A Study of Finite and Linear Elasticity

A Study of Finite and Linear Elasticity

California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 1996 A study of finite and linear elasticity Fen Rui Johnson Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Johnson, Fen Rui, "A study of finite and linear elasticity" (1996). Theses Digitization Project. 1096. https://scholarworks.lib.csusb.edu/etd-project/1096 This Project is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. A STUDY OFHNITEANDLINEARELASTICITY A Project Presented to the Faculty of California State University, San Bernardino In PartialFulfillment ofthe Requirementsfor the Degree Master ofArts in Mathematics by Fen Rui Johnson June 1996 A STUDY OF HNITE AND LINEAR ELATICITY A Project Presented to the Faculty of California State University, San Bernardino by Fen Rui Johnson June 1996 Approved by: Dr. Chgtan Prakash, Mathematics Dr.U. Paul Vicknair, Department Chair, Mathematics ABSTRACT The purpose ofthis paper is to study the basic concepts offinite and linear elasticity such as stress, strain and displacement,and tofind outthe relationship between them.It is importantto know thatlinear elasticity is the special case offinite elasticity.Although linear elasticity was developedfrom finite elasticity,they have very different characters.This can be seenfrom the examples presented in the paper. m TABLEOFCONTENTS ABSTRACT iii CHAPTER 1 FINITE ELASTICITY. ......1 CHAPTER 2 STRESS IN LINEAR ELASTICITY... ......7 CHAPTER3 STRAIN AND DISPLACEMENT... :...... ..........28 CHAPTER4 STRESS - STRAIN RELATION....... ...40 CHAPTERS FORMULATION OFPROBLEMSIN ELASTICITY...... .....52 CHAPTER6 UNIFORM LOADINGIN HNITEELASTICITY 64 BIBLIOGRAPHY. .68 IV CHAPTER 1 FINITE ELASTICITY Finite elasticity is a theory ofelastic materials capable ofundergoing large deformation.This is anon-linear theory.To study the basic concepts offinite elasticity and toleam the difference betweenfinite and linear elasticity, we need tolook atthe following definitions. A body B is a closed ,connected setin with piecewise smooth boundary. Pointsin B are called materialpoints and are denoted by p. A defomiation ofB is a smooth one to one map u: B-^R' u(p)= x Thus,u is a yectorfunction. The displacement is u(p)-p =x-p The pointu(p)is the place occupied by p in the deformation u.F=Vu is the deformation gradient. Vu is a tensorfield with components(Vu)y = dU;/ dp^. Thus,we can think of Vu as the Jacobian matrix,i.e., the classical deriyatiye ofu.In whatfollows we assume detVu >0 WeletF^ denote the transpose ofF. A mapping x-»d>tx')with domain uCBl is called a spatialfield. ' In finite elasticity surface forces can be modeled as smooth functions t which associate a yector in R^ with a giyen point x in u(B)and a giyen unit yector n. i.e. x->t(n,x) t(n,x)representstheforce per unit area at x on any oriented surface through x With positiye unitnormal n.The reason the surface force is defined this way is because we treat X as a pointin a deformed body which the surface goes through.Surface forces arise when there is a physical contact with another body.Body forces may be modeled as continuousfuntions which associate a vectorin with each point x E u(B). i.e. x-^ b(x) where b(x)is theforce exerted atX per unit volume. Thelaws offorce and momentslead to a basic law ofcontinuum mechanics knows as Cauchy's Theorem. Cauchv'sTheorem There exists a smooth^ symmetric spatial tensor field T such that t(ii,x)=T(x)n for every unit vector n and all X Eu(B).Further divT+b =0 where div T is the vectorfield with components(divT)j= dT^/ dx- and the usual summation is assumed. Note:T is called the Cauchv stress which is usuallyjust called stress. Also,the equation t(n,x)= T(x)ii is called the constitutive equation,b is the body force. The main example we wish to work outin this chapter is aboutsimple shears ofa homogeneous,isotropic cube.We say a material is isotropic ifthe properties relating to its behavior under stress are the same in any direction ateach point. LetB be a homogeneous,isotropic body in the shape ofa cube. Consider the deformation X = u(p)defined by Xi=Pi+yP2 ^ ' ■ ; , , X2= p2 X3 = P3 where y =tan0 is called the shearing strain ,which will be precisely defined in chapter3. u 0 B is a homogeneous body,i.e., B is a body in which the corresponding stress T is constant.Also,T satisfies the equation^ ofequilibrium div T=0,for it is knownfrom the general theory that a homogeneous body can be deformed without body force,so b=0 Wecalculate the deformation gradient o O 1 y 0 • 0 1 0 1 o 0 0 1 • 1 0 0 so F^= r 1 0 0 0 ij From these we calculate a quantity known as the left Cauchv-Green strain tensor B=FF^ B= 0 1 0 y 1 0 0 0 1 0 0 1 1+y' y 0 y 1 0 0 0 1 From B we have B" 1 -y 0 B"'=|-y 1+y^ 0 0 0 1 and we calculate 1+3y + y"^ 2y + y' 0| 3 1 2 B'= 2y + y' 1+y' 0 0 0 1 tr(B) —Ajj4-A22"t"A33—3+y^ tr(B^)=3+4y^+y^. From tensor analysis we know we can expressT in terms ofthree principal invariants: lj(B)= tr(B)=3+Y2 II 12(B)=i|(tr(5))'-tr(fi^)]=3+Y^ Jr.2\] 13(B)= detB=l hence the list of principal invariants ofB is Lb =[li(B),12(B), 13(B)]=(3^-Y^ 3+Y^ 1). Therefore by the constitutive equationfor an isotropic material,we have T= PoI + PiB + P2B-' where Pj's are scalarfunctions ofLg. Thisis equivalentto T 1 0 o" 1+ y^ 1 0 ^11 ^12 T^s r 1 -y °1 1 2 T T ^12 ^22 ^3 0 1 +Pi r 1 0 +P2 -Y l+y 0 T T 0 ^13 ^23 7^3Ji 0 ij 0 0 1 0 0 1 HenceTj3=T23=0and Ti2=PiY+p2(-Y) T so — = jU 7 where - P2 is a quantity called the generalized shear modulus. In linear elasticity theory the normal stresses T,i,T22, and T33in simple shear are zero ,which we will leam in chapter 4.Here Tii=y'Pi +T T22 =Y'p2+ ^33=1^ where x is shear stress(which will be defined in Chapter 2)and t=Po + Pj+ pj• This shows there are relations between shear stress and normal stress in finite elasticity.In order to produce a pure shear the normal stresses need to be zero.By the equations we have above if^1=^2=0^ then the normal stresses will vanish. Hencet=Po. This implies p=0. Thus if then it is impossible to produce a simple(pure)shear by applying shear stresses alone because according to the equations we have,the normal stresses exist.This will not be the case in linear elasticity where there is no relationship between normal stresses and shear stresses. This example isfrom"Topicsin Finite Elasiticity" by Morton E.Gurtin. After this briefintroduction offinite elasticity,I willfocus on the linear elasticity in the nextfour chapters,then conclude the project with another example offinite elasticity. CHAPTER2 STRESSIN LINEARELASTICITY In a body that is not deformed,all parts ofthe body are in mechanical equilibrium. This means the resultantforce on the body is zero.When a deformation occurs the body will try to go back to its original state ofequilibrium.Forces therefore arise which tend to return the body to equilibrium.Theseforces are called internalforces.The interrial forces depend on the externalforces,the body forces and the surface forces.Body forces are the forces that are connected with the mass ofthe body and are distributed throughoutthe volume ofa body;they are notthe result ofthe contact oftwo bodies.Forces such as gravitational, magnetic,and inertia forces are body forces.They are measured in terms of force per unit volume.On the other hand surfaceforces arisefrom the physical contact between two bodies. We can picture an imaginary surface within a body acting on an adjacent surface. Pi AA Figure 2.1 External surface forces and intemalforces 7 The internalforces which occur when a body is deformed are call internal stress.This means the intemalforces became internal stresses under deformation.Ifno deformation occurs,there are no intemal stresses.The averageforce per unit area is AF ■ ■ p — • ■ AVE ~ • AA From this we have the definition of stress^ which is the limit value ofaverageforces per unit area as the area of AA^0. .^ p _ lim ^ dF ~ AA^O AA ~ dA Here the stress on the area dA is a vector and has the same direction as the force vector dF. One thing we should be clear on is that stress is not a vector unless a specific plane is given.This means we can only representthe stress as a stress vector on a specific plane.When several stress acton the same plane,at this point we treat stress as a vector and can use rale for vector addition.Therefore we cannot talk aboutthe stress on a point since through a point we can draw infinitely many planes and that will give usinfinitely many vectors on different planes.Thus,the stress does not behave as a vector because a vector quantity associates only a scalar (its component)with each direction in space where as stress does not.

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