorithm is used to prevent material interpenetration. At times, issues in mechanics simply state the space variations in the tensor without creating explicit point to the mapping function. In relation to these concerns, compatibility condition guarantees that the mapping function is present despite it not being utilized explicitly. Moreover, compatibility guarantees that the tensor field can be integrated. In this situation, the continuum totally regains its original shape. However, irreversible deformations continue even after the elimination of stresses. Plastic deformation, for instance, happens in various material bodies following the attainment of a threshold value in stresses known as the yield stress or the elastic limit. The outcome of this is the slip or the mechanism of dislocation at the atomic level.
Viscous deformation is an irreversible type of deformation. In elastic deformations, the reaction function that links strain to the stress is the materials’ compliance tensor. Strain is the relative displacement of body particles that does not take into account the rigid body movements. Different choices may be created for the strain field expression depending on how it is defined in terms of the beginning and final shape of the body and whether there is consideration of the type of metric tensor used.

DEFORMED SHAPE OF A SOLID

The solid configuration is a space filled by the object. When describing motion, convenient solid configuration is considered and utilized as a reference. This is generally the primary undeformed solid; however, it can be a convenient space that could be filled by the solid. The changes in materials’ shape are brought about by the action of outside loads, and at a certain period fill a new space which is the current or deformed solid configuration. For a number of applications such as fluids, issues with development or evolving microstructures, a stable reference shape cannot be recognized. Hence, the deformed material is often used as the reference shape. In Mathematics, a deformation is described by the ratio 1:1 mapping which alters points from the reference shape of a solid to the solid’s deformed shape. In particular, if the symbol is the three numbers that occupy certain position of a certain point in the solid’s undeformed shape; these could include the three elements of a vector position in a Cartesian coordinate system. As the deformation of solid transpires, every single value of the coordinates transform to other numbers. This can be expressed in the following form: or the deformation mapping.
There are certain standards to be considered for an object to be a deformation that is physically admissible. First, the coordinates should specify locations in a Newtonian reference frame. This implies that it should be possible to locate a number of coordinate transformations - , in a way that are elements in an orthogonal ground, which is believed to be ‘motionless’ in Newtonian dynamics. In addition, the functions  should be in a 1:1 ratio on the complete group of real numbers; and should be invertible. Moreover, the mapping should satisfy this formula:  
 

THE DEFORMATION GRADIENT

The deformation gradient pertains to the measure of body’s local deformation. Other measures of deformation that possess suggestive meanings include change of orientation and shape. According to the algebraic theorem called polar decomposition, for any tensor F that is non-singular, there is the presence of distinct symmetric positive definite tensors U and V as well as a distinct orthogonal tensor R. Hence, F = UR = RV. The deformation gradient F holds such decomposition because it is non-singular. In addition, a positive fixed symmetric tensor signifies a condition of pure expanses along three mutually orthogonal axes as well as an orthogonal tensor rotation. Hence, the equation F = UR = RV guarantees that any form of local deformation is a collaboration of rotation and pure stretch. R is known as the rotation tensor. On the other hand V and U are the left and right stretch tensors. Both tensors gauge the local strain, the alteration in shape, while the R tensor elucidates the local rotation as well as alteration of orientation experienced by body’s material elements. This creates a clear equation expressed as follows:
U2 = FT F, V2 = F FT
det U = det V = |det F|
Because V = RURT, V and U have similar eigenvalues and there is variation in their eigenvectors through the rotation R. Their eigenvalues are termed as the principal stretches while the eigenvectors are the principal directions.
Figure 3: Orthogonal grid, body, and differential block in reference shape (Left); deformed body (Right)
Figure 3 presents a reference shape with a grid of straight lines. The different points of the grid are in line with the particles of the material. As time differs, the particles of materials can occupy diverse spatial points and the grid experiences alteration. Straight lines alter length. Right angles either decrease or increase and the figure become curved. In continuum mechanics, the particles of materials are considered differential cubes whose boundaries are no longer orthogonal.
Any cube is described by the three orthonormal vectors that create its edges. In addition, the deformed parallel piped is described by the three vectors that line the edges. The tensor F deformation gradient quantifies the transformations in these vectors by bringing together the three edge vectors that have been deformed. The edges are brought into columns containing a 3x3 matrix. For rectangular Cartesian coordinate systems, the matrix is made up of the deformed edge vectors, stated in relation to the undeformed vectors that delineate the initial cube. Relative means that all length transformations are stated as multiples of the cube’s initial edge lengths and the rest of the directions are stated in relation to the cube’s initial edge directions. In turn, it can be said that beginning infinitesimal cube is described as a unit cube with three edge vectors aligned with the orthonormal laboratory basis. As it undergoes deformation, the edge vectors start to deform into a completely new set of vectors.
Figure 4: (a) Differential block in position state with sizes; (b) Differential block following extension along its limits: (c) Differential block following a shear deformation.
Figure 4a presents a differential cube of object in the reference shape with three edges having similar length. On the other hand, figure 4b presents the differential cube after the configuration when the edges altered the length and has turned into a differential rectangular parallel piped having the boundaries day, daz, and dax. The normal strain is described as the alteration in length for every unit length and is generally denoted by the ǫ symbol. In addition, the block is made of three normal strains, one related to each edge. Shear strains happen because of changes in angle. Figure 4c presents the differential 3D square in a later arrangement when the top surface has displaced distance da in relation to the base surface along these lines bringing on a change between two edges.
The transformation in boundaries, the shear strain, is indicated by the γ image and is described by tan γ xy = da dAy. At the point when the strain is little in extent, it is imperative to utilize the rough guess tan γ xy ≈ γ xy. A differential 3D square for the most part experiences concurrent typical and shear strains in every one of the three bearings. At an altered time, the strains shift all through the body. At an altered block the strains change with time. The strains depict the contortion of the microstructure of the material. At the point when a body distorts from the reference to a later arrangement, parts of the body move with respect to different parts. To keep up its coherence and not be destroyed, the body produces interior strong strengths and minutes.
Figure 5: Body in deformed shape (Right); part of the body presenting the external surface as well as internal planar surface (Left)

THE VELOCITY AND DISPLACEMENT FIELDS

The displacement vector u(x, t) defines the movement of each point in the solid. To make this accurate, imagine a solid that deforms under external loads. Every single point in the solid changes as the load is used: for instance, a point at location x in the undeformed object may transfer to a new location y at time t.  This defines the displacement vector which is states as follows:

It can also be expressed through the use of index notation.

The displacement field fully stipulates the alteration in shape of the object. The velocity field would define its movement, as
The acceleration field is described as
 
 
 

EXAMPLES OF DEFORMATION

 
The physical importance of a homogeneous deformation is the fact that all straight lines in the object still remain straight despite the deformation. Hence, every point in the solid encounters the same configuration change.
Figure 6: Rotation and Deformation
A homogenous deformation is a special type of deformation in which the gradient has similar value everywhere in the body. For homogenous deformations, there is deformation of straight, deformation of planes to planes, cubes deformation to parallelepipeds, as well as deformation of spheres to ellipsoids. A rotation is a special form of deformation where material vectors change position but they do not alter their length. A stretch is a totally distinctive type of deformation with three material vector outlines in the 3D initial arrangement that will alter its length, be that as it may not at the start. For this situation, the rotating angle tensor will be both symmetric as well as a positive unequivocal. The stipulation about positive definiteness is critical. On the other hand, a distortion angle tensor is symmetric and does not necessarily mean it is stretched. Case in point, an untainted pivot results in a symmetric twisting disposition. Being symmetric, a stretch is askew in its key premise. The key qualities break even with the degree of deformation to undeformed lengths of the three non-turning material strands. Material strands that are not adjusted to the key headings of a stretch will change initially, yet for each fiber pivoting edges, there will be an alternate rotation in the other way, making the net pivot of material strands zero for stretch deformation.

CONCLUSION

Continuum mechanics is the examination of the mechanical and kinematic characteristic of materials demonstrated on the continuum. The basic equations involved in continuum mechanics may be formed using the global or integral form and the field or differential approach. In continuum mechanics, deformation is the conversion of certain bodies or objects from a reference shape to a present shape. Forces including electromagnetic forces or gravity, external loads, moisture content, temperature changes, or even chemical responses may cause deformation. The deformation gradient is the measure of body’s local deformation. Orientation and shape are suggestive meanings of deformation. Deformation in continuum mechanics is the change in the shape of the body from the past configuration to the current configuration.

References

Bowen, R. (1989). Introduction to continuum mechanics for engineers. New York: Plenum Press.
Irgens, F. (2008). Continuum mechanics. Berlin: Springer.
Lai, W., Rubin, D. and Krempl, E. (2010). Introduction to continuum mechanics. Amsterdam: Butterworth-Heinemann/Elsevier.
Mase, G. and Mase, G. (1999). Continuum mechanics for engineers. Boca Raton, Fla.: CRC Press.
Narasimhan, M. (1993). Principles of continuum mechanics. New York: Wiley.
Pucci, E. and Saccomandi, G. (1997). On universal relations in continuum mechanics. Continuum Mechanics and Thermodynamics, 9(2), pp.61-72.