Dr. Wang Xingbo Fall ， 2005

Dr. Wang XingboDr. Wang Xingbo

FallFall ，， 20052005

Mathematical & MechanicalMathematical & MechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

1.1. Kinematical relations Kinematical relations

2.2. Stress equilibrium Stress equilibrium

3.3. Constitutive relations Constitutive relations

4.4. Boundary conditions Boundary conditions


Finite Deformation Elasticity Finite Deformation Elasticity

In Mechanical engineering, machine In Mechanical engineering, machine parts contacted each other may have parts contacted each other may have deformations in their shapes. The deformations in their shapes. The deformation depends on the forces deformation depends on the forces exerted on the parts, the properties of exerted on the parts, the properties of the materials and the original shapes of the materials and the original shapes of the parts. The study of part-shape the parts. The study of part-shape deformations is a task of continuum deformations is a task of continuum dynamics. Tensor and tensor field are dynamics. Tensor and tensor field are basic tools in this study.basic tools in this study.


IntroductionIntroduction

Continuum mechanics deals with the movement of Continuum mechanics deals with the movement of materials when subjected to applied forces. The materials when subjected to applied forces. The motion of a continuous and deformable solid can motion of a continuous and deformable solid can be described by a continuous displacement field be described by a continuous displacement field resulting from a set of forces acting on the solid resulting from a set of forces acting on the solid body. In general, the displacements and forces mbody. In general, the displacements and forces may vary continuously with time, but for the preseay vary continuously with time, but for the present purpose a two-state quasi-static analysis will nt purpose a two-state quasi-static analysis will be discussed. be discussed.


Continuum mechanicsContinuum mechanics

The initial unloaded state of the material is referrThe initial unloaded state of the material is referred to as the reference or undeformed state as ted to as the reference or undeformed state as the displacements are zero everywhere. The mahe displacements are zero everywhere. The material then reconfigures due to applied loads anterial then reconfigures due to applied loads and reaches an equilibrium state referred to as thd reaches an equilibrium state referred to as the deformed state. The concepts of strain, a meae deformed state. The concepts of strain, a measure of length change or displacement gradient,sure of length change or displacement gradient, and stress, the force per unit area on an infinit and stress, the force per unit area on an infinitesimally small plane surface within the materiaesimally small plane surface within the material, are of fundamental importance for finite defol, are of fundamental importance for finite deformation elasticity theory.rmation elasticity theory.


Premary ConceptsPremary Concepts

The equations that govern the motion of deformable The equations that govern the motion of deformable materials can be derived in the following four stepsmaterials can be derived in the following four steps


Projection of a VectorProjection of a Vector

1. Kinematical relations, which define strain tensor in terms of displacement gradients.

2. Stress equilibrium, or equations of motion derived from the laws of conservation of linear momentum and conservation of angular momentum.

3. Constitutive relations, which express the relationship between stress and strain

4. Boundary conditions, which specify the external loads or displacement constraints acting on the deforming body.

The key to analyzing strain in a material undergoingThe key to analyzing strain in a material undergoinglarge displacements and deformation is to establishlarge displacements and deformation is to establishtwo coordinate systems and the relationship betweetwo coordinate systems and the relationship betwee

nnthem. The first is a material coordinate system to them. The first is a material coordinate system to effectively tag individual particles in the body.effectively tag individual particles in the body.The second is a fixed spatial coordinate system.The second is a fixed spatial coordinate system.Deformation is quantified by expressing the spatial Deformation is quantified by expressing the spatial coordinates of a material particle in the deformed coordinates of a material particle in the deformed state, as a function of the coordinates of the samestate, as a function of the coordinates of the sameparticle in the undeformed state. Length changes of particle in the undeformed state. Length changes of material segments can then be determined from thematerial segments can then be determined from theknown deformation fields and thus strain tensors maknown deformation fields and thus strain tensors ma

yybe calculated.be calculated.


Kinematical Relations Kinematical Relations

Experiment 1Experiment 1. Consider a pipe with fluid moving thro. Consider a pipe with fluid moving through it, we'd like to model the velocity of the fluid as it ugh it, we'd like to model the velocity of the fluid as it moves through the pipe. An experimentalist may measmoves through the pipe. An experimentalist may measure the rate at which the total mass passes through a paure the rate at which the total mass passes through a particular cross-section rticular cross-section SS. Given the density, the mean ve. Given the density, the mean ve

locity can then be calculatedlocity can then be calculated


Two Coordinate SystemsTwo Coordinate Systems

S

P P

P

Experiment 2. Suppose we are watching the motion of a number of basketballs, each of which is numbered a “coordinate” at the original state, and want to record the motion velocity of one basketball X. In this case we want to measure the velocity of one particle at different locations as function of time.


Two Coordinate SystemsTwo Coordinate Systems

P3 v3

v2

P2

v1

P1

Deformation is defined by the movement of material particles, which can be thought of as small non-overlapping quantities of material that occupy unique points within the undeformed body. For this reason a method of labeling the particles is required. We use the material coordinate system .By convention it is denoted by capital letters X with components X1, X2 and X3. Since a material coordinate is actually the encoding number of a materialparticle, a unique material particle is always identified by the same coordinate values no matter how the body deforms.


Material Particles Material Particles

As the body is deforming, each of its material points is moving in space, thus Eulerian coordinate system is also needed to record the spatial location of the moving particles. Eulerian coordinate is in lower letter x with components x1, x2 and x3. Conversely, a fixed material particle X may move to several spatial positions during the deformation.

Mathematical & mechanicalMathematical & mechanicalMethod in Mechanical EngineeringMethod in Mechanical Engineering

Spatial Position Spatial Position

An infinitesimal material line segment dX in the undeformed body B0


Deformation and Strain Deformation and Strain

In the deformed body B the same material particlesthat constituted dX have reconfigured into dx

X3 x3

dX3

dx3 dx

dX dX2 dx2

O X2 o x2

dX1 dx1

X1 x1

Let the coordinate transformation of X tox complies with

x=x(X) or x=x(X, t)


Deformation and StrainDeformation and Strain

is a tensor of order 2, and we call it deformation gradient tensor

1 2 3

1 2 3

( , , )0

( , , )

x x xJ

X X X

x

Xi

i M i MMM

xdx dX F dX

X

( )i

MFF

Any deformation can be split into two parts: a rigid body rotation and a stretch

F = RU Where rotation tensor R is orthogonal (RT =R-1) and stretch tensor U

is symmetric.


Polar Decomposition

Strain in a deforming body is determined by measuring segment length changes

XCXXFFXXFXF TT dddddddddxdxds TTTii )(2 xx

Green’s deformation tensor or the right Cauchy-Greendeformation tensor, which indicates how each componen

tof the undeformed line segment dX contributes to thesquared length of the deformed line segment dx.


Strain Strain

T

M NX X

C F F k kx x

{ }

( ) ( )T T T T T C F F RU RU U R RU U UWe can see that the deformation tensor C is symmetric and only related to the stretch tensor U

One method for computing the stretch tensor Ufrom the deformation gradient tensor F is tofirst calculate C= FTF, then calculate theeigenvalues (1) 2, (2) 2and (3) 2, and orthogonal eigenvectors s1, s2 and s3 of C,then U may be constructed by:


Vector Calculus Vector Calculus

21 1

2 22 2

23 3

T T T T

C , U


Physical Interpretations

The two orthogonal tensors R and , derived from F, have quite different physical interpretations. R describes the rigid body rotation component of the deformation with no information about the material stretching. On the other hand, the columns of are the orientations of the principal stretch axes relative to the material coordinates.


Principal Invariants Principal Invariants

2 21 2 3

1[( ) ] det

2tr tr tr I C I C C I C

2 2 21 1 2 3

2 2 2 2 2 22 1 2 2 3 3 1

2 2 23 1 2 3

I

I

I2 2 2

3 1 2 3det 1 C Ibe an additional kinematical constraint that must be imposed on the deformation field for incompressible materials


Lagrangian strain tensor1

( )2

E C I

Stress Tensors


Stress Equilibrium Stress Equilibrium

Stress is defined as the force per unit area acting on an infinitesimally small plane surface. If the line of action of the force is normal to the plane then a normal or axial stress results, whereas a shear stress arises when the line of action of the force is tangential to the plane. The quantities of force and area can be referred either to the reference (undeformed) or deformed configurations, which leads to three important ways of representing stress in a deforming body, namely using the Cauchy, first or second Piola-Kirchhoff stress tensors.

denoted ij, represents the force measured per unit deformed area acting on an element of surface in the deformed configuration. The first index indicates the direction of the normal to the surface on which ij acts and the second index indicates the direction of the stress component.

It should be noted that the Cauchy stress tensor is symmetric for non-polar materials and that ij are also the physical components of stress in rectangular Cartesian coordinates.


The Cauchy stress tensor The Cauchy stress tensor

The first Piola-Kirchhoff stress tensor, denoted sMj, represents the force acting on an element of surface in the deformed configuration but measured per unit-undeformed area. The first index is written in uppercaseas it refers to the normal of the surface in the undeformed state, and is thus a material coordinate index. The second index denotes the direction of the force acting on the deformed material and is a spatial coordinate index. For this reason the first Piola-Kirchhoffstress tensor is generally not symmetric. It is sometimes referred to as the Lagrangian stress tensorand is often used in experimental testing where force ismeasured in the deformed tissue, but the area over which it acts is measured in the undeformed tissue.


Piola-Kirchhoff stress tensor

denoted TMN, represents the force measured per unit-undeformed area P, acting on an element of surface in the undeformed configuration. This force may be determined from the actual force p, in the same way that the undeformed material vector dX is determined from the deformed material vector dx. Specifically P = F-1p just as dX = F-1dx. The primary use of the second Piola-Kirchhoff stress tensor is for representing material behavior at a point, independent of rigid body motion.


The second Piola-Kirchhoff stress tensor The second Piola-Kirchhoff stress tensor


Relationships Relationships

-1 -1T T

Mj ij MN Mj ijN NM M

i j i j

X XX Xs s

x x x x

S JF Σ T S F JF Σ F

J T J

-1 -1( ) ( )

or inversely 1 1

1 1

T

j jMj MN ij Mj MNi i

N M M N

x xx xs s

X X X X

S TF Σ FS FTFJ J

T TJ J

T

3 1 2 3det J F I

Conservation of massConservation of mass


Conservation Laws and The Principle of Virtual WorkConservation Laws and The Principle of Virtual Work

00

000

VVV

JdVdVdV

30 IJ


Conservation of linear momentum Conservation of linear momentum

S V V

dt m dS dV dV

dt v t b vF

iji jdS n dSt i

Consider b=bjij and v= vjij

( ) 0j

ij ji

S V

dvn dS b dV

dt

0][

dVfbxV

jj

i

ij


Cauchy’s first law of motion Cauchy’s first law of motion

jj

i

ij

fbx

jj

N

jMN

M

fbX

xT

X 00)(

The conservation of angular momentum equates the time rThe conservation of angular momentum equates the time rate of change of the total angular momentum for a set ate of change of the total angular momentum for a set of particles to the vector sum of the moments of the extof particles to the vector sum of the moments of the external forces acting on the system. For stress equilibriuernal forces acting on the system. For stress equilibrium of non-polar materials, this principle is equivalent to m of non-polar materials, this principle is equivalent to the symmetry condition on the Cauchy stress tensor, nathe symmetry condition on the Cauchy stress tensor, namely mely ijij = = jiji. Note that if the Cauchy stress tensor is s. Note that if the Cauchy stress tensor is symmetric (as is the case for the non-polar materials beiymmetric (as is the case for the non-polar materials being considered here), the second Piola-Kirchhoff stress ng considered here), the second Piola-Kirchhoff stress tensor is also symmetric tensor is also symmetric


Conservation of angular momentum Conservation of angular momentum

Now consider a body of volume Now consider a body of volume VV and surface and surface SS

loaded by a surface traction loaded by a surface traction s s which is inwhich is in equilibrium with the internal equilibrium with the internal stress vectorstress vector tt.. If the body is subjected to an arbitrarilyIf the body is subjected to an arbitrarily small displacement small displacement vv, which satisfies, which satisfies compatibility and any displacement boundarycompatibility and any displacement boundary conditions specified on conditions specified on SS (where (where v v must bemust be zero), then the principle of virtual work canzero), then the principle of virtual work can be expressed by:be expressed by:


Principle of virtual work Principle of virtual work


Principle of Virtual Work Principle of Virtual Work

2

dS dS S S

v t vs

The virtual displacements may be resolved into components dv =dvjij.

2 2

iji j

S S

dS n dS s v v

2 v

vv

i iS

dS dVx x

vs

i ji j j

j[ ]


Principle of Virtual WorkPrinciple of Virtual Work

2

( )jij j jj

i V S

vdV b f v dV dS

x

v

s v

2

1( )jMN j ji

jM NV V S

xxdV b f v dV dS

X X

vT sJ


Constitutive Relations Constitutive Relations

)(2

1

NMMN

MN

E

W

E

WT


Boundary Constraints and Surface Boundary Constraints and Surface TractionsTractions

2 2

( )appl j j

S S

dS p n v dS s v

V S

j(appl)jjj

V N

j

M

iMN dSpdvfbdVX

x

X

x

2

ˆ)(1

jvnJ

VT

Class is Over! Class is Over!

See you Friday Evening!See you Friday Evening!
