Concept of Stress at a Point

Washkeic College of Engineering Section : STRONG FORMULATION Concept of Stress at a Point Consider a point ithin an arbitraril loaded deformable bod Define Normal Stress Shear Stress lim A Fn A lim A FS A

Washkeic College of Engineering Section : STRONG FORMULATION We take or deformable bod and irtall slice arond a point creating a differential olme represented b a cbe. The cbe has faces perpendiclar to the three coordinate aes. With the preios to definitions for stress e are in a position to represent the state of stress at a point graphicall sing this differential olme: Si faces three components of stress per face (= 8?). The state of stress at a point can be represented b nine stress components si of hich are independent as e ill see shortl.

Washkeic College of Engineering Section : STRONG FORMULATION Initiall let s assme that bod forces are minimal relatie to the loa on the component generating the state of stress at the point e are inestigating. The preios figre reflects this. Smming moments abot an ais throgh the centroid of the -face and parallel to the -ais iel M c ddd ddd Similar smmations of moments abot the an ais throgh the centroid of the -face and parallel to the -ais as ell as an ais throgh the centroid of the -face and parallel to the -ais iel

Washkeic College of Engineering Section : STRONG FORMULATION If the stress components on opposite faces are eqal and opposite (8 --> 9) then these eqalities among the shear stresses allos s to represent the state of stress at a point ith a 3 3 smmetric matri i.e. Recall from linear algebra that the eigenales of smmetric matrices are all real. From ndergradate strength of materials e kno that the eigenales of the matri aboe are the principle stresses at that point.

Washkeic College of Engineering Section : STRONG FORMULATION Differential Eqations of Motion P Preiosl e assmed that the stress components do not change from one face of the differential cbe representing stress at a point to the corresponding face on the other sided of the differential cbe. We no rela this assmption and allo stress gradients to eist. Consider the tapered bar sbjected to an aial force depicted in the figre to the left. Aial stresses change along the length. Bod forces are still neglected. In order for the aial stress to change from one cross section to the net along the -ais then

Washkeic College of Engineering Section : STRONG FORMULATION In general e design components so that a stress gradient Focsing on the -faces of the differential olme representing the state of stress at a point e old see the folloing is small. There are sitations here this is not possible e.g. at sharp corners. Here stress ales change rapidl and intensif. d d d B In general e cold allo for bod forces as is shon in the figre.

Washkeic College of Engineering Section : STRONG FORMULATION The stresses acting on the -faces of the differential cbe representing the state of stress at a point is portraed in the figre to the right. d B d A bod force B is shon acting in the -direction d

Washkeic College of Engineering Section : STRONG FORMULATION And the -faces of the differential cbe representing the state of stress at a point generall appears as follos d d A bod force B is shon acting in the - direction d B

Washkeic College of Engineering Section : STRONG FORMULATION Smming forces in the -direction sing the last three figres iel F dd d dd dd d dd dd d dd B ddd F B ddd B

Section : STRONG FORMULATION Washkeic College of Engineering Similar epressions are obtained b smming forces in the -direction and the -direction The bod forces B B and B are per nit olme qantities. From force eqilibrim e no hae a sstem of three copled partial differential eqations in si nknon stress components at a point in a component.. B F B F B B B F F

Section : STRONG FORMULATION Washkeic College of Engineering Si nknons and three eqations is an nderdetermined (inconsistent) sstem of eqations. So e search for more eqations to se in soling for stress at a point a solid mechanics problem. Let s introdce si eqations stiplating the relationship beteen stress and strain (Hooke s La). Hoeer this introdces si more nknons i.e. the components of the strain tensor. So no e no hae nine eqations and tele nknons. We do not appear to be gaining grond here. Since e hae looked into the concept of stress in detail it is time to do the same ith strain. E E E E E E

Washkeic College of Engineering Section : STRONG FORMULATION The Differential Definition of Strain We define strain b monitoring the effect loa and bondar conditions hae on irtal lines segments arbitraril assigned to positions ithin a component. That position is considered a point becase the line segments hae onl a differential length. Consider the points P( ) and Q(+ ) that both lie on a line parallel to the -ais. P Q

Washkeic College of Engineering Section : STRONG FORMULATION As the component deforms the original points P( ) and Q(+ ) moe to ne positions defined b P( ) and Q(+d +d +d) P > Q + d + d + d The initial and final positions of points P and Q can be sed to define the aial strain at a point in terms of the change in length of the differential line segment.

Washkeic College of Engineering Section : STRONG FORMULATION The initial sqared length of the differential line segment PQ is d d The aial strain in the -direction is defined as the change in length oer the original length and e maniplate that relationship as follos

Section : STRONG FORMULATION Washkeic College of Engineering Sqaring both sides of the last epression iel If e designate and (three more nknons the list gros longer) as the displacements at the point e are tring to characterie strain then the folloing relationships can be easil deeloped from the geometr in the preios figres The relationships aboe nderscores that the three ne nknons i.e. the displacements are fnctions of position.

Washkeic College of Engineering Section : STRONG FORMULATION Since = ( ) then b the chain rle of calcls d d d d With a fnction of position then recall that from the definition of the line segment PQ d d Ths d d d d The partial deriatie of ith respect to represents the change in the displacement ith respect to ( and are held constant). This is knon as a displacement gradient.

Section : STRONG FORMULATION Washkeic College of Engineering Similarl here it is assmed This implies the coordinate ariables defining the initiating point for line segment PQ are independent ariables i.e. d d d d d d d f

Section : STRONG FORMULATION Washkeic College of Engineering Since the choice of the location of line segment PQ is arbitrar (bt the initial direction is not) the assmption of independence is reasonable. Similarl B Pthagorean theorem s the sqared length d d d d d d d d d d

Section : STRONG FORMULATION Washkeic College of Engineering Sbstitting d d and d from aboe iel We also note that since in the original figre d d d d d

Section : STRONG FORMULATION Washkeic College of Engineering Ths and from this d d d d d

Section : STRONG FORMULATION Washkeic College of Engineering Strains are qite small in most engineering applications. Taking a qantit that is small and sqaring it prodces something een smaller. So if terms qadratic in strain (left side of the eqal sign) are neglected If displacement gradients are small then sqaring them ill be een smaller. Neglecting qadratic displacement gradient terms iel Note that if strains and displacement gradients are not small none of hat appears on this page can be implemented.

Section : STRONG FORMULATION Washkeic College of Engineering In a similar fashion If displacement gradients are small These are the normal strains. The net step is defining shear strains. Normal strains correspond to changes in length. Shear strains correspond to changes in angles. For angles e need to differential line segments to define an angle.

Washkeic College of Engineering Section : STRONG FORMULATION No consider to differential line segments. Both originall lie in an arbitrar - plane contained ithin the deformable bod and are initiall perpendiclar to one another. The intersection of the to line segments defines or point in the component. Initiall differential line segments and are nit lengths.

Washkeic College of Engineering Section : STRONG FORMULATION If the line segments deform sch that their lengths are nchanged ( = and = ) bt their orientations in Cartesian three space are altered then the original line segments cold assme the folloing locations + d + d + d + d + d + d

Section : STRONG FORMULATION Washkeic College of Engineering d d d d d d d d d d d d Earlier the chain rle from Calcls as emploed and no appling this rle to iel the folloing ector components hen e treat the line segment as a positional ector. Similarl

Section : STRONG FORMULATION Washkeic College of Engineering Net a dot prodct is formed from the to positional ector that iel the folloing scalar ale hich eqal to the angle formed beteen the to ectors i.e. After introdcing a third line segment 3 parallel to the -ais then deriations similar to the one aboe ield cos

Section : STRONG FORMULATION Washkeic College of Engineering Under the assmption of small deformation gradients to go along ith We hae si eqations in terms of three additional nknons i.e. the displacements and. Viola! We hae 5 eqations in terms of 5 nknons hen the stress components embedded in the force eqilibrim are inclded ith the linear relationships beteen stress and strain. A solable sstem.

Section : STRONG FORMULATION Washkeic College of Engineering B B B E E E E E E Hooke s La 6 eqations nknons Force Eqilibrim 3 eqations Strain-Displacement Relationships - 6 eqations 3 nknons

Washkeic College of Engineering Section : STRONG FORMULATION These eqations mst hold at eer point in the component for a alid soltion. When bondar conditions are added (force and/or displacement bondar conditions) these 9 linear somehat non-homogeneos (no bod forces and then all the eqations are homogeneos) linear partial differential eqations mst be soled along ith the 6 algebraic eqations stiplating the relationship beteen stress and strain. This is knon as the strong formlation of the problem. Soling partial differential eqations is not as eas as soling ordinar differential eqations. Yo are inited to take a gradate corse on the topic. Soling a sstem of partial differential eqations at eer point in a component is een more difficlt. A cationar note: finite element analsis does not sole the strong formlation of the problem sing nmerical metho. As e ill see a eak formlation for the solid mechanics problem is posed and finite element analsis soles this formlation at select points (nodes) throghot the component. Eerhere other than these select points the soltion is an approimation.

Section : STRONG FORMULATION Washkeic College of Engineering Naier s Eqations The preios section of notes demonstrated that the strong formlation of a solid mechanics problem inoled soled 5 eqations in 5 nknons. In a corse in Elasticit (CVE 64 MME 64) or Continm Mechanics o ill learn ho to compress the nmber of eqations to three and these eqations ill be in terms of the nknon displacements and. The three eqations are knon as Naier s problem in partial differential eqations Soltion of this sstem is still danting. Bondar conditions for this sstem of eqations? 3 ) ( ) ( ) ( B B B

Washkeic College of Engineering Section : STRONG FORMULATION Strong Formlation Closing Remarks Commonl in engineering problems the goerning partial differential eqations contain deriaties p to and inclding the second order i.e. the most general form in to dimensions is of the form A ( ) B ( ) C ( ) Note the first order and ero order deriaties to the right of the eqal sign. This notation is a mathematics artifact. Eqation formlation are classified as follos B AC < B AC = B AC > Elliptic Parabolic Hperbolic A ide range of problems from elasticit acostics atmospheric science and hdralics are goerned b hperbolic partial differential field eqations.

Washkeic College of Engineering Section : STRONG FORMULATION Parabolic eqations arise in heat condction problems. The strong formlation places continit reqirements on the dependent field ariables in or case the displacement field. Whateer fnction is deeloped for the soltion to the hperbolic partial differential eqation(s) the soltion mst be continosl differentiable throgh second order deriaties. In finite element analsis this is identified as C continit. As noted earlier obtaining eact soltions or an soltions for a strong formlation of a solid mechanics problem can be a difficlt task. One can emplo finite difference metho to sole a sstem of eqations of the strong form and obtain an approimate soltion. Hoeer this approach onl orks ell for problems ith simple and reglar geometr and bondar conditions. Since most solid mechanics problems do not fit into this scheme most engineering problems are soled sing finite element metho applied to a eak formlation of the problem. The eak form can be obtained throgh energ principles. This approach fits into concept of ariational metho.